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best friends forever bff finding lasting dense subgraphs konstantinos semertzidis evaggelia pitoura evimaria terzi panayiotis tsaparas dept computer science engineering university ioannina greece dept computer science boston university usa oct ksemer pitoura tsap evimaria form natural model relationships interactions entities example people social cooperation networks servers computer networks tags words documents tweets relationships interactions lasting ones paper study following problem given set graph snapshots may correspond state evolving graph different time instances identify set nodes densely connected snapshots call problem best friends ever problem provide definitions density multiple graph snapshots capture different semantics connectedness time study corresponding variants problem look problem relaxes requirement nodes connected snapshots asks densest set nodes least given set graph snapshots show problem definitions density propose set efficient algorithms finally present experiments synthetic real datasets show efficiency algorithms usefulness problems ntroduction graphs offer natural model capturing interactions relationships among entities oftentimes multiple snapshots graph available example snapshots may correspond states dynamic graph different time instances states complex system different conditions call sets graph snapshots graph history analysis graph history finds large spectrum applications ranging marketing virus propagation digital forensics central question context interactions relationships graph history lasting ones paper formalize question design algorithms effectively identify relationships particular given graph history introduce problem efficiently finding set nodes remains tightly connected history call problem best friends ever problem formulate problem problem locating set nodes maximum aggregate density graph history provide different definitions aggregate density capture different notions connectedness time result four variants problem extend problem capture cases subsets nodes densely connected subset snapshots consider example set collaborators work intensely together years drift apart set friends social network stop interacting snapshots reconnect identify subsets nodes define problem short problem ask set nodes set snapshots aggregate density nodes snapshots maximized identifying nodes finds many applications example collaboration social networks nodes correspond individuals chosen form teams organize successful professional social events network locate protein complexes densely interacting different states thus indicating possible underlying regulatory mechanism network nodes words tags edges correspond documents tweets published specific period time identifying nodes may serve first step topic identification tag recommendation types analysis computer network locating servers communicate heavily time may useful identifying potential attacks bottlenecks problem identifying dense subgraph static graph received lot attention also work finding dense subgraphs dynamic graphs however line work goal efficiently locate densest subgraph current graph snapshot whereas interested locating subgraphs remain dense whole graph history best knowledge first systematically introduce study density graph history define problems related work authors study one four variants problem context graph databases compare performance algorithms variant algorithm proposed experimentally study complexity different variants problems two variants solved optimally propose generic algorithmic framework solving problems works linear time experimental results real synthetic datasets show efficiency effectiveness algorithms discovering lasting dense subgraphs two case studies bibliographic collaboration networks hashtag networks twitter validate approach summarize main contributions work following minimum density graph minimum degree node introduce novel problems identifying subset nodes define dense subgraphs graph history end extend notion density graph histories provide definitions capture different semantics density time leading four variants problems min degree study complexity variants problems propose appropriate algorithms prove optimality approximation factor algorithms whenever possible extend definitions algorithms identify ffs input set query nodes perform experiments real synthetic datasets demonstrate problem definitions meaningful algorithms work well identifying dense subgraphs practice roadmap section provide definitions aggregate density introduce problem algorithms section iii problem algorithms section section study extensions original problem experimental evaluation presented section comparison related work section vii section viii concludes paper aggregate density assume given input multiple graph snapshots set nodes snapshots may ordered example snapshots correspond states dynamic graph may also unordered collection graphs example snapshots correspond graphs collected result scientific experiments refer graph collections graph history definition raph istory graph history collection graph snapshots snapshot defined set nodes example graph history four snapshots shown figure note definition applicable graph snapshots different set nodes considering union define notion density graph history start reviewing two basic definitions graph density single graph snapshot given undirected graph node use degree denote degree average density graph average degree nodes degree intuitively given graph defined single node one minimum degree accounts degrees thus connectivity nodes example figure clearly lower bound subscript ignored density either also define density subset nodes graph end use induced subgraph define density example snapshot figure highest minimum density whereas highest average density define density set nodes graph history need way aggregate density set nodes multiple graph snapshots aggregating density sequences given graph history use denote sequence density values graph induced set graph snapshots consider two definitions aggregation function aggregates densities snapshots first computes minimum density snapshots min second computes average density snapshots intuitively minimum aggregation function requires high density every snapshot average aggregation function looks snapshots whole use collectively refer define aggregate density definition aggregate ensity given graph history defined set nodes define aggregate density depending choice density function aggregation function following four versions fmm fma fam faa density definition associates different semantics meaning density among nodes graph history large values fmm correspond groups nodes member group connected large number members group snapshot node ceases fig graph history consisting four snapshots considered member group loses touch members even single snapshot large values fma achieved groups high average density snapshot opposed fmm requirement placed member group large values fma indicative group persistently high density whole faa metric takes large values group many connections average thus faa loose terms consistency time terms requirements individual group member level lastly fam takes average minimum degree node snapshot thus less sensitive density single instance example graph history figure aggregate densities equal however faa fma faa faa fma fma due last instance note also fmm value determined one node one snapshot node last snapshot fam average graph finally let define average graph graph history graph weight edge equal fraction snapshots edge appears definition average raph given graph history set nodes average graph weighted undirected graph set nodes usual degree node weighted graph defined degree average graph performs aggregation basis average degree degree node time average graph lose information regarding density individual snapshots algebraic manipulation prove following lemma shows connection average graph faa density function lemma let graph history set nodes subset nodes holds faa iii roblem section introduce problem study hardness propose appropriate algorithms problem definition given snapshots graph history goal locate best friends ever identify subset nodes nodes remain densely connected snapshots formally problem best friends forever problem given graph history aggregate density function find subset nodes maximized considering four choices aggregate density function four variants problem specifically fmm fma fam faa give rise problems respectively algorithms introduce generic algorithm problem algorithm shown algorithm algorithm inspired popular algorithm densest subgraph problem static graph use denote sequence induced subgraphs set nodes algorithm starts set nodes consisting nodes performs steps step produces set removing one nodes set finds set maximum aggregate density algorithm ind algorithm input graph history aggregate density function output subset nodes arg min score return arg max ind algorithm forms basis algorithms propose four variants problem interestingly defining appropriate scoring functions score used line select node remove get efficient algorithms variants algorithm scorem algorithm algorithm scorea algorithm input graph history output node minimum scorem list nodes degree procedure core pdate dmint smallest scorem min dmint arg min dmint scorem scorem degree update degree return solving problem define score node scorem minimum degree sequence scorem min degree iteration ind algorithm select node minimum scorem value call instantiation ind algorithm ind prove ind provides optimal solution problem proposition problem solved optimally polynomial time using ind algorithm proof let iteration ind algorithm first time node belongs optimal solution selected removed let node clearly fact every iteration remove edges graphs scorem scorem since node pick iteration every node satisfies mingt degree scorem scorem scorem since true every node means indeed optimal algorithm find running time ind number snapshots history graph total number edges appear snapshots node minimum scorem value computed procedure core pdate shown algorithm also removes node edges snapshots snapshot keep list nodes degree line algorithm lists constructed time given lists time required input graph history output node minimum scorea construct average graph list nodes degree procedure core pdate scorea smallest scorea scorea degree update degree return find node minimum scorem lines snapshots neighbors removed node need moved position lists lines degree every neighbor removed node decreased one throughout execution algorithm moves happen therefore total running time ind note algorithm iteratively removes graph node minimum degree first studied shown compute densest subgraph problem density optimal density solving solve problem shall use average lemma shows graph thus based results faa charikar goldberg conclude proposition problem solved optimally polynomial time although exists optimal algorithms computational complexity algorithms case algorithm makes hard use real graphs therefore instead algorithm use ind algorithm define score node scorea equal average degree graph history scorea degree iteration select node minimum average degree refer instantiation ind ind using lemma results charikar following proposition ind algorithm problem proof easy see ind removes node charikar shown minimum density algorithm iteratively removes graph node minimum density provides finding subset nodes maximizes average density single weighted graph snapshot given equivalence established lemma ind also algorithm show steps finding node minimum scorea value algorithm uses lists nodes degree average graph achieve total running time ind solving consider application ind ind algorithms two problems following propositions prove two algorithms give poor approximation ratio problems recall problems maximization problems therefore lower approximation ratio worse performance algorithm proposition approximation algorithm ind problem number nodes refer algorithm ind ind requires check nodes choosing node remove step shown algorithm thus leading complexity algorithm scoreg algorithm input graph history aggregate density function output node minimum scoreg procedure core pdate scoreg scoreg arg min scoreg return proof appendix proposition approximation algorithm ind problem number nodes proof appendix proposition approximation ratio algorithm ind problem number nodes roblem section relax requirement nodes connected snapshots graph history instead ask find subset nodes maximum aggregate density least snapshots call problem problem formally define show develop two general types algorithms efficiently solving practice problem definition proof appendix also consider applying ind algorithm selects remove node minimum average degree show ind poor approximation ratio problem problem seek find collection graph snapshots set nodes subgraphs induced high aggregate density formally problem defined follows proposition approximation ratio algorithm ind problem number nodes problem problem given graph history aggregate density function integer find subset nodes subset size maximized proof appendix complexity open problem jethava beerenwinkel conjecture problem yet provide proof given ind ind theoretical guarantees also investigate greedy approach selects node remove based objective function problem hand greedy approach instance iterative algorithm shown algorithm specifically target function either fam fma given set define score scoreg node follows problem depending choice aggregate density function four variants thus fmm fma fam faa give rise problems respectively note subcollection graphs need consist contiguous graph snapshots case problem could solved easily considering possible contiguous subsets outputting one highest density however four variants become drop constraint consecutive graph snapshots scoreg theorem problem definition aggregate density function iteration algorithm selects node causes smallest decrease largest increase target function prove exists clique size least graph exists set nodes algorithm iterative itr ind algorithm input graph history function integer output subset nodes subset snapshots converged false nitialize converged est napshots ind converged true else return subset snapshots forward direction easy exists subset nodes form clique selecting set nodes subset snapshots correspond nodes wield fmm fam follows fact every snapshot complete star prove direction observe snapshots consist star graph collection disconnected nodes given set nodes connected center node zero otherwise therefore fmm fam implies means centers graph snapshots connected nodes hence therefore form clique size graph case faa fma construction proceeds follows given graph edges construct graph history snapshots snapshots defined vertex set snapshot edge consisting single edge prove exists clique size least graph exists set nodes subset snapshots prove exists clique size least graph exists set nodes subset snapshots forward direction easy exists subset nodes form clique selecting set nodes snapshots correspond edges nodes yield faa fma prove direction assume clique size greater equal let subset snapshots let union endpoints edges since clique follows therefore faa fma algorithms consider two general types algorithms iterative incremental algorithms iterative algorithm starts initial size collection graph snapshots improves whereas incremental algorithm builds collection incrementally adding one snapshot time next describe two types algorithms detail note depending whether solving problem use appropriate version ind algorithm algorithms iterative algorithm iterative itr algorithm shown algorithm starts initial collection snapshots set nodes routine nitialize iteration given set finds graph snapshots highest score done est napshots est napshots computes density snapshot outputs snapshots highest density given algorithm finds set maximized step essentially solves problem input aggregate density function using ind algorithm itr algorithm keeps iterating collections dense sets nodes iterations improve score important step iterative ind initialization consider three different alternatives initialization random contiguous random initialization itrr initialization randomly pick snapshots snapshots used solving corresponding problem input produce ind contiguous initialization itrc initialization first find consists best contiguous graph snapshots given contiguous sets snapshots find set snapshots corresponding set nodes maximize intuition behind initialization technique assumes best snapshots going contiguous experiments demonstrate practice true many datasets collaboration networks evolve time expect see temporal locality initialization itrk initialization solve problem independently snapshot results different sets one solution includes nodes appear least sets intuition behind initialization include initial solution nodes appear densely connected many snapshots also experimented natural alternatives union intersection approach seems strike balance two running time iterative ind algorithm number iterations required convergence comes running time ind practice observed algorithm converges iterations algorithm incremental density incd ind algorithm input graph history function integer output subset nodes subset snapshots sij ind arg max sij ind arg max ind return algorithm incremental overlap inco ind algorithm input graph history function integer output subset nodes subset snapshots ind arg max ind arg max ind return incremental algorithm incremental algorithm starts collection two snapshots incrementally adds snapshots collection snapshots formed appropriate ind algorithm used compute dense subset nodes use two different policies selecting snapshots first one termed incremental density incd algorithm shown algorithm selects graph snapshots maximize density whereas second one termed incremental overlap inco algorithm shown algorithm selects graph snapshots maximize overlap among nodes dense subsets incremental density incd select pair snapshots form initial collection solve problem independently pair snapshots gives dense sets sij solutions select pair snapshots whose dense subgraph sij largest density lines algorithm builds solution incrementally iterations iteration construct solution adding solution graph snapshot maximizes density function densest subset sequence arg max lines running time incd algorithm first term due initialization step line look best pair snapshots efficiency important initialize algorithm random pair save time incremental overlap inco form initial collection first solve problem independently snapshot gives different sets dense subgraph algorithm selects sets two similar ones initializes corresponding snapshots lines defining similarity sets nodes use jaccard similarity form algorithm first solves problem let solution selects remaining snapshots adds snapshot whose dense set similar lines running time inco algorithm first term time initialization second term roblem xtensions definitions problem focus identification set nodes aggregate density maximized consider natural extensions problem placing additional constraints dense subgraphs constraint interesting extension introducing set seed query nodes requiring output set nodes high density also contains input seed nodes similar extension introduced static single snapshots graphs practice variant identifies lasting best friends query nodes call problem modify ind algorithms appropriately take consideration additional constraint particular ind stops query node selected removed let call modified algorithm ind prove following proposition omit proof due space constraints proposition ind solves problem optimally polynomial time also modify ind remove seed nodes follows step node minimum average degree happens seed node algorithm selects remove node next smallest degree seed node algorithm stops remaining nodes seed nodes let call modified algorithm ind proposition let optimal solution problem solution ind algorithm holds faa degree proof lemma suffices show ind algorithm provides approximation average density single graph let optimal solution let graph results delete edges two query nodes clearly also optimal solution assume assign edge either node let number edges assigned let amax maxu easy see faa amax since edge optimal solution must assigned node assume assignment edges nodes performed ind algorithm proceeds initially edges unassigned step node deleted assign edges note assignment maintains invariant step edges two nodes current set unassigned edges assigned algorithm stops edges assigned consider single iteration algorithm node umin selected removed let current set let number nodes pbe number query nodes holds faa degree degree degree let degree since smallest degree min among nodes seed nodes faa umin since edges assigned edges assigned node node removed step execution algorithm umin amax thus faa amax faa connectivity constraint another meaningful extension impose restrictions connectivity connectivity graph history may many different interpretations one may consider version induced subgraphs connected another alternative least connected assume definition connectivity given form predicate connected true connected false otherwise problem becomes given graph history set query nodes aggregate density function find subset nodes maximized connected true solve problem modify ind tests connectivity predicate connected stops connectivity constraint longer holds experiments apply simplest test running algorithms connected components query nodes size constraint finally note definition impose constraint size output set nodes respect problem necessary one add additional constraint problem definition imposing cardinality constraint output however cardinality constraint makes problem computationally hard also holds problem simply consider graph history replicas graph xperimental valuation goal experimental evaluation threefold first want evaluate performance algorithms problems terms quality solutions running time second want compare different variants aggregate density functions third want show usefulness problem presenting results two real datasets namely research collaborators dblp hashtags twitter table real dataset characteristics dataset caida twitter nodes edges aver per snapshot snapshots datasets setting evaluate algorithms use number real graph histories snapshots correspond collaboration computer concept networks dataset contains yearly snapshots coauthorship graph interval top database data mining conferences edge two authors graph snapshot coauthored paper corresponding year two papers corresponding interval dataset consists nine graph snapshots peering information inferred oregon march may one snapshot per week dataset consists nine weekly snapshots graphs march may dataset contains caida autonomous systems graphs derived set route views instances twitter dataset nodes hashtags tweets edges represent hashtags tweet dataset contains daily snapshots october november dataset represents communication network bgp border gateway protocol logs dataset contains daily snapshots span interval days november january dataset characteristics summarized table since ground truth information real datasets also use synthetic datasets particular create graph snapshots using forest fire model wellknown model creating evolving networks using default forward backward burning probabilities plant dense subgraphs snapshots randomly selecting set nodes creating additional edges different snapshot http https https http https evaluation since shown section ind ind provably good problems respectively consider algorithms problems problems use three algorithms ind ind ind problem also use dcs algorithm proposed problem similar dcs algorithm also iterative algorithm removes nodes one time step dcs finds subgraphs largest average density snapshots identifies subgraph smallest average density among removes node smallest degree subgraph accuracy ind comparison density definitions start evaluation accuracy algorithms along comparison different aggregate densities since ground truth information real data use first synthetic datasets first create graph snapshots nodes using forest fire model one snapshots plant dense random subgraph nodes inserting extra edges probability vary edge probabilities fig report measure achieved four density definitions trying recover subgraph recall takes values larger value better recall precision solution respect ground truth case sensitive measure since reports dense subgraph even smallest edge probability achieve perfect value edge probability larger edge probability least smaller values three density definitions locate supersets due averaging variations ind algorithms produce results study various density definitions behave second dense subgraph case plant subgraph edge probability snapshots second dense subgraph number nodes edge probability percentage snapshots different values fig depicts two graphs graph shown blue graph shown red output ind algorithms different density definitions report densest subgraph since measures ask high density every snapshot however report dense subgraph appears sufficient number half snapshots density definitions algorithms recover exact set case also run algorithms using real datasets present results table report value ran experiments system intel core ghz processor memory used one core experiments problems fraction planted snapshots dense graph reported problems fig accuracy density definition comparison objective function size solution first observation expected value aggregate density reported solution independently problem variant increases density graphs problem observe solutions usually small cardinality compared solutions problems since fmm objective rather strict solution twitter empty solutions problem datasets appear higher fmm scores may due fact larger groups nodes lasting connections datasets nodes communicate intensely observation period comparison ind alternatives shown table problem ind ind perform overall best datasets producing subgraphs large fma values ind performs slightly worse ind caida dataset caida dataset due probably large number snapshots ind based average degree returns set smallest density ind dcs comparable performance since remove nodes small degrees individual snapshots outperformed ind ind problem ind outperforms ind ind deeper analysis inferior performance ind problem revealed ind often gets trapped local maxima removing nodes graph find good solutions running time table iii report execution times expected response time ind algorithm slowest datasets due quadratic complexity problem ind general faster dcs difference times ind algorithms due differences computation density functions additional experiments including ones synthetic datasets larger graphs intervals show similar behavior depicted figs particular fig show execution time different algorithms problem varying nodes whereas fig shows execution time varying snapshots summary conclusion algorithms successfully discovered planted dense subgraphs even density small sensitive measure minimum aggregation densities requires dense subgraph present snapshots table results different algorithms problem real datasets ind size fmm datasets caida twitter ind size fma ind ind size fma size fma dcs size fma ind size fam ind size fam ind size fam ind size faa table iii execution time sec different algorithms problem real datasets datasets ind ind caida twitter ind ind ind dcs ind time sec time sec ind dcs ind ind ind nodes thousands snapshots fig synthetic dataset execution time different algorithms problem varying number nodes snapshots whereas average aggregation densities asks nodes sufficiently connected average problems ind returns general dense subgraphs alternatives including dcs ind ind scale well perform similarly different density functions differences running time attributed complexity calculating respective functions evaluation set experiments evaluate performance iterative incremental ind algorithms comparison algorithms terms solution quality similar plant dense random graph snapshots run ind algorithms value fig report measure different values expressed percentage total number snapshots iterative ind algorithm initialization itrk outperforms two successfully locates four density definitions appears sufficient number snapshots initializations work equally well average aggregation time incremental ind algorithm dcs ind ind ind ind sity incd slightly outperforms overlap inco overall incremental algorithms achieve highest compared iterative ones also conduct second experiment plant dense random graph edge probability snapshots dense random graph edge probability snapshots fig report measure assuming correct output problem different values expressed percentage total number snapshots comparing different initializations iterative ind algorithm observe among iterative algorithms itrk successfully locates four density definitions appears sufficient number snapshots previous experiment initializations work equally well average aggregation time incremental algorithms outperform iterative ones incd champion since achieve higher measure values even appears snapshots also apply ind algorithms real datasets various values figs report value aggregate density different values expressed percentage total number snapshots input graph history overall observed contradistinction experiments real datasets contiguous initialization itrc iterative algorithm emerges best algorithm many cases slightly outperforming incd indicative temporal locality dense subgraphs datasets datasets dense subgraphs usually alive contiguous snapshots especially evident datasets collaboration networks dblp datasets also notice incremental algorithms find solutions density close iterative algorithms finally also observe increases aggregate density solutions decrease explained fact often dense subgraphs alive snapshots snapshots snapshots snapshots snapshots fig synthetic dataset values problems snapshots snapshots snapshots snapshots fig synthetic dataset values problems faa fam fma fmm snapshots snapshots snapshots snapshots fig dataset scores aggregate density functions faa fam fma fmm snapshots snapshots snapshots snapshots fig dataset scores aggregate density functions fam faa fma fmm snapshots snapshots snapshots fig dataset scores aggregate density functions snapshots table solutions parenthesis dense author subgroups time sec time sec wei fan philip jiawei han charu aggarwal qin jeffrey xuemin lin guoliang jianhua feng craig macdonald iadh ounis wei fan jing gao philip jiawei han charu aggarwal jeffrey xuemin lin ying zhang snapshots snapshots wei fan jing gao philip jiawei han fig execution time different algorithms problem synthetic datasets convergence running time terms convergence iterative ind requires iterations converge datasets fig report execution time algorithms problem synthetic datasets observed iterative incremental inco algorithms scale well comparing incremental algorithms inco faster incd synthetic datasets respectively due quadratic complexity latter additional experiments including ones synthetic datasets larger graphs intervals depicted fig fig respectively particular fig shows execution time different algorithms problem varying number nodes whereas fig shows execution time varying number snapshots summary conclusion algorithms successfully discovered planted dense subgraphs lasted sufficient percentage much less half snapshots incremental ones sensitive among ind algorithms incremental algorithms outperform iterative ones cases among incremental algorithms incd slightly better inco however given slow running time incd inco preferable choice finally datasets consisting dense subgraphs temporal locality itrc good choice detecting graphs time sec time sec nodes thousands snapshots fig synthetic dataset execution time log scale different algorithms problem varying number nodes snapshots wei fan jing gao philip jiawei han charu aggarwal mohammad masud latifur khan bhavani thuraisingham case studies section report indicative results obtained using twitter datasets results identify lasting dense author collaborations hashtag cooccurrences respectively lasting dense table report set nodes output solutions different problem variants dataset first observe three authors wei fan philip jiawei han part four solutions three authors two papers together dataset pairs collaborated frequently last decade solutions contain additional collaborators authors obtain solution authors although group paper subsets authors collaborated many snapshots resulting high value faa solutions contain aforementioned three authors collaborators also new names authors scarce collaborations former group thus case solutions consist one dense subgroups authors grouped parentheses densely connected within sparsely connected others case table report results dataset authors dense collaborators recall years dataset also report corresponding years dense collaborations many new groups authors appear example new groups collaborators tsinghua university cmu rpi among others authors appeared solutions also appear large values also studied experimentally problem table show indicative results three authors paper seed nodes pitoura retrieve group students lasting prolific collaboration terzi close collaborators university tsaparas group collaborators time microsoft research note last case selected set consists researchers table authors output solutions problem years years years years christos faloutsos leman akoglu lei keith henderson hanghang tong tina liu chetan gupta song wang ismail ari elke rundensteiner yong dingyi han zhong lichun yang shengliang shenghua bao liyun min zhang yiqun liu shaoping min zhang liyun bhavani yiqun liu shaoping latifur khan thuraisingham mohammad masud wei fan jing gao philip jiawei han min zhang yiqun liu shaoping wei fan jing gao philip jiawei han charu aggarwal liyun min zhang yiqun liu shaoping table example authors output solutions problem pitoura koloniari drosou stefanidis ishakian erdos bestavros tsaparas fuxman kannan agrawal faloutsos koutra chris ding akoglu huang lei tao tong tsaparas several papers period recorded dataset authors also collaborating amongst finally use one authors appearing dense subgraphs namely faloutsos seed node case obtain dense subgraph similar one reported table finally consider query two authors faloutsos student koutra adding koutra query set changes consistency result focusing authors collaborators query nodes lasting dense hashtag appearances twitter table vii report results problem twitter dataset note results problem dataset shown table small graphs since hashtags appear together days dataset seen table vii able discover interesting dense subgraphs hashtags appearing days hashtags correspond actual events including races wikileaks trending period solution also report selected snapshot dates expected selected dates approach also captures interest fluctuation time example wikileaks topic captured dense hashtag set wikileaks snowden nsa prism best snapshots collections contiguous intervals rather single contiguous interval note also large values get interesting results fact consistent ephemeral nature twitter hashtags especially true fmm fma impose strict density constraints result solutions consist disconnected edges latifur khan bhavani thuraisingham mohammad masud wei fan jing gao philip jiawei han charu aggarwal vii elated ork best knowledge first systematically study variants problems research related recent work jethava beerenwinkel rozenshtein best understanding authors introduce one four variants problem studied namely paper authors conjecture problem propose heuristic algorithm work performs rigorous systematic study general problem multiple variants aggregate density function additionally introduce study problem studied authors study problem considered special case problem particular goal identify subset nodes dense graph consisting union edges appearing selected snapshots weak definition aggregate density furthermore focus finding collections contiguous intervals rather arbitrary snapshots propose algorithm similar iterative algorithm consider shown outperformed incremental algorithms huge literature extracting dense subgraphs single graph snapshot formulations finding subgraphs define often often hard approximate due connection maximumclique problem result problem finding subgraph maximum average minimum degree become particularly popular due computational tractability specifically problem finding subgraph maximum average degree solved optimally polynomial time exists practical greedy algorithm gives guarantee time linear number edges nodes input graph problem identifying subgraph maximum minimum degree solved optimally polynomial time using greedy algorithm proposed charikar work use average minimum degree quantify density subgraph table vii hashtags chosen snapshot dates output solutions problem twitter dates dates dates kimi abudhabigp allowin ozpol mexico malaysia signapore vietnam chile peru tpp japan canada oct nov abudhabigp abudhabi guti pushpush hulk allowin bottas kimi oct oct abudhabigp wikileaks snowden nsa prism oct oct nov abudhabigp abudhabi guti pushpush hulk allowin bottas kimi oct oct nov many tags report wikileaks snowden nsa prism oct nov assange wikileaks snowden nsa prism oct nov many tags report wikileaks snowden nsa oct nov assange wikileaks snowden nsa prism oct nov dates single graph snapshot extend definitions sets snapshots algorithmic techniques use problem inspired techniques proposed charikar sozio gionis however adapting handle multiple snapshots existing work also studies problem identifying dense subgraph graphs graphs new nodes edges may appear time existing ones may disappear goal line work devise streaming algorithm point time reports densest subgraph current version graph work interested dynamic version problem thus algorithmic challenges problem raises orthogonal faced work streaming algorithms recent work focuses detecting dense subgraphs special class temporal weighted networks fixed nodes edges edge weights change time may take positive negative values different problem since consider graphs changing edge sets furthermore density presence edges negative weights different density edges positive weights finally another line research focuses processing queries reachability path distance graph matching etc multiple graph snapshots main goal work devise effective storage indexing retrieving techniques queries sequences graphs answered efficiently paper propose novel problem finding dense subgraphs viii ummary paper introduced systematically studied problem identifying dense subgraphs collection graph snapshots defining graph history showed many definitions aggregate density functions problem identifying subset nodes snapshots problem solved linear time also demonstrated versions problem solved algorithm identify dense subgraphs occur yet snapshots graph history also defined problem variants problem showed devised iterative incremental algorithm solving extensive experimental evaluation datasets diverse domains demonstrated effectiveness efficiency algorithms eferences charikar greedy approximation algorithms finding dense components graph approximation algorithms combinatorial optimization goldberg finding maximum density subgraph university california berkeley khuller saha finding dense subgraphs icalp epasto lattanzi sozio efficient densest subgraph computation evolving graphs www jethava beerenwinkel finding dense subgraphs relational graphs ecml pkdd asahiro iwama tamaki tokuyama greedily finding dense subgraph swat sozio gionis problem plan successful cocktail party acm sigkdd tsantarliotis pitoura topic detection using critical term graph tweets proceedings workshops leskovec kleinberg faloutsos graph evolution densification shrinking diameters tkdd vol rozenshtein tatti gionis discovering dynamic communities interaction networks ecml pkdd dall asta barrat vespignani large scale networks fingerprinting visualization using decomposition nips bourjolly laporte pesant exact algorithm maximum problem undirected graph european journal operational research vol makino uno new algorithms enumerating maximal cliques swat mcclosky hicks combinatorial algorithms maximum problem comb vol tsourakakis bonchi gionis gullo tsiarli denser densest subgraph extracting optimal quality guarantees acm sigkdd bahmani kumar vassilvitskii densest subgraph streaming mapreduce pvldb vol bhattacharya henzinger nanongkai tsourakakis algorithm maintaining dense subgraphs dynamic streams stoc wang lin huai fast computation dense temporal subgraphs icde semertzidis pitoura lillis timereach historical reachability queries evolving graphs edbt moffitt stoyanovich towards distributed infrastructure evolving graph analytics www semertzidis pitoura durable graph pattern queries historical graphs icde khurana deshpande efficient snapshot retrieval historical graph data icde ren kao zhu cheng querying historical evolving graph sequences pvldb vol ppendix section present demonstrate ind ind applied yield solution poor approximation optimal solution following use denote number nodes different snapshots denote number snapshots proof proposition proof order prove claim need construct instance problem ind algorithm produces solution approximation ratio construct graph history follows first snapshots consist full clique nodes plus additional node connected single node clique last snapshot consists edge first iterations ind algorithm node minimum minimum degree one nodes clique node thus nodes clique iteratively removed left edge since node present intermediate subsets minimum degree snapshots therefore solution ind algorithm fam hand clearly optimal solution consists nodes clique minimum degree except last instance minimum degree zero proves claim therefore proof proposition proof order prove claim need construct instance problem ind algorithm produces solution approximation ratio construct graph history even follows snapshot contains nodes nodes form complete bipartite graph let denote additional three nodes node connected nodes graph snapshots except last snapshot connected nodes connected snapshots node connected nodes bipartite graph first snapshots node connected nodes bipartite graph last snapshots throughout assume note optimal set history graph consists nodes bipartite graph fam score scorea every node nodes bipartite graph scorea nodes scorea scorea node score scorea therefore first iteration algorithm remove one nodes bipartite graph without loss generality assume removes one nodes left partition node left partition still scorea node right partition scorea nodes scorea scorea node scorea therefore second iteration algorithm select remove one nodes right partition note resulting graph identical structure nodes therefore procedure repeated nodes bipartite graph removed nodes kept set last iterations result set returned ind fam degree nodes yielding approximation ratio proof proposition proof order prove claim need construct instance problem ind rithm produces solution approximation ratio construct graph history follows snapshots identical consist two sets nodes size respectively nodes form cycle nodes graph snapshot form clique nodes except one node different snapshot optimal set consists nodes average degree ind starts set nodes average degree snapshot also value fma function first iterations algorithm nodes scorem ones removed first nodes removed iterations average degree snapshot remains therefore set returned ind fma approximation ratio since proves claim proof proposition proof order prove claim need construct instance problem ind rithm produces solution approximation ratio construction proof similar construct graph history follows snapshots identical except last snapshot snapshots consist two sets nodes form two complete cliques size respectively last snapshot nodes disconnected optimal set consists nodes fma ind starts set nodes value fma determined last snapshot average degree nodes average degree time nodes average degree therefore algorithm iteratively remove nodes iteration resulting set fma nodes removed fma therefore approximation ratio instance claim follows fact
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dimensionality reduction sdps sketching andreas jul daniel stilck department mathematics technical university munich garching germany august show sketch semidefinite programs sdps using positive maps order reduce dimension precisely use transforms produce smaller sdp whose solution preserves feasibility approximates value original problem high probability techniques allow improve complexity storage space requirements apply problems schatten matrices specifying sdp solution problem constant problem size furthermore provide results clarify limitations positive linear sketches setting finally discuss numerical examples benchmark methods contents introduction preliminaries sketching product positive maps sketching linear matrix inequality feasibility problems approximating value semidefinite programs sketching complexity memory gains dsfranca applications numerical examples estimating value semidefinite packing problem linear matrix inequality feasibility problems conclusion complexifying transforms proof theorem lower bound value sdps sketching random feasibility problems introduction semidefinite programs sdps prominent class optimization problems applications across different areas science mathematics discrete optimization control theory however although many different algorithms solve sdp error time scales polynomially dimension logarithmically solving large instances sdps still remains challenge due fact number cost iterations scale superquadratically dimension algorithms solve sdps also due fact memory required solve large instances beyond current capabilities therefore motivated research algorithms solve sdps least obtain approximate solution less memory requirements one example recent ideas similar applied achieve optimal storage requirements necessary solve certain class sdps work proposes new way solve sdp using linear sketches approach relies standard convex optimization methods work develop algorithms estimate value sdp linear inequality constraints determine given linear matrix inequality lmi feasible algorithms convert original problem one type smaller dimension call sketched problem subsequently new problem solved techniques original one potentially using less memory achieving smaller runtime therefore call black box algorithm high probability optimal solution sketched problem allows infer something original problem case lmis sketched problem infeasibile obtain certificate original problem also infeasibile sketched problem feasible able infer original problem either close feasible feasible high probability technical assumptions case estimating value sdps able give upper bound holds high probability lower bound value sdp value sketched problem technical assumptions certain class sdps includes semidefinite packing problems able find feasible point original problem close optimal point technical aspects simplify significantly class checked whether feasible point indeed optimal algorithms work conjugating matrices define constraints sdp transforms thereby preserving structure problem similar ideas proposed reduce memory usage complexity solving linear programs techniques aim reduce number constraints goal reduce dimension matrices involved unfortunately dimension sketch needed fixed error high probability scales schatten constraints optimal solution sdp significantly restricts class problems methods applied able show one significantly improve scaling one sketch general sdps using linear maps paper organized follows section fix notation recall basic notions matrix analysis transforms semidefinite programs convex analysis need throughout paper proceed show sketch scalar product positive maps section apply techniques section show certify certain lmis infeasible showing infeasibility lmi smaller dimension section apply similar ideas estimate value sdp linear inequality constraints solving sdp lower dimension followed discussion possible gains complexity solving problems memory requirements section furthermore make numerical simulations section benchmark findings applying techniques problem field optimal designs experiments random lmi matrices sampled gaussian unitary ensemble preliminaries begin fixing notation brevity write set set matrices field written often omit underlying field relevant statement denote msym set symmetric matrices real case set hermitian matrices complex case denote transpose real case hermitian conjugate complex case avoid cumbersome notation redundant theorems prove statements real matrices however note statements translate complex case straightforward fashion state definitions real matrices clear generalize complex case msym write positive semidefinite denote cone positive semidefinite matrices interior positive definite matrices definition schatten scalar product let singular values define schatten denoted kakp given spi kakp max schatten induced schmidt scalar product given bihs sometimes refer case norm operator norm definition positive map linear map called positive structure set positive maps still well understood purposes however need maps form sxs easily seen positive adopt standard big notation asymptotic behavior functions two functions write exists constant analogously write exists constant following families matrices play crucial role purposes definition transform random matrix transform jlt parameters probability least subset holds swi note one usually demands norm vectors involved distorted definition jlts equivalent definition chose polarization identity many different examples jlts literature refer references therein details constructions jlts focus real matrices section show lift results cover complex matrices prominent jlt probably following entries standard gaussian random variables log theorem let proof refer lemma proof main drawback using gaussian jlts dense matrices denote nnz number nonzero elements matrix later want compute products form sxs advantageous sparse speed computation product computational cost forming product often bottleneck algorithms fortunately lot recent work sparse jlts particular following almost optimal result theorem sparse jlt section log log nonzero entries per column proof refer section proof remark proof constructive given jlt positive map sxs called sketching map sketching dimension fix notation semidefinite programs semidefinite programs class optimization problems linear functional optimized linear constraints set positive semidefinite matrices refer introduction topic many equivalent ways formulating sdps work assume sdps given following form definition sketchable sdp let msym call constrained optimization problem maximize subject sketchable sdp sometimes also refer sketchable sdp original problem see later approximate value sdps sdps rich duality theory instead optimizing positive semidefinite matrices satisfy certain constraints one optimizes points satisfy linear matrix inequality lmi dual problem sketchable sdp given following minimize subject vector coefficients sdps lmis called feasible least one point satisfying constraints otherwise call infeasible sketchable sdp called strictly feasible point constraints satisfied strict inequality conditions primal problem dual problem value one widely used sufficient condition slater condition asserts strictly feasible point full rank primal problem dual problem feasible primal dual value optimal solution dual problem need standard concepts convex analysis given vector space denote conv convex hull points cone denote cone generated elements convex cone called pointed sketching product positive maps one main ingredients sketch sdp lmi random positive map preserves scalar product high probability demand positivity assure structure sdp lmi preserved first consider example sxs jlt lemma let msym rank sbi sbj kbj proof observe eigenvectors corresponding nonzero eigenvalues form subset cardinality let probability least normalized eigenvectors sbj reverse triangle inequality also ksai ksbj fact jlt inequality follows sbj hence sbj let eigenvalues respectively sas sbs sbj probability least arbitrary claim follows scaling error schatten matrices involved lemma highly undesirable estimates used prove admittedly crude note similar estimate proved moreover using fact hilbert space could use msym isometrically embed resulting vector symmetric matrix denoting transformation obtain kbj demand sketching map map symmetric matrices symmetric matrices clearly obtain better scaling error procedure note however map may positive one requirements later sketch sdps next theorem shows scaling error schatten matrices involved possible positive maps want achieve compression theorem let random positive map strictly positive probability yit kyj proof refer appendix proof one could hope achieve better bound low rank matrices note significantly improve bound rank choosing may ensure inequality holds norm rank matrices involved bounded increases dimension involved jlt matrices factor usual dependence dimension jlts remains open one could achieve better compression sublinear number matrices sketching linear matrix inequality feasibility problems section show use jlts certify certain linear matrix inequalities lmi infeasible showing lmi smaller dimension infeasible consider inequalities like ones following lemma lemma let msym suppose cone exists pointed suppose proof let conv show exists could rescale obtain feasible point contradiction cone would would turn contradict pointed arguments follows set therefore closed may thus find convex compact disjoint convex closed set hyperplane strictly separates msym follows positive semidefinite clear normalizing may choose main idea show conditions may sketch hyperplane way still separates set positive semidefinite matrices sketched version set theorem let msym suppose cone moreover let pointed set min kbm take rank rank rank sbi sas probability least proof first noted exists lemma matrix defines hyperplane strictly separates set show strictly separates sets sbi sas probability least claim follows note choice follows lemma sbi probability least similarly instead therefore follows positive semidefinite matrix follows therefore found strictly separating lmi infeasible hyperplane theorem suggests way sketching feasibility problems form interested case infeasible investigate lmi sbi sas feasible suitably chosen jlt theorem equation infeasible know equation infeasible choice feasible leads feasible equation moreover follows theorem cone spanned enough jlt suitably chosen happens low probability equation feasible equation obtain concrete bounds probability original problem feasible one would need know parameter possible applications emphasize black box algorithm decide whether large instance lmi infeasible showing smaller instance lmi infeasible section discuss implications complexity memory usage last theorems approximating value semidefinite programs sketching show approximate high probability value sketchable sdp conjugating target matrix matrices describe constraints jlts subsequently solving smaller instance sdp next theorem shows general possible approximate value sketchable sdp using linear sketches high probability need make assumptions problem achieve compression using linear maps theorem let random linear map sketchable sdps exists algorithm allows estimate value sdp constant factor given sketch probability least proof duality relations schatten easy see value sdp maximize subject twice operator norm matrix theorem shown algorithm estimates operator norm matrix linear sketch probability larger must sketch dimension sketch would thus allow sketch operator norm assertion follows result remains true even restrict sdps optimal points small schatten low rank follows fact sdp given equation optimal solution rank trace may even restrict sdps whose value scales sublinearly see notice show operator norm sketched constructs two families random matrices whose operator norm order lemma class sdps considered covers problem obtain claim see soon main hurdle sketch sdps thus overcome last theorem also need suppose matrices define constraints target function small schatten one optimal solution end define definition sketched sdp let msym given optimal point sketchable sdp defined matrices satisfies given random matrix call optimization problem maximize sas subject sbi sketched sdp motivation defining sketched sdp given following theorem theorem let msym denote value sketchable sdp assume attained optimal point satisfies moreover let rank rank rank let value sketched sdp defined probability least proof follows lemma feasible point sketched sdp probability least lemma sas follows section discuss implications memory usage complexity approximating value sdp note optimal value sdp necessarily attained could also demand close optimality would slightly increase error made sketch since makes notation cumbersome assume existence optimal point although customary assume bound schatten optimal solution sdp common assume example solution lies given ellipsoid using ellipsoid method solve sdps assumptions straightforward derive bounds norm solution also given optimal solution sdp low rank norm gives good upper bound schatten remarked proof lemma solutions low rank sdps extensively studied past years many results available literature guarantee optimal solution sdp low rank general shown constraints sdp feasible optimal solution rank notice theorem rule possibility value sketched problem much larger sketchable sdp investigate issue introduce following definition relaxed sdp let msym given optimal point sketchable sdp defined matrices satisfies call optimization problem maximize subject relaxed sdp notice given feasible point sketched sdp feasible point relaxed problem cyclicity trace follows values original relaxed close values original sketched problem close well formalize intuition prove following bound appendix theorem setting definition assume exists constraints sketchable sdp strictly satisfied dual problem feasible value sketched sdp bounded max min optimal point sketchable sdp proof refer appendix proof note statement probabilistic holds regardless choice sketching map could also case gives better lower bound value one theorem one therefore say theorem guarantees general value sketched sdp differ significantly value original one feasible points close boundary combining theorem theorem possible pick small enough arbitrarily small additive error structural assumptions sdp need bounds schatten given strictly feasible point relaxed sdp bound schatten optimal solution sketchable sdp case sketchable sdp may obtain bound value approximate solution much simpler way class includes semidefinite packing problems addition note may set dividing obtain theorem sketchable sdp max kbi moreover denoting optimal point sketched sdp feasible point sketchable sdp attains lower bound furthermore kbi checked lower bound optimal value proof lower bound equation follows immediately cyclicity feasible point sketchable sdp given optimal solution trace dual sketched sdp kbi possible check lower bound given sketched sdp indeed optimal follows slater condition holds sketched sketchable sdp strictly feasible point small enough obtained feasible point indeed optimal sketchable sdp strong duality possible relax condition schatten matrices define constraints still obtain lower bound checked whether indeed optimal achieve necessary modify constraints sketched sdp equation instead primal picture find optimal point sketchable sdp optimal sketched sdp whenever point sketchable sdp note semidefinite packing problems possible derive bound schatten optimal solution straightforward way lemma let positive semidefinite matrices smallest strictly positive eigenvalue given sketchable sdp constraints finite value exists optimal point proof assume sdp finite value may restrict solutions whose support contained support denote projection onto support conjugating sides taking trace obtain supposed support contained support feasible point obtain claim assumptions needed results theorem bound schatten optimal solution sdp arguably difficult show bounds form readily available literature moreover sdps constraint would natural candidates apply methods able obtain compression scheme discussed far however one interested obtaining upper bound value sdp still possible constraint target matrix achieve compression theorem let msym let denote value sketchable sdp assume attained optimal point moreover let rank rank rank let value modifed sketched sdp defined given maximize sas subject sbi probability least proof essentially one theorem feasible point probability least main difference modified sketched sdp sketched sdp conjugate identity constraints feasible point relaxed sdp need assumption obtain upper bound clear theorem may also optimize trace without conjugating still upper bound compression may therefore summarize results section follows want obtain upper bound value sketchable sdp techniques necessary upper bounds schatten matrices define constraints target matrix optimal solution may choose jlt suitable dimension solve sketched sdp whose value allow infer upper bound original problem high probability addtionaly given strictly feasible point sketchable sdp solving semidefinite packing problem also obtain lower bound value sketchable sdp terms sketched one case semidefinite packing problems even obtain feasible point sketchable sdp whose value close sketched value given bound schatten optimal solution may impose constraint theorem obtain upper bound value sketchable sdp constrained points schatten bounded although able drop assumption schatten optimal solution able prove upper bound differ significantly true value case complexity memory gains section discuss much gain considering sketched sdp instead sketchable sdp focus results section discussion carries results section throughout section assume guaranteed schatten optimal solution sdp matrices define constraints therefore theorem need sketch appropriate size theorem states need jlt upper bound hold probability least stated theorem choose sketching matrix log log nonzero entries per column theorem cost generating smaller order necessary matrix multiplications therefore take cost account rest analysis one could argue one needs know schatten different matrices define constraints estimating value obtaining concrete bounds feasibility problems however suppose upper bound schatten optimal solution constraints given computed time case example semidefinite packing problem matrices positive semidefinite compute schatten time generate sketched sdp need compute matrices form sbs computations needs max nnz log operations worst case matrices dense full rank becomes log operations generate sketched sdp fixed let collect considerations proposition proposition let msym sketchable sdp given furthermore let max nnz nnz nnz sdp complexity solving sketchable sdp accuracy dimension number max log log sdp log operations needed generate solve sketched sdp defined theorem easy see parallelize computing matrices sbi typically costs forming sketched matrices sbi dominates overall complexity example using ellipsoid method chapter complexity solving sdp becomes sdp max log assuming fixed need max operations solve sketchable sdp compared log operations obtain approximate solution via first forming sketched problem solving admittedly ellipsoid method used using interior point methods still need practice sdp max log operations chapter exponent matrix multiplication best known algorithms achieve sdp sketched gives speedup long complexity solving sdp directly great advantage sketched problem need store matrices size instead collect proposition proposition let msym sketchable sdp need store log entries sketched problem defined theorem applications numerical examples estimating value semidefinite packing problem inspired test techniques sdp stemming field optimal design experiments problem following experimenter wishes estimate quantity unknown parameter given end one given linear measurements parameter centered measurement noise refer details topic find amount effort spend experiment minimize variance given sdp maximize cct subject ati problem always admits optimal solutions rank generated random instances sdp following way sampled four matrices distributed follows first three rows sampled independently uniform distribution unit sphere rows set given generate getting four samples uniform distribution four samples standard normal distribution given row gives matrices rank almost surely schatten equal fact linear combination rows ensures problem bounded easily seen looking dual problem note semidefinite packing problem able use results theorem obtain lower bound exists optimal solution whose schatten bounded high value error error sketchable sketch table combination sketchable dimension dimension sketch generated instances sdp equation stands mean running time stands lower bound upper bound column shows mean sample column value stands optimal value sketchable sdp probability easily follows theorem fact unit vectors almost orthogonal obtain upper bound used results theorem lower bound theorem chosen generate results used sparse jlts sparsity parameter obtain faster matrix multiplications form sketches define error lower bound given upper solve sdp given equation used cvx package specifying solving convex programs see results table show excluding case sketching dimension able find feasible points numerically indistinguishable optimal using sketching methods moreover time needed find optimal solution smaller orders magnitude linear matrix inequality feasibility problems apply techniques lmi feasibility problem let random matrices sampled independently gaussian unitary ensemble gue see example section section definition consider rescaled shifted matrices random isometry test techniques feasibility lmi depending isometry equation clearly feasible lmi feasible using standard techniques random matrix theory show equation equation feasible high probability infeasible high probability case refer appendix section proof claim therefore allows quantify close feasible lmi inequality terms close know whether lmi feasible choose way avoid schatten matrices define lmi order dimension technique used solve equation discussed section check feasibility complex gaussian jlt independent gaussian jlts refer section proof choice random matrices indeed gives jlt scaling parameters real one refer theorem proof lmi defined equation satisfies assumptions theorem high probability solve sdp given equation used cvx package specifying solving convex programs results summarized table observe using methods able show lmi infeasibile much smaller running time even show certain lmi infeasible direct computation possible due memory constraints choices parameters cases however necessary increase sketch dimension show inequality infeasibile conclusion shown obtain sketches product using positive maps obtained jlts apply show certain lmi infeasible obtain approximations value certain sdps cases techniques lead significant improvements runtime necessary solve instances sdps significant gains memory needed solve however class problems techniques applied significantly restricted fact matrices define constraints problems solution must schatten scale dimension advantageous moreover theorems proved show one significantly improve results using positive linear maps sketch norm approximate value sdps acknowledgements would like thank ion nechita helpful discussions acknowledges support isam graduate center technical university munich original sketch error rate table combination dimension image random isometry dimension domain random isometry dimension sketch generated instances random lmi equation stands mean running time error rate gives ratio infeasible problems detected infeasible sketching dash running time means able solve lmi ran memory edges support graduate program topmath elite network bavaria topmath graduate center tum graduate school technical university munich supported technical university munich institute advanced study funded german excellence initiative european union seventh framework programme grant agreement references anderson guionnet zeitouni introduction random matrices cambridge university press barvinok problems distance geometry convex properties quadratic maps discrete computational geometry boyd ghaoui feron balakrishnan linear matrix inequalities system control theory volume studies applied mathematics siam june bhatia matrix analysis volume graduate texts mathematics springer bubeck convex optimization algorithms complexity found trends mach november boyd vandenberghe convex optimization cambridge university press klerk aspects semidefinite programming interior point algorithms selected applications applied optimization springer grant boyd graph implementations nonsmooth convex programs blondel boyd kimura editors recent advances learning control lecture notes control information sciences pages springer http grant boyd cvx matlab software disciplined convex programming version http march schrijver geometric algorithms combinatorial optimization springer iyengar phillips stein approximation algorithms semidefinite packing problems applications maxcut graph coloring pages springer kane nelson sparser transforms acm january lasserre anjos handbook semidefinite conic polynomial optimization international series operations research management science series springer gall powers tensors fast matrix multiplication proceedings international symposium symbolic algebraic computation issac pages acm ledoux rider small deviations beta ensembles electronic journal probability pataki rank extreme matrices semidefinite programs multiplicity optimal eigenvalues mathematics operations research pukelsheim optimal design experiments classics applied mathematics society industrial applied mathematics rudelson vershynin inequality concentration electronic journal probability sagnol class semidefinite programs solutions linear algebra applications stark harrow compressibility positive semidefinite factorizations quantum models isit pages ieee positive linear maps operator algebras springer monographs mathematics springer vershynin compressed sensing chapter introduction nonasymptotic analysis random matrices pages cambridge university press poirion liberti using lemma linear integer programming arxiv july wolkowicz anjos semidefinite programming discrete optimization matrix completion problems discrete appl november wolf quantum channels operations guided tour lecture notes available http woodruff sketching tool numerical linear algebra foundations trends theoretical computer science yurtsever udell tropp cevher sketchy decisions convex matrix optimization optimal storage arxiv february complexifying transforms appendix generalize results need concerning jlts complex vector spaces see constant statements hold real case also hold complex case consider matrices form independent entries show give complex jlts note constructions clearly give sparse jlts real jlts used sparse definition distribution probability distribution random variable called exist random variable norm defined sup see example section details main ingredient show generalize jlts complex case following theorem theorem theorem let fixed consider random vector independent random variables satisfying kxi kakhs exp using different way proving lemma generalizes complex case follow proof theorem lemma let independent entries xij fixed proof define linear operator fixed vector use standard isomorphisms denote map composed isomorphisms linear map real vector space another real vector space moreover observe matrix form mapped isomorphism map play role statement theorem straightforward compute norms involved statement explained section entries vector satisfies assumptions theorem statement follows unfortunately sparse jlts discussed theorem form entries jlts independent one assumptions lemma proof theorem theorem let random positive map positive probability yit kyj proof let orthonormal basis define eti equation satisfied positive probability must exist positive map equation satisfied orthonormal scalar product positivity define matrix clear hermitian entries positive aij aii follows aij let aij follows equation since also write thus infer follows finally obtain notice follows equation may rescale pairwise scalar product still satisfies claim follows therefore suppose hence equation positivity fact positive semidefinite positive semidefinite proposition set positive semidefinite matrices definition set equation follows elementary computations finally obtain lower bound value sdps sketching obtain lower bounds value sketchable sdp terms value sketched sdp continuity bounds relaxed sdp continuity bound use sdps given equality form begin giving equivalent formulation sketchable sdp equality constraints method using duality derive perturbation bounds convex optimization problem used standard refer section similar derivation given sketchable sdp define maps etj orthonormal basis zjj help matrix etj sketchable sdp written equality form maximize subject submatrices relevant discussion dual problem equation may written minimize subject yjj yjj yjj lemma sketchable sdp satisfied strict inequality dual problem feasible primal problem given equation dual equation feasible duality gap dual solution attains optimal value furthermore condition equivalent slater condition proof show equivalence slater condition first statement follows automatically slater condition primal problem following satisfies constraints duality gap zero attains optimal value block matrix also need hence formulate slater condition entries observe hence satisfies constraints strict inequality converse clear bound optimal solution equation denote feasible set relaxed sdp definition notation feasible set primal problem analogously denote optimal value relaxed problem optimal value sketchable sdp lemma assume exists constraints strictly satisfied optimal solution dual problem sketchable sdp proof first inequality obvious since also lemma strong duality holds achieves optimal value hence first line holds duality take supremum infer since corollary assume exists constraints strictly satisfied max min defined lemma proof equation since follows min max corollary follows theorem assume exists constraints strictly satisfied value sketched sdp bounded max min optimal point sketchable sdp proof key step recognize scs equal cyclicity trace thus relaxed sdp gives upper bound sketched sdp theorem follows corollary note result probabilistic holds regardless sketching matrix used random feasibility problems section investigate conditions convex hull random gue section matrices shifted multiple identity contains positive semidefinite matrix cone define pointed used section let random matrices sampled independently gue means hermitian zkl real normal random variable mean variance zkl complex normal random variable mean variance since need similar matrices different variance call distribution gue consider rescaled shifted matrices call convex hull matrices conv lemma let numerical constant independent proof let definition gue gue theorem fact gue invariant unitary transformations holds expression maximized tmin obtain value see remark hence infer tmin tmin tmin tmin implies tmin positive definite assertions follows lemma let numerical constant independent proof take means shown section choose assume implies weyl perturbation theorem theorem follows assume clearly thus estimate probability assumptions met using union bound obtain using theorem infer last estimate used gue element different variance proof lemma combining estimates concerning assertion follows obtain corollary cone generated pointed high probability corollary let denote cone numerical constant independent proof know lemma lower bound probability event note event implies cone pointed cone pointed would exist implies theorem consider lmi let lmi feasible probability least moreover lmi infeasible cone generated pointed probability least proof first assertion lemma contains positive definite element probability lower bounded expression equation feasible large enough second assertion note lmi infeasible equivalent containing positive definite element lemma lower bound probability lmi infeasible moreover fact lower bound holds probability cone pointed follows corollary
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cluster algebras jan chris fraser abstract introduce cluster algebras flexible notion map cluster algebras type different coefficients definition given terms seed orbits smallest equivalence classes seeds mutation rules seeds unambiguous present examples involving familiar cluster algebras cluster structures grassmannians associated marked surfaces boundary explore related notion compare resulting group groups symmetries cluster structures cluster algebras surfaces determine subgroup inside tagged mapping class group surface introduction general structural theory cluster algebras well developed years since inception despite seem consensus right notion homomorphism cluster algebras several notions arisen different mathematical settings see perspective key difficulty defining homomorphisms cluster algebras rooted fact construction cluster algebra involves three operations addition multiplication auxiliary addition used normalization condition preexisting notions cluster homomorphism designed respect three operations rather restrictive requirement suggest instead even ordinary normalized setting fruitful consider maps preserve structures intrinsic cluster algebras ignoring auxiliary addition leads concept seed orbits mutation patterns orbits form morphisms mutation patterns main object interest call paper devoted systematic study related algebraic constructs cluster algebra defined specifying distinguished set generators called cluster variables inside ambient field rational functions variables starting initial cluster cluster variables remaining cluster variables obtained iterating algebraic steps called mutations mutation produces new cluster current one exchanging one cluster variable new one specific rules computing latter encoded two additional ingredients exchange matrix mathematics subject classification key words phrases cluster algebra seed orbit cluster modular group tagged mapping class group work supported graduate fellowship national physical science consortium nsf grant chris fraser coefficient tuple consisting elements fixed coefficient group triple consisting cluster exchange matrix coefficient tuple called seed cluster mutates ingredients also new matrix given explicitly terms new tuple satisfies constraint involving collection seeds related mutations possible directions forms seed pattern general cluster algebra setup seed patterns mutation recipe uniquely specify new coefficient tuple ambiguity propagates iterated mutations consequently set cluster variables uniquely determined initial seed usual way remove ambiguity impose additional assumption coefficient group endowed additional operation auxiliary addition making semifield require corresponding normalization condition hold every seed assumption satisfied important examples cluster algebras arising representation theory paper make use another way removing ambiguity considering seed orbits smallest equivalence classes seeds mutation rules unambiguous gives rise concept mutation pattern seed orbits pattern determined uniquely one constituent seed orbits natural notion homomorphism two mutation patterns seed orbits brings definition rational map precisely semifield homomorphism respects seed orbit structure commutes mutations though appropriate context defining nonnormalized seed patterns see two ways useful structural theory ordinary normalized seed patterns first important understand relationships cluster algebras underlying pattern exchange matrices different choices coefficients one celebrated result kind separation additions formula theorem given mutation pattern exchange matrices formula expresses cluster variables cluster algebra choice coefficients terms cluster algebra special choice principal coefficients proposition puts formula wider context every pair normalized seed patterns witnesses separation additions idea used construct new cluster algebra starting known one precisely using known cluster structure algebra one produce cluster structure another algebra describing appropriate map second naturally defined concept gives rise quasiautomorphism group seed pattern group interpolates previously defined groups either sensitive coefficients groups small refer coefficients groups large cluster algebras arising applications nontrivial coefficients cluster algebras often afford nontrivial cluster structure twist map grassmannian one important example remark forthcoming companion paper construct large group grassmannian cluster cluster algebras algebras whose action cluster variables simple description much abstract setup paper developed application mind paper organized follows section presents background seed patterns mostly standard taken emphasis notion ambient semifield definition section ends motivating example pair seed patterns illustrate various notions subsequent sections reader familiar cluster algebras skim section head directly example sections conceptual core paper define seed orbits smallest equivalence classes seeds mutation rule unambiguous proposition give explicit characterization seed orbits orbits respect rescaling action seeds section introduces seed patterns basic properties end section describing key differences preexisting notions specifically rooted cluster morphisms coefficient specializations section discuss normalized seed patterns seed patterns geometric type relate linear combinations rows extended exchange matrix making connections separation additions formula gradings cluster algebras section introduces easiest way specifying practice checking given semifield map sends cluster variables rescaled cluster variables nerve section define group seed pattern compare cluster modular group group cluster automorphisms sections focus cluster algebras associated bordered marked surfaces main result theorem describing group cluster algebra subgroup tagged mapping class group excluding exceptional surfaces particular establishes regardless choice coefficients cluster algebra group always finite index subgroup cluster modular group concept nerve introduced section new includes special case star neighborhood vertex star neighborhoods show algebraic hartogs principle argument used establish given cluster algebra contained another algebra proposition appendix extend argument star neighborhood arbitrary nerve section illustrates techniques section generalize example describing grassmannian cluster algebras polynomial rings arising coordinate rings band matrices acknowledgements would like thank ian greg muller gregg musiker helpful conversations especially thank sergey fomin many conversations suggestions work supported graduate fellowship national physical science consortium nsf grant midst carrying work learned thomas lam david speyer independently obtained results similar corollary remark chris fraser preliminaries seed patterns cluster algebra constructed set data called nonnormalized seed pattern define data fixing standard notation number let max let sign equal either according whether negative zero positive denote setup begins choice ambient field rational functions coefficients coefficient group coefficient group abelian multiplicative group without torsion ambient field field rational functions variables coefficients set expressions made elements elements using standard arithmetic operations usual notion equivalence rational expressions integer called rank definition seed let nonnormalized seed triple consisting following three ingredients matrix bij coefficient tuple consisting elements cluster whose elements called cluster variables algebraically independent freely generate restrictive notion normalized seed given definition normalized seeds much studied literature usually simply called seeds thus persistently use adjective setting although little clumsy definition labeled tree tree edges labeled integers set labels emanating vertex write indicate vertices joined edge label isomorphism labeled trees sends vertices vertices edges edges preserving incidences edges edge labels isomorhism uniquely determined value single vertex definition seed pattern let collection seeds one seed called seed pattern edge seeds related mutation direction matrices related matrix mutation bij bij sign bik bik bkj coefficient tuples related bkj bkj otherwise bkj bkj cluster algebras clusters related bjk latter called exchange relation rules ambiguous meaning determined uniquely indeed since mentions ratio one rescale common element preserving write indicate two seeds related mutation direction condition symmetric thinking recipe computing crucially observe computation operations needed motivates following definition definition ambient semifield let seed pattern one clusters ambient semifield subset elements given rational expression elements coefficients thus set rational functions built elements using operations since independent choice depends every cluster variable lies recall semifield abelian multiplicative group additional binary operation called auxiliary addition commutative associative distributes multiplication ambient semifield semifield respect multiplication addition operations justifying name homomorphisms semifields defined obvious way ambient semifield following universality property lemma definition let seed pattern coefficient group ambient semifield fix cluster let semifield given multiplicative group homomorphism function exists unique semifield homomorphism agreeing given maps following elements play prominent role section definition hatted variables let seed pattern let denote hatted variables bij obtained taking ratio two terms right hand side hatted variables adjacent seeds determine follows proposition proposition let seed pattern hatted variables edge chris fraser satisfy bkj propagation rule takes place depends matrix preceding discussion need section briefly recall definitions useful presenting examples first exchange graph associated seed pattern graph whose vertices unlabeled seeds whose edges correspond mutations seeds precisely permuting indices seed commutes mutation rules exchange graph graph obtained identifying vertices seeds permutations star neighborhood star vertex set edges adjacent rather indexed data unlabeled seed indexed seeds adjacent elements star second concrete examples paper chosen distinguished finite set elements called frozen variables coefficient group free abelian multiplicative group laurent monomials frozen variables cluster algebra associated seed pattern generated frozen variables cluster variables arising seeds example introduce pair affine algebraic varieties pair seed patterns respective fields rational functions cluster algebras associated seed patterns coordinate rings cluster algebras finite dynkin type let affine cone grassmann manifold planes points decomposable tensors coordinate ring generated coordinates extracting coefficient standard basis representing given matrix maximal minor matrix columns well known cluster structure special case cluster structure arbitrary grassmannians constructed scott frozen variables coordinates consisting cyclically consecutive columns five cluster variables listed cyclically adjacent pairs cluster variables forming clusters clusters exchange relations given figure data seed pattern determined example focusing seed whose cluster cluster algebras first fifth exchange relations figure follows exchange relations written mutating moving clockwise exchange graph mutation moves counterclockwise one swap order two terms exchange relation figure exchange graph vertices clusters edges vertices mutations mutation exchanges two cluster variables via exchange relation listed table top right extra data seed inferred exchange relations second let affine space band matrices form coordinate ring contains minors evaluating returns minor occupying rows columns minors factor figure shows seed pattern whose cluster algebra frozen variables following minors cluster variables listed cyclically adjacent pairs forming clusters seed orbits introduce seed orbits first describing equivalence classes certain equivalence relation seeds proposition gives another characterization orbits explicit rescaling action chris fraser figure exchange graph exchange relations mirroring figure definition let sequence elements choosing base point sequence determines walk say contractible walk starts ends vertex given seeds write contractible sequence mutations contractible sequence seeds clearly equivalence relation seeds furthermore removes ambiguity present mutation seeds lemma mutation rule becomes unambiguous involutive thought rule equivalence classes seeds fixing class direction set seeds class seeds characterize classes explicitly say two elements proportional written emphasize include constants thus proportional etc proposition seed orbits let seeds rank following equivalent cluster algebras exist scalars equations define rescaling action seeds seed proposition denoted says class seeds precisely orbit action henceforth refer equivalence classes seed orbits proof conditions immediate calculation show implies defining seed orbit implication follows fact seed orbits closed precisely seed orbit two seeds satisfying seed orbit proposition know claim follow check obvious mutation says bjk desired returning implication symmetry follows since seed orbits closed related along contractible sequence mutation seed related contractible sequence mutations therefore seed orbit show implies let denote vector entries equal define similarly clearly suffices show since rescalings type generate seeds form equivalent follows mutating twice direction seeds form let seed satisfying follow calculation admissible since already know rescaling preserves equivalence seeds since chris fraser amounts mutating direction twice seed equivalence classes follows desired give definition seed pattern another seed pattern retain notation section data use bars denote analogous quantities second pattern thus coefficient group ambient field seeds hatted variables built second copy tree motivating observation following since mutation rules certain algebraic relations preserved homomorphism semifields definition let seed patterns let semifield homomorphism satisfying case say preserves coefficients say maps seed seed seed way compatible mutation precisely let isomorphism labeled trees triple obtained evaluating motivation definition imagine situation well understood combinatorially would like understand another seed pattern comparing requirement says seeds mutate parallel seeds sense corresponding seeds differ rescalings following propositions show two ways proofs follow immediately observation applying semifield homomorphism commutes mutation proposition let semifield homomorphism satisfying rather checking holds every suffices check single proposition let let seed let seed satisfying proposition says preserves seeds thus denotes seed orbit ditto maps inside therefore natural notion homomorphism respective seed orbit patterns describe pair seed patterns example cluster algebras example given let denote rows surjective map varieties sending determines map cluster algebras sending ijk figure shows seed pattern arises applying figure factoring cluster variables inside seeds figure seed orbit corresponding seeds figure thus figure seed pattern obtained applying seed pattern figure clusters agree clusters figure frozen variables listed cancelling common frozen variable factors sides exchange relations yields exchange relations figure follows values figures definition two called proportional seeds say proportional identity map say one another two seed patterns think essentially coefficients maps directions allows write cluster variables one seed pattern terms cluster variables one chris fraser remark set seed patterns morphisms category proportionality equivalence relation morphisms category equivalence relation respects composition yields quotient category whose objects seed patterns whose morphisms proportionality classes morphism quotient category isomorphism one hence constituent following lemma provides simple method checking candidate map given lemma let semifield map preserves coefficients cluster variables lemma follows general proposition fact suffice merely check lying nerve definition example using lemma describe homomorphism example let morphism sending band matrix whose entries coordinates coordinate ring map sends coordinate entry matrix interesting property minors monomials coordinates particular maximal minors agree multiplicative factor thus cluster variable since preserves coefficients nontrivial check lemma follows remark defined map ambient semifields since maps transparently preserve mutation rules suitable purposes since one mostly interested evaluating cluster variables coefficients however cluster algebra familiar algebraic object associated seed pattern one wants think algebra map cluster algebras one sometimes need first localize frozen variables close section explaining differences preexisting notions homomorphisms cluster algebras specifically consider notion rooted cluster morphism category cluster algebras described assem dupont shiffler also coefficient specialization defined fomin zelevinsky studied reading key difference cluster algebras notions allows cluster variables rescaled element extra flexibility provides freedom constructing new cluster algebras old ones section finding nice cluster algebras giving rise elements cluster modular group section little detail coefficient specialization map whose underlying map coefficients group homomorphism must send cluster variable cluster variable thus coefficient specialization special one since cluster variables allowed rescaled elements rooted cluster morphisms require choosing pair initial seeds sense morphism rooted pair seeds morphism algebra map sends cluster variable either cluster variable integer sends frozen variable either frozen variable cluster variable integer hence flexible allowing cluster variables rescaled frozen variables sent monomials frozen variables also less flexible allow unfreezing frozen variables specializing variables integers probably would hard combine two notions one technicality streamline discussion formulated definition preserve exchange matrices whereas rooted cluster morphisms allow make definition consonant preexisting notions one could modify say either opp see definition opposite seed section without significant changes normalized seed patterns section recall definition normalized seed patterns apply results section case normalized definition normalized seed pattern seed pattern definition called normalized coefficient group semifield coefficient tuple satisfies addition advantage normalization condition makes mutation rule therefore mutation normalized seeds unambiguous indeed specifies ratio terms unique choice ratio satisfying normalization condition namely pair furthermore mutating twice given direction identity time disadvantage computing cluster algebra involves three operations two operations present along definition prioritizes first two operations proposition says case normalized seed patterns separation additions phenomenon separating addition one chris fraser stating proposition say little normalized seed patterns normalized seed pattern tuple ratios determines coefficient tuple accordingly normalized seed patterns one keeps track rather rewriting terms determines recurrence semifield bkj collection quantities satisfying lying semifield called semfield notice concept semifield playing two different roles either ambient semifield exchange relation calculations take place coefficient semifield used remove ambiguity mutation seeds surprising connection two roles lemma recognize saying form ambient semfield important example coefficient semifield arising applications tropical semifield definition tropical semifield free abelian multiplicative group generators auxiliary addition given min normalized seed pattern tropical semifield said geometric type case denoting frozen variables data entirely described matrix bij called extended exchange matrix top submatrix called principal part bottom rows called coefficient rows specified equality mutation rule translates rule seed pattern figure geometric type tropical semifield frozen variables holds seed pattern figure frozen variables hand seed pattern figure normalized first exchange relation satisfies state proposition describing normalized seed patterns arose process writing forthcoming book cluster algebras proving one direction finite type classification namely cluster algebra quiver whose principal part orientation dynkin quiver necessarily finitely many seeds state recipe constructing normalized seed pattern given one since envision useful applications proposition let seed pattern usual notation let fixed initial cluster let semifield semifield rational expressions algebraically independent elements coefficients cluster algebras let semifield map satisfying let composition semifield maps second map composition specializes identity normalized seed pattern seeds satisfying clearly proof formulas follow applying proposition renormalizing scalars massaging formulas given alternatively also straightforward check right hand sides satisfy required recurrences directly carried example beginning seed pattern figure one construct normalized seed pattern figure first applying semifield map obtaining nonnormalized seed pattern figure normalizing semifield map map agrees frozen variables sends cluster variable frozen variable monomial dividing geometric type constructing sends one seed seed orbit another seed matter linear algebra corollary let seeds geometric type frozen variables respectively let respective seed patterns let determines monomial map let denote matrix exponents monomial map thus matrix satisfying extended exchange matrices related particular exists principal parts agree integer row span contains integer row span proof indeed entry left hand side encodes exponent entry right hand side encodes exponent follows final statement follows studying interesting rows bottom rows rows determines particular linear combination rows linear combinations prescribed arbitrarily prescribing exponent using lemma chris fraser remark exchange graphs separation additions formulas show exchange graph covers particular corollary rows span exchange graph corresponding cluster algebra covers exchange graph every cluster algebra underlying exchange matrix natural generalization separation additions formula theorem case quiver principal coefficients whose rows span namely let pair normalized seeds exchange matrix suppose principal coefficients natural choice maps mapping proposition defined choice formula becomes separation additions numerator theorem evaluating polynomial applying semifield homomorphism denominator specializing cluster variables evaluating polynomial applying semifield map remark proportionality gradings let seed pattern geometric type recall briefly concept choosing initial seed choice grading determined grading matrix satisfying ith column determines grading vector condition guarantees every exchange relation homogeneous respect turn defines adjacent cluster variable thereby adjacent grading matrix seen adjacent grading matrices satisfy left kernel condition grading propagates entire cluster algebra cluster variables coefficients homogeneous suppose given two seeds geometric type notation corollary let pair proportional obtain matrices implies defines first rows define trivial grading conversely fixing matrix choice matrix provides proportional whose matrix remark simplicity stated corollary terms row spans similar statement holds row spans one enlarges tropical semifield puiseux tropical semifield consisting puiseux monomials rational exponents frozen variables unpleasant perspective cluster algebras coordinate rings perfectly fine one interested writing algebraic formulas cluster variables etc foreshadowed work sherman zelevinsky section discusses rank cluster algebra exchange matrix authors write cluster variables cluster algebra matrix terms cluster variables formulas involve puiseux monomials frozen variables cluster algebras nerves proposition check given semifield map suffices check pair seeds proposition means checking furthermore holds envision applications checking proportionality condition cluster variables easy done many seeds checking equality exchange matrices inconvenient goal section give criterion guarantees checking proportionality conditions relevant concept nerve seed pattern definition let seed pattern nerve connected subgraph every edge label arises least basic example nerve star neighborhood vertex believe many theorems form property holds nerve holds entire seed give example theorem appendix generalizing starfish lemma proposition star neighborhood nerve stating result section need address mostly unimportant difference seed opposite seed say seed indecomposable underlying graph described exchange matrix vertex set vertices joined edge connected seed opposite seed opp popp xopp seed defined opp popp xopp opp satisfies operations restricting indecomposable component replacing seed opposite seed commute mutation proposition let seed patterns respective ambient semifields suppose seeds indecomposable let semifield homomorphism preserves coefficients satisfies every vertex label opp particular applying proposition star neighborhood vertex check suffices check lemma well checking adjacent edge follows proof choose vertex hypothesis left checking suppose edge scalar exchange relation defining bjk hand applying relation defining rearranging yields bjk chris fraser abbreviating two terms right hand side two terms see algebraic independence seed either refer case case respectively inspection see ratio thus case deduce matrices column opp case deduce thing replace apply lemma lemma let two whose matrices indecomposable let nerve suppose every vertex label edge one following holds bjk bjk ykopp bopp bjk opp accordingly roughly two issues first question whether checked nerve second whether dealing opp relies indecomposability proof given pair necessarily refer column seed equations say either opposite edge pick vertex edge incident necessary replace opp given seek prove let edge nerve incident mutation rules involutive property seed depends seed since given agree see agree agree repeatedly apply observation mutating possible directions nerve preserving fact coincide coincide since nerve connected every edge label shows least conclude either opposite connectedness hypothesis assures fact cluster modular group given seed pattern definition one think choice map describing automorphism pattern seed orbits associated use define variant group automorphisms generalizing group cluster automorphisms defined seed patterns trivial coefficients retaining many properties cluster automorphisms proposition corollary proposition cluster algebras following example illustrates notion general naive notion semifield automorphism preserving seed orbit pattern example automorphism semifields consider composition example proportional identity map rescales variable product frozens ambient semifield grading every variable degree one every homogeneous element image degree multiple thus surjective definition group set proportionality classes automorphism group quotient category discussed remark remark let call trivial proportional identity map set trivial monoid usually group composition composition example bears witness one way construct proportional given compose trivial tempting try define purely terms thse trivial without mentioning proportionality however relation defined neither symmetric transitive one form quotient category using relation write geometric type specified initial matrix remark two proportional ratio defines taking exponents elements obtain elements fixing particular number degrees freedom specifying another proportional therefore corank lemma let seed pattern geometric type thus geometric type every determines element proof lemma two mutation class related pair unimodular integer matrices gln corollary pair vertices sending seed orbit seed orbit principal parts agree row span contains row span unimodularity mutation criterion preserved swapping roles row span contains row span fact two row spans equal submodules remark marsh scott described version twist grassmannian cluster algebras one show using corollary recall definitions preexisting groups automorphisms associated seed pattern namely chris fraser cluster modular group cmg fock goncharov group aut automorphisms category rooted cluster algebras defined assem dupont schiffler first present definitions discuss particular example groups computed compared group definition cluster modular group definition let seed pattern exchange graph cluster modular group cmg group graph automorphisms aut preserve exchange matrices precisely recall unlabeled seed vertex indexed elements star element cluster modular group graph automorphism aut satisfying star graph automorphism determined choosing pair vertices identification star star remark definition terms automorphisms exchange graph cluster modular group appears depend entire seed pattern underlying exchange matrices however widely believed exchange graph therefore cluster modular group fact independent choice coefficients depends exchange matrices therefore prescribed giving single matrix proven exchange matrices group subgroup cluster modular group indeed determines cluster modular group element via proportional determine since preserves exchange matrices evaluating commutes permuting cluster variables seed element produced way indeed element cluster modular group one also consider automorphisms category cluster algebras defined reproduce version definition sake convenience definition let seed pattern say two seeds similar coincides first permuting frozen variables permuting indices appropriately suppose exchange matrices indecomposable let cluster algebra map automorphism every equivalently seed opp similar seed denote group automorphisms aut elements aut similar strong isomorphisms slightly general since one allowed permute frozen variables say definition direct automorphism inverse automorphism according whether opp seed let aut denote subgroup direct automorphisms similar reasoning theorem subgroup index two aut seed otherwise aut important special case definition trivial coefficients case group aut group cluster automorphisms trivial cluster algebras coefficients cmg furthermore direct cluster automorphism case summarize containments preceding groups cmg etriv seed pattern etriv seed pattern obtained trivializing coefficients equality cmg etriv depends belief cmg cmg etriv remark group aut etriv contains groups group aut sit nicely rest containments aut next illustrate differences groups using particular cluster algebra associated bordered marked surface basic notions references concerning class cluster algebras given section example let annulus two marked points boundary component figure colored marked points either black white aid describing automorphism groups figure annulus two marked points boundary component right show flat form annulus obtained cutting along dashed line cluster modular group cmg cluster algebra associated annulus coincides mapping class group annulus see proposition group following explicit description let isotopy class homeomorphism rotates inner boundary annulus clockwise let clockwise outer boundary let homeomorphism represented degree turn flat form annulus swaps inner outer boundary components elements generate cluster modular group group presentation cmg respect generators central extension cmg infinite dihedral group using map figure gives choice lamination triangulation well quivers let corresponding cluster algebra frozen variable seed pattern ambient semifield cluster modular group element permutes arcs induces automorphism quiver therefore element aut chris fraser figure lamination consisting two copies curve annulus determining single frozen variable also drawn triangulation annulus arcs quivers shown right extra arcs values quiver read laurent monomial incoming variables divided outgoing quivers neither isomorphic opposite strong automorphism sending seed seed likewise strong automorphism however relating seeds describe semifield map defined well sends corresponding defining whose map seed orbits simple global description cluster variables checked inductively performing appropriate mutations away namely arc let denote number times crosses two curves example arcs annulus power right hand side always equal also simple describe realizing perhaps surprisingly seeds related indeed values equal top bottom vertices figure equal holds putting together one nontrivial strong automorphism namely element hand group infinite generated direct product index two subgroup cmg namely kernel map cmg computes parity number black marked points sent white marked point example suggests although cluster modular group cmg may strictly larger group gap groups large indeed section establishes seed patterns associated surfaces always finite index subgroup cluster modular group cluster algebras cluster algebras surfaces section place example context via results valid cluster algebra associated marked bordered surface describe quasiautomorphisms cluster algebras terms tagged mapping class group marked surface follow setup notation let denote cluster algebra geometric type determined triple oriented bordered surface nonempty set marked points marked points reside either interior call punctures call cilia set punctures disallow possibilities namely sphere three fewer punctures monogon choice coefficients specified integral unbounded measured laminations lamination consists finite number curves cluster variables indexed tagged arcs set denote seeds indexed tagged triangulations extended exchange matrix seed signed adjacency matrix principal part shear coordinate vector lamination respect ith row coefficients exchange graph resulting cluster algebra independent choice coefficients corollary let cmg denote corresponding cluster modular group closely related following geometrically defined group definition tagged mapping class group let bordered marked surface closed surface exactly one puncture tagged mapping class pair element mapping class group orientiationpreserving homeomorphism mapping setwise considered isotopies fix pointwise function set punctures closed surface one puncture make definition impose since tagged versions arcs cluster algebra tagged mapping classes comprise tagged mapping class group denoted understand action tagged arcs tagged mapping class acts first performing homeomorphism changing tag end incident puncture resulting action tagged triangulations preserves signed adjacency matrices embeds subgroup cmg fact following true proposition tagged mapping class group coincides cluster modular group cmg unless sphere four punctures square digon one two punctures thus barring exceptional cases two tagged triangulations isomorphic quivers precisely related element tagged mapping class group see proposition subsequent discussion see also conjecture chris fraser exceptional cases listed proposition tagged mapping class group proper finite index subgroup cluster modular group motivated proposition set describe various choices coefficients group definition subgroup tagged mapping class group main ingredient answer coloring similar one examples section definition even components punctures well boundary components even number cilia let denote number even components label even components even boundary component color cilia black white colors alternate adjacent cilia opposite colors using coloring definition tagged mapping class determines signed permutation matrix whose entries indexed even components entry unless boundary component entry sends black cilia black cilia sends black cilia white cilia punctures sign entry sign signed permutation matrices arise way since permute components number cilia definition let lamination curve figure shows assign sign end either lands even boundary component spirals around puncture puncture sign chosen according whether spirals counterclockwise clockwise puncture boundary component sign chosen according whether nearest neighboring cilium clockwise direction along black white end odd component zero sign pairing lamination even component sum signs associated sum curves signs two ends let denote vector pairings even components example works signs annulus example figure conventions assigning signs end curve lands even boundary component case boundary component cilia spirals around puncture pairing obtained adding signs addition acting tagged arcs also acts laminations tagged mapping class acts first performing homeomorphism changing direction spiral puncture action preserves sheard coordinates sense triangulation cluster algebras lamination easy see acts vector pairings matrix lamination next theorem main result section describing qaut inside marked mapping class group concrete terms theorem suppose one four exceptional surfaces proposition let let span submodule spanned vectors pairings associated laminations prove theorem section subgroup described depends endpoint behavior laminations mention topology surface much curves wrap around holes handles surface map group homomorphism group signed permutation matrices subgroup inverse image subgroup signed permutation matrices fix therefore always finite index corollary let tagged mapping class plus minus identity matrix choice otherwise choice remark tagged mapping classes corollary fix even components setwise furthermore either preserve coloring ends curves simultaneously swap colors group generated following four types elements see generators mapping class group dehn twists simple closed curves homeomorphisms permute odd components fractional dehn twists rotating cilia given boundary component two units tagged rotation last element one simultaneously changes tags punctures rotates boundary components one unit studied shown coincide shift functor cluster category associated surface proof clearly preserves regardless choice theorem think signed permutation usual way index let lamination consisting curve two black ends satisfying index choose pair indices case let lamination consisting curve connecting even components curve black ends two cases see span theorem example order boundary components figure inner boundary first vector pairings lamination figure act vector parings swapping sign first second component respectively acts permuting first second component description example matches one theorem subgroup elements described corollary direct product generated index cluster modular group chris fraser remark theorem modified case one exceptional surfaces proposition namely left hand side merely describes subgroup consisting tagged mapping classes ignoring exotic symmetries particular choices coefficients extra elements cluster modular group might also inside proofs section key result section proposition describing cluster algebras surfaces theorem follows special case let denote set boundary segments connecting adjacent cilia especially natural choice lboundary one frozen variable boundary segment see remark lemma expresses shear coordinates certain laminations terms extended exchange matrix determined lboundary patterned lemma lemma let lamination none whose curves end spirals puncture given arc transverse measure minimal number intersections curves denote boundary segment similarly let denote number ends curves let row vector containing transverse measures lboundary proof check components left right hand sides equal let quadrilateral containing number clockwise order may boundary segments time shears across quadrilateral crossing contributes left hand side contributing right hand side argument like lemma simpler dealing spirals puncture comparing see none whose curves spiral punctures lboundary version lemma allowing spirals punctures would involve extended exchange matrix fully opened surface lifts opened surface see sections details corresponding version determines next step describe row span signed adjacency matrices strengthens theorem states corank number even components description requires associating sign ends arcs similar fashion done ends curves definition namely endpoint boundary component endpoint gets sign endpoint black white cilium respectively endpoint puncture sign end plain tagged respectively endpoint odd component gets sign pairing sum signs ends reside vector pairings pairing satisfies cluster algebras elementary lamination determined definition see also section lemma tagged triangulation let space row vectors entries indexed let dual space column vectors entries indexed dual basis even component consider column vector vector row span dot product vanishes said differently map even component determines cluster algebra gradings form standard varies standard grading one spans kernel see rely gradings follows lemma proved end section vector residue around dot product shear coordinate lamination residue writing following simple description lemma let lamination even component residue around pairing definition proof residue computed terms shear coordinates arcs adjacent compute shear coordinates rather considering entire surface focus set triangles least one vertex lifting finite cover perhaps order remove interesting topology nearby irrelevant computing residue union triangles either triangulated annulus boundary component puncture call set triangles annnular neighborhood even consists single curve intersection annular neighborhood might consist several curves linearity shear coordinates residues suffices consider case consists single curve annular neighborhood puncture annular neighborhood punctured disc triangulation whose arcs radii joining puncture boundary disc inspection curve contributes nonzero residue spirals value residue according whether spirals counterclockwise clockwise respectively claimed even boundary component compute residue using right hand side split right hand side two terms splitting concatenation performing matrix multiplication block form first term expression zero residue around since linear combination rows left sum chris fraser claim sum evaluates sum nonzero must nonzero end segment clockwise order segment contained unique triangle call two sides triangle whose endpoint respectively cases according whether either sides boundary segment neither according whether white black total contribution degenerate case boundary segment contribute effect cancelled proposition suppose among four listed exceptions proposition let recall submodules theorem let tagged mapping class corresponding signed permutation matrix following equivalent whose map tagged triangulations proof proposition determined pair tagged triangulations related since principal parts matrices agree proposition tagged mapping class furthermore lamination vector must span since lemma equivalent find linear combination zero residue around every even component proposition follows lemma fact acts vector pairings matrix proof lemma restating lemma seek show form basis dual space row begin verifying vectors pair zero row span first check puncture begin case arcs untagged need check vanishes indeed letting lamination consisting tiny simple closed curve contractible shear coordinate vector clearly apply choice component says desired using fact boundary segment argument arcs tagged identical incident exactly two arcs namely plain tagged version arc follows definition calculation digon second check boundary component number cilia let tiny lamination contractible two endpoints two boundary segments adjacent clearly particular black white summing corresponding right hand sides performing matrix multiplication block form argument lemma terms corresponding boundary segments present sum black sum white canceling common cluster algebras terms get equality black white says desired thus pair zero row span show linearly independent completes proof since expected size theorem consider linear relation form define scalars marked points follows puncture cilium residing even component sign consistent coloring cilium odd component set consider vertices forming edge triangulation claim indeed endpoints arc component relation construction holds endpoints boundary segment clearly holds however given triangle vertices way hold pairs varying vertex triangle containing establishes hence desired appendix starfish lemma nerve give appropriate generalization starfish lemma proposition star neighborhood nerve proof follows proof starfish lemma appropriate modifications let domain say two elements coprime contained prime ideal height unique factorization domain every pair irreducible elements coprime proposition let nerve let noetherian normal domain let seed pattern geometric type satisfying following frozen variables vertex cluster cluster variables pairwise coprime elements edge cluster variables pairwise coprime cluster algebra defined satisfies proof relies following two lemmas first standard fact commutative algebra prime ideal let denote localization away lemma theorem normal noetherian domain natural inclusion intersection height one primes equality chris fraser lemma hypotheses proposition let height one prime ideal least one products proof coprimeness cluster one cluster variables product satisfies show least one none cluster variables establishing claim since prime pick vertex suppose cluster variable given edge cluster variable coprimality assumption cluster repeatedly applying assumption mutating along nerve using connectedness hypothesis fact every edge label shows nerve finally arrive vertex edge extended cluster variables coprimeness assumption along edge see cluster one product proof proposition need prove cluster variable lemma suffices show height one prime indeed lemma cluster laurent phenomenon laurent polynomial elements coefficients particular desired appendix grassmannians band matrices illustration proposition extend constructions example example case general let affine cone grassmannian subspaces points decomposable tensors let affine space band matrices width set matrices whose entries zero unless describe coordinate rings particular cluster structure appears new coordinate ring ring generated coordinates ranges cluster structure geometric type frozen variables coordinates consisting cyclically consecutive columns coordinates cluster variables introduce useful sign convention set natural numbers let denote coordinate obtained first reducing elements least positive residue modulo sorting residues taking corresponding coordinate fewer distinct elements modulo identically zero cluster algebras coordinate ring contains minors subsets size denoting row column indices respectively polynomial ring coordinate functions following elements serve frozen variables let denote corresponding tropical semifield frozen variables example morphism varieties sending decomposable tensor rows map coordinate rings defined letting coordinate one sees irreducible minor map bijection coordinates irreducible minors example morphism varieties sending band matrix whose whose entry certain coordinate evaluated since coordinate right hand side nonzero unless indeed point map coordinate rings theorem let varieties let semfield expressions abusing notation let semfield map determined map let map determined cluster algebra geometric type maps irreducible minors cluster frozen variables proof proof relies following properties cluster structure exist clusters consisting entirely coordinates called clusters every coordinate shows least one clusters set clusters connected one another mutations whose exchange relations short relations form sij denotes union certain clusters every neighboring cluster cluster first two facts consequences technology plabic graphs square moves third fact follows considering particular explicit seed grassmannian one whose quiver grid quiver chris fraser state theorem carefully let cluster coordinate let irreducible minor related see algebraically independent generators assuming proved applying construction proposition obtain semifield map normalized seed whose cluster variables semifield map satisfies defined key claim fact holds therefore semifield map depend key claim follows seeds related mutation using proposition furthermore every irreducible minor cluster variable since includes shows indeed seeds seed expected size necessary form transcendence basis seeds related mutation union clearly contains generating set field fractions cluster algebra resulting seed pattern clearly contains opposite containment follows algebraic hartogs argument starfish section using fact thus remains check key claim fact suffices check preserves short relations sik sij sjk verifying direct piecewise check exponent left hand side according whether neither one sik contain interval performing similar computation sij well sjk taking minimum respective answers gives exponent right hand side claim left right hand exponents always equal done case analysis let largest number sijkl sides return otherwise sides return return return otherwise sides return return return otherwise similar calculation checks exponents match sides finally check lemma need see preserves coefficients proportional identity suffices check every follows determinantal identity lemma applied lemma let consecutive subset subset size notice first product right hand side monomial frozen coordinates cluster algebras proof proceed induction clear need relation see section exercise let assuming holds smaller values expand along first row see result follows using remark case construction motivating example considerd yang zelevinsky establish homogeneous coordinate ring certain bruhat cell dynkin type cluster algebra principal coefficients elements double bruhat cell band matrices width example follows setting certain frozen variables equal setting equal latter operation permissible frozen variables isolated vertices quivers also remark already known grassmannian cluster algebras polynomial ring fairly uninteresting indeed realize affine space matrices closed subvariety defined specializing frozen variable specialization resulting cluster structure polynomial ring unrelated one given section references assem dupont schiffler category cluster algebras pure appl algebra assem schiffler shramchenko cluster automorphisms proc lond math soc berenstein fomin zelevinsky cluster algebras iii upper bounds double bruhat cells duke math bridgeland smith quadratic differentials stability conditions publ math inst hautes tudes sci qiu tagged mapping class groups translation math chang zhu cluster automorphism groups automorphism groups exchange graphs chang zhu cluster automorphism groups cluster algebras coefficients farb margalit primer mapping class groups princeton mathematical series princeton university press princeton chris fraser fock goncharov cluster ensembles quantization dilogarithm ann sci norm fock goncharov moduli spaces local systems higher theory publ math inst hautes etudes sci fomin pylyavskyy tensor diagrams cluster appear adv math fomin shapiro thurston cluster algebras triangulated surfaces part cluster complexes acta math fomin thurston cluster algebras triangulated surfaces part lambda lengths fomin williams zelevinsky introduction cluster algebras preparation fomin zelevinsky cluster algebras foundations amer math soc fomin zelevinsky cluster algebras finite type classification invent math fomin zelevinsky cluster algebras coefficients compos math fraser preparation fulton young tableaux applications representation theory geometry london mathematical society student texts cambridge university press cambridge gehktman shapiro vainshtein cluster algebras poisson geometry mosc math grabowski graded cluster algebras grabowski launois graded quantum cluster algebras application quantum grassmannians proc lond math soc cerulli irelli keller plamondon linear independence cluster monomials cluster algebras compos math marsh scott twists coordinates dimer partition functions matsumura commutative ring theory second vol cambridge studies advanced mathematics cambridge university press cambridge lated japanese reid reading universal geometric cluster algebras math reading universal geometric cluster algebras surfaces trans amer math soc scott grassmannians cluster algebras proc london math soc sherman zelevinsky positivity canonical bases rank cluster algebras finite affine types moscow math yang zelevinsky cluster algebras finite type via coxeter elements principal minors transform groups department mathematics university michigan ann arbor usa address cmfra
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enumerating graded ideals graded rings associated free nilpotent lie rings jun seungjai lee christopher voll abstract compute zeta functions enumerating graded ideals graded lie rings associated free lie rings nilpotency class apply computations obtain information reduced topological zeta functions particular pertaining degrees special values introduction enumerating graded ideals graded lie rings let ring integers number field completion ring nonzero prime ideal let nilpotent algebra nilpotency class free finite rank lower central series plqqci set lpiq plq plq associated graded algebra grl lpiq grl finite index grl graded homogeneous generated homogeneous elements equivalently lpiq case write grl define graded ideal zeta function dirichlet generating series psq grl enumerating graded ideals grl finite index grl complex variable assumptions guarantee psq converges complex similar definition slightly general setting see section assume ring integers number field nonzero prime ideal specpoq write completion complete discrete valuation ring characteristic zero residue field cardinality say primary decomposition yields euler product psq psq ppspecpoq psq infinite product local expressing global zeta function ones slight abuse notation denote specpoq set nonzero prime ideals individual euler factor rational function parameter fact euler product cone integrals sense theorem results paper regarding factors product date june mathematics subject classification key words phrases graded ideal zeta functions free nilpotent lie rings local functional equations seungjai lee christopher voll analytic properties apply fact analysis restricted case insubstantial conclusion main results present paper concerned graded ideal zeta functions free nilpotent lie rings finite rank given let free nilpotent lie ring nilpotency class lie generators one may identify quotient free algebra generators term pfd lower central series quotient pfc pfc given witt function piq denotes function satz hence rkz pfc piq given commutative ring paper always form write prq considered algebra graded ideal zeta function free abelian lie ring poq enumerating finite index well known equal dgrpoq psq denotes dedekind zeta function proposition euler product reflects euler product psq ppspecpoq dgrpoq psq ppspecpoq psq say ppspecpoq second author computed ideal zeta functions enumerating ideals finite index rings poq paper discusses case computations carry mutatis mutandis case general number rings compute graded ideal zeta functions dgrpoq psq section section current paper paper involved result computation section graded ideal zeta function poq end compute explicit formula psq valid finite extension integers prime viz local ring form nonzero prime ideal lying theorem exists explicitly determined rational function qpx primes finite extensions residue cardinality psq may written enumerating graded ideals free nilpotent lie rings polynomial degree qrx polynomial degree rational function satisfies functional equation note witt function values conjecture proof theorem yields sum explicitly given summands listed essentially section reproduce final outcome summation numerator fills several pages record however several corollaries explicit formula first corollary concerns analytic properties global zeta function psq corollary global graded ideal zeta function psq converges repsq may continued meromorphically repsq follows two observations firstly product ppspecpoq product finitely many translates dedekind zeta function psq abscissa convergence may continued meromorphically whole complex plain secondly product ppspecpoq may continued meromorphically repsq indeed ipi finite index set ppspecpoq may continued meromorphically repsq maxt lemma follows inspection second corollary concerns reduced graded ideal zeta function red qpy section concept introduced albeit context graded ideal zeta functions expect hoc definition fit general definition reduced graded ideal zeta functions along lines corollary reduced graded ideal zeta function red satisfies red red seungjai lee christopher voll red zry equal seems remarkable simple pole order red nonnegative unimodal coefficients consistent speculation fact series graded algebra associated natural way remark note right hand side lowest terms palindromic symmetry coefficients red implied functional equation third corollary concerns another limiting object local graded ideal zeta functions psq viz topological graded ideal zeta function top psq section rossmann introduced studied topological zeta functions associated range dirichlet generating series informally speaking topological zeta functions may viewed suitably defined limits local zeta functions whilst logical versions zeta functions psq yet studied specifically one may define top psq coefficient expansion psq definition corollary topological graded ideal zeta function top psq satisfies top psq top psq top psq zrss defined number features topological reduced graded ideal zeta functions associated considered seem remarkable numerical values following corollary presented illustrate general conjectures section extracted paper explicit computations degree rational function qpy mean degy degy degy corollary degs top psq rkz enumerating graded ideals free nilpotent lie rings top red top psq psq connection reduced graded ideal zeta function series graded algebras hold rational number part currently interpretation interpreted terms multiplicity associated graded algebra instance section background motivation methodology recall nilpotent lie algebra ring integers number field ideal zeta function dirichlet generating series enumerating finite index viz psq psq ppspecpoq complex variable nonzero prime ideal psq enumerates lpop ideal zeta functions nilpotent lie rings relatives graded ideal zeta functions studied present paper one main results introduced former zeta functions establishes rationality euler factors psq theorem numerous examples ideal zeta functions nilpotent lie rings see mal cev correspondence ideal zeta functions nilpotent lie rings closely related normal subgroup zeta functions enumerating finite index normal subgroups finitely generated nilpotent groups particular given almost finitely many euler factors psq coincide normal subgroup zeta function free group section study normal subgroup growth free nilpotent groups connected enumeration finite isomorphism informally speaking graded ideal zeta functions nilpotent lie rings may seen psq approximations ideal zeta functions indeed almost euler factors actually enumerate sublattice lattice ideals enumerated psq general approximation quite coarse nilpotency class however problems computing ideal zeta functions graded ideal zeta functions closely related following example shows seungjai lee christopher voll example assume nilpotent class isolated commutator ideal rko essentially lemma psq whereas psq special cases proximity explains proximity explicit formulae recorded theorems current paper higher nilpotency classes aware simple parallels realm free nilpotent lie rings class greater two explicit computations ideal zeta functions seem unfeasible particular know formulae ideal zeta functions lie rings poq poq numerous questions regarding ideal zeta functions analogues regarding approximating graded ideal zeta functions one may speculate latter easier answer former grunewald segal smith formulate example conjecture would imply exists qpx almost finite exa rational function consequence known hold tensions poq psq wide open general including cases conjecture verified explicit computations current paper particular may viewed graded version conjecture rossmann formulates number conjectures certain special values padic topological zeta functions pertaining particular ideal zeta functions nilpotent lie rings conjectures graded counterparts theorem second author proved local functional equation generic euler factors ideal zeta functions poq psq upon inversion prime results suggest phenomenon also appears graded ideal zeta functions free nilpotent lie rings conjecture computations owe feasibility fact parameter values considered viz enumeration graded ideals equivalent enumeration various flags lattices free depend linear fashion lattices elementary divisor types computations rely simple polynomial formula due birkhoff numbers given type finite given type proposition obtain closed formulae relevant graded ideal zeta functions need organize enumeration infinitely many values birkhoff formula manageable way meet challenge organizing pairs partitions encoding two lattices elementary divisor types overlap type formally one finitely many multiset permutations viz words two letters multiplicity section instance led consider specific words length alphabet enumerating graded ideals free nilpotent lie rings table restricting relevant enumerations fixed word yields formulae products coefficients igusa functions section theorem compute formulae words question similar methodology applied compute ideal zeta functions heisenberg lie rings number rings free nilpotent lie rings class able reduce computations combinatorial considerations partitions essentially owed fact parameters considered automorphism groups free lie rings act transitively lattices given elementary divisor types relevant sections allows assume relevant lattices generated multiples elements hall bases section lattice may always generated multiples linear basis follows course elementary divisor theorem subsets hall bases may used lucky consequence existence many automorphisms explain automorphism group poq contains copy gld poq allows perform arbitrary invertible transformations lie generators poq observation already exploited proof theorem conjunction birkhoff formula sufficient compute graded ideal zeta function poq section lie commutators rxi linear transformations act via exterior square representation general image natural map gld poq glpdq poq rather small map surjective however isomorphism facilitates computations copy poq even induces full automorphism group generated exploit notation given write rns write given subset write denote given denote interval interval write power set set notation subset indicates similarly denotes partition parts write compact discrete valuation ring characteristic zero finite extension ring integers equivalently ring form completion ring integers number field nonzero prime ideal write cardinality residue field residue characteristic set complex variable dedekind zeta function general preliminaries let set following analogous lemma lemma let prime finite extension let seungjai lee christopher voll psq psq proof graded additive sublattice fpoq determines determined sequence fpoqpiq rcs fpoq proves first equality second follows definition given noting birkhoff formula given pair partitions let denote number torsion type contained fixed torsion type clearly unless notice pnq following explicit general formula attributed birkhoff proposition birkhoff let partitions dual partitions zrqs igusa functions functional equations definition definition let given variables set qpy ipi qpy ipi set qpy note functional equations proposition make repeated use remark symmetry centres resp functional equations depend enumerating graded ideals free nilpotent lie rings throughout paper always substitute hence write pxq instead pxq instead igusa functions ubiquitous theory zeta functions groups rings note instance psq ppq qjprds hall bases free nilpotent lie rings let lie generators hall basis may constructed selecting inductively certain basic commutators see details note piq denotes witt function table record hall bases relevant current paper adopt standard abbreviation xir commutators xir case resp write instead hall basis xyx xzx yzx xyy xzy yzy xzz yzzu xyx xyyu xyx xyy xyxx xyyx xyyyu pwd piqqiprcs pdq table hall bases reduced zeta functions evseev introduced certain limit various local zeta functions informally speaking idea exploit fact coefficients generating functions enumerate points constructible sets desired limit obtained replacing constructible sets characteristics resulting rational functions called reduced zeta functions section paper define reduced graded ideal zeta functions explicitly provides theoretical framework allows quite readily dgrpoq psq dgr rational function dgr qpx primes finite extensions dgr red psq dgr qpy evseev proved section reduced ideal zeta functions lie rings socalled nice simple bases series enumerating integral points rational polyhedral cones hall bases given table instance property comparing one sees nilpotent lie rings red also remark nilpotency class red topological zeta functions another means defining limits zeta tions topological zeta functions grpoq psq dgr seungjai lee christopher voll rational function dgr qpx almost primes finite sions dgr top psq simply coefficient expansion dgr rkz pfc example thus oppq sps dgrpoq psq psq whence dgr top psq sps generally definition applies system local zeta functions denef type number field sense definition examples systems families zeta functions arising families form qqppspecpoq suitable qpx usual specpoq also local graded ideal zeta functions considered current paper fit uniform families expect phenomenon universal context free nilpotent lie rings conjecture formal far general definition topological zeta functions refer section section rossmann collects number intriguing conjectures analytic properties topological zeta functions associated various counting problems expect conjectures analogues realm topological graded ideal zeta functions motivated computation various topological graded ideal zeta functions free nilpotent lie rings made throughout current paper make number conjectures section proof theorem let let free nilpotent lie ring generators nilpotency class prime finite extension uniformizer note order parameterize lattices denote integer partitions parts respectively set proposition psq psq qqq qqq proof starting point lemma case reads psq psq enumerating graded ideals free nilpotent lie rings let partition lattices whose elementary divisor type respect given clearly claim indeed lattices may parametrized elementary divisor type coset poq certain stabilizer subgroup poq index lattice solutions simultaneous congruences mod section details index clearly change generators fpoq necessary may assume point crucially use fact gld poq obtain gld poq xyx xyy xyz xzx xzy xzz yzx yzy yzzyo jacobi identity involving three underlined terms nontrivial relation terms indeed relation xyz zxy yzx implies xyz xzy yzxyo pxzy yzxq xzy yzxyo xzy yzx hence implies xyx xyy xzx xzy xzz yzx yzy yzz whence type respect shows number lattices whose elementary divisor type respect given partition satisfy equal lattice satisfies overlap types words approach computing right hand side similar one taken compute local factors ideal zeta functions lie rings form poq compute right hand side carry case distinction respect finitely many ways partitions may overlap precise let partitions parts respectively satisfying uniquely determined numbers seungjai lee christopher voll define call integer sequence pmi arising overlap type pair set overlap type determines determined word alphabet length eleven occurs three times observe latter condition equivalent eight times case write denote set arising way remark notion related classical one dyck word length latter words featuring occurrences letter may used model overlap types two partitions parts section contrast model overlaps two partitions whilst genuinely parts distinct parts ties blocks respective sizes chose phrase results supporting notation similar relevant material possible leaving reader free concentrate crucial technical differences lemmata instance similar subtly different lemmata theorem analogous theorem set qqq proposition allows write psq psq section main result theorem giving general formula functions table lists words together overlap types indication section explicit formula may found order obtain formulae useful notation successive differences parts set define similarly set definition note also observe set enumerating graded ideals free nilpotent lie rings table overlap types pmi finally rrs define section jsj jrj smi smi rmi rni start immediate consequence proposition lemma let partitions parts respectively satisfying let partitions lemma let pmi overlap type resp dual partitions set also set seungjai lee christopher voll note determines unique lemma rrs proof since indices appearing product left hand side satisfy interval thus rmi observe may case index satisfies condition result see rmi exactly elements segment note case exactly exists follows make use identity gaussian binomial coefficients applying inductively see hence side equal enumerating graded ideals free nilpotent lie rings thus required lemma rrs proof note product left hand side may empty happens case since interval finally observe rni claim follows proof previous lemma preparations place give formula functions theorem let imi ymi numerical data tmi proof given set rrs suppm pvq rmi suppn practice one vectors successive differences one vectors successive differences given pair partitions satisfying recall definitions easy see every rrs suppm pvq suppn seungjai lee christopher voll thus follows suppm suppn pvq let usual kronecker delta function substituting lemma rewriting expressions terms using find formula splits product follows rrs rni smi suppn suppm pvq show factors products igusa functions gaussian binomial coefficients given rrs rmi define piq set vectors smi unless tmi numerical data defined psi piq jppi qytmi pymi qsmi smi ymi imi ymi analogously one shows numerical data defined completes proof theorem explicit formulae functions ppq remark note psq proposition enumerating graded ideals free nilpotent lie rings ppq ppq ppq ppq ppq ppq ppq ppq completion proof first two claims theorem follows explicit formulae given section deduce local functional equation one checks repeatedly using functional equations igusa functions resp fact functions satisfies functional equation follows psq nilpotency class two let section compute local graded ideal zeta functions psq prove functional equations functions establish behaviour use determine abscissae convergence global graded ideal zeta functions grpoq psq properties associated topological reduced graded ideal zeta functions throughout write pdq seungjai lee christopher voll formulae paper determines normal subgroup zeta functions free groups enumerating groups normal subgroups finite index mal cev correspondence ideal zeta functions free nilpotent rings computations generalize case general number rings straightforward manner recall paper main result define function given paper defines total order disjoint union without loss generality may assume sets cardinality set jpiq min ipjq max prd following essentially main result notation compatible current paper underlining terms form meant facilitate comparison graded numerical data theorem may ignored theorem theorem primes finite extensions poq psq psq psq qijr xjr numerical data qqpd jpir following graded analogue theorem theorem primes finite extensions dgrpoq psq psq jgr rhs enumerating graded ideals free nilpotent lie rings jgr defined numerical data qqpjpir rhs proof difference reflects difference corollary poq poq psq poq psq dgrpoq proof theorem proof proceeds establishing symmetry question summands using formulae differ numerical data remark suffices control value cases inspection respectively reveals ideal case term factor larger graded ideal case thus poq psq dgrpoq establishes behaviour zero describe behaviours poq psq grpoq psq theorem primes finite extensions poq psq psq poq psq psq proof prove note poq psq psq inspection see ppq ppq prove note dgrpoq psq psq jgr seungjai lee christopher voll inspection see ppq simple pole contrast jgr pole hence dgrpoq psq ppq psq ppq remark pertinent special cases confirms conjecture form global analytic properties following result compares known analytic properties poq psq theorem grpoq psq poq psq dgrpoq psq theorem abscissae convergence resp jqpd resp max jqj max respective meromorphic continuations zeta functions beyond abscissae convergence simple poles resp proof ensuing claim meromorphic continuation essentially theorem analogous claim proved analogously topological zeta functions degree behaviour zero theorem moreover degs top psq degs dgr top psq top dgr top proof note given rhs tbi oppq pbi summands formula poq psq products igusa functions gaussian binomial coefficients hence exist poq psq pbi oppq sequel sums pairs hence top psq pbi enumerating graded ideals free nilpotent lie rings rational function degree confirming first claim second claim degree dgr top psq follows analogous considerations jgr based indeed exist dgrpoq psq hence jgr pbi dgr top psq oppq jgr pbi rational function degree turning behaviour start observation pole unless case simple thus summand pbi pole unless case simple specifically qiprds ppq ppq whence thus ppd jqs jqjq top jqjq establishes proof goes along similar lines observe psqai jgr simple pole unless case double thus summand pbi simple pole unless case double specifically ppq qiprds ppq whence thus dgr top ppd jqs jqjq qpd jqjq proves remark pertinent special cases theorem first statement confirms conjecture whereas confirms conjecture topological form seungjai lee christopher voll explicit examples record following consequences theorem proposition heisenberg psq red top psq global graded ideal zeta function psq abscissa convergence meromorphic continuation whole complex plane proposition psq top psq red global graded ideal zeta function psq abscissa convergence may continued meromorphically repsq remark formulae resp almost also given table labels resp omit largish formula psq note following consequences proposition exists rational function qrx psq setting qry red top psq global graded ideal zeta function poq psq abscissa convergence may continued meromorphically repsq enumerating graded ideals free nilpotent lie rings two generators compute graded ideal zeta functions psq well reduced topological counterparts see see proposition proposition ppc red top psq psq psq global graded ideal zeta function psq abscissa convergence meromorphic continuation whole complex plane proof sketch let recall hall basis table recall formula zeta function fpoq terms pairs sublattices respectively note satisfies condition proves first claim others trivial consequences remark comparison poq psq theorem yields red red note hall basis nice simple sense formula almost also given table label proposition ppc set xpy qrx red top psq psq psq global graded ideal zeta function poq psq abscissa convergence may continued meromorphically repsq seungjai lee christopher voll proof sketch let recall hall basis table recall formula zeta function fpoq terms triples sublattices respectively note satisfies condition satisfies investigate latter condition may assume xyx xyyyo partition xyxx xyxy xyyx xyyyyo xyxy xyyx jacobi identity underlined term may omitted whence xyxx xyyx xyyy enumerate lattices satisfying distinguish according overlap type lattice may indeed elementary divisor type given partition either two cases covered formula similar one established theorem adding yields formula zeta function viz psq pqt pqt pqt proves first claim others trivial consequences claim meromorphic continuation follows lemma general conjectures let record number general conjectures regarding graded ideal zeta functions dgrpoq psq well topological reduced counterparts write witt function set rkz pfc piq exception conjecture conjectures section confirmed relevant special cases results paper conjecture uniformity exists dgr qpx almost primes finite extensions dgrpoq psq dgr conjecture local functional equations almost primes finite extensions dgrpoq piqs dgrpoq psq remark local functional equations universal feature quite general subring zeta functions theorem ideal zeta functions nilpotent lie rings nilpotency class theorem fact expect proof latter result carry graded setting making functional equations enumerating graded ideals free nilpotent lie rings instance general phenomenon class example conjecture holds conjecture states dgr piq dgr general operation defined rather delicately terms inversion frobenius eigenvalues nilpotency class greater local functional equations expected general ideal zeta functions nilpotent lie rings see section counterexamples ungraded setting table graded setting sufficient criterion local functional equations enumerative setup generalizing ideal zeta functions nilpotent lie rings given theorem applies ideal zeta functions free nilpotent lie rings theorem note notation result iqw piq following conjecture assume conjecture holds case may define reduced graded ideal zeta function dgr red dgr qpy expect technology may adapted define reduced graded ideal zeta function even without assumption conjecture conjecture reduced zeta function reduced graded ideal zeta function dgr red pole order exist rrs well nonnegative integers jqwd pjq red remark palindromic property holds course particular conjecture holds conjecture suggests reduced graded ideal zeta tions dgr red share key properties series graded cohenmacaulay algebras algebra would explain nonnegativity coefficients whereas palindromy would reflect gorensteinness assuming domain validity conjecture may seen facts red red hall basis table nice simple sense hence proposition applicable following graded analogue conjecture conjecture degree topological zeta function degs dgr top psq conjecture behaviour topological zeta function dgr red dgr top seungjai lee christopher voll remark interpretation terms numbers featuring conjecture conjecture hold reasons explained remark related multiplicity associated graded algebra close two conjectures behaviour topological graded ideal zeta functions conjecture behaviour topological zeta function zero topological graded ideal zeta function dgr top psq pole order leading coefficient piq top pjq piq remark conjecture topological form asserts top psq simple pole residue top conjecture behaviour zeta function zero grpoq psq piq piq psq pjq remark conjecture form asserts poq psq psq seems remarkable right hand sides independent note outside domain convergence series defining poq psq resp dgrpoq psq remark simply replacing numbers piq ranks successive quotients upper central series extend conjectures graded ideal zeta functions general nilpotent lie rings consider instance direct product primes finite extensions indeed fundamental graded lie ring whose local graded ideal zeta functions recorded almost section table formula may also easily derived directly formulae ideal zeta functions theorem using comparison identities recorded example case top likewise psq psq psq psq enumerating graded ideals free nilpotent lie rings value however nonnegative rational number independent well coincidence red top seem indicate conjectures made section may generalizations general graded ideal zeta functions acknowledgments work supported dfg sonderforschungsbereich spectral structures topological methods mathematics bielefeld university first author also supported fund national institute mathematical sciences acknowledge numerous helpful conversations tobias rossmann references bruns herzog rings cambridge studies advanced mathematics vol cambridge university press cambridge butler unimodality result enumeration subgroups finite abelian group proc amer math soc sautoy counting nilpotent groups publ math sautoy grunewald analytic properties zeta functions subgroup growth ann math sautoy woodward zeta functions groups rings lecture notes mathematics vol berlin evseev reduced zeta functions lie algebras reine angew math grunewald segal smith subgroups finite index nilpotent groups invent math hall basis free lie rings higher commutators free groups proc amer math soc rossmann computing topological zeta functions groups algebras modules proc lond math soc stability results local zeta functions groups related structures computing local zeta functions groups algebras modules preprint schein voll normal zeta functions heisenberg groups number rings unramified case lond math soc stanley hilbert functions graded algebras adv math voll zeta functions groups enumeration buildings amer math normal subgroup growth free groups math ann functional equations zeta functions groups rings ann math local functional equations submodule zeta functions associated nilpotent algebras endomorphisms preprint witt treue darstellung liescher ringe reine angew math seungjai lee christopher voll mathematical institute oxford university united kingdom current address national institute mathematical sciences daejeon south korea address mathematik bielefeld bielefeld germany 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differential privacy finite computers sep victor salil center research computation society school engineering applied sciences harvard university vbalcer salil vadhan september abstract consider problem designing analyzing differentially private algorithms implemented discrete models computation strict polynomial time motivated known attacks floating point implementations differentially private algorithms mironov ccs potential timing attacks expected polynomialtime algorithms use case study basic problem approximating histogram categorical dataset possibly large data universe classic laplace mechanism dwork mcsherry nissim smith tcc privacy confidentiality satisfy requirements based real arithmetic natural discrete analogues geometric mechanism ghosh roughgarden sundarajan stoc sicomp take time least linear exponential bit length input paper provide strict discrete algorithms approximate histograms whose simultaneous accuracy maximum error bins matches laplace mechanism constant factors retaining pure differential privacy guarantee one algorithms produces sparse histogram output accuracy error individual bins worse laplace mechanism factor log prove lower bound showing necessary algorithm produces sparse histogram second algorithm avoids lower bound matches accuracy laplace mechanism producing compact efficiently computable representation dense histogram based independent implementation appropriately clamped version discrete geometric mechanism supported nsf grant http salil supported nsf grant simons investigator award grant sloan foundation introduction differential privacy framework statistical analysis sensitive datasets much work differential privacy involves interplay statistics computer science statistics provides many analyses wish approximate differentially private algorithms well probabilistic tools useful analyzing algorithms necessarily randomized computer science differential privacy draws upon tradition adversarial modeling strong security definitions techniques designing analyzing randomized algorithms considerations algorithmic resource constraints time memory connection statistics natural much literature differential privacy considers estimation functions data sample mean introduces noise continuous probability distributions laplace distribution obtain privacy however choices incompatible standard computer science models algorithms like turing machine ram model well implementation physical computers use finite approximations real arithmetic via floating point numbers discrepancy theoretical concern mironov strikingly demonstrated common implementations basic differentially private algorithm laplace mechanism vulnerable real attacks mironov shows prevent attack simple modification implementation solution specific single differentially private mechanism particular arithmetic standard solution increases error constant factor likely efficient practice algorithm use replace laplace mechanism however provides bounds asymptotic running time gazeau miller palamidessi provide general conditions implementation real numbers mechanism perturbs correct answer noise maintains differential privacy however provide explicit construction bounds accuracy running time theoretical point view appealing approach resolving issues avoid real arithmetic entirely consider differentially private computations involve discrete inputs outputs rational probabilities algorithms realizable standard discrete models computation however algorithms running times bounded expectation due sampling exponential distribution supported natural numbers raises potential vulnerability timing attacks adversary observe running time algorithm learns something algorithm coin tosses assumed secret definition differential privacy even time directly observed practice adversary determine upper bound running time information implicitly assumed secret privacy definition considerations advocate following principle differential privacy finite computers describe implement differentially private algorithms discrete models computation strict bounds running time ideally polynomial bit length input analyze effects constraints privacy accuracy note strict bound running time prevent timing attacks bound pad executions take amount time also standard discrete models computation randomized turing machines defined terms countable rather finite resources infinite tape strict bound running time fix upper bound input length indeed implemented truly finite computer like randomized boolean circuit many cases goal achieved appropriate discretizations truncations applied standard differentially private algorithm however modifications nontrivial price accuracy privacy thus also call rigorous analysis effects paper carry case study achieving differential privacy finite computers one first tasks studied differential privacy namely approximating histogram categorical dataset even basic problem turns require nontrivial effort particularly maintain strict polynomial time optimal accuracy pure differential privacy data universe large recall definition differential privacy definition let randomized algorithm say differentially private every two datasets differ one row every subset say private algorithm satisfies pure differential privacy say satisfies approximate differential privacy paper study problem estimating histogram dataset vector number rows value histograms approximated satisfying differential privacy using laplace mechanism introduced original paper dwork mcsherry nissim smith specifically obtain differential privacy add independent noise distributed according laplace distribution specifically lap component output resulting vector lap continuous random variable probability density function proportional exp laplace mechanism also achieves high accuracy two respects error bin high probability simultaneous error high probability maxx log note bounds independent number rows dataset fractional error vanishes linearly grows simultaneous error notion differential privacy literature consider error equally natural concept think approximate histogram containing approximate answers different counting queries corresponding bins error captures error experienced analyst may interested one bins advantage considering error significantly smaller simultaneous error case laplace mechanism data universe large known error bounds achieved laplace mechanism optimal constant factors private algorithm histograms achieve smaller error simultaneous error unfortunately laplace mechanism uses real arithmetic thus implemented finite computer avoid real arithmetic could use geometric mechanism adds noise component according geometric distribution geo supported integers probability mass function exp however mechanism uses integers unbounded size thus implemented finite computer indeed algorithm implemented running time bounded expectation reducing hence probabilities rational numbers truncating long executions allowing adversary observe actual running time lead violation differential privacy thus better work truncated geometric mechanism ghosh roughgarden soundararajan clamp noisy count interval observe resulting probability distribution supported described explicitly terms sampled polynomial time using integer arithmetic ensuring rational thus obtain theorem bounded geometric mechanism informal statement thm every finite private algorithm histograms achieving error simultaneous error log strict running time poly bit length input dataset note consider particular definition accuracy namely high probability ghosh proved output bounded geometric mechanism used get optimal expected loss respect extremely general class loss functions arbitrary priors result applies individual noisy count output mechanism since bin distributed according bounded geometric mechanism modification ensure rational probabilities bounded geometric mechanism polynomial time large data universes indeed running time output length linear rather polynomial bit length data elements log achieve truly polynomial time similarly discretize truncate variant histogram bun nissim stemmer mechanism adds lap noise nonzero components retains noisy values larger threshold log thus algorithm outputs partial histogram counts subset bins rest counts treated zero replacing use laplace mechanism rational bounded geometric mechanism implement algorithm strict polynomial time theorem histogram informal statement thm every finite private algorithm histograms achieving error bins true count least log simultaneous error log strict running time poly bit length input dataset notice simultaneous error bound log better achieved laplace mechanism known optimal constant factors range parameters see theorem fact error bound independent data universe size makes tempting apply even infinite data domains however note infinite impossible algorithm strict bound running time needs time read arbitrarily long data elements thus vulnerable timing attacks implementable finite computer note also error bound holds bins large enough true count namely larger threshold discuss point disadvantage histogram sacrifices pure differential privacy natural ask whether achieve polynomial running time retaining pure differential privacy step direction made cormode procopiuc srivastava tran observe appropriate threshold log run bounded geometric mechanism retain noisy counts larger expected number bins remain less indeed expected number bins retain whose true count zero empty bins less describe method directly sample distribution empty bins retained without actually adding noise bins yields algorithm whose output length polynomial expectation however output length strictly polynomial nonzero probability outputting bins clear implement algorithm expected polynomial time even making probabilities rational denominators bit length linear address issues consider slightly different algorithm instead trying retain noisy counts larger fixed threshold retain largest noisy counts since nonzero true counts results mechanism whose output length always polynomial rather expectation however probabilities still denominators bit length linear thus show approximately sample distribution within arbitrarily small statistical distance price poly log increase running time naively would result privacy however significantly smaller range mechanism convert private mechanism private mechanism simply outputting uniformly random element small probability similar idea case used since range exponential size indeed polynomial bit length cost runtime taking polynomial ideas obtain theorem pure histogram polynomial time informal statement thm every finite private algorithm histograms achieving error bins true count least log simultaneous error log strict running time poly bit length input dataset theorems retain error bins large enough true count also prove lower bound showing limitation inherent algorithm outputs sparse histogram algorithms theorem lower bound error sparse histograms theorem suppose private algorithm histograms always outputs histograms nonempty bins error bins min log log provided lower bound similar spirit lower bound shows differentially private pac learner point functions functions exactly one element domain produce sparse functions hypotheses bypass lower bound consider algorithms produce succinct descriptions dense histograms algorithm output description function evaluated polynomial time even though may exponential size show relaxation allows regain error theorem histograms optimal accuracy informal statement thm every finite private algorithm histograms appropriate class succinct descriptions histograms achieving error simultaneous error log strict running time poly bit length input dataset producing description noisy histogram evaluating point algorithm essentially independent instantiation bounded geometric mechanism specifically release function selected independent family hash functions view coin tosses specifying sample bounded geometric distribution let efficient sampling algorithm bounded geometric distribution noisy count hash function chosen randomly family conditioned values nonempty bins obtain running actual bounded geometric mechanism bins independence ensures behavior two neighboring datasets together involve distinct elements indistinguishable way ordinary bounded geometric mechanism accuracy comes fact marginal distributions noisy counts bounded geometric mechanism actually incur small approximation error matching domain sampling procedure range family hash functions far know use limited independence constructing differentially private algorithms use pairwise independence differentially private pac learning algorithms class point functions although problem related one consider releasing histogram amounts query release class point functions discussed design analysis algorithm appears quite different particular analysis seems rely independence essential way another potential interest technique another method bypassing limitations synthetic data query release large family predicates interested differentially private algorithms given dataset output summary allows one approximate answers counting queries associated predicates example family point functions consisting predicates evaluate exactly one point data universe query release problem amounts approximating histogram fundamental result blum ligett roth successors show possible even families data universes size exponential moreover summaries produced algorithms form synthetic dataset dataset every query unfortunately shown even simple families queries correlations pairs binary attributes constructing differentially private synthetic dataset requires time exponential bitlength log data universe elements thus important find ways representing approximate answers natural families counting queries bypass inherent limitations synthetic data progress along lines made variety works algorithm use independence seen yet another representation bypasses limitation synthetic data albeit statistical rather computational one indeed sparse histogram simply synthetic dataset approximates answers point functions theorem algorithm achieves provably better accuracy possible synthetic datasets raises question whether similar ideas also useful bypassing computational limitations synthetic data complex families counting queries preliminaries throughout paper let set set let nonstandard set notice given set finite set define set length vectors indexed elements differential privacy define dataset ordered tuple rows row drawn discrete data universe row corresponding individual two datasets considered neighbors differ exactly one row definition let randomized algorithm say differentially private every pair neighboring datasets every subset say private algorithm satisfies pure differential privacy say satisfies approximate differential privacy intuitively captures upper bound adversary ability determine whether particular individual dataset parameter represents upper bound probability catastrophic privacy breach entire dataset released common setting parameters takes small constant negligible following properties differentially private algorithms used proofs lemma let private randomized function private lemma group privacy let private let datasets obtained changing rows lemma composition let private private define private histograms point function defined count number occurrences given dataset paper focus algorithms privately releasing approximations values point functions also known histogram histogram collection bins one element data universe xth bin consisting label count representations input algorithms always dataset element outputs represent approximate histograms consider following histogram representations algorithms outputs vector use denote histogram approximate count element partial vector element appears pair interpreted element approximate count elements listed partial vector assumed count implicitly algorithm return partial vector releasing bins subset data structure encoded string defines function denoted approximate count efficiently computable given data structure time polynomial length data structure section data structure consists coefficients polynomial along parameters representation able express histogram difference memory used efficiency computing count example computing approximate count using data structure representation bounded time takes compute associated function using partial vectors one needs iterate vector determine approximate count define following class histograms let set histograms integer counts nonzero using partial vectors element stored log log bits shorter vector representation log accuracy order preserve privacy algorithms return histograms noise added counts therefore crucial understand accuracy guarantees given dataset compare noisy count count released algorithm true count focus following two metrics definition histogram algorithm accuracy definition histogram algorithm accuracy respectively metrics capture maximum error one bin maximum error simultaneously bins even though simultaneous accuracy commonly used differential privacy accuracy several advantages histograms one provable achieve smaller error possible simultaneous error indeed optimal simultaneous error private histograms log whereas optimal error log independent accuracy may easier convey end user differential privacy example common interpretation error bars shown graphical depiction histogram figure histogram error bars many algorithms accuracy good enough imply optimal simultaneous accuracy indeed algorithm accuracy also achieves accuracy union bound however may always able achieve good accuracy want also use following relaxation bounds error bins large enough true count definition histogram algorithm accuracy counts larger probability terminology definition let random variable probability mass function denoted function cumulative distribution function denoted function support denoted supp set elements definition let random variables taking values discrete range total variation distance defined max lemma let random variables discrete range total variation distance following properties identically distributed denoted let function discrete let random variables discrete range max sampling interested computational efficiency algorithms need consider efficiency sampling various distributions standard method sampling random variable via inverse transform sampling lemma let uniformly distributed random variable defined min supp random variable wish sample finite support compute inverse cumulative distribution performing binary search supp find minimum method removes need compute inverse function cumulative distribution function used algorithms order statistics definition let random variables order statistic denoted smallest value among lemma let random variables cumulative distribution function support otherwise lemma iteratively sample random variables distributed identically without sample original random variables inverse cumulative distributions order statistics geometric distribution common technique creating differentially private algorithms perturb desired output appropriately scaled laplace noise algorithms outputs counts focus discrete analogue laplace distribution say random variable follows geometric distribution scale parameter centered denoted geo probability mass function proportional verified cumulative distribution function otherwise specified assumed inverse cumulative distribution otherwise equivalently sign model computation analyze running time algorithms respect word ram model taking log bit length algorithms inputs possibly additional parameters model memory accesses basic operations arithmetic comparisons logical words constant time addition assume data universe parameters algorithms rational represent rationals pairs integers algorithms require randomness assume access oracle given number returns uniformly random integer inclusively geometric mechanism section show construct differentially private histogram using laplace mechanism requiring integer computations bounded length shown dwork mcsherry nissim smith privately release histogram adding independent appropriately scaled laplace noise bin state variant uses discrete noise formally studied algorithm geometricmechanism following set geo clamped interval geo otherwise release note output algorithm collection bins represents partial vector case count defines complete vector privacy accuracy properties algorithm similar laplace mechanism theorem geometricmechanism following properties geometricmechanism private geometricmechanism accuracy iii geometricmechanism accuracy proof let geo part iii follows similarly noting independence counts accuracy bounds constant factors match lower bounds releasing differentially private histogram presented algorithm needs store integers unbounded size since geo unbounded magnitude noted restricting generated noise fixed range avoid problem however even generated noise restricted fixed range generating noise via inverse transform sampling may require infinite precision appropriately choosing probabilities noise cumulative distribution function represented finite precision therefore generating noise via inverse transform sampling requires finite precision proposition algorithm geosample output identically distributed geometric random variable scale parameter centered clamped range define morever geosample running time poly chosen cumulative distribution function geometric random variable scale parameter clamped takes rational values common denominator therefore implement inverse transform sampling distribution need choose uniformly random integer rather uniformly random variable algorithm geosample let define function sample uniformly random using binary search find smallest return function obtained clearing denominators cumulative distribution function geo clamped lemma let defined algorithm equals cumulative distribution function geo clamped prove lemma seeing implies proposition proof proposition let drawn uniformly random construction geosample implying geosample geo lemma bound running time binary search takes log rounds largest number used bit length operations polynomial bit length numbers therefore geosample running time poly proof lemma cumulative distribution function geo consider case similar argument holds using algorithm ready construct private histogram algorithm bounded time complexity whose accuracy identical geometricmechanism constant factors algorithm boundedgeometricmechanism rational let following let geosample release theorem let rational boundedgeometricmechanism following properties boundedgeometricmechanism private boundedgeometricmechanism accuracy iii boundedgeometricmechanism accuracy boundedgeometricmechanism running time poly bit length algorithm input proof proposition geosample generates geometric random variable scale parameter centered clamped algorithm identically distributed geometricmechanism parts follows theorem proof computing takes time log proposition geosample takes times poly log log improving running time datasets large domains linear running time algorithm prohibitive present algorithm reduces running time dependence universe size nearly linear based observation counts observation made cormode procopiuc srivastava tran output sparse histograms sparse histograms start reducing output length geometricmechanism release bins heaviest largest counts interpreted partial vector algorithm keepheavy set geo clamped let elements largest counts sorted order max release observe output length improved log log bits compared log bits needed represent outputs geometricmechanism theorem keepheavy following properties keepheavy private keepheavy accuracy counts larger iii keepheavy accuracy note unlike geometricmechanism algorithm log accuracy counts larger log loss necessary algorithm outputs sparse histogram show theorem proof privacy follows privacy geometricmechanism part theorem along differential privacy closure lemma prove remaining parts start following lemma lemma let define event implies proof probability occurring follows part iii theorem identically distributed output geometricmechanism assume event distinct elements proof part theorem let lemma part theorem proof part iii theorem let event lemma occurs probability least assume lemma implies remaining trivially however described keepheavy still requires adding noise count every bin following algorithm simulates keepheavy generating candidate set heavy bins heaviest released candidate set constructed bins nonzero true count sample representing bins true count heaviest noisy counts algorithm let set geo clamped pick uniformly random sequence distinct elements sample joint distribution order statistics distributed geo clamped sort elements release proposition identically distributed keepheavy proof let noisy counts set keepheavy let sorted ordering defined counts identically distributed identically distributed bins heaviest counts let random variable set labels bins heaviest counts therefore shows identically distributed keepheavy order sample order statistics used construct following algorithm similar geosample algorithm previous section proposition let let define random variables identically distributed geo clamped following subroutine ordsample identically distributed order statistic conditioned also ordsample running time poly ensures one use geometricmechanism instead used continuous noise last step equivalent releasing heaviest bins however discrete case ties occur set determine bins count tied heaviest may many noisy counts tied result output bins strictly heavier count boundedgeometricmechanism algorithm chosen cumulative distribution function geo takes rational values common denominator therefore cumulative distribution function conditioned also rational sample finite precision algorithm ordsample let define function sample uniformly random using binary search find smallest return proof lemma cumulative distribution geo clamped therefore lemma let drawn uniformly random construction ordsample implying ordsample binary search takes log iterations iteration running time polynomial bit length numbers used therefore algorithm running time poly log poly replace sampling joint distribution order statistics iterative calls ordsample get following algorithm algorithm rational let let let geosample pick uniformly random sequence distinct elements let ordsample let ordsample sort elements release theorem let rational identically distributed keepheavy therefore private accuracy counts larger iii accuracy proof proposition identically distributed joint distribution used therefore algorithm identically distributed proposition identically distributed keepheavy parts iii follow theorem algorithm output length log log however running time depends polynomially since sampling wth order statistic using ordsample takes time polynomial indeed necessary since distribution order statistic probabilities exponentially small efficient approximation remedy inefficiency consider efficient algorithm approximates output distribution notice geo coins sample distribution need toss theorem exists algorithm nonzerogeometric input datasets rational nonzerogeometric nonzerogeometric private iii moreover running time nonzerogeometric poly bit length algorithm input note algorithm achieves privacy reducing algorithm better approximates improving accuracy cost increasing running time polynomial log contrast private algorithms stability based algorithm section one needs log get meaningful accuracy convert nonzerogeometric pure differentially private algorithm mixing uniformly random output inspired similar technique algorithm rational probability release otherwise release uniformly random element lemma let private discrete range suppose algorithm satisfies input datasets parameter algorithm following properties private whenever running time upper bounded sum bit length represent running time time required sample uniformly random element taking small enough satisfying constraint part algorithm satisfies pure differential privacy nearly utility due statistical distance allowing possibly efficient implementation since need approximately sample output distribution proof neighboring datasets rearranging terms using upper bound yields proof follows directly construction apply lemma nonzerogeometric reasonable settings parameters get accuracy bounds identical constant factors algorithm purenonzerogeometric rational probability release nonzerogeometric otherwise release uniformly random element theorem let rational purenonzerogeometric following properties purenonzerogeometric private purenonzerogeometric accuracy counts larger iii purenonzerogeometric accuracy purenonzerogeometric running time poly bit length algorithm input proof notice constraint needed nonzerogeometric privacy follows lemma taking nonzerogeometric proof define set construction purenonzerogeometric nonzerogeometric defined algorithm notice theorem nonzerogeometric part theorem similarly bound simultaneous accuracy using part iii theorem proof lemma takes poly log time sample uniformly random element compute theorem nonzerogeometric running time poly lemma purenonzerogeometric desired running time construction nonzerogeometric finish section construction nonzerogeometric notice passes arguments ordsample result ordsample exponentiating integer represents numerator fraction power want ensure numbers used ordsample exceed maximum following algorithm approximate using repeated squaring truncating intermediate result keep bit length manageable following lemma provides bound error running time proposition algorithm expapprox expapprox nondecreasing function expapprox expapprox satisfies accuracy bound expapprox iii expapprox running time poly log log log proof algorithm defined follows algorithm expapprox return otherwise let expapprox return even odd proof proceed induction case trivial let assume expapprox nondecreasing function consider odd expapprox expapprox expapprox expapprox likewise result holds even therefore expapprox nondecreasing function trivial expapprox proof ease notation define expapprox parameters change analysis construction even bound error odd solving recurrence gives proof iii running time follows observation call expapprox makes log recursive calls remaining operations polynomial bit lengths numbers used modify ordsample using expapprox keep bit lengths numbers becoming large yielding efficient algorithm whose output distribution close ordsample algorithm approxordsample let define function sample uniformly using binary search find smallest expapprox return proposition let ordsample approxordsample addition approxordsample running time poly log log proof let ordsample approxordsample let define part proposition expapprox increasing function therefore expapprox expapprox therefore triangle inequality part proposition expapprox summing yields desired bound total variation distance definition running time dominated log calls expapprox part iii proposition call takes time poly log log log poly log log log log overall approxordsample desired running time samples joint distribution obtained iterated calls ordsample must also consider accumulated distance iterated calls ordsample iterated calls approxordsample corollary let consider following random variables ordsample ordsample approxordsample yej approxordsample yen proof part iii lemma proposition yej yej max yej ready state mechanism nonzerogeometric show satisfies theorem identical except replace calls ordsample calls approxordsample algorithm nonzerogeometric rational let let let let geosample pick uniformly random sequence distinct elements let approxordsample let approxordsample sort elements release theorem restated algorithm nonzerogeometric satisfies input datasets rational nonzerogeometric nonzerogeometric private iii moreover running time nonzerogeometric poly bit length algorithm input proof let algorithm except instead releasing heaviest bins releases bins elements releases similarly define respect nonzerogeometric notice distribution overs bins nonzero true count bins counts sampled using ordsample approxordsample respectively output distributions differ result apply corollary output distributions consider effect keeping heaviest counts define function sets counts strictly larger count input notice nonzerogeometric part lemma nonzerogeometric proof let neighboring datasets let previous part part theorem nonzerogeometric nonzerogeometric therefore nonzerogeometric private proof iii consider running time step construction true histogram takes log time proposition calls geosample take poly log time sampling random bin labels takes log time nonzerogeometric makes calls approxordsample argument exceeding term respectively proposition calls take time poly log log log poly log log log sorting elements releasing heaviest counts takes log log time therefore overall nonzerogeometric desired running time constructed algorithms releasing differentially private histogram pure approximate running time polynomial log simultaneous accuracy matching geometricmechanism constant factors removing dependence universe size would like accuracy independent use approximate differentially private algorithm based stability techniques proposition present reformulation algorithm using geometric noise instead laplace noise algorithm stabilityhistogram let set geo clamped release note release counts whose true count nonzero namely elements set thus output length log log however releasing set private would distinguish neighboring datasets one count count element thus release noisy counts exceed threshold large enough count kept small probability yielding approximate differential privacy theorem stabilityhistogram following properties stabilityhistogram private provided stabilityhistogram accuracy counts larger iii stabilityhistogram accuracy proof let neighboring datasets let cases consider geo similarly neighboring database noisy count geo differential privacy geometricmechanism part theorem notice distributed geo clamped thus geo therefore case follows similarly previous case considering overall two bins differing counts count computed independently lemma algorithm private proof let part theorem implies thus included output stabilityhistogram giving desired accuracy proof iii notice counts elements trivially accurate therefore need consider counts elements part iii theorem final step increase error additively also therefore using geosample see proposition construct computationally efficient algorithm releasing histogram private accuracies matching algorithm constant factors algorithm boundedstabilityhistogram rational let let let geosample release theorem let rational boundedstabilityhistogram satisfies following properties boundedstabilityhistogram private provided boundedstabilityhistogram accuracy counts larger iii boundedstabilityhistogram accuracy boundedstabilityhistogram running time poly bit length algorithm input proof proposition geosample generates geometric random variable scale parameter centered clamped therefore algorithm identically distributed stabilityhistogram parts follow theorem proof construction true histogram takes log time proposition calls geosample take poly log time counts exceed final step takes poly log time therefore constructed efficient algorithm releasing sparse histogram approximate differential privacy lower bounds section prove lower bound accuracy histogram algorithms whose outputs restricted sparse histograms using packing argument first completeness state reprove existing lower bounds accuracy simultaneous accuracy well generalize case theorem following let private accuracy min accuracy log log min proof assume let define dataset rows define dataset first rows remaining rows notice accuracy lemma fact therefore proof assume let define dataset first rows remaining rows let lemma notice collection disjoint sets therefore implies desired lower bound state lower bound sparse histograms theorem let private accuracy min histogram algorithms sections achieve log accuracy large enough counts however smaller counts guarantee accuracy log log taking threshold log algorithms sections respectively theorem shows bounds best possible poly poly proof assume let define dataset first rows remaining rows definition accuracy fact lemma distance thus linearity expectations hand therefore along implies lower bound min therefore along part theorem min max min min min min better accuracy via compact representations section present histogram algorithm whose running time unlike algorithm able achieve query accuracy log histogram algorithm output histogram properly chosen family family necessarily contains histograms many nonzero counts avoid lower bound theorem lemma let geosample exists multiset histograms satisfying let drawn uniformly random let drawn uniformly random iii let drawn uniformly random histogram represented string length poly log given representation count evaluated time poly log sampling histogram uniformly random done time poly log parts state histogram sampled uniformly random marginal distribution count closely distributed geometric distribution centered clamped proof construction let would like construct independent hash family consisting functions uniformly distributed distinct random variables independent given function family construct histogram using randomness evaluating noisy count via inverse transform sampling similar manner algorithm marginal distribution uniformly distributed set degree polynomial finite field independent hash family ideally would take case map subset use bijection get desired family functions however may larger prime power therefore must pick large enough field mapping resulting marginal distributions approximately uniform let max log family polynomials finite field degree define taking construct histogram polynomial min mod bijection running time poly defined algorithm notice drawn uniformly random drawn uniformly random proof let geosample min mod mod mod lemma similarly part follows choice proof pick proceed splitting probability based whether integer value randomness lies within one intervals length let drawn uniformly random drawn uniformly random geosample min mod mod mod monotonicity independent hash family proof iii mod mod proof represented irreducible polynomial degree encoded binary string length likewise elements represented polynomial degree requires bits encoding defines efficient bijection also interpreting string binary representation element represented coefficients description field representation encoded poly log bits given encoding evaluation done poly poly log time computing min mod get approximate count takes poly log time proof let sample given coefficients uniformly random following steps construct finding irreducible polynomial degree sample uniformly random set min mod sample set observing mod let sample uniformly random take coefficients coefficients interpolating polynomial given set points polynomial exists unique first prove correctness notice procedure return histogram let histogram sampling coefficients therefore steps output uniformly random construction done time integer operations limited numbers exceeding polynomial interpolation takes poly time therefore procedure runs time poly log algorithm boundedgeometricmechanism bin count marginal distribution geosample counts independent using previously defined family following algorithm essentially marginal distributions algorithm counts independent yields efficient algorithm need small number random bits polynomial log compared amount required counts independent linear algorithm compacthistogram rational let let let geosample release drawn uniformly random theorem let rational compacthistogram following properties compacthistogram private compacthistogram accuracy iii compacthistogram accuracy compacthistogram running time poly bit length algorithm input proof let neighboring datasets let similarly define let notice let let drawn uniformly random compacthistogram compacthistogram geosample along proposition part lemma compacthistogram compacthistogram therefore compacthistogram private proof let compacthistogram let construction geosample accuracy follows part theorem let drawn uniformly random notice parts iii lemma geosample thus accuracy follows part theorem proof iii union bound along previous part gives bound accuracy proof compacthistogram makes calls geosample therefore proposition part lemma get desired bound running time therefore constructed histogram algorithm running time polynomial accuracy matching lower bounds releasing private histograms constant factors acknowledgments thank harvard privacy tools differential privacy research group particularly mark bun kobbi nissim informative discussions feedback anonymous tpdp reviewers helpful comments references amos beimel hai brenner shiva prasad kasiviswanathan kobbi nissim bounds sample complexity private learning private data release machine learning avrim blum katrina ligett aaron roth learning theory approach noninteractive database privacy journal acm jacm url https mark bun kobbi nissim uri stemmer simultaneous private learning multiple concepts proceedings acm conference innovations theoretical computer science pages acm url https victor pereyra solution vandermonde systems equations mathematics computation bryan cai constantinos daskalakis gautam kamath priv private sample efficient identity testing arxiv preprint url https mahdi cheraghchi adam klivans pravesh kothari homin lee submodular functions noise stable proceedings annual symposium discrete algorithms pages society industrial applied mathematics url https graham cormode magda procopiuc divesh srivastava thanh tran differentially private publication sparse data arxiv preprint url https karthekeyan chandrasekaran justin thaler jonathan ullman andrew wan faster private release marginals small databases proceedings conference innovations theoretical computer science pages acm url https cynthia dwork jing lei differential privacy robust statistics proceedings annual acm symposium theory computing pages acm cynthia dwork frank mcsherry kobbi nissim adam smith calibrating noise sensitivity private data analysis theory cryptography conference pages springer cynthia dwork aleksandar nikolov kunal talwar efficient algorithms privately releasing marginals via convex relaxations discrete computational geometry url https ivan gazeau dale miller catuscia palamidessi preserving differential privacy semantics theoretical computer science url https arpita ghosh tim roughgarden mukund sundararajan universally utilitymaximizing privacy mechanisms siam journal computing anupam gupta aaron roth jonathan ullman iterative constructions private data release theory cryptography pages url https moritz hardt guy rothblum rocco servedio private data release via learning thresholds proceedings annual symposium discrete algorithms pages society industrial applied mathematics url https moritz hardt kunal talwar geometry differential privacy proceedings acm symposium theory computing pages acm url https shiva prasad kasiviswanathan homin lee kobbi nissim sofya raskhodnikova adam smith learn privately siam journal computing url https donald knuth combinatorial algorithms part volume art computer programming professional ilya mironov significance least significant bits differential privacy proceedings acm conference computer communications security pages acm victor shoup new algorithms finding irreducible polynomials finite fields mathematics computation justin thaler jonathan ullman salil vadhan faster algorithms privately releasing marginals international colloquium automata languages programming pages springer url https jonathan ullman salil vadhan pcps hardness generating private synthetic data tcc volume pages springer url https generating sparse histogram uniformly random able efficiently compute sample uniformly random element needed efficient implementation algorithm lemma calculated uniformly random element sampled time poly log log proof show histsample defined efficiently samples uniformly random element using bijection algorithm histsample pick uniformly random find smallest let use integer division find map corresponding subset size specified sorted elements specifically let sequence representing combinatorial number system degree unique sequence satisfying sequence found greedily decreasing using binary search find largest qii let digits base representation release steps define bijection mapping step defines bijection mapping steps decompositions respective number systems bijections likewise bijection mapping element therefore mapping bijective proving correctness computing takes poly log log time remaining steps done poly log log time number used exceeds numbers used expressible sum numbers steps consist iterations
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mar benchmarks systems modular model library building automation systems extended version nathalie cauchi alessandro abate department computer science university oxford oxford abstract building automation systems bas exemplars systems cps incorporating digital control architectures underlying continuous physical processes provide modular model library bas drawn expertise developed real bas setup library allows build models comprising either physical quantities digital control structure operation dynamics model complex incorporating stochasticity iii numerous continuous variables discrete states various input output signals large number possible discrete configurations modular composition bas components generate useful cps benchmarks display use means three realistic case studies corresponding models built engaged different analysis goals benchmarks model library data collected bas setup university oxford kept https keywords systems building automation systems thermal modelling hybrid models simulation reachability analysis probabilistic safety control synthesis introduction paper describes library models building automation systems bas employed create benchmarks verification control synthesis simulation purposes systems cps models inspired built around experimental setup within department computer science university oxford part research collaboration service engineers industrial partners sector library allows create numerous meaningful models bas examples cps integrating continuous dynamics discrete modes interest bas also colloquially known smart buildings gaining rapid momentum particular means ensuring thermal comfort minimising energy consumption ascertaining reliability quantitative models needed evaluate system performance verify correct behaviour develop specific control algorithms overview different bas modelling techniques used literature presented several simulation tools see devised aide development analysis models bas attempting characterisation broad spectrum existing bas models find either deterministic stochastic ones ones discrete continuous inputs states choice model art craft one must select simplifying assumptions accurately reflect operational performance bas specific environments introduce uncertainty represent components unknown parameters random occupants therefore aim simplify modelling process simulation verification strategy synthesis carried seamlessly different verification policy synthesis tools exist literature typically specific particular type model structure case stochastic hybrid models oftenlimited small number continuous variables use tools also requires expert knowledge specific formalism tool makes use order display versatility library bas models present three case studies built components focus modelling temperature dynamics key element ensuring thermal comfort employ three generated models different analysis goals comprising simulation reachability control synthesis models delineation kept https allow use modification different applications comparison modelling approaches bas repository also contains real data gathered bas lab oxford employed modelling studies article following structure section introduces bas modelling framework cps identify three modelling introduce different complexities model dynamics based develop analyse three case studies section building automation systems bas structure components bas models clearly depend size topology building climate control setup work consider bas setup department computer science university oxford graphical depiction shown figure bas consists two teaching rooms connected system boiler supplies heat heating coil within ahu two radiators valves control rate water flow heating coil radiators ahu supplies air two zones connected back back adjacent outside interior hall figure zone air rooms mix outside air exchanges circulating air ahu return water ahu heating coils radiators collected pumped back boiler figure presents resistor capacitance network circuit two zones underpins dynamics temperature zone component corresponding equations table heat level room modified radiative solar energy absorbed walls occupants iii ahu input supply air radiators ahu return water effect heat stored walls rooms depicted capacitors whereas thermal resistance heat transfer walls depicted resistor elements heating system air handling unit radiators circuit internal thermal dynamics within two zones figure building automation system setup bas dynamics configurations define models individual components bas system based expertise developed bas setup oxford single components intended separate physical structures within bas models built underlying physics improved via industrial feedback existing literature obtain models number unknown parameters estimated validated using data collected bas setup list indices table quantities variables parameters inputs listed table table presents relations among variables model algebraic relations define static couplings whereas differential relations define dynamics corresponding variables structure figure quantities table variables associated dynamics table together allow construct global models complete bas setup refer set models describing individual components table library models one select individual components models library build different bas configurations table indices index adj ref reference ahu adjacent exterior zones mixer individual zones zone walls windows adjacent interior zone outside reference supply air supply water wall water index adj hall occ solar reference adjacent zone boiler hallway zone walls windows zone walls occupants radiator return water solar energy collector zone air global model bas complex comprising algebraic differential relations may affected process noise model also contains number inputs either construed control signals exogenous signals dynamics view continuous variables coupled ahu air duct model table number continuous variables also increase substantially considering bas setup multiple zones employing zone component configuration zones would result model continuous state variables furthermore model features multiple components present switching discrete behaviours affecting dynamics continuous variables discrete modes listed table result discrete configurations order tackle complexity global bas models add level flexibility modelling framework consider bas component separate module characterised inputs output elements internal variables make use individual modules describing component type connect different modules based possible physical couplings coupling different modules also achieved via relationships zone module coupling two zones continuous variable tadj corresponding adjacent zones wall separating two zones figure corresponds individual zone temperatures two zone modules table zone equations modular structure individual components provides added level versatility since connect different components create various new models modularisation also allows perform analysis whole setup executing analysis individual modules extend library models defining new modules connect existing modules via relations symbol ben cpa cpw pout wmax quantity area windows zone boiler switched capacitance medium specific heat capacity air water measurements zone boiler mass air flow rate number zones radiator rated power output associated heat gain thermal resistance heat walls medium temperature associated medium mixing ratio overall transmittance factor medium volume medium water flow rate associated medium maximum permitted valve valve position associated component offset factors associated process noise density medium time constant medium type constant discrete constant constant input constant input constant input constant input constant constant input constant input constants constant constant constant table list variables inputs parameters bas description model library library bas components comes form matlab scripts script represents individual bas component models form two types linear bilinear depending component represent defined using symbolic toolbox parametrised described discrete continuous time provide parameters estimated data gathered bas lab construct individual models however users easily make use parameters construct model different components connected together based relations cascading different symbolic models component done provided scripts allow simulate models plots defined output variables presented case studies next set three case studies present trade discussed elements complexity case studies establish dynamics models constructed library components iii describe results obtained heating setup deterministic stochastic dynamics consider two zones heated one radiator common supply air portrayed figure table select two components corresponding models component boiler continuous variables dtsw relation ben ben component valve continuous variables exp wmax differential relation algebraic component mixer continuous variables tout tzi relation algebraic component ahu heating coil continuous variables dtrw cpw cpw tsw trw trw relation differential component ahu air duct continuous variables dtsai cpa tsai tzi tsai relation differential component radiator continuous variables dtrw cpw vri cpw wri tsw trw tzi trw relation differential component zone continuous variables twjn tzi dtzi czi qrw qocci qsai rij tadj tzi tadjl twjn dtwjn cwjn qrw ajn rout rlj tadj tzi tadjl twjw dtwjw cwjw qsolarjw qrw ajw rout rljw relation differential qrw pradi trw tzi qocci qsai cpa tsai tzi qrw trw twj qsolarjw tout component collector continuous variables trw trw trw relation algebraic table components dynamics functional relations among variables component boiler ahu air duct mixer ahu heating coil radiator heating coil radiator heating coil discrete modes boiler ben fan medium high open closed valve healthy faulty valve healthy faulty valve fully open half open closed table discrete operational modes radiator zone simplify models following assumptions wall temperature constant across zones fixed value boiler switched providing supply temperature tsw bss iii fix mass air flow rate radiator water flow rate include heat gain windows ahu heating coils twss zone obtain model four state variables trw trw common supply temperature tsa input setup consider three different dynamics purely figure bas setup first case study deterministic one deterministic model additive disturbance iii stochastic model thus remove process noise template model models iii include occupancy heat gain also discretise dynamics scheme deterministic models scheme stochastic model using uniform sampling time minutes obtain set linear models one note models considered fully observable since individual zone temperatures variables interest variables provided outputs dynamics deterministic model described matrices properly sized constructed based models table twss twss cpw cpw tsw bss tsw bss cpw cpw mean resistance offered walls deterministic model additive disturbances fda dda qda mda extended additive noise vector qocci table representing different levels zone qda defined qda fda properly sized stochastic model expressed extending include process noise diag encompasses ances process noise state independent gaussian random variables also independent initial condition process simulation run three models depicted figure mda time hours figure first case study simulations two days reachability analysis case study would like perform following verification task decide whether traces generated models remain within specified safe set given time period achieved reachability analysis takes probabilistic flavour stochastic model safe set described interval around temperature tsp constrain input lie within set tsa tsa models employ fixed time horizon hours perform reachability analysis axelerator use perform probabilistic reachability analysis order perform reachability analysis using axelerator models set initial condition trw trw reach tube model whole time horizon shown figure encompasses union reachable states horizon results obtained using axelerator figure conservative results confirm model indeed stay required safe set initial states also exit note coupling two zones zone tends stay higher zone temperatures zone similar reach tube obtained model mda similarly perform probabilistic reachability analysis model defining safe set assuming input set resulting adaptive partition safe set along optimal safety probability partition set depicted figure performing probabilistic reachability analysis using model figure deduce models high probability within required safe set specifically heating setup large number continuous variables second case study focus dynamics zone component table consider two zones shown figure assume central fan pumps air rooms common supply temperature tsa input mass airflow fixed iii return water temperature ahu heating coils fixed trw ass previous case study selected model discretised using sampling time minutes obtain model dci variables figure first case study reach tube whole time horizon prob figure first case study partition safe set model along optimal safety probability partition set common fan supplies two zones supply rate tsa whereas tout thall trw trw pradi trw ass matrices properly sized recall vector corresponds disturbance signals represents constant additive terms within model finally model disturbances random external effects policy synthesis refinement would like synthesise policy ensuring temperature within zone deviate set point time horizon equal four hours requirement translated following pctl property tsp aim synthesising policy maximising safety probability synthesis goal computationally hard due number continuous variables making mitigate limitation perform policy synthesis via abstractions simplify four abstract models using technique abstract models labelled represents number continuous variables corresponding abstract model models take form markov decision processes quantify error output variable introduced different levels abstractions use simulation relations pair represents deviation output trajectories complex abstract models differences probability distribution processes respectively metrics allows designer select considered abstract models provides best trade precision desirable achieve little deviation output trajectories small probability distributions small table second case study error metrics concrete abstract models compute simulation relations set abstract models presented table pair providing optimal trade obtained abstract model corresponds next use perform computation safety probability obtain model size overall accuracy approximation synthesise optimal policy abstract model results safety probability refine obtained policy results one used overall process results satisfied safety probability abstraction error introduced results obtained highlight trading complexity number continuous variables computing relations synthesise policies using simpler models yet achieve high performance still refined policy applied original model single zone heating multiple switching controls third last case study focus mixer ahu air duct zone components table select ahu source heat within zone boiler disconnected pictorial description setup figure mixer operates either two modes open closed ahu air duct recirculates air either internal zone mixer mode outside mixer mode via continuous variable output mixer component rate air pumped zone controlled fan three operating speeds medium high mixer position fan settings used maintain comfortable temperature within zone setup described hybrid model discrete modes set describe possible configurations fan operating speeds mixer position fan switched mixer position air pumped zone continuous variables model zone temperature together supply air temperature tsa pumped zone transitions discrete modes triggered continuous dynamics crossing spatial guards guards denote deviations temperature graphical description figure bas setup third case study overall hybrid model together different guard conditions shown figure continuous dynamics built table follow twss cpa tsa cpa tsa tsa figure hybrid model third case study showing discrete states guard conditions initial discrete state variables take values according discrete mode med high tout else med high correspond air flow rates fan operating medium high speeds reachability analysis interested performing reachability analysis hybrid model run using spaceex notice case discretise time consider continuous time horizon hours consider two different initial conditions first experiment select initial condition equal tsa second set tsa resulting reach tube experiments shown figure tsa initial condition tsa initial condition tsa figure third case study reach tubes obtained two different initial conditions figure see model initially state jumps new state warm outside air pumped zone due low temperature initial conditions switches notice reduction gradient variables tsa eventually switches back order maintain temperature within comfort region figure reach tube shows system remains within initial state whole time horizon conclusions paper presented library cps models bas bas modelling framework comprises three different types complexities stochasticity number continuous variables number discrete modes modular various bas components composed illustrate use bas components library via three case studies highlights different side complexity trade solves different problem simulation probabilistic reachability analysis strategy synthesis respectively current work developed towards obtaining compositional tool able allow easy construction bas models interfacing different verification tools performing analysis synthesis tools computation optimal strategies acknowledgements work funded european commission seventh framework programme project ambi grant part supported alan turing institute malta endeavour scholarships scheme authors would also like thank dario cattaruzza sofie haesaert honeywell laboratories prague fruitful feedback references hamza belkhouane jan hensen shady attia thermal comfort models net zero energy buildings hot climates second international conference energy indoor environment hot climates doha dario cattaruzza alessandro abate peter schrammel daniel kroening analysis guarded lti systems inputs abstract acceleration international static analysis pages springer nathalie cauchi khaza anuarul hoque alessandro abate stoelinga efficient probabilistic model checking smart building maintenance using fault maintenance trees arxiv preprint scott cotton goran frehse olivier lebeltel spaceex modeling language drury crawley jon hand kummert brent griffith contrasting capabilities building energy performance simulation programs building environment iulia dragomir viorel preoteasa stavros tripakis refinement calculus reactive systems toolset arxiv preprint goran frehse colas guernic alexandre scott cotton rajarshi ray olivier lebeltel rodolfo ripado antoine girard thao dang oded maler spaceex scalable verification hybrid systems computer aided verification pages springer sofie haesaert nathalie cauchi alessandro abate certified policy synthesis general markov decision processes application building automation systems performance evaluation sofie haesaert sadegh esmaeil zadeh soudjani alessandro abate verification general markov decision processes approximate similarity relations policy refinement siam journal control optimization ernst moritz hahn arnd hartmanns holger hermanns katoen compositional modelling analysis framework stochastic hybrid systems formal methods system design ondrej holub majid zamani alessandro abate efficient hvac controls symbolic approach control conference ecc european pages ieee woohyun kim srinivas katipamula review fault detection diagnostics methods building systems science technology built environment pages niels rode kristensen henrik madsen sten bay parameter estimation stochastic models automatica february kelman daly borrelli predictive control energy efficient buildings thermal storage modeling stimulation experiments control systems ieee february privara cigler vana oldewurtel sagerschnig zacekova building modeling crucial part building predictive control energy buildings esmaeil zadeh soudjani abate aggregation control populations thermostatically controlled loads formal abstractions control systems technology ieee transactions may sadegh esmaeil zadeh soudjani caspar gevaerts alessandro abate faust formal abstractions stochastic processes tacas volume pages wei baizhan hongyuan jia ming zhang wang application multiobjective genetic algorithm optimize energy efficiency thermal comfort building design energy buildings rongpeng zhang tianzhen hong modeling hvac operational faults building performance simulation applied energy appendix models heating system two zones deterministic stochastic dynamics present corresponding system matrices three models mda described sec given characterised following system matrices mda given characterised system matrices used together fda given characterised system matrices used together cases use twss tsp tsw bss trw trw perform simulation run three models period otherwise depict resulting simulation runs temperature within zone figure reachability analysis using axelarator perform reachability analysis using axelarator need use following commands command line file contains corresponding system matrices defines dimensions state space control inputs underlying model initial conditions defined using force tool stay inside safe region possible defined within commands set conditions problem axelarator makes use incremental search multiple precision intervals process unsound results also provide axelerator safe set defined form polyhedral safe set used reference resulting reach tube computed results obtained using axelarator form polyhedral sets define reach tube underlying system time steps polyhedral sets given trw trw trw trw trw trw trw trw whereas mda probabilistic reachability analysis using solve problem achieve low errors within computationally feasible time frame abstract model states interest fix rest state based model construct stochastic kernel form gaussian conditional distribution diag input space process set defined select pctl safety option compute analysis model abstracted based adaptive partitioning partitioning error obtain discretised transition kernel maximal safety probability partition computed based transition kernel recursively whole time horizon models heating setup large number continuous variables present corresponding system matrices model described subsection model disturbances random external effects affecting room temperature dynamics tout thall trw next present abstract models mca taking form xtc model dtc tout tout tout tout thall trw thall trw trw trw system matrices described using models heating system setup multiple switching controls construct hybrid model use twss tout tsp med high continuous dynamics composed via following simplifying assumptions outside air temperature fixed tout used warm zone temperature tsp maintained fix iii constant wall temperature twss radiators within zone process noise resulting continuous state models discrete mode using corresponding models presented table discrete state continuous space dynamics reachability analysis using spaceex first implement hybrid system delineated using figure within spaceex using spaceex modelling language bound zone temperature lie supply temperature lie tsa corresponding physically feasible states bas operating good condition model file loaded spaceex input model converted flat hybrid automaton representation reachability analysis performed next configuration file defining initial states form loc refers initial state label forbidden states case study none iii direction reach sets select oct similar results achieved direction reach set set box configuration file loaded reachability analysis algorithm run defined time horizon case corresponds two hours reachable sets generated figure
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algebra identities exponential functionals brownian motion apr reda chhaibi memoriam marc yor abstract explicitly compute exit law certain hypoelliptic brownian motion solvable lie group underlying random variable seen multidimensional exponential functional brownian motion consequence obtain hidden identities law gamma random variables probabilistic manifestation braid relations classical algebra identity corresponds braid move root system type ones seem new key ingredient conditional representation theorem relates hypoelliptic brownian motion conditioned exiting fixed point certain deterministic transform brownian motion identities law gamma variables tropicalize identities exponential random variables continuous versions identities geometric random variables related changes parametrizations lusztig canonical basis hence see exit law hypoelliptic brownian motion geometric analogue simple natural measure lusztig canonical basis msc subject classifications keywords algebra identities exponential functionals brownian motion braid relations total positivity brownian motion email contents introduction main results preliminaries conditional representation exit law algebra theorem identities law gamma identities exponential identities geometric identities proof conditional representation theorem review dufresne identity relationship proven matsumoto yor exponential functionals brownian motion proofs relationship brownian motions opposing drifts open questions introduction let complex group rank fix borel subgroup maximal complex torus lower unipotent subgroup denote lie algebra subspace roots real moreover write exp since euclidean space thanks killing form natural notion brownian motion brownian motion drift denoted context geometric crystals correspondence random input performed solving borel subgroup stochastic differential equation driven brownian motion section one obtains lie group valued stochastic process first introduced said preliminary section refer process hypoelliptic brownian motion infinitesimal generator satisfies parabolic condition although make use fact reassuring know smooth transition kernel stochastic process decomposition part multiplicative brownian motion converge focus part plays role multidimensional exponential functional brownian motion let set set simple roots open weyl chamber part converges refer law exit law hypoelliptic brownian motion first result conditional representation theorem characterizes law certain integral transform brownian motion brownian motion conditioned fixed consequence able give explicit formula exit law theorem second main result expression involves independent gamma variables case group one recovers dufresne identity law exponential functional brownian motion drift standard brownian motion identity states random variable distribution gamma random variable parameter groups higher rank presented construction gives explicit law multiple exponential functionals brownian motion almost surely belongs set totally positive matrices study totally positivity reductive groups initiated george lusztig see survey motivated theory canonical bases need fact possesses equivalent charts indexed reduced words longest element weyl group simply noticing law uses gamma variables charts depend choice reduced word find hidden identities law gamma variables primitive identities associated braid moves follows many primitive identities rank root systems content third main result stated theorem fact identities gamma variables tropicalize identities exponential random variables also prove discrete version involving geometric random variables using lusztig parametrization canonical bases structure paper begin stating three main theorems section necessary preliminaries lie theory total positivity illustrate claims thanks examples convenient postpone proof conditional representation theorem explain right away implies two others section show rational identities gamma variables tropicalize identities exponential variables implies discrete version involving geometric random variables section prove conditional representation theorem thanks results reduce problem induction whose base case result matsumoto yor relationship brownian motions opposite drifts diving proof explain theorem case review result sake completeness section along dufresne identity section absent published version finally conclude open questions acknowledgments author grateful marc yor fruitful discussions exponential functionals brownian motion identities paper dedicated memory also thankful philippe bougerol guidance phd thesis article based main results preliminaries lie theory need mostly standard notations terminology groups algebras see example let complex lie algebra rank maximal abelian subalgebra cartan subalgebra cartan decomposition resp denote simple roots resp coroots real part cartan subalgebra subspace simple roots real valued entirely encoded cartan matrix structure coefficients cartan matrix termine relations chevalley generators simple roots form basis dual fundamental coweights let complex lie group lie algebra subgroups lie algebras maximal torus form pair opposite borel subgroups subgroup every exp exp convenient sometimes written identify thanks killing form tion general write reflection respect hyperplane ker weyl group defined norm acts torus conjugation hence coxeter group generated reflections every written product sik sequence reduced word sequence shortest possible length set possible reduced words denoted weyl group unique longest element denoted set bruhat decomposition states disjoint union cells largest opposite bruhat cell every element admits unique gauss decomposition form nau sequel write gauss decomposition also remark reader unfamiliar lie groups mind example sln rank following matrices chosen chevalley generators usual elementary matrices set complex diagonal matrices zero trace identify set diagonal matrices determinant resp set lower resp upper triangular unipotent matrices tei weyl group group permutation matrices acts permuting coordinates identification symmetric group acting elements reflections identified transpositions longest word reorders elements decreasing order case gln second bruhat decomposition known linear algebra lpu decomposition states every invertible matrix decomposed product lower triangular matrix permutation matrix upper triangular matrix unique largest opposite bruhat cell corresponds dense locus principal minors remark classification complex simple lie algebras known classified types type slr slr type type spr spr type exceptional types number subscript indicates rank set roots denoted roots split positive negative roots set roots written positive sum simple roots reduced expressions weyl group elements give convex orderings positive roots see lemma let reduced expression produces positive roots set inversions inv produces positive roots inv call sequence enumeration positive roots chosen reduced expression obvious context drop subscript appendix list positive root enumerations rank systems total positivity classical case gln totally matrix matrix minors thanks classical formula totally matrices gln form lusztig generalized total positivity reductive groups motivated theory canonical bases level generality taking property definition simpler totally part denoted defined generated following sets exp generated generated interested parametrizations lusztig proved totally elements admit gauss decomposition made totally elements theorem lemma element unique gauss decomposition nau therefore one focus parametrizations following theorem generalizes result whitney gln says totally positive element written product elementary matrices depending falls bruhat decomposition theorem proposition proposition every reduced word gives rise parametrization bwb xik also define transition maps play particularly important role purposes paper need case call parametrization lusztig parametrization positive reals called lusztig parameters dependence implicit lusztig canonical basis interested parametrizations canonical basis combinatorics total posivity group related combinatorics lusztig canonical basis dual group tropicalization procedure trop said ization section trop defined let record following theorem see canonical basis lower unipotent part dual quantum group natural parametrization associated every reduced word changes parametrizations given trop brownian motion since made euclidean space thanks killing form natural notion brownian motion indeed scalar product restricted induced norm denoted let canonical probability space brownian motion brownian motion drift denoted brownian motion drift starting position written one obtains brownian motion solving following stochastic differential equation driven path symbol indicates sde understood stratonovitch sense dbt sde explicitly solved clearly decomposition given fik dtk ext remark presence half squared norms account fact time flow fashion roots groups one choose roots length hence choosing remark case opt normalization made previous remark indeed classical choice root hence factor following example example type let brownian motion drift sde dxt dbt solution ext ext example sln type remark let brownian motion drift notational reasons drop superscript put indices exponents sde becomes dxt dbt dxtn solution given ext ext exs ext process multiplicative brownian motion converge main theorems concern part converges drift inside weyl chamber able give explicit formula exit law occasions need consider path driving flow path previous construction carries verbatim conditional representation theorem symbol stands equality law random variables moreover generic gamma random variable parameter denoted gamma euler function let space continuous functions values following theorem defines path transform theorem proposition section totally path transform path gbt transform sense gauss decomposition always exists fix reduced word length call associated positive roots enumeration choose representative longest element norm also introduce shift vector log fundamental coweights remark ade groups roots chosen chosen squared norm state following theorem whose proof postponed section theorem conditional representation theorem let standard brownian motion distributed brownian motion drift initial position conditioned bijective function moreover pick xim random independent lusztig parameters standard brownian motion drift starting remark conditioning approach owes lot baudoin connell spirit ends quite different indeed paper considered conditioning brownian motion respect simple integrals simple root nevertheless geometric path model developed makes natural condition respect random variable contains simple integrals also iterated ones example one choose example one choose exit law thanks conditional representation theorem obtain exit law little effort theorem law brownian motion drift weyl chamber converges almost surely inside open cell xim lusztig parameters independent random variables proof condition entails convergence iterated integrals explicit expression equation fact follows total posivity criterion theorem theorem formulated terms generalized minors basically one adapt proofs theorem lemma case infinite time horizon law comes directly theorem worth noting probability measure smooth density charges entire space open dense cell cells smaller dimension therefore zero measure hence viewed random variable let examine example group smallest rank example know theorem tells exactly dufresne identity law law seen meaningful generalization dufresne identity uses inverse gamma group setting see map plays role inverse map higher dimensional examples give new identities exponential functionals brownian motion example consider brownian motion drift weyl chamber simple roots obtain choose reduced word independent gamma random variables corresponding parameters example theorem tell result seems new restated follows terms two positive parameters two correlated real brownian motions equality equivalent fact triple distribution notice two first marginals consistent dufresne identity order see one perform brownian rescaling time factor invoke classical identity algebra natural notion gamma law inverse gamma law denote independent gamma random variables definition gamma law inverse gamma positive roots enumeration associated reduced expression sim define law positive sense total positivity random variable xim define inverse gamma law laws well defined sense expressions depend choice reduced expression proposition reduced words every following equality law holds independent gamma random variables proof theorem relates laws exit law hypoelliptic brownian motion intrinsically defined formula defining meets eye indeed order law reduced expressions must hidden equalities law qualified identities algebra authors call see references therein said next section remarkable think probabilistic manifestation group structure precisely braid relationships reflections associated simple roots order braid relationship equality terms braid move occurs substituting within reduced word considering two reduced expressions weyl group element well known result due hideya matsumoto tits states one obtained successive braid moves see also using fact saying defined unambiguously equivalent saying proposition holds reduced words differ braid move hence one turns rank cases contain possible hidden identities classical rank reduction fix one root systems consider two reduced words longest element rank system maps computed every group explicitly tabulated berenstein zelevinsky theorem write thanks fundamental coweights using explicit formulas change lusztig parameters root enumerations given tables obtain following theorem theorem algebra identities associated every rank root system following identities law hold type table let type table let type table let type table let identities law gamma identities let make remarks obtained gamma identities classical algebra identity states two independent gamma random variables parameters form pair independent random variables fact easy retrieve case indeed considering independent variables designated laws algebraically defined know independent since independent get independent moreover lukacs proved independence pair characterizes gamma law leads first open question section exponential identities generic exponential random variable parameter denoted consider rational expression variables minus sign tropicalizing trop tantamounts replacing algebraic operations min rational expression operations min commonly referred tropical expression formally rational functions trop min trop trop trop trop trop example trop min less algebraic definition could used using limit always exists proposition analytic tropicalization rational subtractionfree expression variables log trop quantity variables bounded uniformly proof let prove statement induction size expression meaning number operations uses addition multiplication division base case notice monomial constant statement trivially true inductive step product ratio two rational expressions statement true statement follows using properties logarithm sum whose terms satisfy induction hypothesis log log log trop trop min trop trop log trop well known tropicalize changes parametrizations lusztig canonical basis theorem therefore consider following crystallizing procedure rational expressions use logarithm function understood trop lim log probabilistic side one recover tropical version gamma identities involving exponential variables following lemma shows gamma variables degenerate exponential variables lemma yor goes zero log converges law exponential variable parameter proof direct proof using densities possible straightforward let rather present aesthetically pleasing derivation suggested marc yor private communication reduce problem lim log lim log beta random variable parameters write product two independent random variables knowing fact uniform random variable log distributed hence log log proof finished upon noticing log converges zero probability hence tropical version proposition proposition reduced words every following equality law holds independent exponential random variables trop proof tropicalization parameter using previous lemma lim log lim log log log exp lim log trop information contained rank two case finite list identities exponential variables sums results far theorem exponential identities associated every rank root system following identities law hold type table let min min min type table let min min min min fashion one deduce exponential identities types point seems important mention path model coxeter groups developed exponential laws play key role infinima brownian motion appropriate drift hidden identities law found involving general coxeter braid relations goes beyond crystallographic case considered leads second open question section geometric identities let generic geometric random variable parameter denoted let log write log condition log entails every positive coroot allows formulate identities law geometric random variables proposition reduced words every following equality law holds independent geometric random variables trop proof claim true trop thanks elementary fact immediate identities theorem hold geometric random variables corollary allows prove natural analogue exists level lusztig canonical basis corollary associated independent geometric random tive root enumeration variables random variable defined distribution lusztig canonical basis independent choice proof conditional representation theorem let state version matsumoto yor relationship brownian motions opposite drifts based many previous works related exponential functionals brownian motion including dufresne identity theorem theorem let brownian motion euclidean vector space drift linear form denote hyperplane reflection respect ker measure brownian motion conditionally exponential functional equal log motion drift theorem dual version characterises reciprocal transform brownian motion brownian motion conditioned respect exponential functional theorem theorem let brownian motion drift log brownian motion drift conditioned moreover pick random independent brownian motion drift notice compared original formulation used multidimensional setting change sign simply replaced hyperplane reflection let focus proving conditional representation theorem case path transform simple expression properties log continuous path result hiroyuki matsumoto yor reformulated theorem conditional representation brownian motion drift brownian motion drift conditioned moreover pick random independent brownian motion drift remark recall case hence theorem becomes exactly theorem particular case group surprising impressive matsumoto yor fully worked case without starting considerations ready prove conditional representation theorem proof theorem course take let thanks composition property xim xim apply inductively theorem lusztig parameters taken follow right laws order get successive brownian motions end proof follows deterministic inversion lemma lusztig parameters theorem know also valid infinite time horizon subsection concerning notations little precision needs made point flow considered paper conjugation one equation hence little correction formula end concluding proof review dufresne identity relationship proven matsumoto yor section mainly expository nature reviews probabilistic results needed namely dufresne identity proofs theorems exponential functionals brownian motion important fact dufresne identity law proposition dufresne one dimensional brownian motion drift quick proof time inversion fixed random variable let law given ito lemma easily checked diffusion process since satisfies sde dzt ezt hence infinitesimal generator sequence converges law unique invariant measure distribution converges almost surely invariant measure law therefore need prove distribution log invariant measure done easily checking adjoint annihilates density random variable log exp applying adjoint get proofs relationship brownian motions opposing drifts let state version matsumoto yor relationship brownian motions opposite drifts based many previous works related exponential functionals brownian motion including dufresne identity first let start proving theorem using known results expo nential functionals brownian motion let brownian motion drift ftb natural filtration linear form exp lim law comes simple corollary dufresne identity corollary one identity law exp therefore density exp proof define real brownian motion using brownian scaling using change variable choosing result holds using dufresne identity law density bounded measurable function writing letting exp exp initial enlargement filtration using random variable give proof theorem following matsumoto yor original presentation order compute law conditionally ftb following decomposition essential copy independent ftb indeed readability purposes necessary invoke general filtration enlargement theorems give complete proof using usual tools indeed proved law smooth respect lebesgue measure making possible sity ndy following computations let regular version conditional probability bounded measurable function ftb dyf dyf fubini conclude absolutely continuous respect derivative ftb given dqy using expression density corollary get hence exp exp log zero quadratic variation therefore semimartingale bracket log log dns log end using girsanov theorem chapter viii theorem log qit brownian motion completes proof theorem inversion natural question whether recover answer yes argument due matsumoto yor proof theorem follows following inversion lemma lemma inversion lemma let valued paths functions log log moreover case proof immediate see simultaneously true true need prove gives convergence right away ready prove theorem proof theorem consider brownian motion drift ditioned previous filtration enlargement argument brownian motion enlarged filtration log using inversion lemma log following equalities law processes follow log log ends proof first fact second fact consequence knowing law usual disintegration formula continuous functional sample space open questions end paper listing open questions question noticed section algebra identities theorem type characterize gamma random variable thanks true types would like say group theoretic transforms related total positivity give independent random variables input made gamma variables question example proposition sequence mutually independent exponential random variables naturally appears depend choice reduced expression one deduce hidden identities law identical crystalligraphic case however general coxeter setting goes beyond framework simply coxeter group lie group weyl group moreover tropical relations appear irrational coefficients therefore tropicalization rational expressions indeed proof theorem dihedral root system transition maps make use cos tchebicheff polynomials indeed rational crystallographic values true general would interesting gain insight coxeter case able explicit relations possible obtain geometric lifting identities gamma variables references philippe biane philippe bougerol neil connell littelmann paths brownian paths duke math philippe biane philippe bougerol neil connell continuous crystal measure coxeter groups adv fabrice baudoin neil connell exponential functionals brownian motion whittaker functions ann inst henri probab arkady berenstein andrei zelevinsky total positivity schubert varieties comment math reda chhaibi littelmann path model geometric crystals whittaker functions lie groups brownian motion phd thesis paris pages reda chhaibi littelmann path model geometric crystals preprint pages philippe carmona petit marc yor betagamma random variables intertwining relations certain markov processes rev mat iberoamericana daniel dufresne distribution perpetuity applications risk theory pension funding scand actuar sergey fomin andrei zelevinsky double bruhat cells total positivity amer math james humphreys introduction lie algebras representation theory new graduate texts mathematics vol james humphreys reflection groups coxeter groups volume cambridge studies advanced mathematics cambridge university press cambridge eugene lukacs characterization gamma distribution ann math lusztig canonical bases arising quantized enveloping algebras amer math lusztig canonical bases arising quantized enveloping algebras progr theoret phys common trends mathematics quantum field theories kyoto lusztig total positivity reductive groups lie theory geometry volume progr pages boston boston lusztig survey total positivity milan hideya matsumoto relations des groupes weyl acad sci paris hiroyuki matsumoto marc yor relationship brownian motions opposite drifts via certain enlargements brownian filtration osaka springer linear algebraic groups modern classics boston boston second edition jacques tits des mots dans les groupes coxeter symposia mathematica indam rome vol pages academic press london whitney reduction theorem totally positive matrices analyse marc yor private communication appendix positive root enumerations rank systems table positive roots enumerations type table positive roots enumerations type table positive roots enumerations type table positive roots enumerations type
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guarded dependent type theory coinductive types jan hans bugge ranald rasmus lars aarhus university abizjak hbugge birkedal university copenhagen mogel abstract present guarded dependent type theory gdtt extensional dependent type theory later modality clock quantifiers programming proving guarded recursive coinductive types later modality used ensure productivity recursive definitions modular type based way clock quantifiers used controlled elimination later modality encoding coinductive types using guarded recursive types key development gdtt novel type term formers involving call delayed substitutions generalise applicative functor rules later modality considered earlier work crucial programming proving dependent types show soundness type theory respect denotational model technical report version paper appear proceedings fossacs introduction dependent type theory useful programming proving properties elements types modern implementations dependent type theories coq nuprl agda idris used successfully many projects however offer limited support programming proving coinductive types one key challenges ensure functions coinductive types productive unique solutions syntactic guarded recursion used example coq ensures productivity requiring recursive calls nested directly constructor well known syntactic checks exclude many valid definitions particularly presence functions address challenge approach guarded recursion flexible syntactic checks first suggested nakano new modality written called later allows distinguish data access data get later modality must used guard type definitions example guarded streams natural numbers described guarded recursive equation strgn strgn asserting stream heads available tails later types defined via guarded recursion standard coinductive types denotation defined via models based topos trees pragmatically bare addition disallows productive acausal functions every function returns every second element stream atkey mcbride proposed clock quantifiers functions extended dependent types thm shown allow definition types whose denotation precisely standard coinductive types interpreted semantics allow program real coinductive types retaining productivity guarantees paper introduce extensional guarded dependent type theory gdtt provides framework guarded recursion used programming coinductive types also coinductive reasoning types depend terms one key challenges designing gdtt coping elements available later elements types form generalising applicative functor structure dependent setting recall rules applicative functors next first rule allows make later use data second allows example functions applied recursively tails streams suppose type type type intuitively eventually reduce value next resulting type open term may able perform reduction problem occurs coinductive reasoning strgn property streams applications guarded coinduction assumption want apply tail stream type strgn hence must introduce new notion delayed substitution similar allowing give type binding definitional equality rules allow simplify type form next next construction generalises bind list variables delayed substitution essential many examples shown sec surprisingly applicative functor termformer central standard presentation applicative functors turns definable via delayed substitutions shown sec contributions contributions paper introduce extensional guarded dependent type theory gdtt show gives framework programming proving guarded recursive coinductive types key novel feature generalisation later next via delayed substitutions prove soundness gdtt via model similar used earlier work guarded recursive types clock quantifiers focus design soundness type theory restrict attention extensional type theory postpone treatment intensional version theory future work see secs addition examples included paper pleased note preliminary version gdtt already proved crucial formalizing logical relations adequacy proof semantics pcf using guarded recursive types paviotti guarded dependent type theory gdtt type theory base types unit booleans natural numbers along identity types universes space reasons omit definitions standard type theory see jacobs universes tarski distinguish types terms terms represent types called codes types code type recognised circumflex map sending codes types corresponding type follow standard practice often omit examples except important avoid confusion fix countable set clock variables single clock constant necessary define example function sec clock either clock variable clock constant intuitively temporal dimensions types may depend clock context finite set clock variables use judgement express either clock variable set clock constant judgements summarised fig parametrised clock contexts codes types inhabit universes parametrised clock contexts similarly universe clock contexts intuitively contains codes valid clock context type type typing judgment type equality term equality fig judgements gdtt delayed substitution types vary along dimensions universe inclusions whenever examples write explicitly note universes form hierarchy could additionally orthogonal hierarchy universes clock context hierarchy universes judgements closed clock weakening clock substitution former means derivable clock variable judgement also derivable latter means derivable judgement also derivable clock substitution defined obvious rules guarded recursion found figs rules coinductive types postponed sec recall later type former expresses something available later time gdtt clock delay type along different dimensions discussed introduction generalise applicative functor structure via delayed substitutions allow substitution delayed substituent available showed introduction type single delayed substitution work however term one argument example type wish type application applicative functor operation clock may neither available need sequences delayed substitutions define type concrete examples sec show issue arises practice therefore define sequences delayed substitutions new raw types terms delayed substitutions gdtt given grammar note write delayed substitution empty binds variables substituted similarly next three rules used construct type rules formulate generalise types arbitrarily long delayed substitutions type formation rule established introduction rule natural one delayed substitutions define using rules fig derive following typing judgement term form enough information perform substitution term type rule applies substitution equating term result actual substitution rule type using derive basic term equality typical applicative functors often case delayed substitution unnecessary variable substituted occur free express justify simpler typing rule words delayed substitutions type necessary apply function applicative functor identity law follows rule allows simplify term sometimes necessary switch order delayed substitution two substitutions switch places long depend express rule used examples paper implies rule needed paviotti phd work fixed points guarded recursive types gdtt clock valid current clock context combinator differs traditional combinator type recursion variable result type instead type guarded define term using say defined guarded recursion term intuitively proof say proving induction guarded recursive types defined suitably guarded functions universes approach birkedal generality rules gdtt allows define interesting dependent guarded recursive types example predicates sec first illustrate technique defining type guarded streams recall introduction want type guarded streams clock satisfy equation type equal code universe clock variable define code universes type univ type delayed substitutions typing rules type type fig overview new typing rules involving delayed substitutions code simple universe product type via rules gdtt show desired head tail operations simply first second projections conversely construct streams pairing use suggestive notation define asi defining guarded streams also done via guarded recursion example stream consisting ones defined ones rule essential defining guarded recursive types fixedpoints universes also used defining advanced guarded recursive dependent types covectors see sec identity types gdtt standard extensional identity types ida see jacobs two additional type equivalences necessary working guarded dependent types write reflexivity proof ida first type equivalence rule rule validated model sec may thought analogy type equivalences often considered homotopy type theory ida idb definitional type equalities next next next next definitional term equalities next next fig new type term equalities gdtt rules require context without rules assume exchanging allowed type depend vice versa likewise rule assumes exchanging codomains allowed none variables codomains appear type two important differences first using univalence propositional type equality whereas specifices definitional type equality natural extensional type theory second difference terms going directions whereas would term type without rule second novel type equality rule involves clock quantification presented sec examples section present example terms typable gdtt examples use term call type idb term definable type theory strong dependent elimination rule dependent sums second property use simply first second projections also type preserves commutativity gdtt define function type show commutativity implies commutativity inhabited term inhabits type type type term crucially need delayed substitutions example covectors next example sophisticated involves programming proving data type unlike streams dependently typed indeed generalised later carrying delayed substitution necessary type even elementary programs covectors potentially infinite version vectors lists length define guarded covectors first numbers definition gdtt need guarded fix type satisfies using define type family covectors covec case covec inl inr example covectors distinguish covec define ones type con covecn produces covector length consisting ones ones case inl inl inr although one simplest covector programs one imagine without generalised later delayed substitutions map function covectors defined map map case inl inr preserves composition following type inhabited map map map term case inl inr inr coinductive types discussed introduction guarded recursive types disallow productive acausal function definitions capture functions need able remove however eliminations must controlled avoid trivialising unrestricted elimination term elim every type would inhabited via making type theory inconsistent however may eliminate provided term depend clock term typeable context appear intuitively contexts temporal properties along dimension may progress computation without violating guardedness fig extends system fig allow removal clocks setting introducing clock quantifiers binding construct associated term constructor also binds elimination term clock application application term type clock written one may think analogous type polymorphic lambda calculus indeed basic rules precisely additional construct prev called previous allow removal later modality typing new construct prev somewhat complicated requires advancing delayed substitution turns context morphism actual substitution see fig definition judgement expresses context morphism context context use notation extending context morphism mapping variable term illustrate two concrete examples first indeed remove later clock quantier force force prev type correct advancing empty delayed substitution turns identity substitution rules ensure force inverse canonical term type second may see example delayed substitution term prev succ type recall syntactic type type prev fig overview new typing rules coinductive types sugar precisely term succ prev next advancing delayed substitution turns substitution mapping variable term prev succ variable term prev using rule prev rule simplifies substitution mapping succ term equal succ turn equal important property term prev bound hence prev type instead ensures substitution terms types terms need explicit substitutions used example clouston unary used place clocks binding structure ensures instance introduction rule closed substitution rule states type clock appear type matter clock applied resulting term polymorphic lambda calculus corresponding rule universal quantification types would consequence relational parametricity construct rule witness universes closed summarise new raw types terms extending sec prev finally equality rule analogous rule note sec canonical term type without rule term reverse direction derivable type isomorphisms encoding coinductive types using guarded recursive types crucially uses family type isomorphisms commuting type formers prev fig advancing delayed substitution type isomorphism mean two terms types first type isomorphism whenever free terms type type witness isomorphism note used clock constant essential way equality follows using rule clock application equality follows using rule following type isomorphisms follow using laws constructs involved important additional type isomorphism witnessing commutes binary sums however unlike isomorphisms require equality reflection show two functions inverse definitional equality canonical term type using ordinary elimination coproducts using fact encode binary coproducts using universes define term type inverse canonical term particular satisfies following two equalities used inl inl inr inr example programs coinductive types clock context fresh clock variable let type code let stra stra define head tail cons functions stra stra stra cons stra stra prev cons definitional type equalities type definitional term equalities type prev prev fig type term equalities involving clock quantification define acausal every function removes every second element input stream acausal second element output stream third element input therefore type function need input stream always available clock quantification must used function type stra defined stra result guarded stream easily strengthen define type stra stra also work covectors guarded covectors sec dependent coinductive type indexed conatural numbers type con easy define succ inl succ inr next define transport function comcon type comcon con con satisfying comcon inl comcon succ inr function used define type family covectors coveca con term con case comcon inl inr using term equalities derive type isomorphisms coveca coveca succ coveca expected properties type covectors simple function define tail function coveca succ coveca prev note needed type map function type map con coveca covecb defined map con case comcon inl inr lifting guarded functions section show general may lift function guarded recursive types addition guarded streams function coinductive streams moreover show lift proofs properties commutativity addition guarded recursive types coinductive types let context clock context fresh clock suppose types type type finally let function type define satisfying typing judgement assume another term type context proved give term type using type equality rule equal type derived property lifted functions properties guarded versions standard pattern using induction prove property function whose result guarded type derive property lifted function example lift zipwith function guarded streams coinductive streams prove preserves commutativity using result guarded streams sec soundness gdtt shown sound respect denotational model interpreting type theory model refinement bizjak reasons space leave description full model gdtt future work instead provide intuition semantics delayed substitutions describe interpret rule type type case one clock available subsystem gdtt one clock modelled category known topos trees presheaf category first infinite ordinal objects families sets indexed positive integers together families restriction functions rix indexed similarly functor maps object object unique map terminal object model closed type interpreted object type type interpreted indexed family sets together maps rib ria term interpreted morphism element write element type type interpreted object defined notice delayed substitution interpreted substitution reindexing model change index model reindexed along corresponds delayed substitution type theory notice depend interpretation type reduces interpretation defined applied interpretation generalised work general contexts sequences delayed substitutions one validate definitional equality rules indeed hold model related work birkedal introduced dependent type theory modality semantics topos trees guardedness requirement expressed using syntactic check every occurrence type variable lies beneath requirement subsequently refined birkedal showed guarded recursive types could constructed via functions universes however rules considered papers allow one apply terms type applicative functor construction defined simple function spaces therefore less expressive programming consider covector ones function map sec proving noting extensive use delayed substitutions example proofs consider coinductive types restricted causal functions extension coinductive types hence acausal functions due atkey mcbride introduced clock quantifiers simply typed setting guarded recursion extended work dependent types bizjak refined model allow clock synchronisation clouston introduced logic prove properties terms simply typed guarded allowed proofs coinductive types integrated fashion supported dependent type theories moreover relied types total property dependently typed setting would entail strong elimination rule would lead inconsistency sized types combined copatterns alternative approach modular programming coinductive types work mature respect implementation demonstration syntactic properties normalisation development gdtt essential enable proper comparison one advantage gdtt later modality useful examples beyond coinduction beyond utility sized types guarded recursive domain equations used model program logics conclusion future work described dependent type theory gdtt examples detailed show gdtt provides setting programming proving guarded recursive coinductive types future work plan investigate intensional version type theory construct prototype implementation allow experiment larger examples preliminary work suggested path type cubical type theory interacts better new constructs gdtt ordinary identity type finally investigating whether generalisation applicative functors apply dependent function spaces via delayed substitutions might also apply examples quite unconnected later modality acknowledgements research supported part modures sapere aude advanced grant project grant danish council independent research natural sciences fnu bizjak supported part microsoft research phd grant references abel pientka wellfounded recursion copatterns unified approach termination productivity icfp appel richards vouillon modal model modern major general type system popl atkey mcbride productive coprogramming guarded recursion icfp birkedal intensional type theory guarded recursive types qua fixed points universes lics birkedal schwinghammer first steps synthetic guarded domain theory topos trees lmcs bizjak model guarded recursion clock synchronisation mfps brady idris dependently typed programming language design implementation funct programming clouston bizjak grathwohl birkedal programming reasoning guarded recursion coinductive types fossacs cohen coquand huber cubical type theory constructive interpretation univalence axiom unpublished constable allen bromley cleaveland cremer harper howe knoblock mendler panangaden sasaki smith implementing mathematics nuprl proof development system upper saddle river usa coquand infinite objects type theory types codifying guarded definitions recursive schemes types hughes pareto sabry proving correctness reactive systems using sized types popl jacobs categorical logic type theory studies logic foundations mathematics north holland amsterdam krishnaswami benton ultrametric semantics reactive programs lics coq development team coq proof assistant reference manual logical project http version mcbride paterson applicative programming effects funct programming type theory productive coprogramming via guarded recursion nakano modality recursion lics norell towards practical programming language based dependent type theory thesis chalmers university technology paviotti birkedal model pcf guarded type theory mfps shulman univalence inverse diagrams homotopy canonicity mathematical structures computer science svendsen birkedal impredicative concurrent abstract predicates esop univalent foundations program homotopy type theory univalent foundations mathematics http institute advanced study appendix overview appendix sec contains type term equalities fig full detail sec starting page contains detailed explanations examples sec explaining rules gdtt used sec starting page contains detailed explanations examples coinductive types sec starting page contains detailed derivation type isomorphism used sec typing rules definitional type equalities type type free type definitional term equalities free examples section provide detailed explanations typing derivations examples described sec preserves commutativity first proof simplest define standard zipwith function streams show binary function commutative defined guarded recursion note none new generalised rules gdtt needed type function function simple types need dependent types course state prove properties prove example commutativity implies commutativity means must show type inhabited explain construct term typeable gdtt although construction might appear complicated first actual proof term construct simple possible let function say term witnessing commutativity wish construct term type guarded recursion end assume take using proof commutative first type idb definition see type idb show tails equal use induction hypothesis terms type first type note appearance generalised carrying delayed substitution cause variable appear may apply weakening rule derive hence may use derived applicative rule type definitionally equal next next type zipwith zipwith also compute next using exchange rule equality putting together shown term type means term type notice resulting proof term could simpler particular write delayed substitutions terms intermediate types example covectors next example sophisticated involve programming proving data type unlike streams dependently typed particular see generalised later carrying delayed substitution necessary type even elementary programs covectors colists potentially infinite lists vectors lists define guarded covectors first need guarded numbers type satisfying binary sums encoded standard way type theory nition gdtt con fix using con define type covectors type written type satisfying inl inr gdtt first define covec case covec inl inr examples distinguish covec type con covec inside branches type type con evident definition example covectors define ones type produces covector length consisting ones ones case inl inl inr checking type program need generalised later type recursive call type therefore type subterm must aim define function map covectors show preserves composition given two types map function type map defined guarded recursion map case inl inr let see definition correct type first types subterms let write definition inl type inl definition inr analogously inr hence type inr convertible using derived applicative rule second branch type may use simple applicative rule get allows type type inr notice made essential use general applicative rule apply using strong dependent elimination rule binary sums type whole case construct type need give map desired type show map defined satisfies basic property namely preserves composition sense type context types map map map inhabited proof course induction first record definitional equalities follow directly unfolding definitions map inl map inr map map next iterating two equalities get map inl map inl map inr map inr term map map convertible rule term map map similarly map inl map inr convertible map next let get back proving property take assume map map map take write map map map similarly definition map definitional equalities map compute inl type inl branch inr course bit complicated inr take type inr need construct term type map map map first type idc type use induction hypothesis get type map map map using type use applicative rule give type map map map rule next map map map type thus give term inr type inr using dependent elimination rule binary sums get final proof property term case inl inr inr simple could expected lifting predicates streams let predicate type clock variable define lifting predicate predicate streams elements type idea hold precisely holds elements stream however access element stream time holds first element stream holds second element stream one time step later precise definition uses guarded recursion term subterm type type may form type finally type needed see makes sense stream using delayed substitution rules gives rise type equality finally type equality rule gives together give type equality simplify using rule tyeqforce get gives accordance motivation given defined guarded recursion prove properties induction particular may prove holds holds type inhabited context type predicate take since proving induction assume induction hypothesis later let stream definition type equality applying gives first component applying induction hypothesis thus combining previous term proof lifting property term xsi example programs coinductive types let small type clock context fresh clock variable let stra define head tail cons functions stra stra stra prev cons stra stra cons define acausal every function removes every second element input stream acausal second element output stream third element input therefore type function need input stream always available necessitating use clock quantification function stra stra return head immediately recursively call function stream first two elements removed note result guarded stream easily strengthen define type stra stra interesting type type covectors refinement guarded type covectors defined sec first define type numbers con con easy define succ con succ con con inl succ inr next use type isomorphisms define transport function comcon type comcon con con comcon case inl inl inr inr prev function satisfies term equalities comcon inl comcon succ inr using define type covectors coveca coveca con term con case comcon inl inr notice use comcon transport type con term type con case analyse see type satisfies correct type equalities need auxiliary term equalities follow way defined terms using term equalities derive almost expected type equalities coveca coveca succ using type isomorphisms extend type equalities type isomorphisms coveca coveca succ coveca expected type properties covector type simple function define tail function coveca succ coveca prev note used ensure type correct next define map function covectors map con coveca covecb map function type con defined case comcon inl inr let see correct type let analogously type case inl inr type con using abbreviation write type con comcon comcon using straightforward show using dependent elimination rule sums sec correct type indeed inl inr coveca type isomorphisms detail terms rule crucially needed show constitute type isomorphism terms type type terms type converse type terms type prev converse type rules prev ensure pair functions constitutes isomorphism using isomorphisms construct additional type isomorphism witnessing commutes binary sums recall encode binary coproducts using universes standard way given two codes universe define else suppose clock variable write satisfying suppose two codes start auxiliary function comif let term type define else comifb else comifb else typeable due strong elimination rule define function need check types function side condition ensures types see function consider types subterms term type term type term type else get indeed term type contain required equality follows rule clock quantification thus term type else term type else exactly type needed typecheck whole term term derive following definitional term equalities inl inl inr inr also canonical term type defined case inl inl inl inl term inverse although require equality reflection show two functions inverses without equality reflection prove inverses propositional equality isomorphisms defined previously require equality reflection
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static dynamic semantics nosql languages mar giuseppe kim lri orsay france cnrs pps univ paris diderot sorbonne paris paris france ibm watson research hawthorne usa abstract deduced directly structure program even absence explicit type declaration annotation present calculus processing semistructured data spans differences application area among several novel query languages broadly categorized nosql calculus lets users define operators capturing wider range data processing capabilities whilst providing typing precision far typical primitive operators type inference algorithm based semantic type checking resulting type information precise flexible enough handle structured semistructured data illustrate use calculus encoding large fragment jaql including operations iterators json embedded sql expressions show encoding directly yields typing discipline jaql namely without addition type definition type annotation code example use jaql language json developed bigdata analytics illustrate proposed calculus works reason using jaql encompasses features found previously cited query languages includes number original ones well like pig supports sequence iteration filtering grouping operations queries like aql xquery features nested queries furthermore jaql uses rich data model allows arbitrary nesting data works generic sequences json records whose fields contain sequences records languages limited flat data models aql whose similar standard relational model used sql databases tuples scalars lists scalars lastly jaql includes sql embedded relational data reasons although present work focus almost exclusively jaql believe work adapted without effort wide array sequence processing languages following jaql program illustrates features performs one json input containing information departments one relational input containing information employees query returns department name first input number employees second input sql expression used select employees income given value jaql filter used access set departments elements two collections processed group expression jaql denotes current element introduction emergence cloud computing ever growing importance data applications given birth whirlwind new data models languages whether developed banner nosql bigdata analytics cloud computing domain specific languages dsl embedded host language share common subset sql ability handle semistructured data consensus yet precise boundaries class languages share two common traits emphasis sequence operations popular mapreduce paradigm lack types data programs contrary say xml programming relational databases data schemas pervasive meijer argues languages greatly benefit formal foundations suggests comprehensions unifying model although agree meijer need provide unified formal foundations new languages argue foundations account novel features critical various application domains captured comprehensions also languages provide limited type checking ignore altogether believe type checking essential many applications usage ranging error detection optimization understand designers programmers languages averse kind type definition annotation paper propose calculus expressive enough capture languages beyond sql comprehensions show calculus adapts various data models retaining precise type checking exploit flexible way limited type information information group depts filter select employees income dept deptname numemps count query blends jaql expressions filter selects collection depts departments size employees grouping sql statement selecting employees relational table salary relations naturally rendered json collections records example one key difference field access sql requires field present record operation jaql actually field selection jaql expressive since applied also collections effect selection recursively applied components collection collection results returned similarly filter iterators words expression extended abstract work included proceeding popl acm symposium principles programming languages acm press filter work much bound record without size field latter case selection returns null bound collection records arbitrary nested collections thereof accounts semistructured nature json compared relational model calculus express way illustrates difference dynamic semantics static typing calculus selection records whose mandatory field income greater defined let sel nil nil income tail sel tail else sel tail collections encoded lists lisp filtering among records arbitrary nested collections records optional size field present larger let fil nil nil size tail fil tail else fil tail tail fil fil tail tail fil tail terms show nearly basic building blocks calculus composition missing building blocks dub filters filters defined recursively sel tail recursive call perform pattern matching found functional languages filter executes environment resulting matching pattern composed alternation tries apply fails applies spread structure argument sel tail requires argument product type applies corresponding instance filter fil scans collections encoded lists lisp right associative pairs nil denoting empty list argument empty list returns empty list list whose head record size field possibly fields matched captures whole record content field tail list tail keeps discards record according whether field larger head also list recursively applies head tail head list neither list record size field head discarded encoding whole grouping query given section aim propose yet another analytics query language rather show express type languages via encoding core calculus language way preserve execution model obtain free formal semantics type inference system happens prototype implementation type information deduced via encoding without need type annotation used early error detection debugging purposes encoding also yields executable system used rapid prototyping possibilities critical typical usage scenarios languages deployment expensive time resources observed meijer advent big data makes important ever programmers add language system designers single abstraction allows process transform query analyze compute across data presenting utter variability volume structure yielding number new data models query languages execution fabrics framework present claim encompasses goal compilers languages could use type information inferred encoding encoding devise optimizations types pig jaql aql conceived considering execution model type schema manipulated data play role design consequence languages untyped present types optional clearly added afterthought differences data model type discipline particularly important embedded host language since yield impedance mismatch reason types disregarded languages may originate alleged tension type inference data one hand languages conceived work collections data weakly partially structured hand current languages type inference haskell work homogeneous collections typically lists elements type work show two visions coexist type data semantic subtyping type system conceived semistructured data describe computations filters untyped combinators thanks technique weak typing introduced polymorphically type results data query processing high degree precision conception filters driven schema data rather execution model use capture give uniform semantics wide range semi structured data processing capabilities give type system encompasses types defined languages notably pig jaql aql also xml query processing languages see section iii infer precise result types queries written languages without addition explicit type new construct show minimal current syntax languages bring dramatic improvements precision inferred types types propose extensible record types heterogeneous lists whose content described regular expressions types defined following grammar types int char empty null regexp singleton closed record open record sequences base special union difference denotes empty word semantics types expressed terms sets values values either constants true false null latter denoting character records values lists values singleton type type contains value particular null singleton type containing value null closed record type int int contains record values exactly two fields integer values open record type int int contains record values least two fields integer values sequence type set sequences whose content described regular expression example char contains sequences characters use string denote type standard double quote notation denote values int int sake precision comply jaql semantics last pattern rather tail fil tail since field selection fails whenever record list definition would detect possibility failure static type error denotes nonempty lists even length containing record values type int union type contains values contains values difference type shall use bool abbreviation union two singleton types containing true false false empty respectively contain values recursive type definitions also used see section formal details types express types pig jaql aql xml types much instance aql includes homogeneous lists type expressed types jaql documentation one find type long string boolean type arrays whose first element second string booleans easily expressed types bool jaql allows limited use regular expressions kleene star appear tail position types restrictions example char char type strings sequences chars denote email addresses ending either use syntactic sugar make terms previous one readable likewise henceforth use denote field type optional syntactic sugar stating either field undefined contains value type formal definition see appendix coming back initial example filter fil defined expects argument collection following type former limited forms latter finally includes arbitrary deep composition languages offer operator top level whose power nevertheless restrained type system important contribution work directly compares programming language approach tree transducer one calculus implements transformations typical tree transducers several advantages transducer approach transformations expressed formalism immediately intelligible functional programmer calculus untyped version turing complete transformations statically typed expenses turing completeness without annotation yielding precise result types even restrict calculus terms thus losing turing completeness still strictly expressive wellknown widely studied deterministic tree transducer formalisms technical contributions proof turing completeness formalism definition type system copes records computable labels iii definition static type system filters correctness definition static analysis ensures termination proof thereof type inference algorithm complexity bounds expressed size types filters proof terms pass static analysis form language strictly expressive tree transducers outline section present syntax three components system namely minimal set expressions calculus filters used program operators encode operators languages core types types presented encoded section defines operational semantics filters declarative semantics operators type system well type inference algorithm described section section present handle large subset jaql section reports subtler design choices system compare related works section conclude section order avoid blurring presentation proofs secondary results encodings extensions moved separate appendix type depts size int depts possibly empty arbitrary nested list records optional size field type int notice important specify optional field type since size field different type would make expression raise error information deduced structure filter since fil contain type definition annotation define type inference system rejects argument fil type depts deduces arguments type size int addr string sec int depts subtype depts result type size int addr string forget field addr discards field sec replacing recognizes test may fail encoding primitive jaql operations formal core calculus shall provide formal clean semantics well precise typing instance clear applying following dot selection true result shall able deduce applied arbitrary nested lists records optional integer field type int yields arbitrary nested lists int null values type int null finally shall show accept extend current syntax jaql language minimal filter syntax pattern filter obtain huge improvement precision type inference syntax section present syntax three components system minimal set expressions calculus filters used program operators encode operators languages core types types presented introduction encoded core work definition filters types key property development filters grafted host language satisfies minimal requirements simply adding filter application expressions host language minimal requirements host language possible quite simple must constants typically types int char string bool variables either pairs record values necessarily static side host language must least basic products types able assign type expressions given type environment typing assumptions variables addition filter applications host language acquire increase capability define polymorphic iterators query processing expressions enriched powerful precise type system contributions main contribution work definition calculus encompasses structural operators scattered nosql languages possesses characteristics make unique swarm current data processing languages particular parametric though fully embeddable host language uniformly handles width deep nested data recursion languages offer expressions exception bags types include order focus essential features work consider following set expressions definition expressions exprs constants variables pairs records record concatenation field deletion operators filter application ranges filters defined later generic constants string constants record types come two flavors closed record types whose values records exactly fields specified type open record types whose values records least fields specified type product types standard complete set type connectives finite unions intersections negations use empty denote type values type values sometimes denoted used patterns added term recursive types allows encode regular expression types defined introduction generally recursive type definitions used finally use capitalized distinguish expression operators denote host language type operators thus filter applications return values whose type belongs foreign language list functions suppose typing functions given type operators instance succ user defined successor function suppose given type form arrow int int similarly application say apply given type expression presumably int arrow type operator apply expression operator denotational semantics types sets values informally described introduction basis definition subtyping relation types say type subtype type noted set values denoted contained sense set values denoted formal definition decision procedure subtyping relation reader refer work semantic subtyping intuitively expressions represent syntax supplied host language first two one next two really extend missing expressions expression filter application expressions formed constants variables pairs records operation records record concatenation gives priority expression right records contains field label one taken field deletion require record contain field given label though point important metavariable ranges operators well functions constructions belonging defined host language among expressions single set values intuitively results computations formally defined follows use foo character string constants characters integers backquoted words foo atoms constants use three distinguished atoms nil true false double quotes omitted strings labels record fields thus write name john rather name john sequences aka heterogeneous lists ordered collections arrays encoded lisp nested pairs atom nil denotes empty list use syntactic sugar nil patterns filters core untyped operators three different things structurally decompose transform values applied sequentially composed pattern matching order define filters thus first need define patterns definition patterns patterns types expressions particular filter applications typed following set types typically basic product recursive form record types provided host language definition types types type variable pair closed rec open rec subpatterns forming pairs records intersections distinct capture variables forming unions capture variables basic types singleton types products closed records open records union types intersection types negation type empty empty type type recursive types recursion variable foreign type calls patterns essentially types capture variables ranged may occur every position negation recursion pattern used match value matching value pattern noted either fails noted returns substitution variables occurring pattern values substitution used environment expression evaluated pattern type matching fails pattern matched value type otherwise returns empty substitution variable matching always succeeds returns substitution assigns matched value variable pair pattern succeeds matched pair values succeeds corresponding projection value union two substitutions returned record patterns similar product pattern specificity open record pattern matches fields specified every recursion guarded every type variable separated binder least one application type constructor products records types already explained introduction basic types int bool ranged singleton types denoting type contains value pattern intersection pattern succeeds patterns succeed union two substitutions returned union pattern first tries match pattern fails tries pattern succeeds instance pattern int matched value pair values integer case returns substitution fails otherwise finally notice notation used examples introduction syntactic sugar informal semantics matching see formal definition explains reasons restrictions capture variables definition intersections pairs records patterns must matched thus assign distinct variables union patterns one pattern matched hence set variables must assigned whichever alternative selected strength patterns connections types fact pattern matching operator typed exactly entailed following theorems proved expression filter always returns value corresponding evaluation discards argument filter applies filter argument environment obtained matching argument provided matching fail rather powerful feature allows filter perform two critical actions inspect input regular exploring capture part input reused evaluation subfilter argument application returns application product filter argument returns otherwise application fails argument pair fails record filter similar applies specified field corresponding filter stressed leaves fields unchanged fails applications specified fields absent argument record filter returns application argument fails application semantics recursive filter given standard unfolding definition recursive calls real restriction introduce filters recursive calls done arguments given form arguments form values variables may occur restriction practice amounts forbid recursive calls result another recursively defined filter cases easily encoded reason restriction technical since greatly simplifies analysis section ensures termination type inference without hampering expressiveness filters turing complete even restriction see theorem filters composed filter applies result applying argument fails two condition every subterm form contain free recursion variables strictly necessary indeed could allow terms point analysis termination typing would reject terms apart trivial ones result recursive call used composition since restriction restrict expressiveness calculus theorem proves turing completeness restriction addition restriction design rather technical choice prefer forbid programmer write recursive calls side composition systematically reject programs use way finally singled specific filters specifically chose groupby orderby whose semantics generally specified declarative rather operational way bring expressive power calculus proof turing completeness theorem use declarative operators actually encoded remaining filters interesting single yield either simpler encodings precise typing theorem accepted type every pattern set values type call set accepted type note fact exact set values matching succeeds type obvious states every pattern exists syntactic type produced grammar definition whose semantics exactly set values matched existence syntactic type note utmost importance precise typing pattern matching particular given pattern type contained subtype allows compute exact type capture variables matched value theorem type environment exists algorithm every pattern returns type environment vars types filters definition filters filter term generated filters expression pattern product record union recursion recursive call composition declarative operators operators groupby orderby arguments filter grouping filter ordering semantics operational semantics calculus given reduction semantics filter application record operations since former novelty work save space omit latter standard anyhow variables constants pairs record big step semantics every subterm form recursion variable free inf define big step operational semantics filters definition given inference rules figure judgements form describes evaluation application filter argument environment yields object either value latter special value represents runtime error raised rule error either filter match form argument argument filter product pair filters like transducers applied value return another value however unlike transducers possess constructs like ability test input capture subterms recompose intermediary result captured values composition operator first describe informally semantics construct expr eval prod patt comp rec error recd rule applies figure dynamic semantics filters proof sketch encode untyped first applying continuation passing style cps transformations encoding cps term reduction rules substitutions via filters thanks cps eschew restrictions composition pattern matching failed side condition patt hold notice argument filter always value unless filter unfolding recursive call case variables may occurr rule environment used store body recursive definitions semantics filters quite straightforward inspired semantics patterns expression filter discards input evaluates rather asks host language evaluate expression current environment expr thought side branch construct product filter expects pair input applies returns pair results prod filter used particular express sequence mapping first component transforms element list applied tail practice often case recursive call iterates arbitrary lists stops input nil input pair filter fails rule error applies record filter expects input record value least fields specified filter applies value corresponding field leaving contents fields unchanged recd argument record value contain fields specified record filter application subfilter fails whole application record filter fails pattern filter matches input value pattern matching fails filter otherwise evaluates subfilter environment augmented substitution patt alternative filter follows standard policy filter succeeds result returned fails evaluated input value filter particularly useful write alternative two pattern filters making possible conditionally continue computation based shape input composition allows pass result input composition filter paramount importance indeed without way iterate deconstruct input value use product filter always rebuilds pair result finally recursive filter evaluated recording body evaluating rec recursive call replace recursion variable definition concludes presentation semantics nondeclarative filters without groupby orderby form turing complete formalism full proof appendix semantics declarative filters conclude presentation semantics define semantics groupby orderby prefer give semantics declarative form rather operationally order tie particular order keys execution groupby groupby applied sequence reduces sequence sequence vnj vkj partition orderby orderby applied reduces permutation since semantics operators deeply connected notion equality order values host language give operations however illustrate type algebra allows provide precise typing rules specialized particular semantics also possible encode groupby several input sequences combination groupby filters appendix syntactic sugar formally defined semantics filters use introduce syntactic sugar expressions reader may noticed productions expressions definition define destructor projections label selection constructors reason destructors well common expressions encoded filter applications fst snd let else theorem turing completeness language formed constants variables pairs equality applications nondeclarative filters turing complete def def def def def true false match def whether type finite contains finitely many values say bool type finite finiteness regular types seen tree automata decided polynomial time possible write finite union values actually singleton types consider let type type first must subtype string since record labels strings finite expressed means return string therefore type thus union types expressed rule infinite instead say record unknown labels expressed rule possible choice others possible instance jaql dot selection overloaded applied record jaql returns content field field absent argument null jaql returns null fails argument record applied list array jaql terminology recursively applies elements list jaql precisely defined null besides syntactic sugar next section use denote record type formed field types field types whose label already present similarly denote record types formed field types apart one labelled present finally also use expressions types patterns syntactic sugar lists used introduction instance matched lists elements provided element matches type inference onc variables constants pairs straightforwardly typed typing simple foreign expressions onstant string infinite section describe type inference algorithm expressions vars rod records multiple fields handled rule merges result typing single fields using type operator defined cduce record concatenation defined take account undefined unknown fields instance int int bool int unknown fields side may override known fields side instance int bool int int likewise every record type every subtype finally onc deal record concatenation field deletion respectively straightforward way constraint expressions must record type constraints form see appendix formal definitions type operators notice rules ensure record two fields label error detecting error needs sophisticated type systems dependent types beyond scope work rule used type operator case multiple occurring labels since records unordered corresponds randomly choosing one types bound labels field selected would yield error typing ambiguous fine tune rule finite unions record types require pairwise disjoint sets labels since problem would still persist infinite types prefer retain current simpler formulation denotes typing environment function expression variables types denotes constant singleton type containing constant expressions host language typed type function given type environment foreign expression returns type expression suppose given host language oreign type since various contain filter applications thus unknown host language type system rule oreign swaps variables type notice expressions whereas include filter applications include applications expressions expressions therefore host language provides function definitions applications host language must dealt foreign expressions well expression operator apply section typing records typing records novel challenging record expressions may contain string expressions label position type systems record aware labels never computed difficult give type since general statically know value return required form record type must ask value string type record expression thus distinguish two cases according typing filter application filters applied passed around computed therefore assign types filters expression assign types filter applications typing rule filter application function type must able handle type environments types system either subsuming variable specific types types host language host language support singleton types singleton type subsumed int typing foreign expressions using types ilter relies auxiliary deduction system judgments form states environments explained later apply filter value type return result type define auxiliary deduction system core type analysis first need define type accepted filter intuitively type gives necessary condition input filter fail define recursive types order use need define three different environments vars types denoting type environments associate term variables types rvars filters denoting definition environments associate filter recursion variable body definition rvars types tvars denoting memoization environments record call given recursive filter given type yielded introduction fresh recursion type variable typing rules thus work judgments form stating applying expression type environments yields result type judgment derived set rules given figure rules straightforward put side side dynamic semantics filters given section clear type system simulates level types computations carried filters values runtime instance rule xpr calls typing function host language determine type expression rule rod applies product filter recursively first second projection member product decomposition input type returns union result types rule records similar recursively applying member record decomposition returning union resulting record types pattern filter rule subfilter typed environment augmented mapping input type pattern theorem typing rule union filter nion reflects first match policy typing second branch know first taken hence runtime filtered value type notice ensured definition accepted type rough approximation discards grosser errors stressed right definition sufficient ensure evaluation type system premises check induction implies never fail values type ergo values never reach also discard output type contribution branches taken branches whose accepted type empty intersection input type composition rule omp straightforward rule restriction filter open recursion variable ensures output type also type without free recursion variables therefore use input type next three rules work together first introduces recursive filter fresh recursion variable output type also memoize recursive filter associated body input filter input type output type newly introduced recursive type variable dealing recursive call two situations may arise one possibility first time filter applied input type therefore introduce fresh type variable recurse replacing definition otherwise input type already encountered typing filter variable return memoized type type variable finally rule rule handle special cases groupby orderby filters typing explained following section definition accepted type given filter accepted type written set values defined groupby orderby easy show argument included accepted type necessary sufficient cases composition recursion condition evaluation filter fail lemma let filter value every proof straightforward induction structure derivation detailed appendix last two auxiliary definitions need related product record types presence unions general form product type finite union products since intersections distribute products instance consider type int string type denotes set pairs either projections int projections string type less precise since also allows pairs whose first projection int second projection string vice versa see necessary manipulate finite unions products similarly records therefore introduce following notations lemma product decomposition let types product decomposition denoted set types given product decomposition say rank noted rank use notation type tji exist several suitable decompositions whose details scope paper refer interested reader practical algorithms compute decompositions subtype notions decomposition rank projection generalized records lemma record decomposition let types record decomposition denoted finite set types either form tini form tini given record decomposition say rank noted rank use notation type label component typing orderby groupby structural filters enjoy simple compositional typing rules operations orderby groupby need specially crafted rules indeed well known transformation languages ability compare data values also type inference becomes undecidable see therefore provide two typing approximations yield good calculus three different sets variables set vars term variables ranged introduced patterns used expressions arguments calls recursive filters set rvars term recursion variables ranged used define recursive filters set tvars type recursion variables ranged used xpr type rod rank sij rank rank sij sim nion rank omp item orderby orderby fresh type dom fresh type dom item groupby orderby isi ordered figure type inference algorithm filter application compromise precision decidability first define auxiliary function sequence types whose full proof given appendix easy write filter type inference algorithm deduction terminate deduction simulates abstract execution filter type since filters turing complete general possible decide whether deduction given filter terminate every input type reason define static analysis check filters ensures passes analysis every input type deduction terminates space reasons formal definition check relegated appendix behavior easily explained imagine recursive filter applied input type algorithm tracks recursive calls occurring next performs one step reduction recursive call unfolding body finally checks unfolding variable occurs argument recursive call bound type subtree original type words analysis verifies execution derivation every call type pattern always yields type environment variables used recursive calls bound subtrees implies rule new always memoize given types obtained arguments recursive calls replacing variables subtree original type memoized rule since regular finitely many distinct subtrees thus memoize finitely many distinct types therefore algorithm terminates precisely analysis proceeds two passes first pass algorithm tracks recursive filters marks variables occur arguments recursive calls assigns variable abstract identifier representing subtree input type variable bound initial call filter iii returns set types obtained replacing variables associated abstract identifier argument recursive call last set intuitively represents possible ways recursive calls shuffle recompose subtrees forming initial input type second phase analysis first abstractly reduces one step recursive filter applying set types collected first phase analysis checks whether reduction variables marked first phase occur arguments recursive calls still bound subtrees initial input type checks fails filter rejected definition item set let types item set denoted item defined item empty item item item item first second line definition ensure item returns empty set sequence types products namely empty sequence third line handles case sequence type case finite union products whose first components types head sequence second components recursively types tails note also definition since types regular trees number distinct types accumulated item finite defined typing rules orderby groupby operators orderby orderby filter uses argument filter compute key element input sequence returns sequence elements sorted respect key therefore types elements result still known order lost use item compute output type orderby application orderby groupby typing orderby used give rough approximation typing groupby stated rule words obtain list pairs key component result type applied items sequence use orderby shuffle order list far precise typing groupby keeps track relation list elements images via given appendix soundness termination complexity soundness type inference system given property subject reduction filter application theorem subject reduction implies jaql difficult see type inference algorithm converges every input type exists integer recursive calls marked variables bound subtrees initial input type something depend course since deciding whether exists possible analysis checks whether possible input types filter satisfies say every recursive call marked variables satisfy property otherwise rejects filter section show filters used capture popular languages processing data cloud consider jaql query language json developed ibm give translation rules subset jaql filters definition jaql expressions use following simplified grammar jaql distinguish simple expressions ranged core expressions ranged theorem termination check every type deduction furthermore given tree automaton nta exptime size problem proofs see appendix termination appendix complexity complexity result line similar formalisms instance shown non deterministic tree transducers exptime input output types given nta filters defined paper excepted appendix pass analysis example consider filter rotate applied list returns list first element moved last position empty list applied empty list analysis succeeds filter denote abstract subtree bound variable recursive call executed abstract argument unfolding recursive call bound whereas bound two distinct subtrees variables recursive call thus bound subtrees original tree even though argument recursive call subtree original tree therefore filter accepted order appreciate precision inference algorithm consider type type lists formed integers least one followed booleans least one application rotate argument type algorithm statically infers precise type int int apply inferred type int int int bool int bool generic filters turing complete however requiring check holds filter typeable restricts expressive power filters preventing recomposing new value recursive call instance possible typecheck filter reverses elements sequence determining exact class transformations typeable filters express challenging however possible show appendix typeable filters strictly expressive tree transducers regular lookahead formalism tree transformations introduced intuition result conveyed example consider tree tree whose root labeled two children monadic tree height respectively possible write tree transducer regular creates tree concatenation two children root seen sequences transformation easily programmed typeable filters key difference expressive power comes fact filters evaluated environment binds capture variables input feature essential encode sequence concatenation sequence flattening pervasive operations dealing expressed tree transducers regular constants variables current value arrays records field access function call pipe filter filter transform transform expand expand group grouping filters order ease presentation extend syntax adding filter definitions already informally used introduction filters filter calls expressions let filter filter defn call ranges filter names mapping language consider rely following filters let filter filter nil nil true false let filter transform nil nil let filter expand nil nil nil mapping jaql expressions mapped expressions follows distinguished expression variable interpreting jaql jck jxk jen jek jen jop jen nil jek jkkf jaql core expressions mapped filters follows jfilter ekf jfilter ekf jfilter ekf filter jek jtransform ekf jtransform ekf jtransform ekf transform jek jexpandkf jtransform ekf jexpand ekf expand jexpandkf jgroup ekf jgroup ekf jgroup jgroup jgroup jgroup jgroup groupby transform translation defines first knowledge formal semantics jaql translation needed define semantics nosql language bonus endow type inference system described without requiring modification original language action demanded since machinery exploit developed work typing every jaql expression encoded filter ensured terminate check holds filter transform expand provided holds also arguments since perform recursive calls recombinations subtrees input definition encoding introduce new recursion hence always yields composition application filters check holds let illustrate composition filters typed consider instance employees type remp remp dept int income int depts type rdep rbranch rdep depid int name string size int rbranch brid int name string type subtype dept defined introduction global input type therefore line remp rdep rbranch becomes selection filtering line remp rdep note occurrences rbranch ignored fil tagging integer line flattening line yields remp rdep illustrates precise typing products coupled singleton types instead int groupby line introduces approximation dependency tag corresponding type kept int remp rdep lastly transform typed exactly yielding final type dept int deptname numemps int note null retained output type since may employees without department head may applied empty list returning null selection name null returns null instance suppose pipe jaql grouping defined introduction following jaql expression order produce printable representation records result examples show use encoding let encode example introduction sake concision use filter definitions rather expanding details use fil sel defined introduction expand transform defined beginning section encoding jaql field selection defined section finally head returns first element sequence family recursive filters rgrpi defined transform denotes string concatenation conversion operator type string composition three reasons field dept misspelled dep type int must applied concatenation programmer account fact value stored field deptname may null encoding produces following lines appended previous code let filter head nil null let filter rgrpi nil nil tail rgrpi tail rgrpi tail query introduction encoded follows employees depts sel fil transform transform expand groupby transform dept deptname head numemps count transform three errors detected type system subtler example error given following alternative code transform dept deptname string numemps deptname null invalid department corrects previous errors adds new one since detected type system last branch never selected see ensures soundness forcing programmer handle exceptional situations null example also precise enough detect code paths never reached order focus contributions kept language types filters simple however already exists several contributions types expressions used two particular worth mentioning context recursive patterns xml definition defines patterns inductively alternatively consider possibly infinite regular trees coinductively generated productions lines done cduce use recursive patterns obtained encode regular expressions patterns see although enhance words perform selection employees filter departments lines tag element comes employees comes departments line merge two collections line group heterogeneous list according corresponding key line element result grouping capture key line split group employees depts line capture subgroup corresponding variable line return expression specified query lines general definition encoding given appendix latter syntactic sugar defined cduce written pressiveness greatly improves writing programs since makes possible capture distinct subsequences sequence single match instance sequence matched pattern int bool captures list integer elements capture variables regular expression patterns bound lists captures boolean elements remaining elements ignored patterns encoded without rgrp instance transform lines compactly rendered product number name blouse clearly system restrictions merging nesting json xml extension required system define xml query processing expressions introduction syntactic sugar make expressions readable seems helpful transform dept deptname head numemps count concerns xml types used originally defined xml comes surprise seamlessly express xml types values example cduce uses types used encode xml types elements triples first element tag second record representing attributes third heterogeneous sequence content element furthermore adapt results encode forward xpath queries filters therefore requires little effort use filters presented encode languages jsoniq designed integrate json xml precisely type regular expressions xml data xpath queries embedded jaql programs shown section follows system introduced already able reproduce type transformations jsoniq without restrictions drawbacks latter argue better extend nosql languages xml primitives directly derived system rather use system encode jsoniq instance example given jsoniq draft show render xhtml following json data col labels singular plural row labels data spinne spinnen spinnst spinnt spinnt spinnen json xml regex exist various attempts integrate json xml instance jsoniq query language designed allow xml json used query motivation json xml widely used data interchange internet many applications json replacing xml web service apis data feeds applications support formats precisely jsoniq embeds json xml query language xquery stratified way jsoniq allow xml nodes contain json objects arrays result thus similar ocamlduce embedding cduce xml types expressions ocaml drawbacks type system derived type system cduce whereas theory filters originally designed use cduce host language consequence xml types expressions seamlessly integrated work presented without particular restriction end suffices use xml elements types encoding used implementation cduce xml element triple formed tag expression record whose labels attributes element sequence characters xml elements denoting content instance following element encoded filters presented work new syntactic sugar col labels row labels data table transform transform transform expands subsequence containing sequence resulting xhtml document rendered web browser product number name blouse similarly jaql libraries include functions convert manipulate xml data example possible jaql embed sql queries possible evaluate xpath expressions function xpath takes two arguments xml document string containing xpath xpath read seq filters encode forward xpath expressions see precisely type current implementation check type result external query encoded following triple product system number name blouse tegers even greater precision obtained grouping expressions generation key performed filter discriminates types result type keep precise correspondence keys corresponding groups example consider following extended jaql grouping expression xpath sql xml document produced independently jaql encoding forward xpath filters precisely type calls jaql xpath function also feed documents produced jaql expressions finally regular expressions types used describe heterogeneous sequences particular content xml elements used type regular expressions functions working regular expressions regex short form practice yet another domain specific language embedded general purpose languages untyped weakly typed typically every result type string way recursive patterns straightforwardly encode regexp matching therefore combining pattern filter filters possible encode regexp library important advantage stated theorem set values respectively strings accepted pattern respectively regular expression precisely computed expressed type function jaql function extracts substrings match given regex easily implemented filters precisely typed typing rules typing amount intersect type string precise string type accepted pattern encodes regex issue group town roma pisa italia town paris france type town string addr string type inferred system groupby expression italia town roma pisa addr string france town paris addr string town roma pisa paris addr string precisely associates every key type elements groups finally order allow modular usage filters adding filter application expression foreign language suffice parametric filter definitions also needed let filter programming filters however line already said disruption execution model recursive parametric filter definitions probably disallowed since compilation according mapreduce model would require disentangle recursive calls used filters encode operators hardcoded languages order type course possible embed typing technology introduced directly compilers language obtain flexible typing characterizes system however important aspect filters ignored far used directly programmer define operators typed precisely ones therefore possibility extend existing nosql languages adding expressions filter application expression next problem decide far definition filters complete integration taking definitions given far conceivable might disrupt execution model host language since user could define complex iterators fit chosen distributed compilation policy good compromise could add host language filters local effects thus avoiding affect distributed compilation execution model minimal solution consists choosing filters patterns unions expressions commentaries finally let explain subtler design choices system filter design reader may wonder whether products record filters really necessary since first sight filter could encoded similarly records point expressions thus pair closed wihtout free term recursion variables without explicit product filter would possible program filter simple identity map nil since expression free term recursion variable similarly need explicit record filter process recursively defined record types head int tail nil likewise one wonder put filters open record variant copy extra fields closed one reason want filter applied records exactly fields specified filter simply obtained pattern matching filter without trailing simply introduced syntactic sugar adding filters jaql use arrow patterns order avoid confusion jaql pipe operator would allow user define powerful operators use would already dramatically improve type precision instance could define following jaql expression transform sum convention filter occurring expression denotes application current argument syntax inference system able deduce feeding expression returns result argument type int bool bool type int bool sum int precision comes capacity inference system discriminate two branches filter deduce sum field added field present similarly using pattern matching jaql filter expression deduce filter int true false fed sequence elements always returns possibly empty list constructors syntax constructing records pairs exactly patterns types expressions filters reader may wonder distinguish using say product types instead record values combined fact values singletons syntax critical design choice greatly reduces confusion languages since makes possible unique representation syntactically could write nil value would pass since expression must typeable without knowing environment rule ilter beginning section purposedly stratified system order avoid mutual recursion filters expressions related work constructions semantically equivalent consider instance pattern nil syntax nil denotes product type two singletons nil value nil singleton contains value according interpretation choose pattern interpreted pattern matches product pattern matches value differentiated syntax singletons values pairs products pattern could written five different ways point would match exactly sets values chose syntax nested relational sql context many works studied integration nested algebra sql general purpose programming languages among first attempts integration relational model pascal smalltalk also monads comprehensions successfully used design implement query languages including way embed queries within host languages significant efforts done equip languages type systems type checking disciplines recently integration typing aspects however approaches support homogeneous sequences records context specific classes queries practically equivalent nested relational algebra calculus account records computable labels therefore easily transposable setting sequences heterogeneous data queries much expressive present work inspired stems previous works xml iterators targeting nosql languages made filter calculus presented substantially different one dubbed xml filters follows well syntax dynamic static semantics xml filters behave kind tree transducers termination enforced heavy syntactic restrictions less constrained use composition makes type inference challenging requires sometimes cumbersome type annotations xml filters allowed operate composition result recursive call thus simulate tree transformations absence explicit arguments recursive calls makes programs understandable programmers contrast main focus current work make programs immediately intelligible functional programmer make filters effective typing sequence transformations sequence iteration element filtering flattening last two especially difficult write xml filters require type annotations also integration filters record types absent sketched novel much needed encode json transformations record types definition records redundant types patterns instead current definition could used since rest encoded intersections instance opted use redundant defini tion sake clarity order type records computed labels distinguished two cases according whether type record label finite although distinction simple unrealistic labels singleton types cover common case records statically fixed labels dynamic choice label statically known list labels usage pattern seen javascript building object must conform interface based value labels infinite types cover fairly common usage scenario records used dictionaries deduce expression computing label type string thus forcing programmer insert code checks label present accessing rationale behind typing records twofold first foremost work wanted avoid type annotations costs since even notion schema json records collections notion basic type expect jaql programmer put kind type information code sophisticated type systems dependent types would probably preclude type reconstruction dependent types need lot annotations fit requirements second wanted simple yet precise making distinction increases typing precision cost need extra machinery since already singleton types adding heuristics complex analysis gain precision records would blurred main focus paper typing records typing transformations records leave additions future work conclusion work addresses two practical problems namely typing nosql languages comprehensive definition semantics languages add list comprehension sql operators ability work heterogeneous data sets based json instead tuples typing precisely features using best techniques literature would probably yield quite complex mixing row polymorphism records parametric polymorphism form dependent typing skeptical could achieved without using explicit type annotation therefore explored formalization languages scratch defining calculus type system thesis defended operations typical current nosql languages long operate structurally without resorting term equality relations amount combination basic bricks filters structural side claim combining recursive records pairs unions intersections negations suffices capture possible structuring data covering palette ranging comprehensions heterogeneous lists mixing typed untyped data regular expressions types xml schemas therefore calculus provides simple way give formal semantics reciprocally compare combine operators different nosql languages also offers means equip languages current definition without type definition annotation precise type inference record polymorphism reader noticed use row variables type records nevertheless high degree polymorphism row variables useful type functions transformations since keep track record fields modified transformation setting need since type transformations filters application transformations filters terms polymorphic typing via filters see first example given section keeps track field therefore open records suffice record selection languages dynamic ones javascript ruby allow label field selection computed expression considered definition rule type expressions form whenever typed finite unions strings rule would give finite approximation type selection however extension would complex definition type system handle interesting cases finite union type deduced therefore preferred omit study leave future work accounted components according landin constitute design language operators data structures landin considers design terms types independent activities contrary advocate approach design former driven form latter although approaches possible tried convey idea approach nevertheless one yields type system whose precision demonstrated work long comparable precision obtained opposed operators type inference yields surpasses precision systems using parametric polymorphism row variables price pay transformations first class type filters applications however seems advantageous deal world nosql languages selects never passed around least explicitly early error detection critical especially view cost code deployment result filters set untyped terms easily included host language complement typeful framework existing operators ones requirements include filters host language minimal every modern typed programming language satisfies interest resides fact add filter applications language rather filters used define smooth integration calls domain specific languages sql xpath pig regex general purpose ones java python ocaml share set values typing discipline likewise even though filters provide early prototyping platform queries currently used final compilation stage nosql languages operations rely encoding sequences makes correspondence optimized bulk operations lists awkward whether derive efficient compilation filters recovering bulk semantics language challenging question future plans include practical experimentation technique intend benchmark type analysis existing collections jaql programs gauge amount code ill typed verify frequently programmer adopted defensive programming cope potential type errors boag chamberlain florescu robie xquery xml query language recommendation buneman libkin suciu tannen wong comprehension syntax sigmod record buneman nikhil frankel practical functional programming system databases proc conference functional programming architecture acm castagna typed iterators xml proceeding acm sigplan international conference functional programming icfp pages new york usa acm comon dauchet gilleron jacquemard lugiez tison tommasi tree automata techniques applications http copeland maier making smalltalk database system proceedings acm sigmod international conference management data pages editor jsoniq http engelfriet tree transformations comparison theory computing systems engelfriet tree transducers regular mathematical systems theory frisch conception langage programmation fonctionnel xml phd thesis paris denis diderot frisch castagna benzaken semantic subtyping dealing function union intersection negation types journal acm hosoya foundations xml processing tree automata approach cambridge university press jaql http javascript object notation json http martens neven typechecking uniform unranked tree transducers proceedings international conference database theory icdt pages london meijer world according linq acm queue references meijer bierman model data large shared data banks communications acm alon milo neven suciu vianu xml data values typechecking revisited proceedings twentieth acm symposium principles database systems pods pages new york usa acm language combinators xml conception typing implementation phd thesis http odata http alon milo neven suciu vianu typechecking xml views relational databases acm trans comput logic july ohori buneman type inference database programming language acm conference lisp functional programming pages behm borkar carey grover onose vernica deutsch papakonstantinou tsotras asterix towards scalable semistructured data platform evolvingworld models distributed parallel databases ohori buneman tannen database programming machiavelli polymorphic language static type inference proc acm sigmod conference pages ohori ueno making standard practical database programming language chakravarty danvy editors icfp pages acm benzaken castagna frisch cduce general purpose language icfp acm international conference functional programming pages uppsala sweden acm press olston reed srivastava kumar tomkins pig latin language data processing sigmod conference pages beyer ercegovac gemulla balmin eltabakh kanne shekita jaql scripting language large scale semistructured data analysis pvldb hoa beyer balmin liu emerging trends enterprise data analytics connecting hadoop warehouse proceedings international conference management data sigmod pages operators first class host language provides say functions stay typed embedding host type system via foreign type calls sabry wadler reflection proceedings first acm sigplan international conference functional programming icfp pages acm schmidt mall report technical report fachbereich informatik hamburg squeryl scala orm dsl talking databases minimum verbosity maximum type safety http tannen buneman wong naturally embedded query languages icdt pages trinder wadler improving list comprehension database queries fourth ieee region conference tencon pages unql http appendix termination analysis algorithm builds substitution vars ids expression variables variables identifiers thus associating capture variable occurring pattern filter fresh identifier abstract subtree input type bound order deduce result type application filter expression type inference algorithm abstractly executes filter type expression explained section algorithm essentially analyzes may happen original input type subtrees recursive call checks possible cases every subsequent recursive call applied subtrees original input type order track subtrees original input type use infinite set ids variable identifiers ranged possible indexed identifiers used identify subtrees original input type bound variable recursive call consider instance recursive filter uses substitution compute set symbolic arguments recursive calls words recursive calls occur filter treesx returns set symbolic arguments calls obtained hypothesis free variables associated subtrees specified formal definition follows treesx treesx treesx fresh nil treesx treesx treesx treesx treesx treesx treesx treesx treesx treesx treesx treesx treesx treesxs groupby orderby treesx treesx algorithm records first call filter variable bound subtree input type recursive call thus applied abstract argument perform substitutions call see bound bound two subtrees subtree care since used recursive call important bound subtrees original input type respectively subtree therefore filter type inference algorithm terminate every possible input type previous example introduces two important concepts still missing definition analysis denotes application substitution definition mostly straightforward two important cases pattern filter substitution updated associating every capture variable pattern fresh variable identifier one recursive call symbolic argument computed applying actual argument recursive calls applied symbolic arguments obtained arguments replacing variables variable identifiers symbolic arguments ranged formally defined follows symb args second phase second phase implemented function check intuition check must compute application symbolic arguments collected first phase check whether variables occurring arguments recursive calls marked variables actually bound subtrees original type bound either variable identifiers subparts variable identifiers filter composition would relatively easy task would amount compute substitutions matching symbolic arguments patterns apply finally verify marked variables satisfy sought property unfortunately case composition filter analysis complicated imagine want check property application symbolic argument check property recursive calls occurring must compute least approximate set symbolic arguments produced application thus fed compute composition therefore check function rather returning true false return set symbolic arguments result execution fail recursive call satisfy property marked variables precisely function check form variable identifiers constants pairs records indeterminate said care variables disregard others analyze recursive filter variables care occur arguments recursive calls given filter filter recursion variable set variables formally defined follows markx vars denotes subtree containment relation vars set expression variables occur free abuse notation use vars denote capture variables occurring thus vars vars vars vars vars vars rest definition standard aside notice fact given filter type inference algorithm terminates possible input types imply execution terminates possible input values instance analysis correctly detects filter type inference terminates possible input types returning input type although application filter never terminates arguments form formally define two phases analysis algorithm checkv vars stores set marked variables filter substitution least free variables ids symbolic arguments either fail marked variables satisfy property bound subcomponents variable identifiers return result applying hypothesis approximation either set new symbolic arguments latter simply indicates check able compute result application typically result expression first phase first phase implemented function every filter recursion variable function explores filter following two things checkv checkv checkv otherwise checkv checkv checkv checkv checkv checkv checkv checkv checkv checkv otherwise fail checkmarkx fail otherwise checkv check otherwise checkv checkv checkv fresh fresh fail fail checkv fail otherwise figure definition second phase analysis belonging host language application function result recursive filter recursive call filter full definition given figure let comment different cases detail checkv syntactic equivalence case return applied substitution expression case function says able compute result returns filter composition two filters first call checkv analyze compute result applying feed result analysis checkv simply states compute filter set symbolic arguments compute argument return union results course checkv fails checkv next case straightforward know argument given form makes filter fail perform check recursive call never called result returned instance apply filter symbolic argument always fail useless continue checking case formally consider set values form quite simply obtained replacing every occurrence variable identifier check whether value accepted namely whether intersection empty directly return empty set symbolic arguments otherwise perform fine grained analysis according form filter filter expression two possible cases either expression form argument denotes checkv checkv checkv filter recursive recursive call able compute symbolic result however case recursive filter must check whether type inference terminates mark variables recursive calls check whether definition passes static analysis return analysis checkmarkx fail notice quite different failure since allows compute approximation long filter composition bind parts result variables occur arguments recursive calls key example case filter filter defined section parameter filter recursive without filter would rejected static analysis precisely due composition false recursive analysis supposes produces passed pattern recursive call executed environment true empty therefore fail otherwise fresh otherwise otherwise fail fail fresh otherwise fresh otherwise figure pattern matching respect set marked variables variables also checks capture variables also marked actually bound subtree initial input type defined figure enhance readability omitted index since matters first case change along definition check marked variables performed first case definition symbolic argument matched variable variable marked substitution returned marked matching thus analysis fail symbolic argument either value depend input type variable identifier exactly subtree original input type cases definition standard notice product record patterns variable identifier algorithm creates fresh new variable identifiers denote new subtrees decomposed pattern finally reader confound fail former indicates argument match value latter indicates marked variable bound value exactly subtree initial input type notice case happen definition check since called contained result call whatever affect termination subsequent recursive calls even though composition uses result form result make typechecking filter diverge union filter three possible cases always fail return result never executed since never fail return result tell one executed return union two results filter product filter accepted type subtype since case empty two forms either pair variable identifier former case check filters return result product results latter case recall represents subtree original input type application fail means subfilter applied subtree introduce two fresh variable identifiers denote subtrees course subtrees original input type previous case apply check return product results case record filter similar case product filter finally case pattern filter algorithm checks marked variables indeed bound subtrees initial input type check match symbolic argument pattern update substitution check subfilter environment corresponding assignments capture variables notice however pattern matching receives extra argument set marked variables computing substitutions capture two phases together finally put two phases together given recursive filter run analysis marking variables recursive calls running first phase feeding result second phase checkmarkx treesx function fail say passed check checkmarkx treesx fail check able prove theorem let type filter define filters recursive toplevel suffices add def fresh dummy recursion check check support theorem termination check type inference algorithm terminates every input type notice since support finite finitely many different recursion variables occuring filter finite let filter type mentioned statement theorem consider possibly infinite derivation assign every judgment following weight order prove theorem need auxiliary definitions definition plinth plinth types set types following properties wgt size finite contains empty closed boolean connectives denotes cardinality set notice definition finite dom domain set pairs defined size depth syntax tree notice set weights lexicographically ordered form order difficult prove every application rule derivation strictly decreases wgt therefore derivation must finite proved case analysis applied rule must distinguish three cases types types define plinth noted smallest plinth containing intuitively plinth set types obtained possible boolean combination subtrees notice always defined since types regular finitely many distinct subtrees modulo type equivalence thus combined finitely many different ways notice case first component wgt either decreases remains constant second component strictly decreases case lemma ensures first component wgt premise strictly decreases since core proof let expand rule somewhere derivation following rule definition extended plinth support let type extended plinth noted defined ranges values support noted support defined def ids support extended plinth set types obtained possible boolean combination subtrees values occur filter intuition underlying definition support includes possible types arguments recursive calls occurring applied argument type lastly let prove following technical lemma want prove dom dom containment must strict ensure measure decreases first notice dom since side condition application rule order prove containment strict suffices prove consequence lemma ensures support whence result lemma let filter check holds let type every derivation finite infinite every occurrence rule type dom fresh rules first component wgt remains constant second component strictly decreases type dom fresh every vars equivalently type support proof contradiction suppose exists instance rule vars means neither singleton type occurring type obtained applying left projection right projection label selection since check holds must bound either identifier value see figure computation checkmarkx treesx since identifiers check introduced input parameter performing left projection right projection label selection another identifier ensure never bound result expression whose type contradicts improvements although analysis performed algorithm already fine grained refined two simple ways explained section algorithm checks one step reduction capture variables occurring recursive calls bound subtrees initial input type possible improvement consists try check property higher number steps instance consider filter nil nil nil filter pass analysis since identifier bound capture variable unfolding recursive call second branch first recursive call bound pursued abstract execution one step would seen used recursive call thus type inference terminates therefore first improvements modify check stop type type var string string string type var string string let filter eval subst subst eval subst eval let filter subst subst subst subst subst subst var subst subst figure filter encoding one step reduction tries steps determined heuristics based sizes filter input type second opinion promising improvement enhance precision test definition check verifies whether filter fails given symbolic argument current definition information collect type symbolic arguments structure type information currently unexploited provided patterns instance following admittedly stupid filter following reduction rules performed without reduction context order prove turing completeness suffices show encode terms reductions calculus sake readability use mutually recursive types rather encoding use records though pairs would sufficed write recursion variable filter use denote type term productions encoded recursive types given beginning figure next define evaluation filter eval body calls filter subst implements capture free substitution denotes constant defined right straightforward see definitions figure implement reduction semantics cps terms course definition would pass termination condition indeed subst would accepted algorithm eval would fail since possible ensure recursive calls eval receive subst subtrees original input type expected substitution always terminate may return trees subtrees original term int int first pass associate know type int int record information know second pass always match first pattern never argument recursive call words one nested recursive call solution conceptually simple yields cumbersome formalization chose current formulation amounts modify trees introduces fresh variables records type information suffices modify definition obtained replacing every occurrence variable identifier type information rather current definition check rest proof subject reduction theorem first give proof lemma restate let filter value every proof turing completeness theorem proof induction derivation case analysis last rule derivation order prove turing completeness show define filters evaluator untyped allowed recursive calls occur side composition encoding would straightforward implement context reductions goal however show restriction compositions affect expressiveness end avoid context reductions since require recursive calls composition first translate via plotkin cps translation apply steele rabbit administrative reductions obtaining terms latter isomorphic cbv see instance defined follows expr expression therefore rule applied since prod assume either case rule error applies therefore induction hypothesis first premise contradicts side condition little bit sloppy notation used filter parameter pattern strictly speaking allowed filters instance branch var subst rather written var true false var similarly two cases error rule never occur indeed means particular therefore side conditions rules prod pat recd hold similarly second premise therefore rule applied evaluate patt similarly previous case either contradicts induction hypotheis comp induction hypothesis first premise contradicts side condition therefore rule error applied recd similar product type either record therefore rule errro applied one induction hypothesis contradicts side condition since means therefore rule chosen induction hypothesis contradicts side condition rule chosen induction hypothesis gives trivially true since rec apply straightfowardly induction hypothesis premise groupby orderby rule applies error gives complexity typing algorithm proof clarity restrict types without record constructors proof straihgtforwardly extended first remark types isomorphic alternating tree automata ata intersection union complement defined ata possible compute tree automaton size seeing type formalism type generated following grammar const atom const atom ranges negation intersections basic types intuitively recursion variable state nta finite union products set transitions whose products atom productions clear check holds algorithm considers distinct cases thanks memoization set furthermore rule may test subtyping problem exptime thus giving bound equipped prove subject reduction theorem restate general manner every dom dom implies proof induction derivation case analysis rule beside basic case rules straightforward application induction rules must proved simulatneously cases proved direct application induction hypothesis detail case product constructor precise typing groupby process inferring precise type groupby decomposed several steps first compute set discriminant domains filter idea domains types know may give results different types typically corresponds possible branchings filter instance pairwise disjoint types filter set discriminant domains filter since various obtained may different want keep track relation result type input type produced formally defined follows groupby orderby expr suppose host languages enjoys subject reduction hence prod know since value obtain since value rank observation value crucial since general given type true however property holds singleton types therefore values therefore furthermore since suppose typing derivation exists typing rules syntax directed must used prove typing judgement apply induction hypothesis deduce similarly therefore filter evaluates type proves case order high precision typing want compute type intermediate list groupby set disjoint types define normal form set types given set types computes new set formed pairwise disjoint types whose union covers union original types rules must proved together indeed given filter typing derivation judgement either apply induction hypothesis therefore case however might lemma case apply induction hypothesis second premise gives allows conclude recall trying deduce type groupby expression groupby applied argument type types shall use input filter compute type intermediate result surely want every rule types item however enough since would use information descriminant domains filter instance filter gives two different result types positive negative numbers input list integers want compute two different result types positive negatives compute result filter application generic integers indeed item int int idea add set must normalized types set discriminant types however precise enough since domains may much larger item types input list reason take part domain types intersects least possible value list terms consider normal form following set item item qjm xjkm translated jdj jdj jdj jdj xjkm qjm nil wheres otherwise fact dom jtk proved straightforward induction important point remark since deterministic exactly one branch alternate filter selected pattern matching given input check holds sufficient remark recursive call made strict subtree input guarantees check returns true idea normalized set set type grouby type optimization replace know input list empty lemma tdttr filters filters strictly expressive tdttrs comparison tree transducers first show every tree transducers regular encoded filter use definition tree transducers definition tree transducer regular tree transducer regular tdttr input alphabet output alphabet set states set initial states set rules form xim mapping set regular tree languages alphabet tdttr deterministic singleton sides rules pairwise disjoint since filters encode programs make sense compare filters deterministic tdttr first show given deterministic tdttr write equivalent filter furthermore check fail first recall encoding ranked labeled trees filter definition tree encoding let define treeencoding write jtk value defined inductively jak jtn particular clear regular jsk type dom jtk proof encoding follows every state introduce recursion variable formally translation performed function filters defined however impossible write tdttr replaces leaf copy whole input inputs arbitrary size similar example used show tdttr tree transducers comparable lemma tdttr filters let deterministic tdttr exists filter filter works monadic trees form essentially replaces leaf tree copy done tdttr indeed tdttr two ways remember subtree copy one course using variables scope restricted rule therefore arbitrary subtree copied fixed distance original position instance rule form assuming copies input copy original subtree next sibling arbitrary far second way copy subtree remember using states indeed states encode knowledge tdttr accumulated along path however since number states finite thing tdttr copy fixed path instance given exists tdttr performs transformation trees height essentially states remember every possible path taken instance tdttr list notation nil generalize encoding tree languages types let write jsk set values jtk jsk proof even restrict filters domain tdttrs meaning used fixed input alphabet atomic types define filter expressed tdttr instance consider filter nil nil operators record types use theory records defined cduce summarize main definitions adapted given chapter alain frisch phd thesis interested may undefined type take union two corresponding fields since results either according whether record typed undefined reader also find detailed definitions semantic interpretation record types subtyping relation induces modifications must done algorithm decide finer details pattern matching definition compilation techniques record types expressions let denote set function exists set finite case denote set dom element def use denote set functions notation denote function defined although notation univocal unless require largely sufficient purposes section let distinguished constant sets string types string values denote set record types expressions record values respectively constant represents value fields record undefined ease presentation use notation constant singleton type contains occurs values denotes value string types denotes singleton type contains value given definitions clear record types definition nothing specific notations quasiconstant functions string types precisely open record type expression denotes function closed record type expression denotes function similarly optional field notation denotes record type expressions mapped either type let type two record type expressions string types merge respect noted used infix record type expression defined follows empty def otherwise explains examples gave main text particular int int bool int since may undefined right record undefined int since right record defined therefore priority corresponding definition left record encoding shown section groupby operator encode jaql group computes grouping key distinct key evaluated environment bound key bound sequence elements input key expressed group encoded following composition filters jln transform transform expand groupby jen transform rgrpn jek let filter rgrpi nil nil tail rgrpi tail rgrpi tail essentially encoding takes argument sequence sequences values sequences sequence tagged integer flatten sequences tagged values single sequence apply groupby operator modify key selector applied value tagged integer done obtain sequence pairs commen grouping key sequence tagged values apply auxiliary regrp filter extracts subsequences tagged removes integer tag value lastly call recombining expression scope bound current grouping key bound sequences values input whose grouping key recall lemma record type subtype equivalent finite union record type expressions quasiconstant functions string types definition merge easily extended record types follows def finally operators used typing records rules section defined terms merge operator def def constant different semantics operator depend choice long different notice particular result concatenation two record type expressions may result field three different outcomes contain field surely defined take corresponding field undefined take corresponding field figure available png format http
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joint secrecy using uncoded schemes towards secure source broadcast lei houqiang senior member ieee weiping fellow ieee jul abstract paper investigates joint secrecy problem shannon cipher broadcast system suppose list secrecy applied wiretapper allowed produce list reconstruction sequences secrecy measured minimum distortion entire list discrete communication cases propose uncoded scheme cascades random permutation mapping using scheme derive inner bound admissible region secret key rate list rate wiretapper distortion distortions legitimate users converse part easily obtain outer bound admissible region existing result comparing outer bound inner bound shows proposed scheme optimal certain conditions besides extend proposed scheme scalar vector gaussian communication scenarios characterize corresponding performance well two cases also propose another uncoded scheme scheme achieves performance permutationbased scheme interestingly introducing random permutation random orthogonal transform traditional uncoded scheme proposed uncoded schemes one hand provide certain level secrecy hand lose performance terms distortions legitimate users index terms uncoded scheme secrecy permutation orthogonal transform shannon cipher system ntroduction investigations joint coding jscc could trace back shannon pioneering work geometric method developed design communication system jscc transmitting gaussian source gaussian broadcast channel goblick observed source channel bandwidths matched one channel use per source sample directly sending scaled version source samples channel linear scheme fact optimal case separation scheme cascades source coding channel coding indeed suffers performance loss vector gaussian communication cases optimal linear coding studied general schemes consist mappings limited linear one named uncoded schemes optimality uncoded schemes general pair department electrical computer engineering national university singapore singapore leiyu work done university science technology china department electronic engineering information science university science technology china hefei china lihq wpli investigated showed shannon limit achieved uncoded schemes source channel satisfy certain probabilistic matching condition improve performance mismatched pairs hybrid coding hybrid coding studied combines traditional digital coding mapping together converse part jscc problem reznic tian derived nontrivial converse results gaussian source broadcast problem besides generalized achievability converse results gaussian communication general case security shannon cipher system noisy broadcast version depicted fig first investigated shannon pioneering work sender communicates legitimate receiver secretly exploiting shared secret key lossy source communication wiretapper might want decrypt lossy version source schieler studied secrecy measure shannon cipher system around assumption wiretapper ability conduct list decoding fixed list size secrecy measured minimum distortion entire list showed systems secrecy measure equivalent secrecy measured new quantity could considered lossy extension traditional equivocation hence list secrecy closely related traditional equivocation well furthermore used secrecy measure study problem secrecy shannon cipher system showed pair satisfying certain conditions uncoded scheme could outperform separate one jscc improves robustness communication performance broadcast secrecy coding improves security communication exploiting secret key wiretap channel therefore intuitively robustness security could obtained simultaneously combine jscc secrecy coding together joint secrecy jscs problem considered several works already yamamoto studied secure lossy transmission noisy wiretap channel secrecy measured wiretapper best reconstruction distortion however shown secrecy measure cheap fragile since one bit secret key suffices achieve optimality secrecy meanwhile one bit additional information wiretapper suffices decrypt optimal encryption scheme different formulation problem considered authors assumed fixed information leakage wiretapper wish minimize distortion legitimate receiver time providing graceful distortion degradation snr signal noise ratio mismatch showed positive leakage achieved hybrid coding scenario extended consider side information receiver side information sender analog encryption analog scrambling technologies based scheme permutation based scheme scheme seen uncoded jscs schemes well although designed specified pair based scheme improves secrecy changing sign sample according secret key owing one bit secret key used per sample scheme could provide higher secrecy even higher key rate available permutation based scheme improves secrecy shuffling positions samples unlike based scheme supports arbitrarily high key rate furthermore kang liu recently applied permutation operation digital encryption scheme showed permutation another powerful encryption technique besides pad achieve optimality secrecy contributions paper consider joint secrecy problem secure source broadcast bandwidthmatched shannon cipher system see fig list secrecy used measure secrecy wiretapper allowed conduct list decoding fixed list size secrecy measured minimum distortion entire list study achievable region secret key rate list rate wiretapper distortion distortions legitimate users show optimality certain conditions contributions follows discrete source case propose uncoded scheme cascades random permutation mapping scheme differs permutation based scheme proposed two main aspects scheme coupling permutation operation traditional uncoded scheme designed secrecy problem however scheme couples permutation operation digital scheme designed coding problem addition finite alphabet case also extend scheme pairs countably infinite alphabets gaussian pairs require use powerful techniques including unified typicality information geometric analysis analyzing proposed scheme provide inner bound admissible region converse part give outer bound using recent result comparing outer bound inner bound shows proposed scheme optimal certain conditions extend proposed scheme scalar vector gaussian communication scenarios two cases also propose another uncoded scheme scheme achieves inner bounds one achieved scheme interestingly introducing random permutation random orthogonal transform traditional uncoded scheme proposed uncoded schemes matter discrete source case gaussian case one hand provide certain level secrecy hand lose performance terms distortions legitimate users schieler cuff studied list secrecy problem noiseless version shannon cipher system showed digital scheme secret key used choose source codebook code source sequence optimal problem separate coding cascading source coding pad proven optimal well extended problem noisy channel case showed separate strategy cascading source coding pad channel coding suboptimal general uncoded scheme could outperform separate scheme paper extend system word noiseless means wiretap channel noiseless word means one legitimate user key eve fig shannon cipher broadcast system problem noisy broadcast scenarios propose two kind uncoded schemes adopt two different encryption permutation random instead traditional pad encryption show proposed uncoded schemes could achieve optimality certain cases rest paper organized follows section formulates joint secrecy problem section iii proposes uncoded scheme discrete communication analyzed corresponding performance sections extend proposed scheme scalar vector gaussian communications respectively another scheme based scheme also proposed two sections finally section concludes paper roblem ormulation problem setup consider shannon cipher broadcast system two legitimate shown fig sender two legitimate receivers share secret key uniformly distributed independent source sender observes discrete memoryless source sequence element independent identically distributed according transmits legitimate users wiretap broadcast channel confidentially utilizing secret key wiretap channel finally legitimate users produce source reconstructions respectively definition block code consists encoder although consider communication system results section easy extended mismatched system since system bandwidth ratio converted system considering source symbols channel symbols source supersymbol channel supersymbol respectively although consider system two legitimate users results derived paper easily extended cases legitimate users paper set sometimes denoted rms decoders yin spin encoder decoders stochastic another output channel accessed wiretapper eve based wiretapper produces list lpz sqn induced distortion set minimum one entire list minsqn plpz sqn psn sqn pst sqt distortion measure wiretapper given distortion levels nodes want communicate source within distortions respectively exploiting secret key wiretap channel ensuring wiretapper strategy always suffers distortion high probability definition tuple prk achievable exists sequence codes distortion constraint spin psn spn pst spt distortion measure legitimate users secrecy constraint minn log lim min plpz definition admissible region tachievable prk assume wiretapper knows code distributions henchman problem problem equivalent henchman problem wiretapper reconstructs single sequence help henchman access source wiretapper observation depicted fig wiretapper receives best possible nrn bits henchman assist producing reconstruction sequence sqn definition henchman code hcode block code consists encoder decoder sqn assume wiretapper henchman aware block code adopted nodes cooperate design henchman code based block code definition tuple prk achievable henchman problem exists sequence codes simplicity consider legitimate users distortion measure note results derived paper still hold case different distortion measures key eve henchman fig henchman problem distortion constraint secrecy constraint min hcodes lim sqn equivalence list secrecy problem henchman problem shown following proposition proven schieler cuff prop proposition tuple prk achievable list reconstruction problem achievable henchman problem furthermore list secrecy problem henchman problem also equivalent secrecy problem see addition system also consider shannon cipher system gaussian source transmitted gaussian wiretap broadcast channel additive gaussian noises variances independent case constraint channel input power added definitions pxn system involving channel power constraint proposition still holds iii iscrete ommunication permutation based scheme finite alphabets section propose secure uncoded scheme coupling permutation operation traditional uncoded jscc scheme uncoded scheme jscc system two receivers consists three symbol mappings psq induced distortions edb spi pyi easy show benefit replacing encoder psq stochastic one secrecy considered system hand observe psn spni pst spi etsn spn spi tsn psni denotes joint type empirical distribution psn spni induced distortions depend joint type source reconstruction sequences therefore want improve secrecy scheme time retain induced distortions unchanged need require encryption decryption operations change joint type source reconstruction sequences encryption psn tsn psni end consider random permutation decryption spni require encryption operation inverse permutation decryption operation obviously permutation inverse operation change joint type source sequence reconstructions codebook public key generation generate permutation set element uniformly random independently selected set permutations rns denoted public key codebook revealed sender receivers including wiretapper encoding upon observing source sequence key encoder first generates psn generates according pxt permutation sequence psn denotes permutation operation precisely indices respect permutation sequence decoding legitimate users legitimate user upon received sequence yin key decoder first reconstructs spi pyi rns using mapping spi produces spni using inverse permutation operation proposed scheme cascades random permutation operation traditional uncoded jscc scheme uncoded jscc part provide graceful degradation source legitimate users different channel qualities random permutation operation part shifts sequence type provides certain level secrecy next analyze asymptotic performance proposed scheme blocklength goes infinity first need introduce basic properties random codebook observe permutation sequence mapping bijective hence following lemma paper permutation sequence termed permutation sequence distinguish permutation mapping one sequence another sequence termed permutation operation disambiguation call permutation lemma suppose permutation sequence uniformly random selected set permutations rns also uniformly distributed moreover permutation sequence also uniform distribution utilizing lemma establish following lemma lemma suppose permutation sequence uniformly random selected psn transforms arbitrary sequence random sequence uniformly distributed set sequences type tsn type class tsn moreover finite set sequences type tsn cardinality hence psn tsn denotes term tending zero proof psn pntsn psqq tsn ttsn tsn sps follows tsn sps pnts psqq implies psn transforms arbitrary sequence random sequence uniformly distributed set sequences type tsn type counting lemma lem finite tsn hence tsn combining gives psn tsn lemma shows nice property random permutation operation resulting sequence uniformly distributed set sequences type tsn input sequence permutation randomly uniformly chosen set permutations rns utilizing property characterize performance proposed scheme shown following theorem proof theorem given appendix theorem permutation based scheme finite alphabets communication finite alphabets spi finite permutation based scheme achieves region rpiq rpiq prk minspi edb spi pyi min pdq min ips ips ede denoting function pdq min ede denoting conditional function given information note rpiq components components prk depend observe given minspi edb spi pyi minimal distortions legitimate users achieve even communication case hand min larger optimal achieved uncoded schemes key hence compared traditional uncoded schemes proposed scheme one hand improves improve performance secrecy certain extent hand lose performance terms distortions legitimate users first constraint rpiq consistent performance traditional uncoded schemes second constraint rpiq roughly speaking follows following argument one hand henchman wiretapper ignore signal altogether use source code describe within distortion hand proposed scheme forces wiretapper optimal strategy indirect guessing strategy first wiretapper decrypts secret key using rate upon observation wiretapper reconstructs sequence within distortion using rate denote reconstruction finally upon secret key wiretapper reconstructs source sqn since obviously distortion average distortion depends joint type sequences hence wiretapper needs rate achieve distortion consider special case sending binary source binary wiretap broadcast channel binary communication source bernoulli source bern hamming distortion measure spq spq otherwise binary wiretap broadcast channel bern ppi bern set theorem get following corollary corollary binary communication binary communication rpiq piq min rxs max denoting binary convolution xqy denoting binary entropy function ppq log permutation based scheme general alphabets theorem extended general alphabets cases shown following theorem proof theorem given appendix theorem permutation based scheme general alphabets assume countable finite spi assume psq finite satisfies nps log log log nps psq denotes number probability values smaller psq denotes sum probability values smaller denotes minimum number sum probability values except largest ones larger theorem still holds remark conditions equivalent nps pxq log pxq log pxq remark conditions require sequence psq vanish fast possible obviously hold finite besides countably infinite easy verify distribution psq satisfies well however alphabet countable means either finite countably infinite alphabet general means either countable uncountable continuous claim holds ignore integer without loss generality countably infinite converted bijective mapping psq converges slower slow psq converge hence probability distribution implies theorem holds almost probability distributions note countably infinite alphabet need conditions guarantee existence set unified typicality set sequence lemma still holds makes theorem hold done finite alphabets case outer bound system single legitimate user remove legitimate user system fig following outer bound admissible region prk proven recently lemma communication one legitimate user prk ips edb poq min ips psp denotes channel capacity legitimate user min max denoting mutual information distribution function specified wiretap channel first two constraints rpoq follow coding theorem last constraint follows indirect decryption strategy wiretapper roughly speaking wiretapper first reconstructs using rate next decrypts secret key using rate upon secret key produces legitimate user reconstruction finally upon produces final reconstruction sqn using rate details seen applying lemma system two legitimate users system considered paper following outer bound immediate theorem outer bound communication two legitimate users prk ips spi rpoq edb spi psp min denotes channel capacity legitimate user ips spi pyi specialized binary communication following corollary corollary binary communication binary communication prk poq min ppi pdi comparing theorem corollary identify optimality proposed scheme binary communication theorem optimality proposed scheme binary communication legitimate users proposed uncoded scheme optimal holds remark theorem implies conditions compared one legitimate users wiretapper better channel wants produce worse reconstruction legitimate user distortion restricted shannon limit meanwhile wiretapper worse channel wants produce better reconstruction proposed uncoded scheme optimal worth noting optimality conditions include practical scenario wiretapper worse channel legitimate users higher distortion requirement mean scheme optimal practical scenario believe binary broadcast communication without secrecy requirement proposed uncoded scheme permutation operation unique scheme achieve shannon limits legitimate users secrecy requirement involved proposed scheme optimal well matter wiretapper channel condition desired distortion level rate secret key could increase scheme satisfies point course need rigorous proof claim unfortunately idea prove know secrecy constraint traditional uncoded scheme could outperform separate scheme broadcast communication scenarios surprising secrecy constraint involved proposed uncoded scheme still could outperform separate scheme however surprisingly example given shows proposed uncoded scheme may strictly outperform separate coding even secure communication one legitimate user calar aussian ommunication section consider gaussian source transmitted powerconstrained gaussian wiretap broadcast channel average input power constrained distortion measures set spq spq spq spq although also convert gaussian system system done remark results section easily extended system since linear coding used schemes specified one communication system provide two uncoded schemes first one scheme proposed previous section next show permutation based scheme also works gaussian communication case one based scheme cascades random orthogonal transform instead random permutation operation mapping permutation based scheme shown linear coding optimal gaussian broadcast communication secrecy requirement hence set spi pyi linear functions spi proposed scheme communications apply permutation based scheme gaussian communication performance scheme provided following theorem proof given appendix theorem permutation based scheme gaussian communication proposed permutation based scheme achieves region rpiq prk piq min log log log max log remark rpiq one given theorem spi pyi set spi respectively achieved scheme remark first constraint rpiq consistent performance linear coding second constraint rpiq follows similar argument case note rpiq variable moreover region region prk depend satisfies finding similar discrete communication case given prk minimum achievable maximum achievable decreasing implies proposed scheme transmitting source using larger power results smaller distortions legitimate users also leads decrypting source easily wiretapper proposed scheme one hand provides certain level secrecy hand achieves shannon distortion limits legitimate users region theorem illustrated fig given effect prk tradeoff based scheme proposed scheme uses random permutation operation shuffles sequence within type class improve level secrecy works discrete communication also continuous communication gaussian communication subsection propose another secure uncoded scheme gaussian communication case designed geometric point view fig region theorem give interpretation motivation proposed scheme consider special case wiretapper noiseless channel apply linear coding gaussian communication know given euclidean norm sequence gaussian source uniformly distributed sphere sequences channel input outputs source reconstructions assume generate set bijective transforms codebook randomly choose one according key transform source sequence applying linear coding keep power unchanged transforms required map sphere hand using secret key legitimate users could transform back hence induced distortions legitimate users change well furthermore without knowing secret key knowing norm source sequence codebook view wiretapper source sequence uniformly distributed vectors possible generate channel output wiretapper observation key values make wiretapper guess source difficultly possible vectors uniformly equal distance located sphere wiretapper cover either vectors whole sphere meet decryption requirement shown orthogonal transform one transforms hence adopted second scheme codebook public key generation generate random matrices independently whose elements generated according apply orthonormalization process columns matrix hence resulting matrices orthogonal constitute subset orthogonal matrices public key codebook revealed sender receivers including wiretapper encoding upon observing source sequence key encoder generates follows decoding legitimate users legitimate user upon received sequence yin key decoder reconstructs source follows spni yin denotes transpose matrix next analyze asymptotic performance scheme similar case permutation based scheme need first introduce basic properties random codebook lemma suppose random matrix element independently distributed according gaussian distribution let columns let random matrix whose columns obtained applying orthonormalization procedure uniform distribution haar measure orthogonal transform set orthogonal matrices pnq moreover orthogonal matrix also uniform distribution pnq utilizing lemma establish following lemma lemma random orthogonal transform uniformly distributed orthogonal matrices set pnq transforms arbitrary vector random vector uniformly distributed radius proof lemma without loss generality assume obtained manner described lemma let columns orthonormalization know rotation matrix generally orthogonal matrix hand random vector element easy verify rotation matrix distribution normally distributed random vector invariant rotation therefore distribution also invariant rotation implies uniformly distributed unit addition observe hence random matrix transforms vector random vector uniformly distributed arbitrary vector easily find orthogonal matrix first column hence expressed lemma distribution hence also uniformly distributed unit implies uniformly distributed radius lemma implies resulting vector uniformly distributed sphere input vector transform matrix randomly uniformly chosen set orthogonal matrices nice property random orthogonal transform similar property random permutation operation utilizing properties establish following theorem proof given appendix theorem based scheme gaussian communication inner bound rpiq given theorem achieved scheme well inner bound rpiq understood geometric point view random orthogonal transform proposed scheme guarantees given uniform distribution small whose centers uniformly distributed center origin radius radius however owing uniform conditional distribution source given lack balls radius cover secret key wiretapper needs least hand unconditional case source uniform distribution center radius hence ignoring wiretapper needs least balls radius cover sphere results inner bound rpiq seems somewhat counterintuitive permutation based scheme achieves performance based scheme shown theorems easy observe cases case see fig permutations always transform source sequence vectors uniformly equal distance distributed sphere property hold high probability dimension goes infinity actually indeed dimension increases bad source sequences occur vanishing probability seen prssn prssqq besides sphere source sequence also high probability appear neighborhoods vectors prssq moreover prssq consists set good source sequences hence good source sequences occur high probability dimension increases permutations transform arbitrary source sequence high probability set vectors uniformly distributed sphere comparison based scheme previous two subsections give analysis asymptotic performance permutation based scheme based scheme however necessary let blocklength infinity set finite value subsection study simplest finite blocklength case codes case permutation based scheme obviously inferior asymptotic case since dimension case permutation exists except source sequence hence following mainly consider based scheme based scheme reduces based scheme next compare proposed schemes based scheme assume secret key uniformly distributed encoding upon observing source sequence key encoder generates follows source sequence said good permutations uniformly distributed sphere otherwise bad obviously permutations good source sequence also good rss round prssq typical set prss see proof fig illustration permutations source sequence case decoding legitimate receivers legitimate receiver upon received sequence key decoder reconstructs source follows spi easy verify pst pzq psq denotes probability distribution function pdf wiretapper channel noise denotes pdf given regarded gaussian mixture two components equal weight variance scheme shown maximum achievable equivalently minimum rate needed code within distortion information equals conditional function performance based scheme given following theorem theorem based scheme gaussian communication piq based scheme achieves region rsign prk piq rsign denoting conditional function given information defined since hard even possible express closed form ease comparison derive upper bound result shown following lemma proof given appendix lemma follows distribution pubq min log pubq log since denotes minimum rate needed code within distortion available encoder decoder give interpretation upper bound perspective source coding first ignoring side information log log minimum rate needed code without side information second consider following timesharing coding code secret key bit per symbol using linear decoder similar legitimate users reconstruct source within distortion hand code anything results rate distortion using timesharing strategy two schemes qpp need rate reconstruct source within distortion finally using rate code secret key upon reconstruction code residual error within distortion using rate log reconstruct source within distortion combining theorem lemma gives following result piq theorem outer bound rsign gaussian communication region achieved piq poq based scheme satisfies rsign rsign prk poq rsign min rpubq log remark observe key exploited based scheme even hence case performance still given theorem outer bounded pubq lemma observed log pubq poq hence rsign rpiq rpiq log note argument available inequality apply secrecy problem considered paper secrecy problem wiretapper henchman benefit adopting timesharing strategy since constraint restrict probability instead average distortion proposed jscs sign change upper bound fig comparison achievable proposed infinite blocklength schemes based scheme noiseless wiretap channel given theorem denotes achievable region permutation based scheme based scheme implies based scheme strictly inferior proposed schemes condition version based scheme inferior corresponding infinite blocklength version see clearer achieved proposed infinite blocklength schemes given theorem upper bound achieved based scheme given theorem illustrated fig outer bound gaussian communication following outer bound proven system one legitimate user lemma gaussian communication one legitimate user prk poq min log log log using result following outer bound system legitimate users system considered paper theorem outer bound gaussian communication legitimate users prk poq min log log log comparing theorem corollary identify optimality proposed schemes gaussian communication result similar theorem binary communication theorem optimality proposed schemes gaussian communication legitimate users proposed scheme optimal holds similar remark remark applies theorem ector aussian ommunication proposed schemes easily extended vector gaussian communication scenarios consider gaussian source diag transmitted gaussian broadcast channel channel output vector observed legitimate user additive gaussian noise vector wiretapper eve accesses another channel output channel diag additive gaussian noise vector well distortion measures set spq spq psj channel cost function set pxq proposed permutation based scheme consider vectors applied vector gaussian case directly performance scheme proven following similar steps proof scalar gaussian case furthermore apply proposed based scheme pair shown following codebook public key generation generate random matrices rms independently whose elements generated according apply orthonormalization process every matrix hence resulting matrices orthogonal constitute subset orthogonal matrices rms public key codebook revealed sender receivers including wiretapper paper use bold font denote vector matrix denoted respectively encoding upon observing source sequence snm key encoder generates xnm follows xnj snj rms transmitting power decoding legitimate users legitimate user upon received sequence yin key decoder reconstructs source follows spni rms achievable regions proposed schemes permutation based scheme based scheme given following theorem proof given appendix theorem performance proposed schemes vector gaussian communication permutation based scheme based scheme achieves region rpiq piq min log log ppj min min remark actually theorem denotes function source denotes function source side information available encoder decoder rms diag independent oncluding emarks paper studied joint secrecy problem secure source broadcast shannon cipher system list secrecy used measure secrecy communication proposed two secure uncoded schemes permutation based scheme discrete scalar gaussian vector gaussian communications based scheme latter two communications two uncoded schemes random permutation random orthogonal transform cascaded traditional uncoded jscc scheme analysis showed proposed schemes outperform based scheme interestingly adding random permutation operation random orthogonal transform traditional uncoded scheme proposed uncoded schemes one hand provide certain level secrecy hand lose performance terms distortions legitimate users although proposed schemes adopt two different random transforms permutation operation orthogonal transform consistent two aspects first actually permutation operation one kind orthogonal transform second gaussian communication orthogonal transform also considered shift operation shifts sequence another type treat euclidean norm source sequence type furthermore worth noting different common construction codebook information theory including spherical codes one used codebooks proposed schemes constructed generating sequence random permutations random matrices instead sequence random samples words codebooks used specify sequence bijective operations transforms hence apply uncoded schemes common codebooks information theory specify sequence samples hence used quantization operation digital schemes furthermore based codebook construction also found used design digital schemes communication secrecy communication antijamming communication problems different works case used design uncoded schemes instead digital schemes worth noting proofs used paper follow basic outline proofs different besides finite alphabet case also considered countably infinite alphabet continuous gaussian alphabet cases hence powerful techniques including unified typicality information geometric analysis discretization used proofs furthermore unified typicality used proofs different existing one defined unified typical set defined good property sequences nearly number types property coincides finite alphabet case crucial importance proofs believe definition unified typicality could used extend method types countably infinite alphabet cases besides extension ppendix roof heorem denote pyi kind type called weak type since relationship weak typicality similar traditional type empirical distribution strong typicality fact permutation operation bijective pcs spn pyin pspn pspn pyin pspn pyin similarly pcs spn also expressed pcs spn pyin pspn hence forms markov chain furthermore since permutation operation change joint distribution sequences pspn pyin pyi pspi pspi spi pyqu denotes conditional distribution induced decoder pyi pspi distribution given since spin sequence law large numbers spin edb spi hence distortion constraints legitimate users satisfied next prove secrecy constraint also satisfied lim sup min lim ecz maxrn hcodes sqn end need following lemma lemma sequence random variables txn sequence events tan pan pan sequence lemma prove secrecy constraint need show satisfies lim pcz max sqn hcodes sequence next prove define event set defined according notion strong typicality see psq tsn psq sps tsn denotes type empirical distribution simplicity psq also shortly denoted since sequence fact typical set total probability close one following lemma lemma ras consider optimal henchman code maximizes sqn adopted need show lim pcz sqn codes utilizing lemmas pcz sqn sqn rac rac sqn rac rac sqn vanish choosing proper set sequence converges zero fast since vanishes guarantees also vanishes owing rate constraint given reconstruction sqn take values denote set possible values cpc sqn min sqn pcpc sqn apply union bound side write min sqn pcpc sqn max sqn pcpc pcpc max sqn sqn psqn nrn max nrk rde sqn sqn psqn max nrk sqn psqn rde sqn follows markov chain ckz see combine pcz sqn max nrk sqn psqn rde sqn nrk max pcz sqn psqn rde sqn therefore show probability decays doubly exponentially fast proof complete random variables mean consider given sqn rde sqn rde sqn complete proof need introduce following lemmas proof lemma given appendix lemma assume according type sequences sqn prde sqn denotes type term vanishes lemma fix distributed according sqn prde sqn term vanishes lemma sequence random variables interval erxi hence lemma implies hand prde prs prde sqn prs sprde sqn prs sprde sqn using bounds apply lemma probability identifying nrk small enough large enough positive bounded away zero vanishes doubly exponentially fast therefore expression vanishes completes proof theorem ppendix roof emma psq prde sqn hence need consider sps psq sps consider prts hptqq type sequences denotes relative entropy follows moreover thm psq log log min min min minsps psq therefore prts utilizing get prde sqn prde sqn prts prde complete proof need following lemma lemma assume according sqn prde sqn lemma implies prde sqn ppendix roof heorem define distribution pcs spn also satisfies similar finite alphabet case easy show distortion constraints legitimate users satisfied next following similar steps proof finite alphabet case prove secrecy constraint also satisfied case proving need introduce information conditional information first let distribution achieves function pdq necessarily unique information defined follows definition information dmin inf pdq information defined log exp dps expectation respect unconditional distribution reproduction random variable achieves pdq pdq follow distribution define min ips let distribution achieves define pzq dps expectation taken respect definition conditional information pzq pzq inf conditional information condition defined pzqq log exp pzq pzq pzq dps expectation respect margin distribution pzq pzqq next prove secrecy constraint end need pzi typical set defined psq psq psq log psq defined denotes log log psq typical set log psn denotes typical set simplicity psq also shortly denoted since sequence following lemma lemma lem lem assume satisfies log ras derivation still holds pcz sqn nrk max pcz sqn psqn therefore show probability decays doubly exponentially fast proof complete end need introduce following lemmas proof lemma given appendix lemma assume satisfies nps log log according type sequences sqn sqn sqn psi pdq term vanishes lemma fix psz assume given distributed according sqn sqn sqn pzi pdq pzi typical set different one defined definition benefit makes following property hold sequence psq psq tsn equivalently prts tsn term vanishes property coincides finite alphabet case crucial importance proof see term vanishes apply lemmas probability decays doubly exponentially fast completes proof theorem ppendix roof emma log hold prde sqn hence need consider satisfying sps psq sps psq lemma says type sequences prts supp ptq psq denotes suppose log holds term vanishes end divide two parts psq psq psq psq psq psq psq psq psq nps psq psq nps prove follows definition nps follows fact psq psq since sps psq psq psq psqq sps psq log psq psq log log therefore nps log since nps logn logn therefore implies prts furthermore implies obtained following part proof steps thm replaced log respectively hence holds prts term vanishes utilizing get psi pdq prde sqn psi pdq prts prde psi pdq prde sqn complete proof need following lemma lemma assume general necessarily countable alphabets drawn according dmin dmin defined definition sqn sqn prde sqn psi pdq hence lemma implies prde sqn psi pdq ppendix roof heorem similar cases easy show distortion constraints legitimate users satisfied next following similar steps proof cases prove secrecy constraint also satisfied let end first need discretize source reconstruction rss rss obtained mapping closest quantization point rss quantized versions round round rss psi rsi furthermore holds ndpsn sqn sqn sqn rss sqn rss ndprss follows triangle inequality utilizing inequality rdps prss prssn reorder probabilities prss prssq rss decreasing order denote result prss prss obviously ppj hence gaussian sources remark prss satisfies conditions define event rssn prssq prssi rrss prssi pzi observe distribution pcs spn also satisfies implies sequence hence lemma still holds case following similar steps proof theorem get pcz sqn pcz max pcz max nrk dprssn nrk max pcz dprssn nrk prq ssn ssn prssn prq follows appears ball radius implies rssn appears instead whole set ball radius hence sufficient consider furthermore observe volume radius log log log logp log log log therefore show probability decays doubly exponentially fast proof complete apply lemmas probability decays doubly exponentially fast hence lim ecz maxrn hcodes sqn lim sup min rrss complete proof need show rrss suppose prss achieves iprss rss edprss rss since prssrss prss rss rss rss edps rss rssq rss rss rss rss edprss rss edprss rss follows inequality hand defined hence ips rss minimum ips edps let similarly iprss rss prove rrss completes proof theorem ppendix roof heorem denote denotes orthogonal transform instead permutation operation verified gaussian pair distribution pcs spn also satisfies hence forms markov chain similar scheme easy show power constraint distortion constraints legitimate users satisfied next following similar steps proof scheme prove secrecy constraint also satisfied slight difference use argument geometric point view instead one view theory method types used theorems need discretize source still need discretize reconstruction since enable take maximizing operation probability done define event jointly gaussian variables independent typical set jointly typical set become pxq nnx pxn nnx nnz nnx nnu respectively denote variances respectively since sequence fact weakly typical set total probability close one following lemma lemma rac following similar steps proof theorem get pcz dps sqn max pcz nrk prq given prq since shown upper bounded exponential function need show probability decays doubly exponentially fast end need introduce following lemmas proofs lemmas given appendixes respectively lemma assume independent prd pdq log term vanishes denotes vector denotes identity matrix lemma assume uniformly distributed orthogonal matrices set independent prd pdq log term vanishes prd prq prd sfs sfs follows lemma furthermore lemma implies prq follows applying lemmas probability decays doubly exponentially fast completes proof theorem ppendix roof emma consider prd prd nnz nnz nnu prd nnu denote prd prd sdr given hand gaussian distribution isotropic invariant rotation hence condition uniformly distributed sphere center radius prd sin fig cap cut cone unit sphere arcsin solid angle space cone area spherical cap unit sphere see fig approximate need following lemma lemma let solid angle space cone holds cos lemma implies sin logp sin cos qqq sin combine log prd log log term vanishes combining gives prd log completes proof lemma ppendix roof emma observe prd nns nns prd since uniformly distributed orthogonal matrices set stated lemma lemma implies uniformly distributed sphere center origin radius hence described previous section denotes solid angle space cone arcsin lemma nns log log log term vanishes combining gives prd log completes proof lemma ppendix roof emma choosing independent min ips min ips edps edps pubq log need prove first consider case assume probability probability independent denotes timesharing random variable also assume sqq defined edps sqq dps sqq therefore satisfy distortion constraint edps sqq sufficient set qpp substituting sqq ips sqq ips sqq ips sqq pips sqq ips sqq pips sqq pips phpkq follows independent next consider case observe ips min edps min ips ips edps min edps ips min ips edps hpkq min ips edps follows since one hand setting hand given pszk constraint optimization objective depend hence suffices optimize first term satisfies hpkq jointly gaussian defined independent hence also write independent also independent therefore bound second term ips ips min edps hps log log log hps log follows since satisfies constraint edps follows since forms markov chain independent pubq combining gives completes proof lemma ppendix roof heorem following similar steps proof scalar gaussian case easy prove rpiq achievable permutation based scheme however based scheme proof scalar gaussian case applied vector gaussian case directly details need treated specially next give proof case following similar steps proof theorem shown vector gaussian case distortion constraints power constraint satisfied tuples given theorem next prove secrecy constraint also satisfied define events psj jprms similar lemma shown acj derivation still holds vector gaussian case hence pcz max dps hcodes pcz max prmn nrk dps rms observe rdps rms dpsn rqsj snj zjn dsnj one zjn may expect exchange order write expression dpsjn rms however obviously feasible address problem need discretize source eliminate operation since discretization becomes operation number summands polynomial discretize rss round dps rms prss rms addition observe prss rms rms dprsj zjn dprsj dpd dpsjn zjn dpd dmn dprss rss prss follows implies rsj hence sufficient consider case obtained using triangle inequality combining gives pcz max dps hcodes max pcz nrk prq dpsjn zjn prq dpd given prq mean prq zjn dpsjn dpd dpd zjn dpsjn bound need bound first similar shown log oplog using triangle inequality rsj addition rsj dprsj rsj combine prsi sqi round round integers dprsj addition hence bounded polynomial term show probability decays doubly exponentially fast proof complete end using lemma prq rsj rsj pdj pdq given denotes conditional function source side information encoder decoder term vanishes term vanishes follows grows polynomially fast lemma prq pdq given denotes function using bounds applying lemmas probability decays doubly exponentially fast completes proof theorem eferences shannon communication presence noise proc ire vol goblick theoretical limitations transmission data analog sources ieee trans inf theory vol gastpar rimoldi vetterli code code lossy communication revisited ieee trans inf theory vol may lee petersen optimal linear coding vector channels ieee trans vol shamai zamir systematic lossy coding ieee trans inf theory vol mar prabhakaran puri ramchandran hybrid codes broadcast gaussian sources gaussian channels ieee trans inf theory vol wireless scalable video coding using hybrid scheme ieee trans circuits syst video vol wireless cooperative video coding using hybrid scheme ieee trans circuits syst video vol mar distortion bounds source broadcast degraded channel ieee int symp inf theory isit distortion bounds source broadcast problem submitted ieee trans inf theory reznic feder zamir distortion bounds broadcasting bandwidth expansion ieee trans inf theory vol tian diggavi shamai approximate characterizations gaussian source broadcasting distortion region ieee trans inf theory vol shannon communication theory secrecy systems bell syst tech vol schieler cuff henchman problem measuring secrecy minimum distortion list ieee trans inf theory vol jun secrecy shannon cipher system ieee trans inf theory vol apr yamamoto theory shannon cipher system ieee trans inf theory vol may wilson narayanan transmitting analog gaussian source gaussian wiretap channel snr mismatch ieee international conference telecommunications april bagherikaram plataniotis secure hybrid coding ieee international symposium personal indoor mobile radio communications bagherikaram plataniotis secure joint coding interference known transmitter iet communications vol kak jayant speech encryption using waveform scrambling bell syst tech vol wyner analog scrambling scheme expand bandwidth part discrete time ieee trans inf theory vol may ahlswede dueck good codes produced permutations ieee trans inf theory vol may kang liu compressing encrypted data achieving optimality strong secrecy via permutations ieee trans inf theory vol slepian permutation modulation proc ieee vol mar slepian group codes gaussian channel bell syst tech vol apr ericson theorem antijamming group codes ieee trans inf theory vol lapidoth tinguely sending bivariate gaussian gaussian mac ieee trans inf theory vol jun cover thomas elements information theory wiley new york information theory coding theorems discrete memoryless systems cambridge university press gamal kim network information theory cambridge university press eaton multivariate statistics vector space approach wiley sons new york sason inequalities ieee trans inf theory vol shannon probability error optimal codes gaussian channel bell syst tech vol yeung information divergence measures unified typicality ieee trans inf theory vol
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sample complexity neural networks jan noah golowich harvard university alexander rakhlin university pennsylvania ohad shamir weizmann institute science microsoft research abstract study sample complexity learning neural networks providing new bounds rademacher complexity assuming norm constraints parameter matrix layer compared previous work complexity bounds improved dependence network depth additional assumptions fully independent network size depth width results derived using novel techniques may independent interest introduction one major challenges involving neural networks explaining ability generalize well even large potential overfit training data neyshabur zhang learning theory teaches must due inductive bias constrains one learn networks specific configurations either explicitly via regularization implicitly via algorithm used train however understanding nature inductive bias still largely open problem useful starting point consider much restricted class linear predictors class good understanding generalization behavior dictated norm particular assuming kwk signifies euclidean norm distribution kxk almost surely generalization error lipschitz losses given training examples scales completely independent dimension thus natural ask whether general case neural networks one obtain similar sizeindependent results independent networks depth width appropriate norm constraints parameters also natural question considering size modern neural networks often much larger number training examples classical results sample complexity neural networks see anthony bartlett satisfy desideratum strong explicit dependence network size particular trivial number parameters exceeds number training examples recently several works aiming improving sample complexity bounds assuming various norm constraints parameter matrices example neyshabur use rademacher complexity tools show parameter matrices layers frobenius norms upperbounded respectively suitable assumptions activation functions generalization error scales high probability although bound explicit dependence network width dimensions strong exponential dependence network depth even neyshabur also showed dependence sometimes avoided activations unfortunately assumption satisfied common activations relu bartlett use covering numbers argument show bound scaling kwj kwj kwj denotes spectral norm denotes rows ignore factors logarithmic network width unlike explicit exponential dependence depth however still strong unavoidable polynomial dependence see note bound never smaller kwj kwj kwj kwj particular even assume kwj constant bound becomes trivial finally using notation neyshabur utilize analysis prove bound scaling kwj denotes network since kwjj parameter matrix bound never smaller becomes trivial kwj summarize although bounds logarithmic dependence network width aware bound literature avoids strong dependence depth even various norms controlled depth dependency avoided assuming norms sufficiently constrained argue cases must true see let return case linear predictors consider generalized linear predictors form kwk max relu function like linear predictors generalization error class assuming inputs satisfy kxk almost surely bartlett note bound never better bound derived covering numbers argument however difficult show class equivalently written class relu networks form vector scalars depth arbitrary therefore sample complexity class must also scale depends norm product completely independent network depth well dimension argue satisfactory sample complexity analysis similar independence properties applied class general neural networks vector scalars become matrices simple observation longer applies however using intuition natural try derive generalization bounds controlling kwj suitable matrix norm perhaps simplest choice spectral norm indeed product spectral norms utilized previous results mentioned earlier however formally show sec spectral norm alone weak get bounds even network depth small instead show controlling suitable norms indeed lead better depth dependence even fully bounds improving earlier works specifically make following contributions sec show exponential depth dependence rademacher analysis neyshabur avoided applying contraction slightly different object become standard since work bartlett mendelson example networks parameter matrices frobenius norm bound improved technique also applied types norm constraints example consider setup corresponding class networks row attain bound log input dimension dependence polynomial quite mild contrast neyshabur studied similar setup managed obtain exponential dependence sec develop generic technique convert bounds bounds assuming control schatten norm parameter matrices includes instance frobenius norm trace norm special cases key observation utilize prediction function computed networks approximated composition shallow network univariate lipschitz functions example assuming frobenius norms layers bounded improve min ignoring logarithmic factors note upper bounded best knowledge first explicit bound standard neural networks fully assuming suitable norm constraints moreover captures depthindependent sample complexity behavior network class discussed earlier also apply technique get version bound bartlett specifkw ically assume kwj maxj provided bartlett becomes bound contrast show following bound ignoring logarithmic factors log min bound schatten largest possible euclidean norm network output input vectors norm upper bounding min first argument get bound independent depth assuming norms suitably constrained sec provide lower bound showing class neural networks parameter matrix schatten rademacher complexity least max somewhat improves bartlett theorem showed result spectral norm control without term matches upper bound terms norm dependencies moreover establishes controlling spectral norm alone indeed schatten control lead bounds independent size network finally bound shows similar bartlett dependence products norms across layers generally inevitable besides provide additional remarks observations sec technical proofs presented sec preliminaries notation use letters denote vectors capital letters denote matrices fixed parameters clear context given vector kwk refer euclidean norm kwkp refer norm matrix use denote schatten spectrum written vector example refers spectral norm refers frobenius norm refers trace norm case spectral norm drop subscript use also order follow standard convention use denote frobenius norm finally given matrix denote reals let columns neural networks given domain kxk euclidean space consider scalar standard neural networks form parameter matrix fixed dimensions fixed lipschitz continuous function euclidean spaces satisfying denote depth network width defined maximal row column dimensions without loss generality assume lipschitz constant otherwise lipschitz constant absorbed norm constraint neighboring parameter matrix say written application univariate function coordinate input case somewhat abusing notation also use denote univariate function say satisfies important example relu networks every corresponds applying relu function max element simplify notation let wbr shorthand matrix tuple nwbr denote function computed composed layers rademacher complexity results paper focus rademacher complexity standard tool control uniform convergence hence sample complexity given classes predictors see bartlett mendelson details formally given function class set data points define empirical rademacher complexity sup vector uniformly distributed main results provide bounds rademacher complexity sometimes independent long assumed norm respect classes neural networks various norm constraints using standard arguments bounds converted bounds generalization error assuming access sample training examples exponential polynomial depth dependence get bounds rademacher complexity deep neural networks reasonable approach employed neyshabur use peeling argument complexity bound depth networks reduced complexity bound depth networks applying reduction times example consider class relu neural networks layer parameter matrix frobenius norm using straightforward manipulations possible show definition equals sup sup sup kwd upper bounded sup sup iterating inequality times one ends bound scaling neyshabur see also exponential factor follows factor turn follows applying rademacher contraction principle get rid function unfortunately factor generally unavoidable see discussion ledoux talagrand following theorem section point simple trick used reduce exponential depth dependencies polynomial ones nutshell using jensen inequality rewrite scaled rademacher complexity log sup exp log exp sup arbitrary parameter perform peeling argument similar resulting multiplicative factor every peeling step crucially factors accumulate inside log factor end result contains log factor appropriate tuning reduced formalization argument depends matrix norm using begin case frobenius norm key technical condition argument work perform peeling inside exp function captured following lemma lemma let activation function applied relu class convex monotonically increasing function sup sup proof letting rows matrix kwj kwj supremum kwj must attained kwj kwi therefore sup sup since upper bounded sup sup sup equality follows symmetry distribution random variables right hand side turn upper bounded sup kwk sup sup see equation ledoux talagrand lemma hand provide bound rademacher complexity bounded neural networks clean factor replaced theorem let class networks depth domain parameter matrix frobenius norm activation functions satisfying lemma log kxi log proof fix chosen later rademacher complexity upper bounded sup log sup exp log sup exp write last expression log sup exp log sup exp ranges possible functions applied lemma exp repeating process arrive log exp define random variable random function random variables log log exp log exp jensen inequality upper bounded kxi handle log exp term note deterministic function random variables satisfies kxi means satisfies condition proof theorem boucheron implies variance factor kxi kxi satisfies choosing kxi kxi log exp log using get upper bounded follows kxi log kxi kxi log exp log result follows remark note simplicity bound thm stated networks argument easily carries networks composed lipschitz loss function case one uses variant lemma peel losses proceed manner proof thm omit precise details brevity result similar also derived matrix norms example given matrix let denote maximal rows consider class depthd networks parameter matrix satisfies kwj corresponds setting also studied neyshabur weights neuron network bounded case derive variant lemma fact require activation function lemma let activation function applied class convex monotonically increasing function sup sup denotes vector infinity norm using technique use lemma get bound rademacher complexity theorem let class networks depth domain kwj activation functions satisfying condition lemma log log max coordinate vector proofs theorem well lemma appear sec constructions used results section use function exp together inverse log get depth dependencies scaling thus might tempting try improve depth dependence using functions increases sublogarithmically unfortunately argument still requires control difficult increases exponentially next section introduce different idea suitable assumptions allows get rid depth dependence altogether depth dependence depth independence section develop general result allows one convert bound rademacher complexity neural networks one assuming schatten parameter matrices controlled develop formalize main result subsection provide applications subsection proofs results section appear sec general result motivate approach let consider special case networks parameter matrix constrained diagonal size frobenius norm every activation functions identity network computes linear function letting diagonal networks equivalent denotes product therefore would like network compute function clearly need bounded away zero exponentially small still satisfying constraint kwj fact way satisfy requirements simultaneously close unit vector implies matrices must close turns intuition holds much generally even restrict identity activations diagonal parameter matrices essentially show network computes function product schatten bounded must least one parameter matrix far therefore replace parameter matrix appropriate matrix function computed network change much captured following theorem theorem network knw kwj exists another network depth layer dimensions following properties identical except parameter matrix layer rank equals leading singular value singular vectors pairs log knw make following crucial observation network parameter matrix computes function seen composition network univariate function sux moreover norm constraints imply latter function lipschitz therefore class networks considering subset class networks composed univariate lipschitz functions fortunately given class bounded complexity one effectively bound rademacher complexity composition univariate lipschitz functions formalized following theorem theorem let class functions euclidean space let class llipschitz functions fixed letting rademacher complexity satisfies log universal constant remark replaced log empirical gaussian complexity see proof sec details combining ideas plan attack following given class networks arbitrary parameter use thm relate rademacher complexity complexity similar networks one first parameter matrices use thm bound complexity turn using rademacher complexity networks crucially resulting bound explicit dependence original depth new parameter formally following theorem main result section theorem consider following hypothesis class networks kxk knw kwj kwj max parameters also define maps kwj kwj max finally let loss functions satisfy rademacher complexity upper bounded log min universal constant particular one upper bound result choice tuning appropriately get bounds independent depth next subsection provide concrete applications specific choices remark parameters divide norm terms thm closely related notion margin indeed consider binary classification bounds losses converted bound misclassification error rate terms average error training data see bartlett section discussion also viewed maximal margin attainable input domain applications thm section exemplify thm used obtain bounds sample complexity various classes neural networks general technique follows first prove bound generally depends depth scales plug thm utilize following lemma tune appropriately lemma holds min min min begin proving version thm theorem implies class neural networks frobenius norm bounds including plugging thm using lemma straightforward derive following corollary see sec formal derivation corollary let class neural networks parameter matrix satisfies kwj activation functions assuming loss function satisfy conditions thm sets unconstrained holds log log min ignoring logarithmic factors replacing min first argument bound corollary bound completely independent either width depth networks words possible make bound smaller fixed sample size independent network size long bounded hand bound corollary also bounded bound one would get immediate application thm implies asymptotic rate function still maintained next apply thm results bartlett discussed introduction provide bound using different set norms specifically obtain following intermediary result deriving generalization bound theorem bartlett let hypothesis class networks kxk using activation functions given kwjt kwj kwj fixed parameters rademacher complexity log log discussed introduction never smaller hence bound scales least however using bound together thm lemma get following corollary simplicity assume uniformly bounded corollary let class networks activation functions assuming loss function satisfy conditions thm kwjt kwj holds rademacher complexity log log log log min replacing min first argument get bound fully independent network size assuming norms suitably bounded give concrete example take constraints correspond frobenius norm ignore logarithmic factors get bound scaling log min contrast direct application thm setting leads bound finally note since based analysis always weaker noted bartlett corollary also gives version lower bound schatten norms section present lower bound rademacher complexity class neural networks parameter matrices bounded schatten norms formal result following theorem let class neural networks parameter matrix satisfies kwj schatten use convention refers spectral norm exists choice loss data points respect max theorem strengthens theorem bartlett considered case dependence hand consider bounds hold choice consider bounds uniform simplicity furthermore following implications like bartlett theorem implies controlling norms parameter matrix dependence product norms generally inevitable see inevitable factor bound implies controlling spectral norm insufficient get bounds least independent width generally schatten control insufficient get size independence frobenius norm bounds lower bound becomes matches upper bound corollary logarithmic facand order tors except worse polynomial dependence additional remarks guarantees far proved upper bounds empirical rademacher complexity fixed class neural networks form compl complexity measure compl parameter imply learning guarantees algorithms return predictors however context constraints practical algorithms neural networks usually perform unconstrained optimization therefore guaranteed return predictor fixed fortunately straightforward convert bounds probabilistic guarantees neural network bound scaling appropriately complexity compl particular network note guarantees also stated context previous sample complexity bounds neural networks bartlett neyshabur achieved instance union bound say doubling scale complexity refer proof margin bound koltchinskii panchenko theorem example technique complexity lipschitz networks proving results sec key element observation appropriate norm constraints neural network must layer parameter matrix close therefore network viewed composition shallower network univariate lipschitz function fact generalized whenever network parameter matrix close view composition shallow network lipschitz function although develop idea paper observation might useful analyzing types neural network classes taking extreme also bound complexity neural networks computing lipschitz functions studying complexity lipschitz functions domain easily verified setting consider class networks parameter matrix spectral norm network must using estimates covering numbers lipschitz functions get loss assumed dim dimensionality course bound bad dependence input dimension equivalently width first layer network hand dependence network depth matrix norm spectral norm discussed previous subsection also possible use bound get guarantees without constraining lipschitz parameter learned network advance proofs proofs lemma thm first prove lemma letting denote row matrix sup sup max kwj sup since side upper bounded sup sup proof concluded exactly lemma appealing ledoux talagrand turn thm whose proof rather similar thm fixing chosen later rademacher complexity upper bounded sup log sup exp log sup exp applying argument proof thm using lemma upper bound log exp letting denote coordinate using symmetry expectation inside log exp exp max exp exp exp exp exp exp exp exp upper bounding last step used fact exp maxj exp plugging back get log max exp choosing maxj log max upper bound log max result follows proof thm proof build following technical lemmas lemma matrix schatten exists matrix size kpp proof let denote svd decomposition diag choose top singular vectors values first two inequalities lemma easy verify third inequality using unitarial invariance spectral norm spj spj diag equals kpp lemma given network parameters let parameters parameter matrix layer fixed changed matrix kwr kwj sup knw kwr proof simple calculation lipschitz constant function nwbr assume definition kwj lipschitz constant function kwj norm kxk kwj therefore knw knw kwj kwj kwr kwj kwr kwj kwr kwj kxk result follows simplification cases handled exactly manner lemma suppose kwj kwj kwj min kwj proof fixing using stated assumptions well fact kwj kwj kwj kwj min kwj kwj kwj kwj taking root sides result follows lemmas hand turn prove thm combining lemma lemma indeed exists network matrix layer kwr kpp kwr sup knw kwj kwr kwr kpp kwj kwr definition knw kxk kwj since kwj assume exists norm knw follows using assumption kwj plugging lemma follows kwj min kwj substituting get sup knw kwj kwj exp log suppose log using fact exp follows log kwj remains consider case log however regime theorem trivially holds let network matrix rank zero ensures definition log kwj kwj sup knw sup knw proof thm prove theorem use straightforward covering number argument beginning definitions given function class metric elements let covering number denote minimal number functions particular fix set data points define empirical distance also given function class let sup denote empirical gaussian complexity arep standard gaussian random variables well known equivalent log factor ledoux talagrand sudakov minoration theorem see theorem ledoux talagrand log universal constant definitions hand turn prove theorem first note equivalent class therefore sup sup sup sup therefore enough consider simplify notation follows drop subscript first argue log numerical constant prove note functions holds supx enough upper bound first notice range discretize grid given construct function follows input let point nearest breaking ties arbitrarily let rest constructed linear interpolation points easily verified moreover note two neighboring points points must neighboring therefore function parameterized vector form specifies whether starting origin goes remains linear segments number functions therefore recalling majorizes get next argue see pick let respective closest functions cover scale triangle inequality easily verified fact sup therefore cover scale taking possible choices covers scale leading combining get log numerical constant finally use dudley entropy integral together equation implies following numerical constant possibly changing row row log inf inf log inf inf inf inf log choosing particular get upper bound log plugging upper bounding log see ledoux talagrand result follows proof thm enough prove bound fixed take infimum given construct new hypothesis class replacing network network defined thm namely parameter matrix layer replaced matrix use notation clarify dependence according theorem well definition rademacher complexity sup sup sup sup sup sup log reach crucial observation lies heart proof consider network let svd decomposition parameter matrix layer also leading singular value vectors construction definition composition loss equals function equivalent composition function univariate function note since kwr kwr former function contained defined theorem whereas latter function lipschitz constant kwj maps input fixed output denote therefore obtain contained composition qdfunctions whose output bounded class functions consisting result apply thm obtain log plugging back simplifying bit also noting upper bounded universal constant get upper bounded log appropriate constant mentioned beginning proof upper bound holds fixed result follows proof lemma show stated lemma always exists choice min since left hand side also trivially result follows prove inequality case analysis pick case follows min proof corollary direct application thm well fact loss lipschitz implies hand plugging thm using letting space matrices kwj choosing noting mfj get log min upper bounding minimum using lemma result follows log proof corollary consider class maps kwj kwj max since definition maps vector meaning kwrt kwr kwr therefore use thm bound particular log log therefore follows thm log log min log hand direct application thm also implies log log combining two bounds applying lemma get log log proof thm log log min definition rademacher complexity enough lower bound complexity subset particular consider class neural networks form diag kwk diagonal matrix satisfying kwkp refers vector furthermore suppose max finally choose mod standard basis vector letting mod holds sup particular choosing sign sup kwkp sup kwkp max max lower bound max since definition get lower bound alternative bound better obtained considering class neural networks form furthermore supposing identity function times holds sup taking best lower bound lower bound result follows references martin anthony peter bartlett neural network learning theoretical foundations cambridge university press peter bartlett dylan foster matus telgarsky margin bounds neural networks arxiv preprint peter bartlett shahar mendelson rademacher gaussian complexities risk bounds structural results journal machine learning research nov boucheron lugosi pascal massart concentration inequalities nonasymptotic theory independence oxford university press vladimir koltchinskii dmitry panchenko empirical margin distributions bounding generalization error combined classifiers annals statistics pages michel ledoux michel talagrand probability banach spaces springer behnam neyshabur ryota tomioka nathan srebro search real inductive bias role implicit regularization deep learning arxiv preprint behnam neyshabur ryota tomioka nathan srebro capacity control neural networks conference learning theory pages behnam neyshabur srinadh bhojanapalli david mcallester nathan srebro approach margin bounds neural networks arxiv preprint shai shai understanding machine learning theory algorithms cambridge university press chiyuan zhang samy bengio moritz hardt benjamin recht oriol vinyals understanding deep learning requires rethinking generalization arxiv preprint
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robocupsimdata robocup soccer research dataset olivia oliver falk frieder nov western sydney university centre research mathematics harz university applied sciences automation computer sciences department corresponding author abstract robocup international scientific robot competition teams multiple robots compete different leagues provide many sources robotics data used analysis application machine learning paper describes large dataset games top teams robocup soccer simulation league teams robots agents compete overall used different teams play resulting unique pairings pairing ran matches mins leading matches hours game play generated csv files data zipped unzipped dataset unique sense contains ground truth data global complete information objects field well noisy local incomplete percepts robot data made available csv files well original soccer simulator formats lator input asynchronously sensor input also decides next action sent back simulator several times second complexity environment continuous state action spaces together opportunity compete makes robocup soccer interesting testbed learning among many applications introduction robocup international scientific robot competition teams multiple robots compete robocup soccer leagues provide platforms number challenges robotics research including locomotion vision decision making dealing partial information coordination teamwork robocup several different leagues exist emphasize specific research problems using different kinds robots rules different soccer leagues robocup different types sizes hardware software small size middle size standard platform league humanoid simulation kitano soccer simulation leagues akiyama emphasis team work partial noisy information robots controlled separate program receives sensor information assist automated learning team behavior provide large dataset generated using top participants robocup possible use simulator robot learning also generate additional data normally available playing teams directly modified simulator record data robots local perspective restricted views depend robots situation actions also include sensor noise addition every step game recorded ground truth information positions velocities receives additional status information including energy levels referee decisions state game every robot issue parameterized actions every control locomotion direction view field view detailed description information transmitted found simulator manual chen objects soccer field well basic actions robot ground truth information usually recorded logfile available teams match create dataset ran pairings selected teams repetitions game games total robots team single game dataset consists local views plus global view views made available csv files values also provide original logfiles include additional sensors actions robot recorded text files software created patch simulator convert recordings csv provided data useful various different tasks including imitation learning ben amor learning testing olson predictive modeling behavior transfer learning reinforcement learning taylor stone representation learning time series data michael next sections describe environment robots data detail overview provided data robotics data collections often comprise lidar data recorded laser scans useful many applications field robotics simultaneous localization mapping slam tong specific purposes agriculture chebrolu however many contexts one several robots may observed data robocup consider includes information robots environment hence whole system data systems like robocup strategy video game starcraft lin provide information simulated environments robotics however addition contain data agents thus lay basis machine learning research analyze predict agent behavior strategies important applications service robotics systems general provide diverse dataset include several teams last two robocup competitions allowing different behaviors strategies perception behavior robot game depends behavior robots field game logfiles files containing ground truth information obtained recording games produced simulator recorded binary format access individual player percepts however possible within player code learn behavior teams useful use exact information individual players receive rather global noisefree information recorded logfiles therefore modified simulator additionally also record local noisy information received robots field individual files description environment robocup soccer simulation server rcssserver noda software used annual robocup competitions held since hosted used rcssserver version create data paper simulator implements physics rules game also handles interface programs controlling player default players use degree field view receive visual information every throughout game frequency actively changed player individually changing field view degrees degrees degrees respectively visual information transmitted form lists identified objects level detail information depending object distances potential objects include players field ball landmarks like goal posts flags side lines player also csu yunlu gliders helios cyrus helios hfutengine oxsy tion robot players pitch filled several landmark objects depicted fig flags punctual objects lines goal penalty area origin coordinate system center point pitch divided horizontally left right half vertically ydirection top bottom half additional numbers indifigure teams robocup cate distance origin meters since cer simulation world championships every soccer game takes place pitch leipzig germany one file infomation nagoya japan selected landmarks games lists coordataset dinates given table csv format name example row player information stored says right top flag format sent players also pitch table files provide information provide code translate individual logs respective game names csv files contain relative positions files naming conventions summarized velocities fig contain names competing chose teams robocup teams final scores team possibly soccer simulation world championships extended result penalty shootout leipzig germany nagoya time stamp game recorded japan see figure team binaries includsome identifier ing descriptions downloaded central soccerserver chen controls every virtual game physics played team team rules game starts server may times resulting data configured several parameters col zipped csv files game record lected one file identifier parameters original logfiles including message logs also example row denotes generate files ground truth data well ball speed decreases specified faclocal player data format fitor stolzenburg however nally made generating scripts available robotics point view information used reproduce results file relevant like stamina produce additional datasets using robots noise model format robotic soccer teams also smaller coach instructions thus skip details subset games teams play csv files plus soccer simulation game robocup original logfiles data available simulation league lasts mins total divided cycles length cycle logfiles comprise information game particular current description ground truth data tions players ball including according rules world soccer association velocity orientation cycle infifa soccer pitch size formation collected whole game adopted robocup soccer table identifier groundtruth tion league nevertheless physical boundary time point play mode kickoff curof area may sensed robots rent ball position coordinates velocity overall size listed furthermore positions velocities figure flags lines robotic soccer simulation chen landmarks static information games file game data time team left score left team right score right parameters server configuration parameters file groundtruth logfile information game file team name player number suffix suffix landmarks relative distances angles landmarks files moving relative distances angles ball players files figure name conventions data files described frequency messages dependent width view player selects one message every default every players also receive status information body messages individual messages recorded simulator logfiles developers teams implement recording messages agents receive order record messages sent team simulator software modified instead code contains patches simulator allow recording visual body messages individual files player messages stored original format keep amount additional processing game minimal playing game modified simulator result number recorded files player left right team including goalkeeper stated example column head contains velocity left goalkeeper finally information robots body head orientation view angle quality included absolute direction player facing respect pitch coordinate system sum body head direction player description local player data visual sensor players reports objects currently seen information automatically sent players every sense step frequency depending player view width quality default set thus addition three files mentioned files available game altogether robots ten field players one goalkeeper per team two files local player data provided hosting information respective player sees landmarks moving objects respectively file final identifier landmarks provides distances meters angles degrees respective landmarks relative robot head orientation step analogously file final identifier moving provides actual relative distances angles ball players sometimes player number even team name visible hence unknown robot case respective piece information left data available marked nan respective table element server also provides information velocity stamina yellow red cards commands dash turn kick robots cases also information observed state robots available particular whether kicking tackling goalkeeper visual body messages two files player file names follow naming convention game data csv files fig use suffix visual messages body messages recording game ground truth binary format using suffix commands players received simulator plain text using suffix convert visual messages csv files provide python program translates player visual messages two files csv file moving objects players ball csv file perceived landmarks convert simulator logfile ground truth csv file provide program built using open source librcsc library see logfiles recorded regular intervals optionally also store simulation parameters additional csv file code conclusions soccer simulator communicates players released large unique dataset using text messages form lists via udp robocupsimdata creating using ment top robocup simulation league teams later robocup federation creating dataset required modification url https simulation software ran repetitions matches last mins dataset allow number problems investigated learning testing approaches kitano asada kuniyoshi noda osawa matsubara robocup predictive reinchallenge problem magazine forcement learning respect dataset shall useful whole robotics community lin gehring khalidov synnaeve stardata starcraft research funding dataset corr computing research reposthe research reported paper itory cornell university libeen supported german academic exbrary url http change service daad funds man federal ministry education research bmbf programmes michael obst schmidsberger stolzenburg analysing soccer games related personal exchange ppp grant clustering conceptors akyama universities australia obst sammut tonidandel eds joint research cooperarobocup robot soccer world cup xxi tion scheme within project deep conceptors robocup international symposium nagoya temporal data mining decorating japan springer appear references noda matsubara hiraki frank soccer server tool research akiyama dorer lau multiagent systems applied artificial intellithe progress soccer simulation leagues gence bianchi rac akin ramamoorthy sugiura eds robocup robot world olson probabilistic mobile robots ieee transactions cup xviii robocup international symporobotics automation sium lecture notes computer science volume springer stolzenburg obst murray qualitative velocity ball interception ben amor vogt ewerton berger jarke lakemeyer eds jung peters learning advances artificial intelligence sive robot behavior imitation proceedings annual german connational conference intelligent robots ference artificial intelligence lnai systems aachen springer chebrolu lottes schaefer winterhalter burgard stachniss taylor stone transfer learning reinforcement learning domains suragricultural robot dataset plant classificavey journal machine learning research tion localization mapping sugar beet fields international journal robotics research tong gingras larose barfoot dupuis canadian chen dorer foroughi heintz huang planetary emulation terrain mapping kapetanakis kostiadis kummeneje dataset international journal robotics remurray noda obst riley steffens search url http wang yin robocup soccer server soccer server version
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local andreas thomas andreas christoph michael hannes ali ana helmut jun google ist austria austria university toronto canada university salzburg austria university cambridge vienna university technology austria forever hearts abstract semantics concurrent data structures usually given sequential specification consistency condition linearizability popular consistency condition due simplicity general applicability nevertheless applications require guarantees offered linearizability recent research focused improving performance scalability concurrent data structures relaxing semantics paper present local linearizability relaxed consistency condition applicable concurrent data structures like pools queues stacks linearizability requires effect operation observed threads time local linearizability requires thread effects local insertion operations effects removal operations remove values inserted observed threads time investigate theoretical practical properties local linearizability relationship many existing consistency conditions present generic implementation method locally linearizable data structures uses existing linearizable data structures building blocks implementations show performance scalability improvements original building blocks outperform fastest existing implementations acm subject classification programming languages formal definitions data structures lists stacks queues software programming programming keywords phrases concurrent data structures relaxed semantics linearizability introduction concurrent data structures pervasive along software stack operating system code application software beyond correctness performance imperative concurrent data structure implementations correctness usually specified relating concurrent executions admitted implementation sequential executions admitted sequential version data structure latter form sequential specification data structure relationship formally captured consistency conditions linearizability sequential consistency quiescent consistency linearizability accepted consistency condition concurrent data structures due simplicity general applicability guarantees effects paper extended version licensed creative commons license leibniz international proceedings informatics schloss dagstuhl informatik dagstuhl publishing germany local linearizability operations threads observed consistently global visibility requirement imposes need extensive synchronization among threads may turn jeopardize performance scalability order enhance performance scalability implementations recent research explored relaxed sequential specifications resulting implementations concurrent data structures except space alternative consistency conditions relax linearizability left unexplored large extent paper explore part gap investigating local linearizability novel consistency condition applicable large class concurrent data structures call data structures containers short containers include pools queues stacks spectrum consistency conditions enables describe semantics concurrent implementations precisely show appendix work stealing queues could proven linearizable wrt pool actually locally linearizable wrt queue local linearizability local consistency condition guarantees insertions per thread enq deq observed consistently linearizability requires enq deq consistent view insertions require projections global called history thread induced linearizable induced enclosed dashed line history thread tory thread projection program enclosed solid line cution combined remove values inserted irfigure local linearizability respective whether happen program execution locally linearizable iff history linearizable consider example sequential history depicted figure linearizable wrt queue since values dequeued order enqueued however history linearizable wrt queue therefore overall execution locally linearizable wrt queue contrast semantic relaxations based relaxing sequential semantics local linearizability coincides sequential correctness histories therefore sequential history locally linearizable wrt given sequential specification admitted sequential specification local linearizability linearizability coherence sequential consistency coherence almost universally accepted absolute minimum shared memory system satisfy requirement exists unique global order per shared memory location thus accesses threads given memory location conform unique order consistent program order relative ordering accesses multiple memory locations words coherence sequential consistency per memory location similarly local linearizability linearizability per local history view local linearizability offers enough consistency correctness many applications local view client often matters example locally linearizable queue client thread impression using perfect reordering ever observed among values inserted single thread guarantees suffice many cloud applications implementations locally linearizable data structures successfully applied managing free lists design fast scalable memory allocator scalloc moreover except fairness locally linearizable queues guarantee properties required dispatch queues common concurrency programming mechanism mobile devices haas paper study theoretical practical properties local linearizability local linearizability history multiple concurrent objects locally linearizable iff histories locally linearizable see thm locally linearizable data structures including queues stacks admit sane duplicated values values returned thin air values lost see prop local linearizability weakening linearizability natural class data structures including pools queues stacks see sec compare local linearizability linearizability sequential quiescent consistency many consistency conditions finally local linearizability leads new efficient implementations present generic implementation scheme given linearizable implementation sequential specification produces implementation locally linearizable wrt see sec implementations show dramatic improvements performance scalability cases locally linearizable implementations scale almost linearly even outperform pool implementations produced locally linearizable variants concurrent queues stacks well relaxed data structures latter relaxed two dimensions locally linearizable consistency condition relaxed sequential specification relaxed speedup locally linearizable implementation fastest linearizable queue lcrq stack stack implementation threads respectively verification local linearizability proving correctness new locally linearizable implementations immediate given starting implementations linearizable semantics concurrent objects common approach define semantics implementation concurrent data structure specify set valid sequential sequential specification relate admissible concurrent executions sequential executions specified sequential consistency condition means implementation concurrent data structure actually corresponds several sequential data structures vice versa depending consistency condition used sequential data structure object set method calls assume method calls include parameters input output values given set values sequential specification subset elements called sequences ease presentation assume value data structure inserted removed without loss generality may see set values consisting pairs elements core values version numbers note technical assumption makes presentation proofs simpler needed done locally linearizable implementations elements may inserted removed multiple times version numbers provide uniqueness values assumption ensures whenever sequence part sequential specification method call appears exactly additional core value element empty returned remove method calls find element return denote emp set values versions empty emp empty definition order relation given sequence method call appears exactly denote total order method calls given method call write appears throughout paper use pool queue stack typical examples containers specify sequential specifications axiomatic way sets local linearizability emp table pool axioms queue order axiom stack order axiom axioms exactly define valid sequences definition pool queue stack pool queue stack values set sets methods ins rem rem emp enq deq deq emp push pop pop emp respectively denote sequential specification pool sequential specification queue sequential specification stack sequence belongs iff satisfies axioms table pool instantiating ins rem keep axiom completeness although subsumed assumption value inserted removed specification contains sequences satisfy pool axioms axiom queue order instantiating enq deq finally contains sequences satisfy pool axioms axiom stack order instantiating push pop represent concurrent executions via concurrent histories example history shown figure thread executes sequence method calls method calls executed different threads may overlap happen figure duration method calls irrelevant semantics concurrent objects matters whether method calls overlap given abstraction concurrent history fully determined sequence invocation response events method calls distinguish method invocation response events augmenting alphabet let denote sets events events respectively method calls moreover let set thread identifiers let mki mkr denote sets events augmented identifiers executing threads example mki invocation method call thread proceed mention standard notion need several occasions definition projection let sequence alphabet denote projection symbols sequence obtained removing symbols definition history concurrent history sequence invocation response event appears response event mkr appears corresponding invocation event mki also appears example queue history left formal representation sequence right enq deq enq enq enq enq deq enq deq haas history sequential every response event immediately preceded matching invocation event vice versa hence may ignore thread identifiers identify sequential history sequence enq enq deq deq identifies sequential history figure history sequential every thread identifier denotes projection set mki mkr events local thread use term history history also may omit thread identifiers essential discussion history determines partial order set method calls precedence order definition relation precedence order set method calls history method call appears notation precedence order partial order iff denote subset precedence order relates pairs method calls thread program order thread characterize sequential history history whose precedence order total particular precedence order sequential history coincides order total order history fig enq enq deq deq definition projection set method calls let history mii mki mri mkr write mii mri note inherits precedence order history complete response every invocation event appears given history complete denotes set completions set complete histories obtained appending missing response events removing pending invocation events note complete iff complete history concurrent data structure set methods set concurrent histories history may involve several concurrent objects let set concurrent objects individual sets method calls sequential specifications object history history disjoint union method calls objects set method calls added prefix ensures union disjoint projection object denoted history set method calls obtained removing prefix every method call definition linearizability history linearizable wrt sequential specification sequential history completion complete permutation preserves precedence order refer linearization concurrent data structure linearizable wrt every history linearizable wrt history set concurrent objects linearizable wrt sequential specifications exists linearization object local linearizability local linearizability applicable containers whose set method calls disjoint union ins rem dob sob insertion method calls ins removal method calls rem dataobservation method calls dob global method calls sob insertions removals insert remove single value data set empty data observations return local linearizability single value shape observations return value necessarily provides information shape state example size data structure examples data observations head queue top stack peek pool examples shape observations empty returns true data structure empty false otherwise size returns number elements data structure even though refrain formal definitions want stress valid sequence container remains valid deleting observer method calls ins rem also containers multiple methods queue deque container methods local linearizability also applicable however local linearizability requires method call either insertion removal observation consequence set container according definition set ins acts global observer first checking whether version already set inserts also hash tables containers similar reason note arity method call container one excludes data structures like snapshot objects possible deal higher arities fairly natural way however cost complicated presentation chose present local linearizability simple containers present definition local linearizability without shape observations discuss shape observations appendix definition let container history thread define two subsets methods called thread respectively ins rem ins rem emp dob ins hence thread insertions performed removals data observers return values inserted removals remove value empty also automatically added thread hence also could cause inserting empty way removals empty serve means global synchronization without thread could perform operations locally without ever communicating threads note thread need performed return values inserted definition history let history history projection thread definition local linearizability history locally linearizable wrt sequential specification history linearizable wrt histories form decomposition thread data structure locally linearizable wrt every history locally linearizable wrt history set concurrent objects locally linearizable wrt sequential specifications history linearizable histories form decomposition thread local linearizability sequentially correct necessarily sequential history locally linearizable wrt sequential specification iff like haas ity local linearizability compositional complete proof following theorem missing extended proofs following properties found appendix theorem compositionality history set objects sequential specifications locally linearizable iff locally linearizable wrt every proof sketch property follows compositionality linearizability fact every thread object choices made splitting global history subhistories requiring consistency central local linearizability common consistency conditions study local linearizability first step exploring consistency conditions concurrent objects chose subhistories since reduces contention concurrent objects known lead high performance confirmed experiments assign method calls histories took point view associating data values threads gathering method calls data value subhistory associated thread def associate data values thread inserts one think alternative approaches example associate thread values removed view advantages choice clear first assigning inserted values threads every value history assigned thread contrast alternative approach clear assign values inserted removed second assigning inserted values inserting thread enables eager removals ensures progress locally linearizable data structures alternative approach seems like semantics removing empty local orthogonal issue assign values shape observations threads appendix discuss two meaningful approaches show local linearizability extended towards shape data observations appear insertion operations sets finally choose consistency condition required subhistories chose linearizability best strong consistency condition concurrent objects local linearizability linearizability investigate connection local linearizability linearizability proposition lin general linearizability imply local linearizability proof provide example data structure linearizable locally linearizable consider sequential specification snearlyq behaves like queue except first two insertions performed without removal first two elements removed order formally snearlyq iff enq enq deq deq enq enq deq deq deq emp enq enq deq emp example linearizable wrt snearlyq however induced history enq enq deq deq enq enq deq enq deq deq following condition data structure specification sufficient linearizability imply local linearizability satisfied pool queue stack local linearizability empty figure pool queue stack empty figure pool queue stack definition closure seq specification closed iff emp enq enq enq deq deq deq snearlyq enq deq emp snearlyq snearlyq closed proposition lin linearizability implies local linearizability sequential specifications closed proof sketch property follows definition equation exist corner cases local linearizability coincides linearizability histories turn attention pool queue stack proposition seq specifications closed proof sketch let let ins rem emp suffices check axioms pool definition table hold theorem pool queue stack lin pool queue stack local linearizability strictly weaker linearizability proof linearizability implies local linearizability pool queue stack consequence proposition proposition history figure locally linearizable linearizable wrt pool queue stack suitable renaming method calls although local linearizability wrt pool imply linearizability wrt pool theorem still guarantees several properties ensure sane behavior stated next proposition loclin pool let locally linearizable history wrt pool value duplicated every remove method appears values rem ins rem ins value lost emp rem rem ins rem emp ins rem rem rem rem proof direct unfolding definitions note history linearizable wrt pool three stated properties hold consequence linearizability definition local linearizability relaxed consistency conditions compare local linearizability classical consistency conditions better understand guarantees implications notion used name closure projection haas sequential consistency history sequentially consistent wrt sequential specification exists sequential history completion complete permutation preserves thread program order thread refer sequential witness data structure sequentially consistent wrt every history sequentially consistent wrt sequential consistency useful consistency condition shared memory really suitable data structures allows behavior excludes coordination threads implementation data structure every thread uses dedicated copy sequential data structure without synchronization sequentially consistent sequentially consistent queue might always return empty one consumer thread point time operation moved see figure producerconsumer scenario queue might end threads work theorem pool queue stack pool queue stack local linearizability incomparable sequential consistency figures give example histories show statement theorem contrast local linearizability sequential consistency compositional quantitative quiescent consistency qqc like linearizability sequential consistency quiescent consistency also requires existence sequential history quiescent witness satisfies sequential specification three consistency conditions impose order method calls concurrent history witness preserve quiescent consistency uses concept quiescent states relax requirement preserving precedence order imposed linearizability quiescent state point history pending invocation events invoked method calls already responded quiescent witness method call appear method call quiescent state method calls two consecutive quiescent states ordered arbitrarily quantitative quiescent consistency refines quiescent consistency bounding number reorderings operations two quiescent states based concurrent behavior two states next result quiescent consistency pool needed establish connection quiescent consistency local linearizability proposition pool history satisfying prop quiescently consistent prop follows local linearizability implies quiescent consistency pool theorem pool queue stack pool local linearizability strictly stronger quiescent consistency queue stack local linearizability incomparable quiescent consistency local linearizability also imply stronger condition quantitative quiescent consistency like local linearizability quiescent consistency quantitative quiescent consistency compositional details please see appendix consistency conditions distributed shared memory extensive research consistency conditions distributed shared memory appendix compare local linearizability coherence pram consistency processor consistency causal consistency local consistency conditions split history subhistories require consistency subhistories comparison local linearizability ins ins head head head head figure problematic history first define sequential specification single memory location assume memory location preinitialized value vinit returns value last performed memory location vinit denote ins head formally define head vinit ins head note data observations value read multiple times brevity consider histories involve single memory location following summarize comparison details please see appendix local linearizability concurrent data structures necessarily true mentioned consistency conditions hand local linearizability appears problematic shared memory consider locally linearizable history figure read values oscillate different values written different threads therefore local linearizability imply consistency conditions appendix show local linearizability incomparable considered conditions locally linearizable implementations section focus locally linearizable data structure implementations generic follows choose linearizable implementation data structure wrt sequential specification turn distributed data structure called lld locally linearizable wrt lld implementation takes several copies call backends assigns thread backend thread inserts element lld element inserted arbitrary thread removes element lld element removed eagerly element found attempted backend search element continues backends element found one round backends return empty proposition lld correctness let data structure implementation linearizable wrt sequential specification lld locally linearizable wrt proof let history lld crucial observation history backend history hence linearizable wrt number copies backends allowed generic implementation lld take one copy end linearizable implementation also way choosing backend removals fine however number backends backend selection strategy upon removals affect performance significantly lld implementations use one backend per thread resulting contention insertions always attempt local remove first return element continue search backends starting randomly chosen backend lld implementation closely related distributed queues dqs linearizable pool organized single segment length holding backends dqs come different flavours depending insert remove methods distributed haas across segment accessing backends variant follows lld approach described moreover algorithms implemented fixed number backends lld implementations manage segment variable size one backend per active thread note strategy selecting backends lld implementations similar work work stealing however contrast work data structures neither duplicate lose elements lld stack implementations successfully applied managing free lists fast scalable memory allocator scalloc guarantees provided local linearizability needed correctness scalloc free lists could also use weak pool pool without linearizable emptiness check however lld stack implementations provide good caching behavior threads operate local stacks whereas weak pool would potentially negatively impact performance implemented lld variants strict relaxed queue stack implementations none implementations involves observation methods lld algorithm easily extended support observation methods details please see app finally let note also experimented locally linearizable implementations lacked genericity lld implementations whose performance evaluation show promising results see app shown sec locally linearizable pool linearizable pool lacks linearizable emptiness check indeed lld implementations provide linearizable emptiness check despite eager removes provide variant lld provides linearizable emptiness check mild conditions starting implementation see app details experimental evaluation experiments ran uniform memory architecture uma machine four intel xeon processors supporting two hardware threads hyperthreads per core main memory linux kernel version also ran experiments without resulting noticeable difference cpu governor disabled measurements obtained artifactevaluated scal benchmarking framework also find code involved data structures scal uses preallocated memory without freeing avoid memory management artifacts measurements report arithmetic mean confidence interval sample corrected sample standard deviation experiments consider linearizable queues queue lcrq improved version linearizable stacks treiber stack treiber stack relaxed queue linearizable pools based distributed queues using random balancing queue stack implementations pools provide lld variants lld lcrq lld stack lld lld possible variants queue treiber stack making pools locally linearizable promising already distributed whenever achievable data structure implementation present results workloads lld implementations perform visible difference evaluate data structures scal benchmark producer consumer configured execute operations control contention add busy wait operations important high contention results measuring hardware operating system scheduling artifacts number threads ranges number hardware threads half producers half consumers relate performance scalability report number data million operations per sec better local linearizability lcrq lld lcrq lld number threads data structures million operations per sec better treiber stack treiber lld stack lld number threads data structures figure performance scalability microbenchmarks increasing number threads hyperthreads per core machine structure operations per second data structures require parameters set configured allow maximum parallelism workload threads results variants producers consumers parallel single segment producers consumers parallel different backends stack algorithm also needs configured delay parameter use optimal delay stack zero delay lld stack delays degrade performance lld implementation figure shows results benchmarks similar experiments performed elsewhere algorithms treiber scale threads linearizable queue stack algorithms lcrq stack either perform competitively relaxed counter parts even outperform outscale implementation lld available perform scale significantly better even slightly better pool compare best improvement show lld variants queue treiber stack speedup locally linearizable implementation fastest linearizable queue lcrq stack stack haas implementation threads respectively performance degradation lcrq threads aligns performance cpu instruction atomically retrieves modifies contents memory benchmarking machine different original benchmarking machine lcrq uses key atomic instruction conclusion future work local linearizability splits history set histories requires consistency yields intuitive consistency condition concurrent objects enables new data structure implementations superior performance scalability local linearizability desirable properties like compositionality data structures future work interesting investigate guarantees local linearizability provides client programs along line acknowledgments work supported national research network rise rigorous systems engineering austrian science fund fwf google phd fellowship erwin fellowship austrian science fund fwf epsrc grants vienna science technology fund wwtf trough grant proseed european research council erc grant quarem austrian science fund fwf grant wittgenstein award references url https html afek korland yanovsky relaxed consistency improved concurrency opodis pages ahamad bazzi john kohli neiger power processor consistency spaa pages ahamad neiger burns kohli hutto causal memory definitions implementation programming distributed computing aigner kirsch lippautz sokolova fast lowfragmentation memory allocation large virtual memory global data structures oopsla pages alistarh kopinsky shavit spraylist scalable relaxed priority queue ppopp pages bouajjani emmi enea hamza reducing linearizability state reachability icalp pages burckhardt gotsman yang zawirski replicated data types specification verification optimality popl pages cerone gotsman yang parameterised linearisability icalp pages chakraborty henzinger sezgin vafeiadis linearizability proofs logical methods computer science local linearizability popl artifact evaluation committee popl artifact evaluation accessed url http computational systems group university salzburg scal multicorescalable computing url http derrick dongol schellhorn tofan travkin wehrheim quiescent consistency defining verifying relaxed linearizability pages dodds haas kirsch scalable correct stack popl pages filipovic hearn rinetzky yang abstraction concurrent objects theor comput goodman cache consistency sequential consistency university wisconsinmadison computer sciences department haas henzinger holzer kirsch lippautz payer sezgin sokolova veith local linearizability concurrent data structures concur haas henzinger kirsch lippautz payer sezgin sokolova distributed queues shared memory multicore performance scalability quantitative relaxation haas kirsch lippautz preishuber sokolova scal benchmarking suite concurrent data structures netys pages heddaya sinha coherence local consistency distributed shared memory parallel computing technical report computer science department boston university heller herlihy luchangco moir scherer shavit lazy concurrent set algorithm opodis hennessy patterson computer architecture fifth edition quantitative approach morgan kaufmann publishers san francisco usa edition henzinger kirsch payer sezgin sokolova quantitative relaxation concurrent data structures popl pages henzinger sezgin vafeiadis linearizability proofs concur pages herlihy shavit art multiprocessor programming morgan kaufmann publishers san francisco usa herlihy wing linearizability correctness condition concurrent objects acm trans program lang jagadeesan riely linearizability quiescent consistency quantitative quiescent consistency icalp pages kirsch lippautz payer fast scalable queues pact pages kogan petrank queues multiple enqueuers dequeuers ppopp pages lamport make multiprocessor computer correctly executes multiprocess programs ieee trans september lipton sandberg pram scalable shared memory technical report princeton university department computer science lipton sandberg oblivious memory computer networking september patent michael hazard pointers safe memory reclamation objects ieee trans parallel distrib haas michael scott simple fast practical blocking concurrent queue algorithms podc pages michael vechev saraswat idempotent work stealing ppopp pages morrison afek fast concurrent queues processors ppopp pages multicore computing group tel aviv university fast concurrent queues processors accessed url http rihani sanders dementiev multiqueues simpler faster better relaxed concurrent priority queues corr sezgin sequential consistency concurrent data structures corr shavit data structures multicore age cacm march steinke nutt unified theory shared memory consistency acm september treiber systems programming coping parallelism technical report ibm research center local linearizability local linearizability shape observers two possible ways deal shape observers treat locally threadinduced history performing thread treat globally local treatment immediate natural local consistency condition global treatment requires care present solutions next definition local linearizability lso history locally linearizable local shape observers lso wrt sequential specification locally linearizable according definition difference definition also contain shape observers performed thread ins sob global observations require notation auxiliary notions let collection sequences alphabet pairwise disjoint sets symbols sequence interleaving write set interleavings given history method call write incomplete history prefix without response event hence contains invocation response events appear definition let denote sequential specification container shape observer history witness exists sequence linearization history informally definition states global shape observer must justified global witness global witness sequence extended belongs sequential specification interleaving linearizations histories definition local linearizability gso history locally linearizable global shape observers gso wrt sequential specification locally linearizable shape observer sob witness illustrate difference local global approach shape observers following example example consider following queue history global observer size enq enq deq size placeholder concrete natural number history locally linearizable lso locally linearizable gso history locally linearizable gso locally linearizable lso global observers operations expected negative impact performance one cares global consistency local linearizability consistency condition used restriction containers disjoint operations specifies informal way minimal requirements local consistency acceptable neither sets maps containers according definition however possible extend treatment sets maps similar treatment global observers locally haas linearizable sets maps weaker linearizable counterparts due tight coupling mutator observer effects gain performance unlikely substantial one observed data structures technicalities needed extend local linearizability sets maps would complicate theoretical development without considerable benefits therefore excluded data structures additional results proofs theorem compositionality history set objects sequential specifications locally linearizable locally linearizable respect every proof property follows compositionality linearizability fact every thread object assume locally linearizable means histories linearizable hence since linearizability compositional object history linearizable respect every object history linearizable every thread similarly assume every object history locally linearizable every linearizable every thread compositionality linearizability linearizable every thread proves locally linearizable proposition lin loclin linearizability implies local linearizability sequential specifications closed proof assume given history linearizable respect sequential specification closed assume without loss generality complete exists sequential history permutation also given thread consider history let permutation since consist events furthermore since closed dataprojection since equation holds containers finally also since therefore implies thereby shown linearizable respect arbitrary thread hence locally linearizable respect proposition closedness queue stack closed sequential specifications pool proof let let ins rem emp suffices check axioms pool definition table hold clearly methods appear rem rem since ins rem also rem hence ins rem finally ins rem emp ins rem implying rem rem rem rem well rem rem shows closed local linearizability assume enq deq ins rem respectively closed satisfies pool axioms moreover axiom definition table also holds assume enq enq deq enq enq deq since get deq deq deq means deq deq deq hence closed finally push pop ins rem respectively need check axiom definition table holds assume push push pop implies push push pop since get pop pop pop pop pop pop closed proposition loclin pool let locally linearizable history wrt pool value duplicated every remove method appears values rem ins rem ins value lost emp ins rem rem rem rem emp rem rem ins rem proof note history linearizable wrt pool three stated properties hold consequence linearizability definition assume locally linearizable wrt pool rem appears twice also appears twice history contradicting linearizable respect pool shows value duplicated rem rem since linearizable respect pool ins rem ins yields ins rem ins hence values finally rem emp rem let ins rem let ins ins rem since linearizable respect pool rem rem rem yields rem rem rem similarly condition holds hence value lost theorem queue local linearizability queue concurrent history locally linearizable respect queue sequential specification locally linearizable respect pool sequential specification enq enq deq deq deq deq proof assume locally linearizable respect since suitably renamed method calls locally linearizable respect moreover since linearizable respect theorem enq enq deq deq deq deq assume enq enq deq enq enq deq deq deq deq implies deq deq deq opposite assume conditions hold history need show form decomposition clear queue linearizable respect haas linearizable respect pool assume enq enq deq enq enq deq hence deq deq deq enq deq get deq deq deq according theorem enough conclude linearizable respect theorem pool queue stack pool queue stack local linearizability incomparable sequential consistency proof following histories instantiating ins enq push respectively instantiating rem deq pop respectively sequentially consistent locally linearizable wrt pool queue stack pool empty queue stack history already locally linearizable wrt pool queue stack respectively histories provide interesting examples history figure locally linearizable sequentially consistent wrt pool following histories locally linearizable sequentially consistent wrt queue stack respectively queue two histories linearizable respect queue however overall history sequential witness therefore sequentially consistent maintain queue behavior order operations changed however implies value instead value would removed directly stack local linearizability ins rem ins rem rem rem rem empty ins ins figure sequential history two histories linearizable respect stack operations prevent reordering operations therefore overall history sequential witness hence sequentially consistent proposition pool let pool history data duplicated values returned data lost satisfies proposition quiescently consistent proof assume pool history satisfies proposition let histories form sequential decomposition quiescent states beginning end note decomposition nothing decomposition let mhi set methods note sanity conditions ensure none following two situations happen rem ins ins rem empty rem let denote set values ordered way ins rem rem ins moreover let number occurrences rem empty construct sequential history form sequential history permutation shown figure using observations easy check indeed quiescent witness theorem pool queue stack pool local linearizability stronger quiescent consistency queue stack local linearizability incomparable quiescent consistency proof following histories quiescently consistent locally linearizable wrt pool queue stack respectively pool rem empty ins rem empty rem haas queue enq enq enq deq deq stack push push push pop pop three histories quiescent states longest operation therefore operations thread reordered arbitrarily particular way satisfy sequential specification respective concurrent data structure however histories thread linearizable respect pool queue stack respectively therefore none histories locally linearizable also history suffices hand following histories quiescently consistent locally linearizable wrt queue stack respectively queue enq deq enq deq stack push pop push pop histories two operations concurrent data structure quiescent state therefore none operations reordered hence sequential witness exists however histories linearable therefore overall histories locally linearizable particular history pair operations separated quiescent state overlap operations quiescent consistent data structure behaves would linearizable respect sequential specification see semantic differences local linearizability see linearizability local linearizability case study work stealing queues consider data structure admits two operation types ins inserts element container rem returns removes element container imagine implementation uses work stealing queue wsq every thread uses unique designated buffer wsq whenever thread calls ins appended tail calls rem wsq first local linearizability enq enq enq deq deq deq figure history qqc checks whether returns element tail lifo semantics removes otherwise chooses tries return element buffer time different thread buffer checked element removed taken head fifo semantics trying access buffer time usual synchronization measures taken ensure exactly one thread removes one element given implementation developer wants write specification potential users since essentially collection deques developer tempted state deque particular consistency condition however linearizable deque ins followed ins followed rem returns either depending whether calls rem ambiguous semantics seen sequentially consistent deque allow many behaviors deque would allow capture behaviors tightly relaxed sequential specifications work either since converge sequential semantics lifo stack single thread uses short developer fail capture semantics satisfactory manner hand locally linearizable deque rem treated fifo removal whenever lifo removal whenever words local linearizability provides succinct clean representation implementation framework wsq hiding away implementation details compare fact even though wsq queue argue correctness proved linearizable pool even though stronger semantics pool linearizable pool semantics weak observe also since described example essentially providing illusion using monolithic structure implemented terms distributed components shared memory typically implemented message passing expect local linearizability widely applicable quiescent consistency quantitative quiescent consistency without going details definition quantitative quiescent consistency give history figure quantitatively quiescently consistent locally linearizable wrt queue quantitative quiescent consistency allows reorder two thread thereby violates local linearizability consistency conditions distributed shared memory decomposition per thread coherence pram shs ccfsh lod ins head vinit head ins lin memory location ins head yes thread ins head yes yes thread ins head thread ins head scb yes head yes thread memory location ins ins head haas consistency condition sca order witness scb ordered transitive closure thread program orders pairs scc threads reordered even thread logical contradictions local history considered consistency number threads number memory locations shs number subhistories ccfsh consistency condition subhistories lod loss data table comparison consistency conditions single distributed shared memory location local linearizability table compare local linearizability consistency conditions coherence pipelined ram pram consistency processor consistency causal consistency local consistency local linearizability shares consistency conditions idea decomposing concurrent history several subhistories coherence projects concurrent history operations single memory location resulting history sequentially consistent since sequential consistency compositional coherence imply sequential consistency overall history whereas local linearizability single memory location implies local linearizability overall history contrast coherence local consistency local linearizability pram consistency decompose history subhistories threads conditions consider subhistories need sequential witnesses coherence requires one witness per memory location local consistency requires one witness per thread memory location determining subhistory thread coherence pram consistency consider given history ins contrast local linearizability considers thread ins local consistency considers thread well whose values read thread ins ins head regarding pram consistency consider thread coherence considers given history local linearizability considers read initial value vinit read values written thread reading initial value analogous returning empty data structure local linearizability requires subhistory history linearizable respect sequential specification consideration contrast coherence pram consistency require subhistory sequentially consistent variant thereof respect sequential specification however variants sequential consistency used consistency conditions vulnerable loss data discussed section therefore make consistency conditions unsuitable concurrent data structures considering pram consistency sequentialization different threads might observed differently different threads thread might observe write operations thread write operations thread thread might observe write operations write operations contrast threadinduced histories defined local linearizability involve threads involve performed threads like pram consistency processor consistency requires thread performed seen program order performed threads seen respective program order furthermore processor consistency also requires two memory location appear order sequential witness thread even different threads additional condition makes processor consistency strictly stronger pram consistency condition also creates similar effect consideration different threads forming history local linearizability causal consistency considers causal order instead thread program orders alone like local haas backend max add new node adjust alive initial state figure segment modifications throughout linearizability causal consistency matches pairs across different threads particular causal order transitive closure thread program orders pairs considering causal order writes different threads become ordered case local linearizability lld implementation details already mentioned thread inserts elements local backend removes elements either local backend preferred backends accessed single segment array effectively managing backends varying number threads segment dynamic length predefined maximum slot segment refers node consists backend flag indicating whether corresponding thread alive terminated similar work flag used logically removing node segment stays segment backend empty additionally global version number keeps track changes segment algorithm divided two parts maintaining segment adding removing elements backends following refer segment thread local node version number segment current length segment range indices defined maintaining segment provide two methods node used add remove nodes segment upon removal node segment also compacted hole created removing node pointer filled last node pointer segment nodes added removed length segment thus range valid indices segment updated changes segment involve incrementing version number detailed operations maintaining segment compacting nodes cleaned allocates node thread follows searches existing node terminated thread reuses finds one otherwise creates new node adds node adjusts cases increments returns node creation new node illustrated figure node searches node using linear search finds slot copies pointer decrements increments resets null using new found concurrent thread already performed cleanup operation returns figure illustrates example local linearizability backend dead backend dead empty dead write adjust figure segment modifications throughout node initially thread owning node dead corresponding backend empty note updating segment state needed threads joining backends terminated threads become empty consider scenarios infrequent implement corresponding operations using locks alternatively operations implemented using helping approaches similar algorithms also note although operations segments protected locks partial changes observed remove operation defined observe segment intermediate state two pointers pointing node cleanup invariant change destroy integrity segment within valid range slots within range either point valid node nothing null actual algorithm adding removing elements defined follows ins upon first insertion thread gets assigned node containing backend using element inserted subsequent insertions thread use throughout lifetime thread rem remove operation consists two parts finding removing element cleaning nodes terminated threads thread tries get element backend exist thread yet performed single ins operation corresponding backend empty different node selected randomly within valid range backend contained empty operation scans nodes backends linear fashion however version number changed round scanning backends operation restarted immediately note since dynamic remove operation may operate range longer valid checking version number ensures operation restarted case thread calls upon encountering node set false dead contains empty backend cleanup also triggers restart remove operation terminate upon termination thread changes alive flag false dead dynamic memory used nodes susceptible aba problem requires proper handling free memory implementations use aba counters avoid aba problem refrain freeing memory hazard pointers used solving aba problem well freeing memory lld linearizable emptiness check call data structure implementation stateful remove methods modified return state changes upon insert remove element haas change two removes return empty unless element inserted data structure meantime stateful implementations create locally linearizable version linearizable emptiness check queue treiber stack stateful implementations whereas lcrq also stack stateful implementations notion state data structures huge making unsuitable implementations linearizable emptiness checks achieved via atomic snapshot like dqs detailed description lld implementations well pseudo code found appendix present results experimental performance evaluation correctness proposition lld let stateful data structure implementation linearizable respect sequential specification linearizable respect pool proof proving linearizable respect pool particular linearizable emptiness check follows proof general see emptiness check performed creating atomic snapshot states backends stored states array using first loop lines atomic snapshot valid checked via second loop lines particular line backends empty atomic snapshot existed point time creation atomic snapshot backends indeed empty notice since segment dynamic length happen backends contained atomic snapshot guarantee elements missed emptiness check atomic snapshot extended version number segment new backend added segment generation atomic snapshot version number increased atomic snapshot becomes invalid line linearization point remove operation returns empty inbetween two loops last remove attempt first loop version check second loop lld pseudo code implementations use interfaces depicted listing simplicity interface mentions pool queue stack highlighted code refers linearizable emptiness check part implementations methods retrieving elements rem assumed modified possible also return state object uniquely identifies state data structure respect methods inserting elements ins state accessed via observer method listing illustrates maintaining segment backend line either declared stack queue defined listing linearizable data structure listing shows removing highlighted code obtain code lld thread maintains backend enclosed node line insertion local backend always accessed line method also makes sure thread announced line upon first insertion acquires node ins operation always uses thread local backend local linearizability pool element state rem ins element state queue pool element state dequeue enqueue element ins element enqueue element state rem dequeue stack pool element state pop push element ins element push element state rem pop listing pool queue stack interfaces line insertion removing element rem thread tries remove element local backend first line element found backends valid range searched linear fashion starting random index highlighted code lines illustrates checking atomic snapshot lld observer methods implemented lld variants strict relaxed queue stack implementations none lld implementations involves observer methods lld algorithm easily extended support observer methods data observer lld independently thread performs amounts data observer local shape observer lld performed thread executes shape observer global shape observer lld executes shape observer backend produces aggregate value additional implementations present evaluate additional algorithms provide locally linearizable variants queues stacks obtained modifying relaxed order queues stacks way makes sequentially correct also tried another generic implementation related construction implements wrapper sequential precise backends initial experiments performance implementation particularly promising locally linearizable queue queues relaxed queues stacks based lists segments segment holds slots elements effectively allowing reorderings elements list segments implemented variant queue haas node pool backend bool alive linearizable data structure segment node nodes int int version returns indexes exclusive random order int range announces node buffer effectively adding adjusting changing version node protecting concurrent announce cleanup operations node null node backend nodes alive true version segme removes node buffer effectively removing adjusting changing version clea node int protecting concurrent announce cleanup operations error error alive version segm return nodes nodes version nodes null segme listing node segment structure lld queue stack variant treiber stack insert remove methods operate segments ignoring order elements within segment segments used insertion removal identified insertion removal pointers respectively queues elements removed oldest segment inserted mostrecent segment upon trying remove element empty segment segment removed removal pointer advanced next segment upon trying insert element full segment new segment appended insertion pointer advanced new segment similarly different stack removal insertion operate segment removal insertion pointer synonyms identify segment times upon trying remove element empty segment segment removed removal pointer advanced next segment upon trying insert element full segment new segment prepended insertion pointer set new segment queues relaxed queues stacks linearizable respect queue stack respectively linearizable respect pool locally linearizable respect queue stack respectively since reordering elements inserted segment even sequentially local linearizability dynamiclocallylinearizabledq segment node node bool null ins element true backend ins element rem fast path retrieving element thread local backend false null state backend rem null true retry false version range range range nodes version retry true bool alive alive state backend rem null states state alive thread retry true else retry version range nodes null backend states retry true break retry null empty case called upon thread termination terminate false null alive false listing lld queue stack single thread allowed see histories proof theorem sequentially consistent respect queue stack shown histories proof theorem histories respectively haas present modifications enforce local linearizability ensuring thread inserts single segment assuming segments unique tagging pointers remembers last used insertion pointer per thread situation subtle due stack semantics segments reached multiple times insertion removal figure illustrates example top segment reached multiple times thread since general case segments could reached multiple times single thread required maintain full history thread insertions assuming maximum number threads known advance bitmap used maintain information segment thread already pushed value one similarly implement locally linearizable version segment queue ins ins ins ins rem rem segment state figure run insert uncolored segments needs prepend new segment insertion otherwise queue queue pseudo code listing shows pseudo code queue highlight code added original pseudo code similar locally linearizable thread inserts one element segment however queue need flags segment achieve property sufficient remember last segment used insertion thread line enqueue algorithm checks whether executing thread already used segment enqueueing element line segment already used thread tries append new segment effectively adding new tail correctness proof queue theorem proof correctness queue easy theorem correctness queue presented listing locally linearizable proof using theorem first proof obligation show history queue locally linearizable respect pool sequential specification proof analogous proof history locally linearizable respect pool sequential specification therefore postponed corresponding theorem remains show enq enq deq deq deq deq local linearizability locallylinearizablekfifoqueue enqueue item true restart loop index value empty item version cas segment index committed index true else bool committed index segment index true empty version true cas segment index true queue head value version cas head true cas segment index true false item dequeue true index value empty value value empty version cas index value else value value value null listing locally linearizable queue assume enq enq means enqueued thread therefore inserted different segments moreover segment closer head list segment deq method call remove segment head segment segment become head segment segments closer head list get empty means also segment become empty therefore exist deq method call removes segment deq deq haas pseudo code listing shows pseudo code highlighted code code added original pseudo code achieve local linearizability difference original algorithm thread inserts one element segment achieve property segment contains flag per thread set element inserted segment line line thread encounters segment flag already set thread insert element segment tries prepend new segment line otherwise element inserted existing segment flag thread segment set locallylinearizablekstack segmentptr top init calloc ksegment top bool item top calloc ksegment next item use first slot item ver cas top true false top next null remove empty next ver cas top return remove bool committed index index true remove true remove empty ver top cas index true else val ver cas top true cas index true false push item true top item true restart loop index top val empty item ver cas index committed index local linearizability true else item true item pop true top index top val empty empty ver cas index val else empty top null else listing locally linearizable correctness proof local linearizability proof involved interesting use theorem published artifact mechanically proved isabelle hol theorem prover theorem empty returns stack let history let projection pop empty linearizable respect sequential specification pool see definition linearizable respect sequential specification stack see definition linearizable respect proof repeat key insights proof leave technical details complete mechanized version proof available published artifact linearizable respect linearizable respect exists sequential history linearization exists sequential history linearization show construct sequential history linearization linearization constructed follows position pop empty preserved means method call pop empty also pop empty pop empty also pop empty moreover two method calls ordered pop empty therefore transitivity holds also method calls order preserved means two method calls pop empty holds pop empty pop empty construction history sequential permutation next show linearization showing preserves precedence order also construction holds two method calls also either linearizations respectively therefore also since sequential means preserves precedence order next show according definition haas every method call pop empty appears guaranteed since permutation pop appears also push push pop since permutation pop also push since push pop push pop also holds push pop argued already push pop empty pop pop empty property satisfied trivially pop empty operations ordered remains check elements removed stack fashion show following push push pop pop pop pop first show push push pop also push push pop showing exist pop empty push pop empty push push pop empty pop assume towards contradiction push pop empty push transitivity implies push pop empty pop contradicts observation therefore push pop empty push possible reason also push pop empty pop possible push push pop exist pop pop pop reason pop pop empty pop therefore pop pop ordered pop pop therefore theorem correctness algorithm presented listing locally linearizable proof show every history locally linearizable respect sequential specification defined definition means show every history linearizable respect thread theorem show linearizable respect sequential specification pool defined definition projection pop empty linearizable respect sequential specification stack start proof linearizable respect construct sequential history identifying linearization points push pop method calls means two method calls ordered linearization point executed linearization point linearization point push method calls either successful insertion new segment line last successful cas writes element segment slot line linearization point pop method calls successful cas removes element segment slot line linearization point pop empty take linearization point call empty line empty method creates atomic snapshot top segment atomic snapshot state top segment point linearization point empty within execution empty empty returns true exists element atomic snapshot segment local linearizability next show defined definition since exists exactly one linearization point per method call every method call rem empty appears pop appears reads slot top segment linearization point since push method calls write elements segment slots exist push wrote slot therefore linearization point push always linearization point pop therefore push pop segments removed list segments become empty call committed guarantees elements inserted segments removed pop method calls empty single segment left element found segment assume push method call inserts element missed push wrote segment linearization point pop empty segment last segment top segment changed since pop empty searched element therefore check line would fail push wrote last segment pop method call removed segment otherwise would atomic snapshot empty therefore empty would return false therefore push pop empty also pop pop empty therefore sequential specification pool next show linearizable respect construct sequential history identifying linearization points push pop method calls linearization point push operations successful insertion new segment line executed reading empty slot line last therefore successful iteration main loop linearization point pop operation reading slot line last therefore successful iteration main loop exist pop empty method calls since assume sequentially consistent memory model read operations define total order method calls first show sequential specification pool defined definition since exists exactly one linearization point per method call every method call appears pop appears read slot top segment linearization point since push operations write elements segment slots exist push wrote slot linearization point push always written segment slot therefore push pop since exist pop empty operations third pool condition trivially correct next show also provides stack order means show push push pop pop pop pop start observing invariants haas thread never inserts elements segment twice guaranteed call linearization point push time writes element segment segment element gets written removed push operation inserts new segment trivially correct push operation writes element existing segment call committed line guarantees segment removed time linearization point pop time pop reads slot line last therefore successful iteration pop reads slot top segment guaranteed check line assume exist operations push push pop push push pop since push push means operations executed thread therefore according invariant get inserted different segments segment top segment linearization point pop written segment according invariant segment gets inserted get removed linearization point push time written segment invariant means unaccessible pop gets written segment also third invariant top segment changes insertion linearization point pop next observe long removed segment segment become top segment therefore segment become top segment pop remove removed first pop remove therefore exists pop linearization point pop linearization point pop hence sequential specification stack using theorem means listing respect sequential specification stack additional experiments also evaluate implementations another scal workload sequential alternating workload however note workload locally linearizable implementations threads access local backends wonder perform perfectly well mixed workload order evaluate performance scalability mixed workloads workloads threads produce consume values exercise sequential alternating workload scal thread configured execute pairs insert remove operations insert operation followed remove operation workload contention controlled adding busy wait number threads configured range report number data structure operations per second data structures require parameters set configured like producerconsumer benchmark figure shows results mixed workload benchmark considered data structures queue treiber stack perform scale threads benchmark lcrq stack either perform competitively million operations per sec better local linearizability million operations per sec better number threads number threads lcrq lld lcrq lld queues queues pools treiber stack treiber lld stack lld stacks stacks pools figure performance scalability sequential alternating microbenchmarks increasing number threads hyperthreads per core machine relaxed counter parts even outperform outscale case lcrq even outperforms pool queue lld lcrq treiber stack perform well scale nearly linearly number threads surprising result lld performs poorly experiment reason performs poorly almost empty experiment backend instance lld contains one element point time performs better state benefit trying perform local operation first lld algorithms visible comparing utilize local fast path verifying local linearizability general verifying local linearizability amounts verifying linearizability set smaller histories might enable verification way aside important mention locally linearizable data structures section built linearizable building blocks correctness proofs straightforward assuming building blocks proven linearizable addition queue state axiomatic verification theorem local linearizability style whose main theorem recall next slight reformulation theorem queue linearizability queue concurrent history linearizable wrt queue sequential specification linearizable wrt pool sequential specification suitable renaming method calls enq enq deq deq deq deq note analogous change axioms sequential specification pool stack lead characterisation linearizability pools stacks axiomatic characterisation linearizability pools stacks would involve infinite number axioms due need prohibit infinitely many problematic shapes able state result haas theorem queue local linearizability queue concurrent history locally linearizable wrt queue sequential specification locally linearizable wrt pool sequential specification suitable renaming method calls enq enq deq deq deq deq
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simulating dynamics cell subsets throughout lifetime stephanie foan andrew jackson ian spendlove uwe aickelin academic unit clinical oncology intelligent modelling analysis research group university nottingham abstract widely accepted immune system undergoes changes correlating increased disease elderly cell subsets implicated aim work firstly implement validate simulation regulatory cell treg dynamics throughout lifetime based model baltcheva show initial simulation produces inversion precursor mature treys around years age though output differs significantly original laboratory dataset secondly report discusses development model incorporate new data study healthy blood donors addressing balance treys cells novel markers treg potential simulation add insight immune aging discussed introduction system dynamics modelling immunity simulation defined methods applications mimicking behaviour real system benefits simulation immunology include costeffectiveness well less resulting removal biological environment vitro experimentation useful investigating individual interactions far removed whole picture vivo experimentation useful whole picture unlikely answer specific questions using simulation flexibility available systematically generating hypotheses conducting experiments impossible practically yet informed robust data literature system dynamics simulations useful looking complex systems time characterised stocks entity flows stocks immune system examples stocks include precursor mature cell pools flows might represent transition cells precursor mature technique useful modelling relationships defined differential equations example differential equation describing cell dynamics change number precursor cells equated proliferation precursors minus death maturation rates ongoing work apply system dynamics simulation technique complement vitro studies treg throughout lifetime need balance treys throughout lifetime immune system maintains balance mounting adequate immune response protect infection restricting size immune response prevent damage self evidence suggest tendency proinflammatory environment contributing collateral damage autoimmune diseases work addresses hypothesis important contributors state imbalance cells amplifying immune responses tregs dampening immune responses although studies shown increase number tregs human peripheral blood also shown homeostasis maintained one study concluded upon oscillatory nature treg numbers life peaks adolescence year olds recent study cells age showed small decrease frequency cells memory population elderly donors relative young however balance tregs cells currently published literature evidence balance tregs altered agerelated diseases acute coronary syndrome thus intuitive balance examined cross sectional study healthy donors different ages laboratory experimentation begin using flow cytometry enumerate peripheral blood cells expressing signature transcription factors subsets helios tregs rorc cells method simulation treg dynamics ultimately wish build model dynamics tregs cells throughout life data currently collected preliminary work involved building system dynamics simulation anylogic university edition based mathematical model baltcheva model selected comprehensively incorporates functional dynamics tregs terms homeostasis acute immune response characterises changing precursor mature treg populations throughout human lifetime key assumptions include change function responsiveness throughout lifetime change influential factors dynamics dendritic cell number function also immune response considered includes expansion contraction phase one response occur given timepoint original model based numbers precursor mature populations peripheral blood samples donors aged although total numbers cells remained constant ratio precursor mature inverted early adulthood represents important dimension observed homeostasis treg numbers throughout lifespan especially considering thymic involution adolescence reducing number new cells entering system ordinary differential equations describe dynamics mentioned cells stochastic processes control frequency duration nature primary secondary immune responses different cell compartments work simple scenario chosen order test hypothesis model could implemented anylogic scenario assumes lack proliferation death precursor mature tregs proliferation death thymic output external input various treg subsets parameter values used correspond means distributions scenario given baltcheva work simulation shown immuneresponse day year gpt ydonep gqt sigm sigmap pin quiescentreactivation preoursoractivaton alpha ydoner ydoneq preoursordeath preoursorprolf ation actlyeprolf eration activedeath activetoquescent quiescentdeath figure main view events shown time days years flow variables given involve rate conversion multiplied number cells stocks named clonep corresponds precursors active matures quiescent matures stocks prefixed clone represent total tregs determined array homeostatic parameters apply total treg population whereas immune response parameters applied proportion given pin example flow specific precursors active mature stock given specific precursor stock multiplied maturation rate time point anylogic recalculates stock using flows defined additional class immuneresponse controls immune system functional status using irstatechart irstatechart nopesponse day year figure immuneresponse view beginning run immune system state every time steps immune system mounts primary immune response probability defaults secondary response primaryl secondaryl represent expansion phase primaryi secondaryi represent contraction phase continues new response instigated primaryl parameter applied primaryll set zero ters applied secondary parameters applied secondaryi set zero applied total precursors total quiescent matures age years figure generated using matlab output data anylogic complete run total precursor quiescent mature trey stocks total active tregs peak corresponds clones experiencing either primary secondary immune responses figure shows simulation output compared original dataset data collected stock complete replications maximum standard deviation three runs total precursors maximum standard deviation total quiescent matures age years age years figure output data compared baltcheva dataset proportion total tregs precursor stock proportion total tregs quiescent stock order quantify similar simulation output laboratory data datasets split age groups median calculated groups difference medians simulation output data laboratory data documented mann whitney test performed null hypothesis difference laboratory output data values given age years median difference mann whitney test proportion proportion proportion proportion precursors matures precursors matures table comparison median output laboratory data age group discussion concluding remarks implementation baltcheva model system dynamics simulation documented compared experimental evidence shown simulation mimics key feature inversion precursor memory cells early adulthood lack statistical similarity simulation output laboratory data indicates validation model necessary involve comparison scenarios proposed baltcheva work ultimately develop validate simulation novel dataset treg cells using sort approach cell numbers instead treg also collected arguably specific markers terms improving simulation alternatives continually reactivating single treg clone required simulation one immune stimulus time baltcheva discloses various assumptions including difference treg function may possible improve model considering functional well numerical changes treg subsets age abstract research question whether simplistic model immunosenescence lend useful insight biological problem argued process simulation alone might allow researchers address assumptions allow systematic generation hypotheses also hypotheses difficult test laboratory might testable simulation example might introduce intervention mimic ablative chemotherapy depleting stock single time point total values stock might compared simulation runs without intervention make hypotheses treg recovery simulating dynamics cells parallel treg may also allow make predictions maintenance balance throughout life would allow extreme parameter values tested may indicate maximum length time homeostasis maintained primary hypothesis age alters treg cells consequences health older age aim conduct cross sectional study obtain distribution particular changes anticipate strategy laboratory investigation system dynamics simulation exemplified baltcheva work useful address relationships cell subsets time model might also developed consider new questions response interventions length time immune system might able maintain treg cell homeostasis thanks irina baltcheva providing raw data references kelton sadowski swets simulation arena edition international edition kim levy lee modeling simulation immune system network volume methods enzymology academic press figueredo aickelin investigating immune system aging system dynamics modelling proceedings summer computer simulation conference boren gershwin autoimmunity phenotype autoimmunity reviews rosenkranz weyer tolosa gaenslen berg leyhe gasser stoltze higher frequency regulatory cells elderly increased suppressive activity neurodegeneration journal neuroimmunology gregg smith clark dunnion khan chakraverty nayak moss number human peripheral blood regulatory cells increases age clinical experimental immunology hwang kim kang aging human regulatory cells clinical immunology faria moraes freitas speziali soares figueiredoneves martins barbosa soares sathleravelar cardoso comin teixeira queiroz bauer martinsfilho variation rhythms lymphocyte subsets healthy aging neuroimmunomodulation lee lee kim kang lee shin kang kang ageassociated alteration naive memory cell response humans clinical immunology press corrected proof wang chen zhou wei wang wang role oxidized lipoprotein breaking peripheral balance patients acute coronary syndrome biochemical biophysical research communications baltcheva codarri pantaleo boudec lifelong dynamics human regulatory cells insights vivo data mathematical modeling journal theoretical biology thornton korty tran wohlfert murray belkaid shevach expression helios ikaros transcription factor family member differentiates peripherally induced regulatory cells journal immunology ziegler buckner regulation differentiation microbes infection
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uniform null controllability degenerating system approximating simplified cardiac model sep felipe wallison mostafa bendahmane abstract paper devoted analysis uniform null controllability family nonlinear systems approximating system models electrical activity heart uniform respect degenerating parameter null controllability approximating system means single control shown proof based combination carleman estimates weighted energy inequalities introduction let bounded connected open set whose boundary sufficiently regular let let two small nonempty subsets refer control domains use notation main objective paper study properties controllability observability family nonlinear systems degenerates nonlinear system models electrical activity cardiac tissue state model let represent intracellular extracellular electric potentials respectively difference called transmembrane potential anisotropic properties media modeled intracellular extracellular conductivity tensors widely accepted model see describing electrical activity cardiac tissue reads follows div div surface capacitance membrane nonlinear function transmembrane ionic current interesting case cubic polynomial stimulation currents applied respectively system known bidomain model completed dirichlet boundary conditions extracellular electric potentials mathematics subject classification key words phrases system monodomain model carleman estimates uniform null controllability observability supported erc project semi classical analysis partial differential equations project number grant basque government partially supported grant micinn spain erc advanced grant numeriwaves esf research networking programme optpde grant basque government bendahmane initial data transmembrane potential point realistic models describing electrical activities heart also include system ode computing ionic current function transmembrane potential series additional gating variables aim model ionic transfer across cell membrane see case constant bidomain model simplified following system div div system known monodomain model interesting model implementation point view since conserves essential features bidomain model excitability phenomena see main difference bidomain model monodomain model fact first model system two coupled parabolic equations second one system type therefore control point view one could expect two systems least priori different control properties work show properties controllability observability monodomain model seen limit process controllability properties family coupled parabolic systems indeed given approximate monodomain model following family parabolic systems div div div paper give positive answer following question question exists control drives solution zero time true control sequence converges function drives associated solution zero time question approximating equation another different physical properties used several times case parabolic equations degenerating hyperbolic ones see example hyperbolic equations degenerating parabolic ones see example however far know first time controllability parabolic systems degenerating systems studied also important mention families parabolic systems degenerate ones arise many areas biology chemistry astrophysics see uniform controllability degenerating system usual control theory dealing controllability nonlinear problem first consider linearized version div div div bounded function given first obstacle answering positively question drive solution zero time means control way sequence controls converges shown convergent sequence control linear system exists employ fixed point argument conclude true nonlinear system thus introduce adjoint system div div div using duality arguments easy prove task building convergent sequence controls equivalent prove following uniform observability inequality solutions dxdt constant remains bounded prove inequality consequence appropriate carleman inequality solution see section notice due fact control acting first equation carleman inequality need bound global integrals terms local integral uniformly respect two main difficulties appear first coupling first equation div second must show constant get carleman inequality blow first difficulty hard overcome indeed fixed inequality known true system see however main novelty fact obtain boundedness observability constant respect see carleman inequalities alone enough task need combine sharp carleman estimates respect weighted energy inequalities far controllability non degenerate coupled parabolic systems concerned situation fairly well understood instance controllability quite general linear coupled parabolic system studied null controllability result obtained means carleman inequalities using different strategy controllability nonlinear system two coupled parabolic equations analyzed authors prove null controllability linear system local null controllability nonlinear one another relevant work concerning controllability coupled systems authors analyze null controllability cascade system coupled parabolic equations authors able obtain null controllability bendahmane cascade system whenever good coupling structure also worth mentioning works local global controllability results phase field systems studied general discussion controllability coupled parabolic systems see survey paper concerning controllability results bidomain model since equations couplings given time derivatives electrical potentials seems difficult study controllability properties model best knowledge bidomain model problems null approximate controllability still open even two controls regarding null controllability monodomain model since solution parabolic equation enters source term elliptic one following controllability result holds theorem globally lipschitz every exists control solution satisfies moreover control satisfies following estimate constant exists every exists control solution satisfies moreover control satisfies following estimate constant theorem case follows theorem case follows theorem see also theorem paper organized follows section state main results section prove uniform carleman inequality adjoint system next show section uniform null controllability section deal uniform null controllability nonlinear system main results throughout paper assume matrices bounded symmetric positive semidefinite first main result uniform carleman estimate adjoint system theorem given exist positive constants solution satisfies dxdt dxdt dxdt uniform controllability degenerating system every div weight functions defined respectively proof theorem follows combination carleman inequalities heat equation precise dependence degenerating parameter energy inequality adjoint system prove theorem section remark direct consequence carleman inequality unique continuation property solutions given unique continuation property adjoint system implies approximate controllability time system control acting first equation second main result paper gives global null controllability linear system theorem exists control associated solution driven zero time say associated solution satisfies moreover control satisfies estimate constant theorem proof theorem standard however sake completeness prove theorem section third main result paper concerned uniform null controllability nonlinear parabolic system theorem given globally lipschitz every exists control solution satisfies moreover control satisfies estimate constant exists depending every exists control solution satisfies moreover control satisfies estimate constant proof theorem achieved fixed point arguments done section bendahmane remark paper restrict dimension bidomain model makes sense dimensions nevertheless mathematical point view systems make sense case corresponding cable equation taking initial data appropriate space results paper extended higher dimensions carleman inequality section prove theorem simplify notation neglect index since constant matters analysis assume constants normalized unity case adjoint system reads div div div notice regular enough taking div pair satisfies div div prove carleman inequality using system starting proof carleman inequality let first define several weight functions usefull sequel lemma let arbitrary nonempty open set exists function proof see using lemma introduce weight functions min max parameter constant max remark definition follows large enough moreover uniform controllability degenerating system proof theorem better comprehension divide proof several steps step first estimate parabolic system step obtain first carleman estimate adjoint system apply sharp carleman inequalities respect system get global estimate terms local integral another consider set apply sharp carleman inequality respectively get dxdt dxdt dxdt dxdt dxdt dxdt dxdt dxdt dxdt dxdt dxdt adding absorbing lower order terms side get dxdt dxdt dxdt dxdt dxdt dxdt dxdt dxdt dxdt dxdt remark trying drive solution zero means controls equations inequality would sufficient step estimate local integral bendahmane step estimate local integral side done using equation indeed consider function satisfying write div sequel estimate parcel expression first difficult see dxdt dxdt dxdt dxdt next integrating parts get meij meij meij meij show dxdt dxdt dxdt dxdt uniform controllability degenerating system finally dxdt dxdt putting together obtain dxdt dxdt dxdt dxdt dxdt dxdt dxdt dxdt using prove every system null controllable however sequence controls obtained way bounded therefore need step improve estimate goal next step step weighted energy inequality reason get bounded sequence controls step term side step prove weighted energy inequality equation used compensate term let introduce function new function satisfies div multiplying integrating get integrating last inequality form using young inequalities difficult see dxdt dxdt finally obtain dxdt dxdt bendahmane last estimate gives global estimate terms local integral constant bounded respect step last estimates conclusion order finish proof theorem combine inequality slightly different carleman inequality equation indeed following carleman inequality holds dxdt dxdt dxdt dxdt dxdt dxdt together solution notice changed weight proof exactly proof theorem taking appropriate change variable next since dxdt dxdt dxdt dxdt follows dxdt dxdt dxdt exactly density show remains true consider initial data therefore proof theorem finished null controllability linearized system section devoted proving null controllability linearized equation done showing observability inequality adjoint system solving minimization problem arguments used classical control theory linear pde hence give sketch proof proof theorem combining standard energy inequalities system carleman inequality given theorem show following observability inequality solutions dxdt positive constant next since div uniform controllability degenerating system therefore follows dxdt observability inequality density smooth solutions space solutions initial data see observability inequality satisfied solutions initial data order obtain null controllability linear system solve following minimization problem minimize solution adjoint problem initital data easy matter check strictly convex continuous order guarantee existence minimizer thing remaining prove coercivity using observability inequality adjoint system coercivity straightfoward therefore exists unique minimizer let denote corresponding solution associated minimizer taking control duality gives also gives solution associated control get control weak limit subsequence drives solution zero time following estimate control finishes proof theorem nonlinear system section prove theorem proof achieved fixed point arguments proof theorem case consider following linearization system div div div bendahmane follows theorem exists control function solution satisfies said idea use fixed point argument need following generalized version kakutani fixed point theorem due glicksberg theorem let convex compact subset locally convex topological vector space convex mapping closed graph closed fixed point order apply glicksberg theorem define mapping follows solution control satisfying ball easy see well defined convex compact subset let prove convex compact closed graph let since satisfies following inequality holds way take closed let fixed let prove fact definition together function control solution therefore extract subsequence denoted index weakly since bounded argue previous section show hence weakly strongly weakly uniform controllability degenerating system using converges see exists function weakly strongly weakly follows controlled solution associated control hence closed compact closed graph need prove strongly using two previous steps easy show therefore apply glicksberg theorem conclude fixed point proves theorem case nonlinearity globally lipschitz function proof theorem case consider linear system div div div arguing proof theorem show null controllability controls however controls sufficient apply fixed point arguments obtain null controllability nonlinear system reason modify little functional obtaining controls allow employ schauder fixed point theorem indeed consider problem minimize solution adjoint system initital data section show problem unique minimizer defining using fact together solution see solution parabolic equation homogeneous dirichlet boundary conditions null initial data side hence particular arguing lemma easily show independent bendahmane moreover solution satisfies associated taking limit get control weak limit subsequence associated solution satisfies next define map associates solution together corresponding control built note application well defined since regularity theory parabolic equations see instance let consider set defined clear convex closed subset still holds smallness assumption initial data easily show continuous finally since space compactly embedded compact therefore fixed point proof case theorem finished appendix technical results section prove two sharp carleman inequalities used proof theorem consider parabolic equation aij assume matrix aij form aij mij mij elliptic matrix exists mij degenerating carleman inequality first sharp carleman inequality prove following theorem exist every solution satisfies dxdt dxdt dxdt dxdt dxdt dxdt depending uniform controllability degenerating system proof consider change variable implies using fact solution write aij aij aij rest proof devoted analyze terms appearing first write iij iij inner product ith term expression jth term long straightforward calculation show following estimate holds dxdt dxdt dxdt dxdt take follows remark dxdt dxdt dxdt dxdt bendahmane remark since compact exists putting get dxdt dxdt dxdt dxdt deal local integral involving side end introduce function using ellipticity condition aij prove dxdt wdxdt dxdt wdxdt therefore young inequality dxdt dxdt dxdt thus inequality gives dxdt dxdt dxdt let use first two terms side order add integrals side done using expressions indeed uniform controllability degenerating system dxdt mij dxdt dxdt dxdt dxdt using term mij side elliptic regularity easy show dxdt dxdt estimate gives dxdt dxdt dxdt dxdt dxdt fact finish proof theorem slightly changed carleman inequality second sharp carleman inequality following theorem exist every solution satisfies dxdt dxdt dxdt dxdt dxdt dxdt depending proof starting point application carleman inequality given theorem equation indeed bendahmane dxdt dxdt dxdt dxdt dxdt dxdt next introduce function new function satisfies div applying carleman inequality given theorem time obtain large enough dxdt dxdt dxdt dxdt dxdt dxdt definition easy show dxdt dxdt dxdt using inequality becomes dxdt dxdt dxdt dxdt dxdt dxdt remark result follows acknowledgments paper partially established visit bendahmane basque center applied mathematics authors thank professor enrique zuazua various fruitful discussions work erich foster careful reading paper uniform controllability degenerating system references benabdallah teresa recents results controllability linear coupled parabolic problems survey mathematical control related fields benabdallah dupaix systems one control force math anal benabdallah dupaix kostine controllability trajectories models one control force siam control barbu local controllability phase field system nonlinear anal ser theory methods bendahmane finite volume scheme cardiac propagation media isotropic conductivities math comp bendahmane null controllability degenerated system cardiac electrophysiology math acad sci paris bendahmane karlsen analysis class degenerate systems bidomain model cardiac tissue netw heterog media bendahmane karlsen convergence finite volume scheme bidomain model cardiac tissue appl numer biler existence asymptotics solutions system nonlinear boundary conditions nonilinear analysis biler hebisch nadzieja debye system existence long time behavior solutions nonlinear analysis colli franzone degenerate evolution systems modeling cardiac electric field macroscopic level evolution equations semigroups functional analysis milano progr nonlinear differential equations appl basel coron guerrero singular optimal control linear parabolic hyperbolic example asymptot analisys zuazua null approximate controllability weakly blowing semilinear heat equations ann inst henri analise non fursikov imanuvilov controllability evolution equations lecture notes series research institute mathematics seoul national university seoul teresa controllability results cascade systems coupled parabolic pdes one control force port glass approach problem uniform controllability transport equation vanishing viscosity limit funct analysis glicksberg generalization kakutani fixed point theorem applications nash equilibrium points proc amer math controllability results nonlinear coupled parabolic systems one control force asymptot guerrero introduction controllability partial differential equations lectures advances studies institute santiago chile guerrero null controllability systems two parabolic equations one control force siam control guerrero lebeau singular optimal control equation comm partial diff equations henriquez simulating electrical behavior cardiac tissue using bidomain model crit rev biomed hodgkin huxley quantitative description membrane current application conduction excitation nerve imanuvilov yamamoto carleman estimate parabolic equation sobolev space negative order applications control nonlinear distributed parameter systems college station tex lectures notes pure appl math marcel dekker new york keener sneyd mathematical physiology interdisciplinary applied mathematics new york keller segel initiation slime mold aggregation viewed instability theor kunisch wagner optimal control bidomain system monodomain approximation rogersmcculloch model nonlinear analysis real world applications bendahmane ladyzhenskaja solonnikov ural ceva linear quasilinear equations parabolic type translated russian smith translations mathematical monographs american mathematical society providence lopez zhang zuazua null controllability heat equation singular limit exact controllability dissipative wave equations math pures lopez zuazua null controllability heat equation singular limit controllability damped wave equations acad sci paris luo rudy model ventricular cardiac action potential depolarization repolarization interaction circ noble modification equation applicable purkinje fibre action pacemaker potentials puel controllability partial differential equations lectures universidade federal rio janeiro rio janeiro brazil sanfelici convergence galerkin approximation degenerate evolution problem numer methods partial differential equations tung bidomain model describing ischemic myocardial potentials phd thesis mit systems microscopic cellular model cardiac electric field math methods zuazua exact boundary controllability semilinear wave equation nonlinear partial differential equations applications vol pitman res notes math longman harlow nice laboratoire jean umr cnrs parc valrose nice cedex france address fchaves bendahmane institut bordeaux victor segalen bordeaux ter place victoire bordeaux france address
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state art review applying computational intelligence machine learning techniques portfolio optimisation evan hurwitz school electrical information engineering university johannesburg johannesburg gauteng south africa hurwitze tshilidzi marwala faculty engineering university johannesburg johannesburg gauteng south africa tmarwala abstract computational techniques shown much promise field finance owing ability extract sense dauntingly complex systems paper reviews promising techniques traditional computational intelligence methods machine learning siblings particular view application optimising management portfolio financial instruments current state art assessed prospective work assessed recommended keywords reinforcement learning temporal difference neural network portfolio optimisation genetic algorithm genetic programming markowitz portfolio theory investment theory introduction paper serve literature review phd thesis carried university johannesburg paper examine state art machine learning computational intelligence techniques task portfolio optimisation first background precisely constitutes computational intelligence machine learning techniques explored addition successful implementations applications techniques facets make promising following problem portfolio optimisation defined including current objections standard model current applications research evaluated limitations examined technical limitations industry limitations finally research recommended based findings detailed paper background field portfolio optimisation field immense importance economy country indeed global economy general like majority stored money world ranging money funds banks insurance schemes stored portfolios sort another therefore subject risks financial markets pose management portfolios therefore critical area research ever since inception marketplace importance research increasing financial instruments become ever complex recent crisis illustrates understanding systems created need examination understanding navigate waters without bouncing reef reef fields computational intelligence machine learning concerned modelling systems particular modelling systems prove difficult impossible model conventional mathematical means case computational intelligence done observing optimising hueristics based upon gathered data conversely field machine learning relies learning observation oftentimes involving online learning learning process modelled action operation disciplines require relatively intensive computing power coming prime owing advances made past two decades computing power particular said computing power ready availability researchers computational intelligence number computational intelligence techniques examined section including strengths weaknesses typical applications broad purpose techniques solve either classification regression problems order model given system models used prediction purposes system control purposes genetic algorithms genetic algorithms attempt imitate evolutionary process genetic evolution order optimise given problem problem formalised defining fitness function determines given value set parameters optimised way maximising value fitness function returns optimal set parameters algorithm derives name manner optimises parameters done generating initial population possible solutions evaluating fitness function possible solution solutions used generate new population new population termed generation new individual solutions generated inheriting properties specific parents choosing parents done many ways inheriting features invariably individuals better fitness values favoured chosen process precise choices feature inheritance well type mutation random changing given value used explore optimisation parameters optimise process genetic algorithms found much success optimising large searching systems difficult quantify particular successful solving scheduling routing problems addition financial applications relatively recent offshoot genetic algorithms genetic programming method policy evaluated optimised means genetic algorithm particular shows much promise portfolio optimisation realm able combine multiple portfolio optimisation strategies one coherent optimised strategy optimisation pso optimisation another method solving types problems genetic algorithms based paths swarms rather workings genetics fitness function defined way optimisation instead follows following formulae velocity position particle solution lbi lbi learning constants random vectors typically represent global best local best respectively owing similarity gas terms function psos typically applied similar problems gas also applied portfolio optimisation problems various forms neural networks fundamental neural networks neurons neurons simply singleoutput mathematical function neuron number weights connecting inputs another layer added together possibly bias result passed neuron activation function activation function function represents way neural network thinks different activation functions lend different problem types ranging decisions linear nonlinear mathematical relationships layer neural network comprised finite number neurons network may consist number layers layer may contain number neurons neural network run neuron consecutive layer sums inputs multiplies input respective weight treats weighted sum input activation function output passed input next layer final output layer reached hence input data passed network neurons order arrive output figure illustrates interconnected network input neurons three hidden layer neurons two output neurons hidden layer output layer neurons possible activation functions type neural network referred perceptron configuration neural network widely used configuration problems figure sample connectionist network neural networks number characteristic properties mark ideal modelling specifically characteristics universal approximators generalisation pattern recognition neural networks applied multitude differing topologies wide variety problems many attempts made apply neural networks problem predicting prices although limited success one problem often overlooked applying neural networks implicit assumptions data structuring underlying problem example many attempts assumed simple approach making implicit assumption historical prices single equity sufficient predict future value machine learning machine learning subset computational intelligence field specifically focusing allowing system maximise reward signal thereby learn operate within environment complexity modelling system vary greatly allowing researcher high degree customisable freedom appropriate inherent specific choice reinforcement learning reinforcement learning specific form machine learning system react reward signal given told engineer react given specific result typical reinforcement learning problem following topology figure reinforcement learning topology reward signal indicates success failure attempted policy action within environment well degree thereof learning algorithm defines either policy parameters updated based reward signal environment model updated respect way policy learns exploit environment interacts fact bypassing act modelling system instead directly modelling control system working within system versus learning depending level generalisation built model reinforcement learning system may learn either learning may specific policy tested order acquire given reward signal alternatively learning may generalise situations beyond specific policy used advantage offpolicy learning one achieve far swifter learning since every single scenario need encountered order learn handle downside learning learning unintended consequences since applies scenarios beyond observed addition models learning algorithms require greater complexity order successfully learn tabular learning dynamic programming form reinforcement learning simplest form scenario evaluated iteratively updated using appropriate equation bellman equation order create perfect model environment policy constructed take advantage environment picking choice highest expected return temporal difference learning reinforcement learning involves training artificial intelligence system means reflecting manner learning living beings exhibit reinforcement learning well suited episodic tasks reference figure policy would represented appropriate function approximator would updated according chosen learning algorithm methodology allows learning also eliminates need expert knowledge recent work sutton svepesvari maei proven algorithm learning linear function approximator using temporal difference learning allows linear complexity memory computation current learning algorithms still struggle remain stable dealing nonlinear function approximators portfolio theory portfolio theory deals problem allocating funds within portfolio problem generalised include defined universe financial instruments goal portfolio optimisation problem achieve maximum return investment minimum amount risk optimising forms heart portfolio theory quite appropriate key question portfolio management industry namely managing clients funds way keep appropriate level risk maximising clients return investment modern portfolio theory mpt centered around markowitz theories forms central tenet modern investment theory price moves within market assumed independent stochastic events treated problems literature referred instead markov decision processes following typical brownian motion model portfolios examined statistical framework modelled two parameters namely mean return standard deviation latter interpreted inherent riskiness examined asset quantifying assets according two parameters assets strategies compared reward basis efficient strategies highest reward given amount risk alternatively fund managers fund attempt achieve specific return striving lowest possible risk given portfolio optimal efficient strategies lie along efficient horizon indicated figure figure efficient horizon strategies plotted according return risk unfortunately assumptions within markowitz framework seen invalid within investment arena particular assumptions independence internal historical moves external moves seen false similarly assumption brownian motion price moves hold scrutiny cases grossly underestimates extent possible price swings industry standard approach discard events outliers even though occur high frequency validly treated portfolio theory one accepts premise many investors investment houses returns expected return risky indication risk underlying instrument use variance measure risk fact inappropriate markowitz fact recommended use semivariance measure risk opinion computational difficulties formidable unsurprisingly certainly longer case semivariance measure downside variance dataset excluding values mean risk calculation different measure risk formulated one apply mpt designed treat losses risks gains quantifying risk task quantifying risk one extreme importance financial industry yet also extremely difficult field portfolio management risk specific amount money one expect lose market act favourably traditional measure risk mentioned previous sections statistically measurable standard deviation portfolio application risk measure however uses assumption normally distributed price moves shown historically inaccurate insufficient method measuring risk one suggested method recommended mandelbrot use fractal index measurement risk conceivably reasonable due possessing scaling property indicative real price moves static gaussian distribution note applies volatility risk encompass entirety risk involved investing portfolio optimisation portfolio optimisation typically done choosing specific strategy mid cap fund targeted return etc optimising ratio within limitations particular strategy strategy chosen constrain available instruments portfolio thereafter optimisation select available create portfolio example consider fund investing money behalf pension fund money required grow legally prohibited selling short mandated clients relatively low level risk grounds preserving capital important superior growth lastly assume fund large fund also liquidity limitation grounds fund manager decide asset needs sold buyers found amounts held small penny stocks unlikely traded quantities required hence would disqualified fund universe stocks manager would plot remaining instruments strategies graph risk versus reward figure compile portfolio based upon research order comply given mandate process need regular basis various instruments performance changes changing ratings profits losses change composition portfolio affecting overall ratio applied combinations owing promising properties machine learning computational intelligence techniques numerous attempts made apply techniques investment arena widely varying rates success predictive methods attempted numerous different formats using genetic algorithms neural networks others attempts also made utilise techniques option pricing alternative blackscholes modelling approaches current industry norm recently rosen saunders examined using analytical techniques pricing collaterolised debt obligations cdos although also fall old trap assuming normally distributed data set case creditworthiness constituents portfolio need utilising systems include dependant nature market initial assumptions markowitz denies brought comte looking specifically memory volatility models neural network applications within finance even attempted far afield marketing campaigns work done notably neural approach used assign discounts marketing campaign order maximise effectiveness notably researchers utilised machine learning optimise policy still kept neural network model static trained neely used genetic programming approach find optimum trading rules working technical analaysis basis found results inferior simple strategy allen karjalienen attempted similar approach using genetic programming tune technical trading rules case found approach actually lost money backtested notable approaches utilise historical data actual security inputs thus adhere assumption individual price moves independent events assumption quite vigorously challenged indicated earlier lin attempted similar approach using genetic algorithms parameters trading strategies filter rules research evaluated using simulated trading australian stock market method compared greedy algorithm providing optimal returns efficiency comparison proved far efficient greedy algorithm accomplishing seconds greedy algorithm achieved hours testing appears also limited selection stocks thus somewhat suspect cura turn investigated using particle swarm optimisation approach compared tabular search simulated annealing aproaches using markowitzmean model pso results promising testing done data recently freitas investigated using moving reference neural networks order predict price moves trading worked within markowitz framework aiming create efficiently diversified portfolio trading using predictor based manage beat model also fact beat market something traditional portfolio theory states impossible based challenged assumptions detailed earlier worth noting even highly promising result still used relatively simple structure limits ability network necessity adapting changing market condition explored nagayama yoshii explored forgetting appropriate data binary classifier order adapt classifier changing conditions forgetting data longer relevant explored active passive agent forgetting trials promising results noted however test data used limited results could well feature data application limitations field machine learning still remarkably difficult implement reinforcement learning using nonlinear function approximators neural networks limitation makes application nonstationary systems financial markets shown challenging endeavour sutton recently managed develop algorithm learning using temporal difference learning linear function approximator allows linear complexity memory computation almost quite enough handle nonlinear case neural networks limitation become apparent inadequacy traditional risk measurement benchmarks unrealistic assumptions taint underlying analysis assumptions implicit markowitz model fact longer convenient actually hindrance meaningful work much research also touch realities investment world concentrating arbitrary measures reward paying superficial attention issue risk reality dominates recommended research field much scope meaningful work current tumbling financial markets indication desperately needed potential computational intelligence techniques perform predictionbased portfolio optimisation readily apparent owing various properties various niches beckoning development algorithm perform temporal difference updates remains stable nonlinear function approximators would invaluable advance tasks predicting modelling classifying markets space open research far greater input space turn making fewer inherent assumptions market unlikely hold scrutiny particular astonishingly common assumption historical prices price moves given financial instrument index sufficient model behaviour ludicrous assumption information necessary contained within rigid interpretation efficient market hypothesis one bear scrutiny information may reflected price obfuscated overlying effects account dynamics system ignored meaningful useful model developed research risk measurement required fractal indexing particularly promising field currently modifying traditional risk measures make fewer assumptions market also promises fruitful area research lastly using computational intelligence machine learning approaches adaptive portfolio optimiser theoretically feasible combining multiple trading strategies create maintain optimised portfolio according modern portfolio theory measures utilising objective measures risk without unrealistic assumptions references beasley bull martin overview genetic algorithms part fundamentals university computing beasley bull martin overview genetic algorithms part fundamentals university computing func dets chen genetic algorithms genetic programming computational finance boston kluwer koza genetic programming programming computers means natural selection mit press cambridge kennedy eberhart particle swarm optimization proc ieee int conf neural networks vol margarita carlos particle swarm optimizers survery state art international journal computational intelligence research vol issue tunchan cura particle swarm optimization approach portfolio optimization nonlinear analysis real world applications elsevier april wesley hines matlab supplement fuzzy neural approaches engineering john wiley sons new york miller sutton werbos neural networks control mit press cambridge massecheussettes guoqiang zhang eddy patuwo michael forecasting artificial neural networks state art elsevier international journal forecasting vol fernando christian profitability technical trading rules based artificial neural networks evidence madrid stock market elsevier leke betechuoh marwala optimization stock market input using bayesian neural networks ieee international conference service operations logistics informatics beijing peoples republic china august sutton learning predict methods temporal differences mach learning sutton barto reinforcement learning introduction mit press sutton szepesvari maei convergent algorithm learning linear function approximation advances neural information processing systems vancouver december mit press hurwitz marwala optimising reinforcement learning neural networks eurosis tessauro temporal difference learning communications acm march vol markowitz portfolio selection efficient diversification investments blackwell cambridge second edition edition malkiel efficient market hypothesis critics journal economic perspectives fama efficient capital markets journal finance benoit mandlebrot mis behaviour markets fractal view risk ruin reward investment analysis portfolio management eighth edition frank reilly cfa keith brown konno yamazaki deviation portfolio optimization model application tokyo stock market management science balzer measuring investment risk review journal investing fritelli rosazza gianin putting order risk measures journal banking finance elton gruber brown goetzmann modern portfolio theory investment analysis edition johnwiley sons bauer genetic algorithms investment strategies wiley new york kimoto asakawa yoda takeoka stock market prediction system modular neural networks international joint conference neural networks vol june pires marwala american option pricing using bayesian perceptrons bayesian support vector machines ieee international conference computational cybernetics mauritius april dan rosen david saunders analytical methods hedging systematic credit risk linear factor portfolios journal economic dynamics control elsevier fabienne comte long memoryin stochastic volatility models mathematical finance gabriel emilio emili alberto palomares casariego assigning discounts marketing campaign using reinforcement learning neural nets expert systems applications elsevier neely christopher using genetic algorithms find technical trading rules comment risk adjustment october federal reserve bank louis working paper available ssrn http doi franklin allen risto karjalainen using genetic algorithms find technical trading rules journal financial economics fabio freitas alberto souza ailson almeida portfolio optimisation using neural networks neurocomputing accepted august elsevier hirotaka nagayama kengo yoshii active forgetting machine learning application financial problems ieee lin longbing cao jiaqi wang chengqi zhang applications genetic algorithms stock market data mining optimisation faculty information technology university technology sydney nsw australia capital market crc sydney nsw australia tunchan cura particle swarm optimization approach portfolio optimization nonlinear analysis real world applications elsevier naomi miller andrej ruszcynski probability measures portfolio optimization coherent measures risk european journal operational research elsevier
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robust large margin deep neural networks jure student member ieee raja giryes member ieee may guillermo sapiro fellow ieee miguel rodrigues senior member ieee abstract generalization error deep neural networks via classification margin studied work approach based jacobian matrix deep neural network applied networks arbitrary pooling layers networks different architectures feed forward networks residual networks analysis leads conclusion bounded spectral norm network jacobian matrix neighbourhood training samples crucial deep neural network arbitrary depth width generalize well significant improvement current bounds literature imply generalization error grows either width depth network moreover shows recently proposed batch normalization weight normalization enjoy good generalization properties leads novel network regularizer based network jacobian matrix analysis supported experimental results mnist lared imagenet datasets index terms deep learning deep neural networks generalization error robustness ntroduction recent years deep neural networks dnns achieved results image recognition speech recognition many applications dnns constructed series rodrigues department electronic electrical engineering univeristy college london london giryes school electrical engineering faculty engineering university tel aviv israel raja sapiro department electrical computer engineering duke university usa work jure miguel rodrigues supported part epsrc grant work raja giryes supported part gif foundation scientific research development work guillermo sapiro supported part nsf onr aro nga may draft signal transformations applied sequentially parameters layer estimated data typically layer applies input linear affine transformation followed sigmoid function hyperbolic tangent function rectified linear unit relu many dnns also include pooling layers act operators may also provide invariance various input transformations translation may linear average pooling various attempts provide theoretical foundation representation power optimization generalization dnns example works showed neural networks single hidden layer shallow networks approximate measurable borel function hand shown deep network divide space exponential number sets achieved shallow networks use number parameters similarly authors conclude functions implemented dnns exponentially expressive functions implemented shallow networks work shows given number parameters given depth always exists dnn approximated shallower network number parameters shallow network exponential number layers deep network scattering transform convolutional dnn like transform based wavelet transform pointwise provides insights translation invariance stability deformations convolutional dnns dnns random weights studied shown networks perform distance preserving embedding data manifolds authors model loss function dnn model show large networks local optima loss function close global optima optimization aspects dnns studied perspective tensor factorization shown network large possible find global minima initialization gradient descent algorithm role dnns improving convergence speed various iterative algorithms studied optimization dynamics deep linear network studied shown learning speed deep networks may independent depth reparametrization dnn efficient learning studied depth modified version stochastic gradient descent optimization dnns invariant weight rescaling different layers proposed shown optimization may lead smaller generalization error difference empirical error expected error one achieved classical stochastic gradient descent authors propose batch normalization technique normalizes output layer leads faster training also smaller similar technique based normalization may draft weight matrix rows proposed shown empirically reparametrization leads faster training smaller learning dnn bounding spectral norm weight matrices proposed methods dnn regularization include weight decay dropout constraining jacobian matrix encoder regularization enforcing dnn partial isometry important theoretical aspect dnns effect architecture depth width various measures rademacher gaussian complexities algorithmic robustness used bound context dnns example dnn equal number parameters network implies sample complexity linear number parameters network also bounded independently number parameters provided norms weight matrices network linear components constrained appropriately constraints usually enforced training networks weight decay regularization simply weights network example work studies dnn relus constraints norms weight matrices however provides bounds scale exponentially network depth similar behaviour also depicted authors show dnns robust provided weights layer bounded bounds exponential weights norm greater bounds suggest dnn bounded number training samples grows dnn depth size however practice increasing network depth size often leads lower moreover recent work shows layer dnn relus may fit function samples dimensions provided parameters often case practice show nature depends nature data architecture network network able fit structured data random data first low latter large authors conclude data agnostic measures rademacher complexity adequate explain good generalization properties modern dnn work complements previous works dnns bounding terms dnn classification margin independent dnn depth size takes account structure data considering covering number therefore avoids issues presented extension results invariant dnn provided may draft contributions work focus dnn classifier general establish new bounds dnn classifiers via classification margin distance training sample decision boundary induced dnn classifier sample space work capitalizes algorithmic robustness framework cast insight onto generalization properties dnns particular use framework understand operation dnns involves various innovations include derive bounds dnns lower bounding classification margin lower bound classification margin expressed function network jacobian matrix approach includes large class dnns example consider dnns softmax layer network output dnns various rectified linear unit relu sigmoid hyperbolic tangent dnns pooling average pooling networks shortcut connections residual networks analysis shows dnn bounded independently depth width provided spectral norm jacobian matrix neighbourhood training samples bounded argue result gives justification low dnns practice moreover also provides explanation training recently proposed weight normalization batch normalization lead small networks weight matrices fixed regularization apply analysis also leads novel jacobian regularizer applied weight normalized batch normalized networks provide series examples mnist lared imagenet datasets validate analysis demonstrate effectiveness jacobian regularizer contributions differ existing works many ways particular dnns studied via algorithmic robustness framework bounds based weight matrices studied loss relevant classification analysis much broader aims bounding loss directly also considers dnns pooling moreover bounds function network jacobian matrix tighter bounds based norms weight matrices work shows learning transformations locally isometric robust leads small though apply proposed technique dnns show dnn architecture affects work authors observed contractive dnns relus trained hinge loss lead may draft large classification margin however provide bounds moreover results limited dnns relus whereas analysis holds arbitrary dnns pooling dnns softmax layer work related sense proposes regularize constraining frobenious norm encoder jacobian matrix however work empirical less concerned classification margin bounds use jacobian matrix regularize encoder whereas use jacobian matrix regularize entire dnn finally dnn analysis based network jacobian matrix also related concept sensitivity analysis applied feature selection svm neural networks construction radial basis function networks since spectral norm jacobian matrix quantifies sensitivity dnn output respect input perturbation paper organization section introduces problem generalization error including elements algorithmic robustness framework introduces dnn classifiers properties dnns described section iii bounds classification margin dnns implication dnns discussed section generalizations results discussed section section presents experimental results paper concluded section vii proofs deferred appendix notation use following notation sequel matrices column vectors scalars sets denoted boldface letters boldface letters italic letters calligraphic letters respectively convex hull denoted conv denotes identity matrix denotes zero matrix denotes vector ones subscripts omitted dimensions clear context denotes basis vector standard basis denotes euclidean norm denotes spectral norm kxkf denotes frobenious norm element vector denoted element row column denoted covering number balls radius denoted roblem tatement start describing framework statistical learning dwell bounds based robustness framework manor finally present dnn architectures studied paper may draft classification problem consider classification problem observe vector corresponding class label set called input space called label space denotes number classes samples space denoted element denoted assume samples drawn according probability distribution defined training set samples drawn denoted goal learning leverage training set find classifier provides label estimate given input vector work classifier dnn described detail section quality classifier output measured loss function measures discrepancy true label estimated label provided classifier take loss indicator function losses hinge loss categorical cross entropy loss possible empirical loss classifier associated training set expected loss classifier defined emp exp respectively important question occupies throughout work well lemp predicts lexp measure use quantifying prediction quality difference lexp lemp called generalization error exp emp algorithmic robustness framework order provide bounds dnn classifiers leverage robustness framework described next algorithmic robustness framework provides bounds based robustness learning algorithm learns classifier leveraging training set may draft definition let training set sample space learning algorithm robust sample space partitioned disjoint sets denoted note element training set arbitrary element sample space therefore robust learning algorithm chooses classifier losses partition close following theorem provides bound robust theorem theorem learning algorithm probability least log log first term bound constant depends training set second term behaves vanishes size training set approaches infinity case loss corresponds number partitions samples space bound number partitions found covering number samples space covering number smallest number metric balls radius needed cover denoted denotes space cartesian product continuous input space discrete label space write corresponds number classes choice metric determines efficiently one may cover common choice euclidean metric also use paper covering number many structured data models bounded terms intrinsic properties example additional variants theorem provided note always obtain set disjoint partitions set metric balls used construct covering may draft gaussian mixture model gmm gaussians covariance matrices rank leads covering number signals dictionary atoms covering number regular manifold constant captures intrinsic properties covering number large margin classifier example robust learning algorithm large margin classifiers consider work classification margin defined follows definition classification margin classification margin training sample measured metric defined sup classification margin training sample radius largest metric ball induced centered contained decision region associated class label robustness large margin classifiers given following theorem theorem adapted example exists classifier theorems imply classifier margin upper bounded neglecting log term log note case large margin classifier constant equal approaches zero rate number training samples grows also increases number classes finally depends complexity input space classification margin via covering number example take regular manifold upper bound behaves may draft fig dnn transforms input vector feature vector series transforms corollary assume subset regular manifold assume also classifier achieves classification margin take loss probability least log log proof proof follows directly theorems note role classifier captured via achieved classification margin always ensure classification margin bound depends dimension manifold manifold constant relate bound context dnns bounds literature section deep neural network classifier dnn classifier defined arg max element dimensional output dnn rny assume composed layers represents layer parameters output layer denoted rml input layer corresponds output last layer denoted dnn visualized fig next define various layers used modern dnns linear softmax layers start describing last layer dnn maps output previous layer rny corresponds number layer linear assuming classifiers may draft table wise non linearities name function relu max sigmoid hyperbolic tangent tanh derivative derivative bound supx rny weight matrix associated last layer rny bias vector associated last layer note according row interpreted normal hyperplane separates class others last layer linear usual choice learning objective hinge loss common choice last layer softmax layer softmax function note exponential applied elements range often interpreted probabilites associated corresponding class labels decision boundary class class corresponds hyperplane softmax layer usually coupled categorical training objective remainder work take softmax layer last layer dnn note results still apply linear layer used layers layer defined represents applied element rml represents linear transformation layer input rml weight matrix rml bias vector typical relu sigmoid hyperbolic tangent listed table choice usually layers network note layer includes convolutional layers used convolutional neural networks case weight matrix may draft class class class class input space output space fig decision boundaries input space output space plot shows samples class decision regions produced network projected input space plot shows samples transformed network corresponding decision boundary network output pooling layers pooling layer reduces dimension intermediate representation defined pooling matrix usual choices pooling average pooling denote pli row assume pooling regions case pli epi first element pooling region case pli arg maxj case average pooling pli iii eometrical roperties eep eural etworks classification margin introduced section function decision boundary input space visualized fig however training algorithm usually optimizes decision boundary network output fig necessarily imply large classification margin section introduce general approach allows bound expansion distances network input output section use establish bounds classification margin bounds independent network depth width start defining jacobian matrix dnn may draft note properties chain rule computed product jms individual network layers evaluated appropriate values layer inputs use establish relation pair vectors input space output space theorem dnn average jacobian line segment proof proof appears appendix direct consequence theorem bound distance expansion output network corollary dnn kjx sup proof proof appears appendix note established corresponds linear operator maps vector vector implies maximum distance expansion network bounded maximum spectral norm network moreover corresponds product jms layers shown possible calculate jms layers defined section jacobian matrix linear softmax layers linear layer defined equal weight matrix similarly case softmax layer defined diag note diag corresponds softmax function may draft jacobian matrix layers layer derived way softmax layer first define diagonal dzl derivatives associated various provided table layer expressed dzl dzl jacobian matrix pooling layers pooling operator defined linear linear operator corresponding therefore also linear linear equal following lemma collects bounds spectral norm jms layers defined section lemma following statements hold spectral norm jms linear layer softmax layer layer relu sigmoid hyperbolic tangent upper bounded dzl kwl kwl assume pooling regions average pooling operators spectral norm jms upper bounded dzl proof proof appears appendix lemma shows spectral norms layers bounded terms weight matrices consequence spectral norm bounded product spectral norms weight matrices leverage facts provide bounds next section note case relu derivative max defined need use subderivatives subgradients define avoid technical complication simply take derivative max note change results way subset derivatives defined zero measure may draft also briefly explore relationship jacobian matrix fisher information matrix simplify derivations assume model parameter deterministic fisher information measures much information parameter contained random variable represents dnn particular case fisher information given log log log setup parameter interpreted magnitude input perturbation clear small norm jacobian matrix leads small fisher information indicates distribution informative parameters ensuring norm jacobian small naturally endow network robustness perturbations input eneralization error eep eural etwork lassifier section provide classification margin bounds dnn classifiers allow bound follow common practice assume networks trained loss promotes separation different classes network output categorical cross entropy loss hinge loss words training aims maximizing score training sample score defined follows definition score take score training sample min rny kronecker delta vector recall definition classifier note decision boundary class class feature space given hyperplane positive score indicates network output classes separated margin corresponds score however large score necessarily imply large classification margin theorem provides classification margin bounds expressed function score properties network may draft theorem assume dnn classifier defined classifies training sample score classification margin bounded supx set weight matrices proof proof appears appendix given bounds classification margin specialize corollary dnn classifiers corollary assume subset regular manifold assume also dnn classifier achieves lower bound classification margin take loss probability least log log proof proof follows theorems corollary suggests bounded log provided classification margin bounds satisfy leverage classification margin bounds theorem construct constraint sets ensures bounded may draft using obtain sup sup kwl kwl note want maximize score also need constrain network jacobian matrix following weight matrices following stands line common training rationale dnn aim maximizing score training samples ensure correct classification training set also regularization constrains network parameters combination eventually leads lower constraint sets impose different regularization techniques term supx considers supremum spectral norm jacobian matrix evaluated points within classification margin training sample see definition compute margin still obtain rationale regularization long spectral norm jacobian matrix bounded neighbourhood training sample given guarantees constraint jacobian matrix restrictive requires bounded spectral norm samples convex hull input space constraints similar form kwl kwl respectively note weight decay aims bounding frobenious norms weight matrices might used satisfy constrains however note also bound based spectral norm tighter one based frobenious norm example take orthonormal rows dimension constraint based spectral norm form constraint based frobenious norm former case may draft constraint score independent network width depth latter constraint output score exponential network depth polynomial network width difference frobenious norm take account correlation angles rows weight matrix spectral norm therefore bound based frobenious norm corresponds worst case rows aligned case kwl kwl hand rows orthonormal kwl kwl remark put results perspective compare bounds bounds based rademacher complexity hold dnns relus work shows kwi energy training samples bounded although bounds directly comparable since bounds based robustness framework rely underlying assumption data covering number still remarkable difference behaviour suggests grows exponentially network depth even product frobenious norms weight matrices fixed due term bound constraint sets hand imply increase number layers provided norms weight matrices bounded moreover take dnn weight matrices orthonormal rows behaves assuming therefore relies complexity underlying data manifold network depth provides possible answer open question depth independent capacity control possible dnns relus remark important value bounds provide additional explanation success dnn training techniques batch normalization eight normalization weight normalized dnns weight matrices normalized rows diag diag denotes diagonal part matrix main motivation method faster training authors also show empirically networks achieve good generalization note weight matrices kwl therefore bounds based may draft frobenious norm explain good generalization networks adding layers making larger lead larger bound however bound constraint sets show small frobenious norm weight matrices crucial small supporting experiment presented section also note batch normalization also leads weight matrices dnns theorem assume layers dnn relus batch normalized zli denotes relu diag wzi zti normalization matrix weight matrices row normalized exception weight matrix last layer form proof proof appears appendix jacobian regularizer constraint set suggests regularize dnn bounding norm network inputs close therefore propose penalize norm network evaluated training sample implementation regularizer requires computation gradients subgradients case computation subgradient spectral norm requires calculation svd decomposition makes proposed regularizer inefficient circumvent propose surrogate regularizer based frobenious norm jacobian matrix simplify derivation omit bias vectors therefore also centering applied batch normalization affect generality result also follow omit batch normalization scaling included weight matrix layer following batch normalization also omit regularization term assume matrices invertible may draft note frobenious norm spectral norm related follows justifies using surrogate regularizer refer jacobian regularizer computation gradients efficient implementation note row corresponds gradient respect input evaluated denoted write regularizer minimized gradient descent algorithm need compute gradient respect dnn parameters first express gkl gkl gradient respect evaluated input layer output evaluated input gradient respcet given gkl computation gradient regularizer layer requires computation gradients gkl computation jacobian matrices computation gradient typical loss used training dnn usually involves computation gradients computational complexity similar computational complexity gkl therefore computation gradients required implementation jacobian regularizer expensive avoid excessive computational complexity propose simplified version regularizer name jacobian regularizer jacobian regularizer layer defined rfl random index compared made two simplifications first assumed input layer fixed way need compute output layer input second choosing one index per training sample compute one additional gradient per training sample significantly reduces computational complexity gradient simply demonstrate effectiveness regularizers section may draft iscussion preceding sections analysed standard dnns classification margin measured euclidean norm briefly discuss results extend dnn architectures different margin metrics beyond feed forward dnn various dnn architectures residual networks resnets recurrent neural networks rnns long memory lstm networks used frequently practice turns analysis based network also easily extended dnn architectures fact proposed framework encompasses dnn architectures compute resnet resnets introduce shortcut connection layers particular let denote concatenation several layers see block residual network given denote block dzl resnet form jsm jsm denotes layer particular right element product expanded sum jms possible resnet particular elements sum consiting one one element sum consisting observation consistent claims states resnets resemble ensemble relatively shallow networks may draft beyond euclidean metric moreover also consider geodesic distance manifold measure margin instead euclidean distance geodesic distance appropriate euclidean distance since natural metric manifold moreover covering number manifold may smaller use covering based geodesic metric balls lead tighter bounds outline approach assume riemannian manifold take continuous piecewise continuously differentiable curve set curves denoted geodesic distance defined inf similarly section iii show dnn central bounding distance expansion signals dnn input signals dnn output theorem take riemmanian manifold take continuous piecewise continuously differentiable curve connecting sup proof proof appears appendix note established relationship euclidean distance two points output space corresponding geodesic distance input space important implies promoting large euclidean distance points lead large geodesic distance points input space moreover ratio upper bounded maximum value spectral norm network evaluated line result analogous results theorem corollary also implies regularizing network proposed section beneficial also case classification margin measured euclidean metric finally note practice training data may balanced provided bounds still valid cases however classification error may best measure performance cases dominated classification error class highest prior probability therefore alternative performance measures need considered leave detailed study training dnn unbalanced training sets possible future work may draft accuracy accuracy number layers mnist number layers fig classification accuracy dnns trained jacobian regularization solid lines weight decay dashed lines different numbers training samples used red blue black xperiments validate theory series experiments mnist lared imagenet datasets jacobian regularizer applied various dnn architectures dnn fully connected layers convolutional dnn resnet use relus considered dnns currently popular fully connected dnn section compare performance fully connected dnns regularized jacobian regularization weight decay analyse behaviour fully connected dnns various depth width comparison jacobian regularization weight decay first compare standard dnn fully connected layers trained weight decay jacobian regularization mnist datasets different number training samples used consider dnns fully connected layers layers except last one dimension equal input signal dimension case mnist case last layer always softmax layer objective cee loss networks trained using stochastic gradient descent sgd momentum set batch size set learning rate set reduced factor every epochs networks trained epochs total weight decay jacobian regularization factors chosen separate validation set experiments repeated regularization parameters random draws training sets weight matrix initializations classification accuracies averaged different experimental runs shown fig observe proposed jacobian regularization always outperforms weight decay validates theoretical results section predict may draft jacobian matrix crucial control bound interestingly case mnist layer dnn trained training samples jacobian regularization solid blue line fig performs par dnn trained training samples weight decay dashed black line fig means jacobian regularization lead performance significantly less training samples analysis weight normalized deep neural networks next explore weight normalized dnns described section use mnist dataset train dnns different number fully connected layers different sizes weight matrices last layer always softmax layer objective cce loss networks trained using stochastic gradient descent sgd momentum set batch size set learning rate set reduced factor every epochs networks trained epochs total experiments repeated times different random draws training set different random weight initializations employ additional regularization goal explore effects weight normalization dnn behaviour always use training samples classification accuracies shown fig smallest classification score obtained training set shown fig observed configurations training accuracies exception case training accuracy therefore testing set classification accuracies increasing network depth weight matrix size directly imply smaller deeper wider dnns note also score increases network depth width obvious layer dnns whereas layer dnns score close network widths since dnns weight normalized frobenious norms weight matrices equal square root weight matrix dimension product frobenious norms weight matrices grows network depth weight matrix size increase score network depth network width offset product frobenious norms clearly bound based margin bound bound leverage frobenious norms weight matrices predict increase network depth weight matrix size scenario therefore experiment indicates bounds pessimistic also inspected spectral norms weight matrices trained networks cases spectral norms greater one argue bound based margin bound predicts increase network depth product spectral norms grows network depth similar way previous paragraph note however spectral may draft norms weight matrices much smaller frobenious norms weight matrices finally look possible explanation success weight normalization bounds based margin bounds function largest value spectral norm network evaluated training set shown fig largest value spectral norm network evaluated testing set shown fig observe interesting phenomena maximum value spectral norm training set decreases network depth width hand maximum value spectral norm testing set increases network depth slightly network width perspective constraint sets note case latter take account worst case spectral norm inputs conv maximum value spectral norm testing set indicates value increases network depth implies bound based still loose hand bound implies consider neighbourhood training samples approximation take spectral norms jms evaluated training set shown fig values decrease network depth width argue provides reasonable explanation good generalization deeper wider weight normalized dnns convolutional dnn section compare performance convolutional dnns regularized jacobian regularizer weight decay also show jacobian regularization applied batch normalized dnns use standard mnist dataset lared dateset briefly described lared dataset contains depth images distinct hand gestures performed subjects approximately images gesture per subject extracted depth images hands using masks provided resized images images first subjects used create training testing sets addition also constructed testing set composed images last subjects dataset order test generalization across different subjects goal classification gestures based depth image comparison jacobian regularization weight decay use layer convolutional dnn following architecture followed softmax layer denotes convolutional layer filters size denotes pooling regions size training procedure follows one described previous paragraphs results reported table may draft max train set mini accuracy layer width classification accuracy layer width layer width smallest training set largest training set max test set layer width largest test set fig weight normalized dnn layers different sizes weight matrices layer width plot shows classification accuracy plot shows smallest score training samples plot shows largest spectral norm network evaluated training set plot shows largest spectral norm network evaluated testing set table lassification acc convolutional dnn mnist lared subject minst red lared different subject train samples weight jac reg train samples weight jac reg train samples weight jac reg observe training jacobian regularization outperforms weight decay cases obvious smaller training set sizes example mnist dataset dnn trained using training samples regularized weight decay achieves classification accuracy dnn trained jacobian regularization achieves classification accuracy similarly lared dataset jacobian regularization outperforms weight decay difference obvious smallest number training samples note also generalization may draft table iii lassification acc convolutional dnn train samples batch norm batch norm jac reg network subjects outside training set good using training samples classification accuracy testing set containing subjects higher whereas classification accuracy testing set containing different subjects nevertheless jacobian regularization outperforms weight decay also testing set small margin batch normalization jacobian regularization show jacobian regularization also applied batch normalized dnn note shown section batch normalization effect normalizing rows weight matrices dataset use proposed convolutional layers average pooling layer softmax layer convolutional layers batch normalized softmax layer weight normalized networks trained using stochastic gradient descent sgd momentum set batch size set learning rate set reduced factor every epochs networks trained epochs total classification accuracy results presented table iii different sizes training sets observe jacobian regularization also leads smaller case residual networks demonstrate jacobian regularizer also effective applied resnets use imagenet datasets use jacobian regularization experiments section wide resnet architecture proposed follows proposes wider shallower networks leads better performance deeper thinner networks used particular use resnet layers width follow data normalization process also follow training procedure except learning rate use learning rate sequence first number parenthesis corresponds learning rate second may draft table lassification acc train samples resnet resnet jac reg number corresponds number epochs train resnet small training sets training samples without augmentation full training set data augmentation regularization factor set smaller training sets full augmented training set respectively results presented table cases resnet jacobian regularization outperforms standard resnet effect regularization strongest smaller number training samples expected imagenet use layer resnet identity connection training procedure follows learning rate sequence jacobian regularization factor set images dataset resized run experiment without data augmentation data augmentation following includes random cropping images size original image color augmentation classification accuracies training shown fig final results reported table first focus training without data augmentation resnet trained using jacobian regularization much smaller compared baseline resnet demonstrates jacobian regularization decreases theory predicts note smaller jacobian regularized resnet partially transfers higher classification accuracy testing set however practice dnns often trained data augmentation case baseline resnet much lower close resnet jacobian regularization clear data augmentation reduces need strong regularization nevertheless note resnet trained jacobian regularization achieves slightly higher testing set accuracy compared baseline resnet may draft table train test baseline baseline jac reg baseline baseline aug jac reg epoch accuracy setup accuracy accuracy lassification acc magenet epoch data augmentation accuracy data augmentation epoch data augmentation train baseline test baseline train jac reg test jac reg epoch data augmentation fig training set dashed testing set solid classification accuracies training blue curves correspond resnet jacobian regularization red curves correspond baseline resnet classification accuracies reported training without data augmentation training data augmentation computational time finally measure use jacobian regularization affects training time dnns implemented dnns theano includes automatic differentiation computation graph optimization experiments run titan gpu average computational time per batch convolutional dnn mnist dataset section resnet imagenet dataset section reported table note case mnist regularizer used case imagenet regularizer used results also representative datasets network architectures may draft table average computation time batch experiment reg jac reg increase factor mnist imagenet sec observe using jacobian regularizer introduces additional computational time may critical number training samples small training computational time critical hand jacobian regularizer much smaller cost shown experiments regularizer still effective leads increase computation time imagenet dataset due efficiency jacobian regularizer might appropriate large scale experiments computational time important vii onclusion paper studies dnns based classification margin particular bounds express generalization error function classification margin bounded terms achieved separation training samples network output network one hallmarks bounds relates fact characterization behaviour generalization error tighter associated bounds literature bounds predict generalization error deep neural networks independent depth size whereas bounds say generalization error exponential network width size bounds also suggest new regularization strategies regularization network jacobian matrix applied top modern dnn training strategies weight normalization batch normalization standard weight decay applied regularization strategies especially effective limited training data regime comparison approaches moderate increase computational complexity ppendix proof theorem first note line given define function observe may generalized draft fundamental theorem calculus lebesgue differentiation theorem write concludes proof proof corollary first note kjx kjx integral addition notice may always apply following upper bound kjx sup since conv get proof lemma proofs leverage fact two matrices appropriate dimensions holds also leverage bound kakf start proof statement layer note product diagonal matrix weight matrix note considered diagonal elements bounded see derivatives table implies spectral norm matrix bounded therefore spectral norm upper bounded kwl proof linear layer trivial case softmax layer show spectral norm softmax function diag bounded use gershgorin disc theorem states eigenvalues diag bounded max noticing leads upper bound maxi since trivial show upper bounded proof statement straightforward pooling regions straightforward verify rows defined pooling operators orthonormal therefore spectral norm equal may draft proof theorem throughout proof use notation vij start proving inequality assume classification margin training sample given take arg min vyti take point lies decision boundary vyti vyti jxi kjxi kxi note choice kxi similarly kjxi supx therefore write sup leads next prove recall definition classification margin sup kxi sup kxi leverage definition observe vyti min vyti min vyti vyti note min vyti vyti min vyti min vyti min vyti therefore min vyti leads bound classification margin sup kxi min vyti may draft note min vyti max vyti max vyti moreover max vyti kxi sup leveraged fact kvij inequality corollary may write sup kxi sup kxi attains supremum obtain easily get proves bounds follow bounds provided lemma fact spectral norm matrix product upper bounded product spectral norms concludes proof proof theorem denote wln row normalized matrix obtained way noting relu diagonal matrices commute straight forward verify zli note consider zli part weight matrix therefore conclude layer row normalized weight matrix batch normalization applied layers weight matrices row normalized exception weight matrix last layer form may draft proof theorem begin noting first equality follows generalized fundamental theorem calculus following idea presented proof theorem second equality follows chain rule differentiation finally note norm integral always smaller equal integral norm obtain sup sup noted eferences krizhevsky sutskever hinton imagenet classification deep convolutional neural networks advances neural information processing systems nips hinton deng dahl mohamed jaitly senior vanhoucke nguyen sainath kingsbury deep neural networks acoustic modeling speech recognition shared views four research groups ieee signal processing magazine vol lecun bengio hinton deep learning nature vol may zhang ren sun deep residual learning image recognition ieee conference computer vision pattern recognition cvpr nair hinton rectified linear units improve restricted boltzmann machines proceedings international conference machine learning icml bruna szlam lecun learning stable group invariant representations convolutional networks international conference learning representations iclr boureau ponce lecun theoretical analysis feature pooling visual recognition proceedings international conference machine learning icml cybenko approximation superpositions sigmoidal function mathematics control signals systems vol hornik approximation capabilities multilayer feedforward networks neural networks vol pascanu cho bengio number linear regions deep neural networks advances neural information processing systems nips may draft cohen sharir shashua expressive power deep learning tensor analysis annual conference learning theory colt telgarsky benefits depth neural networks annual conference learning theory colt mallat group invariant scattering communications pure applied mathematics vol bruna mallat invariant scattering convolution networks ieee transactions pattern analysis machine intellignce vol mar wiatowski mathematical theory deep convolutional neural networks feature extraction giryes sapiro bronstein deep neural networks random gaussian weights universal classification strategy ieee transactions signal processing vol jul choromanska henaff mathieu arous lecun loss surfaces multilayer networks international conference artificial intelligence statistics aistats haeffele vidal global optimality tensor factorization deep learning beyond giryes eldar bronstein sapiro tradeoffs convergence speed reconstruction accuracy inverse problems saxe mcclelland ganguli exact solutions nonlinear dynamics learning deep linear neural networks international conference learning representations iclr ollivier riemannian metrics neural networks feedforward networks information inference vol jun neyshabur salakhutdinov optimization deep neural networks advances neural information processing systems nips ioffe szegedy batch normalization accelerating deep network training reducing internal covariate shift proceedings international conference machine learning icml salimans kingma weight normalization simple reparameterization accelerate training deep neural networks advances neural information processing systems nips hayat khan bennamoun boussaid sohel contractive rectifier networks nonlinear maximum margin classification proceedings ieee international conference computer vision srivastava hinton krizhevsky sutskever salakhutdinov dropout simple way prevent neural networks overfitting journal machine learning research jmlr vol jun rifai vincent muller glorot bengio contractive explicit invariance feature extraction proceedings international conference machine learning icml huang qiu sapiro calderbank discriminative robust transformation learning advances neural information processing systems nips vapnik overview statistical learning theory ieee transactions neural networks vol understanding machine learning theory algorithms cambridge university press may draft bartlett mendelson rademacher gaussian complexities risk bounds structural results journal machine learning research jmlr vol mannor robustness generalization machine learning vol neyshabur tomioka srebro capacity control neural networks proceedings conference learning theory colt sun chen wang liu large margin deep neural networks theory algorithms zagoruyko komodakis wide residual networks zhang bengio hardt recht understanding deep learning requires rethinking generalization giryes sapiro rodrigues generalization error invariant classifiers international conference artificial intelligence statistics aistats shen ong feature selection via sensitivity analysis svm probabilistic outputs machine learning vol yang shen ong feature selection via sensitivity analysis mlp probabilistic outputs ieee international conference systems man cybernetics shi yeung gao sensitivity analysis applied construction radial basis function networks neural networks vol mar mendelson pajor uniform uncertainty principle bernoulli subgaussian ensembles constructive approximation vol verma distance preserving embeddings general journal machine learning research jmlr vol neyshabur tomioka salakhutdinov srebro path normalization neural networks international conference learning representations iclr watson characterization subdifferential matrix norms linear algebra applications vol jun petersen pedersen matrix cookbook technical university denmark zhang ren sun identity mappings deep residual networks hochreiter schmidhuber long memory neural computation vol bengio learning deep architectures foundations machine learning vol veit wilber belongie residual networks exponential ensembles relatively shallow networks lecun bottou bengio haffner learning applied document recognition proceedings ieee vol krizhevsky hinton learning multiple layers features tiny images computer science department university toronto tech rep apr hsiao lim hua cheng lared large extensible hand gesture dataset proceedings acm multimedia systems conference russakovsky deng krause satheesh huang karpathy khosla bernstein may draft berg imagenet large scale visual recognition challenge international journal computer vision ijcv vol springenberg dosovitskiy brox riedmiller striving simplicity convolutional net international conference learning representations iclr workshop track theano development team theano python framework fast computation mathematical expressions may draft
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jun evolutionary algorithm mutation benjamin doerr laboratoire informatique lix polytechnique palaiseau france christian dtu compute technical university denmark kgs lyngby denmark carsten witt dtu compute technical university denmark kgs lyngby denmark jing yang laboratoire informatique lix polytechnique palaiseau france june abstract propose new way mutation rate evolutionary algorithms discrete search spaces roughly speaking consists creating half offspring mutation rate twice current mutation rate half half current rate mutation rate updated rate used subpopulation contains best offspring analyze evolutionary algorithm mutation rate optimizes onemax test function prove dynamic version finds optimum expected optimization time number fitness evaluations log time asymptotically smaller optimization time classic previous work shows performance among unbiased algorithms result shows new way adjusting mutation rate find optimal dynamic parameter values fly since adjustment mechanism simpler ones previously used adjusting mutation rate parameters optimistic find applications introduction evolutionary algorithms eas shown remarkable performance broad range applications however often observed performance depends crucially use right parameter settings parameter optimization parameter extended abstract report appear proceedings genetic evolutionary computation conference gecco control therefore key topics research since different characteristics discrete continuous search spaces discuss work evolutionary algorithms discrete search spaces theoretical research contributed understanding algorithms mathematically founded runtime analyses many show runtime determined parameters majority works investigate static parameter settings parameters fixed start algorithm changed execution recently number results shown prove advantage dynamic parameter settings parameters algorithm changed execution many rely making parameters functionally dependent current state search process fitness individual provably lead better performances leaves algorithm designer even greater parameter setting task namely inventing suitable functional dependence instead fixing numerical values parameters problem solved theoretical means small number easy benchmark problems highly unclear find functional relations general case way work dynamic parameters modify parameters based simple rules taking account recent performance number recent results shows fly parameter settings give equally good performance optimal parameter setting however much less input algorithm designer example good results obtained increasing decreasing parameter depending whether current iteration improved solution way resembling rule continuous optimization parameter settings work well simple monotonic relation success parameter value one speculates increasing size population helps progress made parameters like mutation rate clear rule look like since low success rate either stem small mutation rate regenerating parent high probability destructive high mutation rate relatively complicated learning mechanism presented tries learn right mutation strength computing average past performance stemming different parameter values learning mechanism needed careful exploiting currently profitably mutation strength experimenting parameter values careful choice parameter controlling much older experience taken less account recent observations new mechanism eas work propose alternative way adjust mutation rate fly algorithms using larger offspring populations aims overcoming difficulties learning mechanism described simple idea create half offspring twice current mutation rate half using half current rate mutation rate modified rate used create best offspring choosing winning offspring randomly among best case ambiguity allow mutation rate leave interval rates used subpopulations always interval add one modification basic idea described first paragraph section instead always modifying mutation rate rate best offspring shall take winner rate probability half else modify mutation rate random one two possible values twice half current rate motivation modification feel additional random noise prevent algorithm adjusting mutation rate direction profitable however increased amount randomness may allow algorithm leave possible basin attraction locally optimal mutation rate observe probability sequence random modification direction appears hence rate algorithm jump mutation rate restriction discrete set mutation rates appear note existence random modifications also exploited runtime analysis show new mechanism selects mutation rates good enough lead asymptotically optimal runtime among dynamic choices mutation rate first work proposing mechanism shall spend much effort finetuning rather show manner find good mutation rates real application likely better results obtained working three subpopulations namely additional one using exploiting current mutation rate also seems natural modest adjustments mutation rate multiplying dividing rate number smaller value used mechanism profitable conduct elementary experiments supporting intuition section runtime analysis onemax prove mechanism presented indeed find good dynamic mutation rates analyse purest possible setting namely optimization classic test function onemax via see algorithm runtime fixed mutation rates onemax well understood particular show expected runtime witt number generations mutation rate constant used thus large mutation rate determines leading constant runtime rate gives asymptotically best runtime consequence work parallel complexities badkobeh lehre sudholt showed suitable mutation rate finds optimum onemax asymptotically better runtime logn log improvement factor log log runtime bestpossible among unary unbiased optimization algorithms particular dynamic choice mutation rate achieve asymptotically better runtime way mutation rate depends fitness result however trivial parent individual fitness distance mutation rate employed max main technical result adjusting mutation rate according mechanism described optimal asymptotic runtime consequently mechanism able find fly mutation rate sufficiently close one proposed achieve asymptotically expected runtime theorem let let denote number generations mutation rate onemax log log corresponds expected number functions evaluations log log best knowledge first time simple achieves via choice mutation rate interesting side remark proofs reveal quite fixed mutation rate also achieves log log improvement implies bound generations small hence constant choice studied yield asymptotically optimal number generations unless small log dominates lemma let let denote number generations fixed mutation rate log log corresponds expected number functions evaluations log log paper structured follows section give overview previous analyses parameter control mechanism eas theoretical perspective section give algorithm mutation scheme convenience also state key theorems frequently use rest paper next three sections deal runtime analysis expected time spent onemax three regions fitness distance label regions far region middle region near region dealt separate section proof main theorem lemma given section finally conclude section related work since theoretically oriented work dynamic parameter choice speeds runtime test function onemax let briefly review known theory dynamic parameter choices general first conduct rigorous runtime analysis jansen jong wegener proved among results optimizing onemax linear exists population size log log log log log log log log log log finding optimal solution takes expected number log generations whereas larger least log generations necessary picture completed proof exn log log pected number generations taken find optimum log log imln plicit constants determined giving bound constant mentioned introduction aside optimization behavior onemax much known least made explicit easy see waiting times improvement larger reduce factor compared offspring populations results made explicit log expected runtime number generations leadingones log expected runtime linear functions log log wmax runtime estimate minimum spanning trees valid dynamic parameter choices clear eas parameters changing run algorithm dynamic parameter settings powerful using static parameter settings recently considerable advantages dynamic choices could demonstrated mathematical means discrete optimization problems continuous optimization step size adaptation obviously necessary approach arbitrarily closely target point describe different ways dynamically control parameters use following language proposed eiben hinterding michalewicz extension deterministic parameter control language deterministic parameter control means dynamic choice parameter depend fitness landscape first rigorously analyze deterministic parameter control scheme jansen wegener regard performance uses iteration mutation rate chosen mod words cyclically use mutation rates largest power two less jansen wegener demonstrate exists example function dynamic significantly outperforms static mutation rate however also observe many classic problems slower factor log mutation rate analyzed previous experimental study suggested profitable approach mathematical runtime analysis rather indicates opposite artificial examples huge runtime gain could shown also runtime reduces essentially rigorous analysis onemax function rather suggests high rate offspring generated mutation rate much higher brings significant risk slowing optimization process two settings evolutionary computation deterministic parameter control mechanisms also gave interesting results problems solution length known precisely number set bits relevant solution quality unknown random mutation rates gave good results however scheme used rather one based slowly decreasing summable sequences problems discrete variables take many values search space large question change value individual variable results suggest harmonic mutation strength changing variable value chosen randomly probability proportional beneficial distribution analyzed earlier case also shown give asymptotically best performance onemax type problem randomized search heuristics outside evolutionary computation wegener showed simulated annealing using temperature beat metropolis algorithm using static temperature adaptive parameter control parameter control scheme called adaptive used kind feedback optimization process functionally dependent mutation rate depends fitness parent rule first conduct runtime analysis adaptive parameter control mechanism show small advantage static choices doerr neumann proposed use mutation rate leadingones optimization leadingones test function proved choice runtime improves roughly compared time stemming classic mutation rate runtime stemming asymptotically optimal static rate approximately offspring population size order suggested parent individual optimum choice improves optimization timep number fitness evaluations optimum found onemax log log log log log stemming optimal static parameter choice since adaptive algorithm mutation rate functionally dependent offspring population size namely via dynamic choice equivalent mutation rate aforementioned work badkobeh mutation log max shown improve classic runtime log log log logn log using flip mutation operator together choice shown give performance onemax close theoretical optimum among unary unbiased algorithms however differs lower order terms performance simple randomized local search heuristic rls algorithms evolutionary ones zarges proved mutation rates beneficial artificial immune systems parameter control results show advantage adaptive parameter setting remains questionable algorithm user would able find functional dependence parameter fitness difficulty overcome via parameter choices parameter modified according simple rule often based success progress previous iterations via parameter encoded genome thus subject variation selection understanding still limited theoretical work topic however promising shows examples lead significant evolutionary algorithms contrast last years produced profound understanding selfadjusting parameter choices first perform mathematical analysis sudholt considered simple parallel island model together two mechanisms population size island number including halving doubling depending whether current iteration led improvement mechanisms proven give significant improvements parallel runtime number generations various test functions without increasing significantly sequential runtime number fitness evaluations shown choice described also found way aim another successbased mechanism proposed imitates rule evolution strategies modifications mechanism also works random satisfiability problems problem optimizing onemax function step size inspired rule found find asymptotically best possible runtime results indicate dynamics work well adjusting parameters monotonic relation like progress difficult increase population size holds adjusting parameter like mutation rate less obvious example search space large mutation rate creating stronger drift towards hamming distance optimum small mutation rate giving small radius exploration detrimental reason obtain version result optimal number optimize onemax via flips learning mechanism proposed past estimates efficiency different parameter values shown find optimal mutation strength sufficiently well obtain essentially runtime stemming mutation strength exhibited light works result methodological perspective shows difficulties learning mechanism whole bookkeeping part also setting parameters regulating discount information time overcome mechanism proposed work sense use larger populations enables adjust mutation rate solely information learned current iteration however also use idea intentionally use parameter settings appear slightly current optimum gain additional insight preliminaries algorithm consider mutation rate minimization functions defined algorithm general idea mutation scheme adjust mutation strength according success population perform mutation applying standard bit mutation two different mutation probabilities call mutation rate precisely even number algorithm creates offspring mutation rate mutation rate adjusted selection probability half new rate taken mutation rate best individual one lowest fitness ties broken uniformly random created adjustment probability mutation rate adjusted random value random adjustment note mutation rate adjusted iteration also offspring worse parent thus parent kept next iteration adjustment rate results new rate outside interval replace rate corresponding boundary value note case subpopulation rate less would generated means flipping less one bit expectation rate subpopulation rate larger would created useful choice formulate algorithm start initial mutation rate rinit assumption rinit greater equal selfadjusting choice mutation rate given pseudocode algorithm algorithm standard bit mutation select uniformly random set rinit create flipping bit copy independently probability probability otherwise arg minxi breaking ties randomly perform one following two actions prob replace mutation rate created replace either probability replace min max let explain motivation random adjustments rate without random adjustments rate changed direction winning offspring generated rate simple functions like onemax likely sufficient however fitness best offspring viewed function rate unimodal several adjustments direction first yielding good offspring might needed reach good values rate random adjustments enable algorithm cross valley unfavorable rate values note ideas uncommon evolutionary computation mutation prominent example allowing perform several steps one iteration cross fitness valleys different way implement mechanism allowing larger changes rate cross unfavorable regions would generate offspring rates allow larger deviations current rate small probability one idea could choosing offspring independently rate probability give similar results process appears chaotic number individuals produced rates runtime also called optimization time smallest individual minimum found note corresponds number iterations also called generations generation creates offspring since offspring evaluated number function evaluations classical cost measure factor larger runtime defined however assuming massively parallel architecture allows parallel evaluation offspring counting number generations seems also valid cost measure particular function onemax increasing observed terms number generations note reasons symmetry makes difference whether onemax minimized present paper maximized several previous research papers throughout paper asymptotic notation respect problem size drift theorems results obtained drift analysis also used previous analyses without onemax linear functions first theorems stating upper bounds hitting time using variable drift back take formulation simplify markov processes notational convenience theorem variable drift upper bound let random variables describing markov process finite state space xmin xmax xmin let random variable denotes earliest point time exists monotone increasing function xmin xmax integrable xmin xmax xmin xmin xmin variable drift theorem often applied special case additive drift discrete spaces assuming constant one obtains since make frequent use following sections well also give version multiplicative drift theorem upper bounds due implied previous variable drift theorem theorem multiplicative drift let random variables describing markov process finite state space let xmin min let random variable denotes earliest point time exist xmin chernoff bounds reasons courtesy reader state two wellknown multiplicative chernoff bounds lesser known additive chernoff bound also known literature bennett inequality theorem bennett inequality chernoff bounds theorem theorem let independent random variables let furthermore let var moreover take values exp exp occupation probabilities mentioned analyzing two depending stochastic processes random decrease fitness random change mutation rate often prove drift analysis rate drifting towards values yield fitness decrease however rate drifted towards values would also like rates stay vicinity values subsequent steps end apply following theorem note paper slightly general version including probability stated need theorem theorem let markov process additive drift least towards given starting readily apply theorem following lemma used throughout paper bound rate lemma point constant min holds proof apply theorem process max note process moves absolute value drift use theorem estimate far region first three technical sections analyze optimization behavior selfadjusting regime fitness distance least since relatively far optimum relatively easy make progress hand regime spans largest number fitness levels namely need exhibit sufficient progress iteration also regime optimal mutation rate varies remark drastically without proof optimal rate log log quickly drops log despite difficulties manages find sufficiently good mutation rates able reach fitness distance expected number log iterations lemma let sufficiently large define probability best offspring created rate least let probability best offspring created rate least probability best offspring worse parent least proof let probability standard bit mutation mutation rate creates parent fitness distance offspring fitness distance comparing component obtain notice exp inequality applies therefore since monotone decreasing let largest second inequality involves uses fact means exp exp factor compensated decreasing large enough notice since let notice increasing obtain results exp also need upper bound since implies chernoff bounds shows probability bits flipped bits less exp means use compute upper bound let probability fitness distance decreased regard terms increase terms change factor consider factor sum factors less geometric series ratio therefore sum contributes least total sum consequently look first terms since finally factor otherwise prove probability less least one offspring achieve probability least best offspring created first regard large define factor becomes exp exp exp exp otherwise exp exp bound exp exp prove probability best progress attains least conditioning obvious obtained probability least since remaining probability less likely best offspring cancel first compute following way since large enough means best progress among offspring attains least according definition offspring moreover use fact also therefore easy bound look function since therefore means best offspring made progress conditional probability least finally bound probability best offspring second statement let denote random decrease fitness distance apply standard bit mutation probability individual ones according bennett inequality theorem exp var compare constant factor let difference means apply bound var exp var exp exp notice second inequality therefore probability less best offspring rate better expectation rate rate let number flipped flipped respectively follow binomial distribution know median means large enough use normal distribution approximate otherwise notice let real number denote denote clear dte dte dte dte see dte dte dte dte dte dte dte dte dte obtain therefore probability least one offspring beats therefore best offspring probability least proves second statement lemma offspring mutation rate worse parent using bennett inequality exp var exp exp therefore probability proves third statement lemma shows rate attracted interval unfortunately show obtain sufficient progress fitness exactly range however range smaller constant factors large values lemma smaller values case distinction motivated fact becomes large approaches good drift smaller range problem since random movements let enter smaller range constant probability see theorem proof let denote fitness gain selection among best offspring generated rate parent fitness distance onemax let independent offspring generated flipping bit independently probability random variable defined max min onemax next show narrow region contained provides least logarithmic drift fitness first proof large close lemma let large enough let min let proof note look number flips ones random variable follows binomial distribution bin assume use normal distribution approximate hard see probability hitting mean satisfies specifically otherwise worst case estimate comparing first notice means comparing see take otherwise bound using stirling approximation exp exp therefore exp exp take otherwise let satisfying required assumption notice therefore probability least one offspring flips least ones shows compare notice otherwise ifp require means lower order compared hence extend lemma whole region situation becomes easier every smaller range provides least expected logarithmic fitness increase together previous lemma obtain following statement drift whole region lemma let large enough assume defined lemma proof consider probability creating parent distance offspring fitness distance least via standard bit mutation probability second greater equal symbol apply exp exp satisfying factor compensated using estimate hence consequently note occurs case apply lemma obtain consider generations use rate within right region bound expected runtime reach since drift fitness order log following theorem shows additional time spent adjusting rate towards right region change bound expected runtime theorem needs generations expectation reach initialization proof first argue quickly takes expected number iterations reach fitness distance end note probability offspring strictly better fitness parent least consequently expected fitness gain one iteration least constant initial fitness distance deviates expectation hence takes generations obtain without loss generality assume initial state intuition begin use rate bounded distance level considerable drift strong drift rate keeps within far away bounds make progress decreases new level corresponding decrease algorithm takes time readjust new bounds consider stochastic process current according lemma pessimistically assume iterations adjusting make progress let taken hits according additive drift analysis theorem takes log iterations bound matter set initial rate consider last time means makes progress referring lemma know probability least according additive drift theorem takes expected number log iterations reach sum different expectation increase log neglect comparing probability decrease thus bound runtime log total number iterations readjustment process satisfy upper bound log log log consider iterations readjustment process satisfy lower bound since decreases along hit lower bound condition obtained following compute expected number generations decrease choose large enough holds positive constant note constant applying lemma obtain decreases another bound level happens probability least see steps range even smaller range described lemma reaches wider region takes constant number iterations expectation reach narrow region mutation scheme employs chance perform random step mutation rate based lemma narrow region rate ensures drift fitness distance contributes average drift least log random rates distance applying theorem estimate runtime log log log log details compute integral found proof theorem notice number different values must bounded runtime combining expected number log iterations adjust expected number iterations hit total runtime expectation middle region section estimate expected number generations number onebits decreased first claim right region hence sensitive choice intuitively due fact total fitness improvement suffices cross middle region whereas improvement needed far region estimate drift fitness lemma apply result afterwards estimate number generations cross region lemma let min proof probability flipped single mutation regard number offspring flipped zeros expectation least applying chernoff theorem observe exceeds probability least exp since assuming happen look first offspring without flipped zeros let number flipped ones offspring bin let max applying result order statistics binomially distributed random variables witt lemma obtain following therefore implies otherwise implies thus min hence using law total probability obtain lower bound drift middle region use result drift estimate time spent region notice means frequently often provides drift need theorem let assume current expected number generations proof upper bound lemma according lemma additive drift theorem yields log time choose large enough holds positive constant note constant applying lemma obtain happens probability least takes constant number iterations expectation draw less according lemma ensures min implies average drift least min random rates distance minimum taken first argument second interested expected time reduce ease application drift analysis artificially modify process make create optimum state strictly less clearly first hitting time state change modification applying variable drift theorem theorem xmin min expected number generations reach state bounded log log overall expected number generations spent log since assumption near region near region hence fitness low expect constant number offspring flip least one remaining assumes constant rate however higher rates detrimental since likely destroy individuals flipping hence expect rate drift towards constant values shown following lemma lemma let probability least proof prove claim exploit fact flipped subpopulations using shall argue follows sufficiently high constant probability contains individual strictly better parent fitness less fitness identical parent conditional either contains individuals fitness less winning individual surely stems subpopulation contains better offspring latter case argue many individuals fitness exactly gives sufficiently high probability taking winning individual side chose uniformly random offspring fitness let number offspring flip using rate respectively since used constant using get fact discriminate using theorem following way exp exp sufficiently large since similarly obtain exp note holds since offspring generated independently events happen together probability least conditioning using union bound probability least one offspring flip flips least one upper bounded using first last inequality using union bound find probability least one flips exactly one exactly one sufficiently large using first inequality third inequality due last inequality stems second inequality follows using constant let number offspring zeroes ones require offspring flip least two probability offspring created using fact decreasing events event offspring created sufficient ensure best individual either surely chosen uniformly random offspring conditioning events probability best offspring chosen least hence using union bound error probabilities unconditional probability least note restriction lemma strictly necessary also smaller probability winning individual chosen additive constant larger showing however would need additional proof arguments smaller event subpopulations contain individuals fitness becomes likely avoid additional technicality arguing enough since constant sufficient fitness drift need since aim making leading constant precise following proof analysis near region use lemma quite additional arguments argue quickly reaches less regularly returns region allows argue near region factor compared since every offspring probability making progress see also theorem assume current expected number generations optimum reached log log proof aim estimate points time generations bound expected number generations mutation rate entered region basically consider stochastic process max lower bound lemma however proved drift towards smaller values region upper bound lemma use potential function dlog dlog otherwise assuming rounded closest power respectively potential function slope lemma gives drift function satisfies corresponds region probability decreasing bounded due random steps still region due concavity potential function finally lemma hence altogether constant log additive drift analysis yields expected number log generations first time holds corresponding denote hitting time consider arbitrary point time aim show drift depending current satisfies probability end use lemma choose large enough holds positive constant note constant consider two cases happens probability least obtain probability least improve using bound improvement using law total probability obtain straightforward multiplicative drift analysis theorem using gives expected number log log generations optimum found together expected number log becomes proves theorem putting everything together section put together analyses different regimes prove main result proof theorem lower bound actually holds unbiased parallel algorithms shown add bounds expected number generations spent three regimes precisely add bounds theorem theorem theorem gives log generations due assumption bound dominated log suggested proof lemma basically revisit regions different analyzed paper bound time spent regions assumption far region lemmas applied value imply fitness drift log per generation expected number generations spent far region computed variable drift analysis proof theorem middle region shortened lower end lemma gives fitness drift implying additive drift analysis generations reduce fitness near region starts argue slightly differently note every offspring probability least flipping hence expect offspring pessimistically assume yield fitness improvement conceptually reduces population size offspring guaranteed flip average runtime runs random steps static static using max figure static average runtime comparison onemax adapting arguments proof theorem probability least one individuals flips least least lower bound fitness drift using multiplicative drift analysis expected number generations near region log putting times regions together obtain lemma experiments since analysis asymptotic nature performed elementary experiments order see whether besides asymptotic runtime improvement showing improvement unspecified large problem size also see improvement realistic problem sizes purpose implemented using gnu scientific library gsl generation numbers plot figure displays average runtime runs onemax given algorithm set initial mutation rate minimum mutation rate algorithm attain moreover plot displays average runtime classic using static mutation probability average runtimes algorithms profit higher offspring population sizes leading lower average runtimes increases interestingly classic outperforms small values higher offspring population sizes outperforms classic one indicating theoretical performance gain fact relevant practice furthermore implemented without random steps rate always adjusted according best offspring distributed two subpopulations experiments show variant performs generally slightly better onemax since onemax fitness landscape structurally simple result totally surprising seems natural fitness best individuals viewed function rate unimodal function case advantage random steps able leave local optima function needed hand course observation suggests try prove performance bound rigorously also case without random rate adjustments currently see lastly implemented using mutation rate max presented experiments suggest scheme outperforms variants considered additionally implemented another variant using three subpopulations additional one using exploiting current mutation rate compared variant without using random steps results shown figure experiments suggest variant using three subpopulations outperforms selfadjusting slightly small population sizes high population sizes using two subpopulations seems better choice gain understanding parameters influence runtime implemented using different mutation rate update factors consider given algorithm mutation rate increased decreased factor instead choice made algorithm note change rule use rates create subpopulations furthermore initialization algorithm starts rate rate capped run accordingly results shown figure plot displays average runtime runs onemax using update factors plot suggests lower values yield better performance result immediately obvious clearly large factor implies rate changes lot generation generation namely factor changes prevent algorithm using good rate several iterations row hand small value implies takes longer adjust rate value far current one subpop random steps subpop random steps average runtime runs average runtime runs figure average runtime two three subpopulations without random steps onemax figure average runtime different mutation rate update factors onemax rate fitness figure development rate fitness three example runs selfadjusting onemax todo forgot check home using different factors finally illustrate nontrivial development rate run algorithm plotted rate three single runs using different factors fitness figure since algorithm initialized rate rate increases initialization decreases decreasing fitnessdistance optimum plot suggests higher values rate unsteady due greater impact rate adjustments smaller rate updates yield stable development rate interestingly three values rates seem correspond rate initial increasing phase note illustration indicate actual runtime fact specific runtimes similar pronounced behaviour seen chose particular values illustrative purposes since variance rate visually confusing reasons given would draw experiment conclusion smaller choice preferable practical application algorithm influence parameter runtime large might worth optimizing rather view algorithm algorithm conclusions proposed analyzed new simple mutation scheme consists creating half offspring slightly larger rest slightly smaller mutation rate based success subpopulations mutation rate adjusted simple scheme overcomes difficulties previous choices careful choice balance forgetting rate learning scheme proved rigorously optimizes onemax test function expected number log log fitness evaluations matches runtime shown careful choice mutation rate also shown asymptotically optimal among blackbox optimization algorithms hence runtime result indicates mechanism developed work able find good mutation rates best knowledge first time choice mutation rate speeds algorithm onemax test function constant factor main technical challenge work analyze quality best offspring contrast previous runtime analyses asymptotic order fitness gain relevant needed much higher degree precision needed make statements best offspring case multiple best offspring distributed two subpopulations note quality best offspring strongly concentrated around expectation average quality analyses observed using fixed rate gives bound log also asymptotically optimal unless small however setting far usual constant choice first time significantly larger mutation rate shown useful simple algorithm simple fitness landscape previously observed larger mutation rates helpful leave local optima work number open problems arise technical challenge prove algorithm also without random rate adjustments performs well requires even precise analysis qualities offspring two subpopulations currently methods understanding mutation rate algorithms two interesting question extent observation larger mutation rates beneficial onemax generalizes algorithms problems problems choice mutation rate gives improvement classic choice static choices acknowledgments work supported public grant part investissement avenir project reference labex lmh grant danish council independent research references golnaz badkobeh per kristian lehre dirk sudholt unbiased complexity parallel search proc ppsn pages springer benjamin doerr frank neumann optimal fixed adaptive mutation rates leadingones problem proc ppsn pages springer maxim buzdalov benjamin doerr runtime analysis genetic algorithm random satisfiable formulas proc gecco acm appear full version available http stephan cathabard per kristian lehre xin yao mutation rates problems unknown solution lengths proc foga pages acm jorge cervantes christopher stephens rank based variation operators genetic algorithms proc gecco pages acm dang per kristian lehre mutation rates populations proc ppsn pages springer martin dietzfelbinger jonathan rowe ingo wegener philipp woelfel tight bounds blind search integers reals combinatorics probability computing benjamin doerr analyzing randomized search heuristics tools probability theory anne auger benjamin doerr editors theory randomized search heuristics pages world scientific publishing benjamin doerr optimal parameter settings genetic algorithm proc gecco pages acm benjamin doerr carola doerr optimal parameter choices selfadjustment applying rule discrete settings proc gecco pages acm benjamin doerr marvin optimizing linear functions evolutionary algorithm different asymptotic runtimes different instances theoretical computer science benjamin doerr daniel johannsen carola winzen multiplicative drift analysis algorithmica benjamin doerr carola doerr franziska ebel complexity designing new genetic algorithms theoretical computer science benjamin doerr carola doerr timo solving problems unknown solution length almost extra cost proc gecco pages acm benjamin doerr carola doerr timo right mutation strength decision variables proc gecco pages acm benjamin doerr carola doerr timo provably optimal step sizes decision variables proc ppsn pages springer benjamin doerr carola doerr jing yang optimal parameter choices via precise analysis proc gecco pages acm benjamin doerr carola doerr jing yang mutation outperforms standard bit mutation proc ppsn pages springer benjamin doerr carola doerr timo unknown solution length problems asymptotically optimal run time proc gecco acm appear benjamin doerr huu phuoc makhmara duy nguyen fast genetic algorithms proc gecco acm appear full version available http agoston endre eiben robert hinterding zbigniew michalewicz parameter control evolutionary algorithms ieee transactions evolutionary computation christian carsten witt interplay population size mutation probability onemax algorithmica press available online http thomas jansen ingo wegener analysis dynamic evolutionary algorithm journal discrete algorithms thomas jansen kenneth jong ingo wegener choice offspring population size evolutionary algorithms evolutionary computation daniel johannsen random combinatorial structures randomized search heuristics phd thesis saarland university rob kaas jan buhrman mean median mode binomial distributions statistica neerlandica timo andrei lissovoi carsten witt generalized dynamic onemax proc foga pages acm dirk sudholt adaptive population models offspring populations parallel evolutionary algorithms proc foga pages acm per kristian lehre carsten witt concentrated hitting times randomized search heuristics variable drift proc isaac pages springer boris mitavskiy jonathan rowe chris cannings theoretical analysis local search strategies optimize network communication subject preserving total number links international journal intelligent computing cybernetics frank neumann ingo wegener randomized local search evolutionary algorithms minimum spanning tree problem theoretical computer science pietro simone oliveto per kristian lehre frank neumann theoretical analysis mutation combining exploration exploitation proc cec pages ieee ingo wegener simulated annealing beats metropolis combinatorial optimization proc icalp pages springer christine zarges rigorous runtime analysis inversely fitness proportional mutation rates proc ppsn pages springer christine zarges utility population size inversely fitness proportional mutation rates proc foga pages acm
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jul table imit heorems mpirical rocesses onditional eighborhood ependence hyung lee kyungchul song university illinois university british columbia bstract paper introduces new concept stochastic dependence among many random variables call conditional neighborhood dependence cnd suppose set random variables set sigma algebras sets indexed set endowed neighborhood system set random variables satisfies cnd two sets random variables conditionally independent given sigma algebras indices one two sets neighborhood random variables cnd include conditional dependency graphs class markov random fields global markov property cnd property useful modeling dependence governed complex large network paper provides two main results first result stable central limit theorem sum random variables cnd second result result stable convergence empirical processes indexed class functions satisfying certain bracketing entropy condition random variables satisfy cnd words conditional neighborhood dependence dependency graphs markov random fields empirical processes maximal inequalities stable central limit theorem ams msc jel lassification date july thank denis kojevnikov valuable comments errors corresponding address kyungchul song vancouver school economics university british columbia iona drive vancouver canada email address kysong address hyung lee department economics university illinois gregory david kinley hall urbana email address jihyung introduction empirical processes indexed class functions arise many applications particular developing asymptotic inference nonparametric semiparametric models specification tests see andrews chapter van der vaart wellner review applications empirical process theory statistics econometrics predominant body literature empirical process theory focuses independent observations time series observations relatively little research empirical processes spatial dependence paper aims contribute literature providing limit theorems empirical processes consist random variables flexible complex crosssectional dependence structure paper introduce new notion stochastic dependence among set random variables suppose given set random variables indexed set set endowed neighborhood system associated subset called neighborhood paper call map neighborhood system given neighborhood system set say conditionally neighborhood dependent cnd respect two subsets conditionally independent given union neighborhoods set removed cnd property generalization dependency graphs markov random fields global markov property lauritzen dawid larsen leimer dependency graphs introduced stein study normal approximation see chen shao rinott rotar general local dependence notion different set random variables graph dependency graph two sets random variables allowed dependent two sets adjacent graph dependence viewed restrictive many applications requires random variables independent even indices indirectly connected graph contrast cnd random variables allowed dependent even adjacent graph cnd property captures notion two random variables independent condition source joint dependence sense cnd property closely related markov property literature random fields however contrast markov property cnd property require generated paper provides two main results first main result bound sum cnd random variables bound comparable bounds established sum random variables dependency graph generic situations literature baldi rinott chen shao penrose name latter literature typically uses stein method establish bound best knowledge existing proofs using stein method dependency graphs seem immediately extendable sum cnd random variables due flexible form conditioning involved cnd property paper use traditional characteristic method derive bound typical form bound including involves maximum degree neighborhood system maximum degree high bound little use however many social networks observed removing small number vertices tends reduce maximum degree neighborhood system substantially exploiting insight provide general version berryesseen bound uses conditioning random variables associated high degrees second main result paper stable limit theorem empirical process indexed class functions empirical process constituted cnd random variables stable convergence stronger notion convergence weak convergence useful asymptotic theory statistics whose normalizing sequence random limit obtain stable limit theorem first extend exponential inequality janson dependency graphs cnd random variables using obtain maximal inequality empirical process type bound maximal inequality useful various purposes especially one needs obtain limit theorems uniform given class functions indexing empirical process using maximal inequality establish asymptotic equicontinuity empirical process combination central limit theorem comes previously established bound gives stable limit theorem enables stable convergence empirical process mixture gaussian process turns stable limit theorem empirical process requires maximum degree neighborhood system bounded however many networks maximum degree substantial especially networks behave like preferential attachment network albert thus following spirit extending bound case conditional high degree vertices extend stable limit theory relies observations relatively low degrees conditioning random variables associated high degree vertices extension enables obtain stable limit theorem empirical processes maximum degree neighborhood system increases infinity size system increases stable convergence extensively studied context martingale central limit theorems see hall heyde see luschgy stable limit theorems markov kernels related topics recent studies kuersteiner prucha hahn kuersteiner mazzocco established stable central limit theorem sum random variables dependence time series dependence utilizing martingale difference array formulation random variables dependence graph received attention literature lauritzen particular pairwise markov property random variables says two random variables conditionally independent given variables captured precision matrix high dimensional gaussian model see meinshausen cai liu zhou references paper cnd property stronger pairwise markov property conditioning generated random variables however cnd property encompasses case latter condition hold thus includes dependency graphs special case unlike dependence mentioned best knowledge stable convergence empirical processes indexed class functions studied either dependency graphs markov random fields graph remainder paper proceeds follows section formally introduce notion conditional neighborhood dependence cnd study basic properties section provide stable central limit theorems sum cnd random variables also present stable convergence empirical process mixture gaussian process mathematical proofs results found appendix conditional neighborhood dependence definition let infinite countable set let finite set let collection subsets assume proper subset call element vertex call map neighborhood system let define simply write suppressing curly brackets let call includes vertices boundary around excludes implies say neighborhood system undirected exists pair say neighborhood system directed often useful compare different dependence structures governed different neighborhood systems two neighborhood systems say strictly finer strictly coarser say weakly finer weakly coarser let introduce notion dependence among triangular array let given probability space let given triangular array indexed proper notation triangular array fin suppress subscript simplicity let smallest contains smallest contains simply take trivial apply notation triangular arrays triangular array similarly define given two say write represent smallest contains given triangular array let introduce sub defined many applications used accommodate random variables common shock example suppose generated random vector set random variables conditionally independent given common random equivalently one might view neighborhood system graph identifying neighborhood neighborhood graph however seems natural think stochastic dependence structure among random variables terms neighborhoods rather terms edges graph variable take generated discuss examples cnd random vectors later section study properties let introduce notion dependence array central focus paper definition given neighborhood system array say given subset conditionally neighborhood dependent cnd respect conditionally independent given generated random vectors subset cnd respect simply say random vectors cnd respect conditional neighborhood dependence specifies conditional independence arises conditional dependence arises conditional neighborhood dependence specify independence dependence furthermore conditional neighborhood dependence accommodate situation neighborhoods system generated random graph long random graph situation results paper continue hold minor modifications take care randomness monotonicity invariance conditioning general conditional independence property monotone conditioning words conditionally independent given imply conditional independence given given contains however cnd partially obeys monotonicity neighborhood systems specifically cnd property finer neighborhood system implies conditional independence restrictions cnd property coarser neighborhood system implies introduce lemma makes precise monotonicity property cnd suppose given two neighborhood systems weakly finer following lemma shows cnd property triangular array respect given neighborhood system carries respect undirected lemma given two neighborhood systems suppose weakly finer triangular array cnd respect igure conditional neighborhood dependence notes suppose given random variables figure depicts neighborhood system cnd respect neighborhood system illustrated implies conditionally independent given suppose undirected cnd respect proof take well let undirectedness implies therefore cnd property conditionally independent given lemma dawid implies conditionally independent given proves lemma follows due lemma requirement undirected eliminated see consider following counterexample lemma taken directed neighborhood system example let take let standard normal random variables let take let neighborhood system given hence directed take denotes generated hard see cnd respect let introduce another neighborhood system weakly coarser let take note certainly conditionally independent given involves shall see later using conditioning cnd property one may obtain better normal approximation sum cnd random variables situations give preliminary result addresses question whether cnd property still holds increase certain way precisely suppose cnd respect let choose define neighborhood system let mnn obtain following result lemma suppose cnd respect fix define cnd respect proof take note let cnd conditionally independent given hence conditionally independent given hence write conditionally independent condition implies conditionally independent given definition choice examples conditional dependency graphs let undirected graph denotes set edges define suppose conditional dependency graph ynn conditionally independent given dependency graphs introduced stein received attention literature see example janson baldi rinott rinott rotar bounds sum random variables dependency graph janson exponential inequality see song application permutation inference cnd respect taken functional local dependence let given directed graph represents edge neighborhood nno set vertices edge vertex vertex similarly nni define nno nni suppose generated following way independent across independent functions nonstochastic way modeling local dependence among base random variables useful many contexts applications particular outcome may arise consequence local interactions among individual variables locality determined given graph see leung canen schwartz song applications economics let define new undirected graph words adjacent overlap define hard see cnd respect triangular arrays long generated nonstochastic measurable map note take trivial case functional local dependence essentially equivalent dependency graph assumption however functional local dependence much richer implications dependency graph assumption alone generates lots conditional independence restrictions implied dependency graph assumption alone using restrictions obtain conditional neighborhood dependence follows let generated hard see triangular array cnd respect nni note neighborhood system nni weakly finer hence graph undirected lemma cnd respect nni expresses richer conditional independence restrictions cnd respect notion functional local dependence related physical dependence main difference physical dependence mainly intended time series dependence involves past present values common function whereas functional local dependence captures local dependence system neighborhoods random variables markov random fields neighborhood system suppose triangular array random vectors undirected neighborhood system path two vertices defined sequence distinct vertices let say set separates sets every path vertex vertex intersects set let consider following two notions markov properties see lauritzen definition say satisfies local markov property vertex conditionally independent given say satisfies global markov property two subsets separated set conditionally independent given suppose generated hard see satisfies global markov property cnd respect cnd respect satisfies local markov property hence notion cnd intermediate concept local global markov stable limit theorems stable central limit theorems basic result section give bound conditional triangular array random variables cnd respect given neighborhood system define dmx max dav denotes cardinality set call dmx maximum degree dav average degree neighborhood system use dmx dav express conditions neighborhood system define max let theorem suppose triangular array cnd respect furthermore assume ndmx dav exists absolute constant ndmx dav log ndmx dav ndmx dav distribution function standard normal distribution lauritzen dawid larsen leimer proposed markov fields directed acyclic graphs defined local markov property global markov property provided sufficient condition equivalent use bound requires good bound observe therefore locally dependent proper sense expect order term dav following corollary gives set conditions true lemma suppose conditions theorem hold furthermore conditionally independent given dav focusing special case satisfying additional condition obtain improved version theorem condition conditionally independent given condition accommodates conditional dependency graphs excludes markov random fields corollary suppose triangular array cnd respect condition holds suppose ndmx dav exists absolute constant ndmx dav log ndmx dav ndmx dav define theorem improvement result due condition two fold first condition weakened second bound involve cumbersome compute reliable bound corollary case dependency graphs much research establishing esseen bound confine attention special case dmx constant bound corollary rate baldi rinott corollary chen shao theorem penrose theorem among others papers adopted stein method obtain bound however best knowledge special case conditional dependency graphs one follow proof theorem penrose obtain slightly improved bound logarithmic factor appears improvement marginal many applications example quantity dav asymptotically dominated ndmx dav dmx increases slower rate straightforward extend results cnd variables main reason conditioning conditional two sets random variables independent varies depending set thus example apply equation penrose context paper resort traditional fourier analytic method combination esseen inequality bound gives stable convergence sum random vectors mixture normal distribution specifically suppose triangular array random variables uniformly continuous bounded map standard normal random variable independent denotes indicator event conditional neighborhood dependence conditional vertices let extend theorem considering normal approximation conditioning random variables associated high degree vertices let given subset neighborhood system given let hence maximum average degrees restriction moreover define mnn write mind choosing set consists vertices neighborhood system lemma cnd respect cnd respect main idea difference two sums asymptotically negligible use bound first sum using theorem deal remainder term comes difference let present extended version theorem let max mnn note supremum definition whereas also define theorem suppose triangular array cnd respect theorem exists absolute constant dmx dav log dmx dav dav compared theorem bound involves additional term additional term arises sum may centered around zero condition third term bound order conditional dependency graph case hence long third term bound vanishes clt theorem restored case converges zero faster dmx theorem improved rate theorem approximation captures situation neighborhood system small fraction high degree vertices improvement still arise generally even conditional dependency graph see note conditionally independent mnn given hence suppose dav dmx rate theorem improves theorem dmx dmx shall see later section approach cnd conditional vertices useful obtaining stable central limit theorem empirical processes random variables cnd respect neighborhood system maximum degree increasing infinity empirical processes stable convergence stable convergence metric spaces let first introduce preliminary results stable convergence metric spaces let given metric space define borel let given probability space sub recall map called markov kernel borel probability measure following notation luschgy let define marginal given finite collection finite dimensional projection markov kernel defined markov kernel spirit approach consider following definition stable convergence empirical processes see berti pratelli rigo similar definition definition suppose given sub sequence stochastic processes markov kernel borel measurable random element markov kernel suppose bounded lipschitz functional denotes indicator function event say converges equivalently converges write equivalently stable convergence according definition implies weak convergence sense borel measurable definition equivalent weak convergence markov kernels many existing results stable convergence carry however equivalence extend case proper definition markov kernels nonmeasurable stochastic processes nevertheless definition still useful one needs deal random norming shown following lemma generalizes theorem aldous eagleson part theorem luschgy lemma suppose random variables gstably borel measurable let denote distribution following holds denotes outer probability continuous first result analogue lemma second result continuous mapping theorem stable convergence empirical process suppose given triangular array random variables cnd respect let given class real measurable functions consider following empirical process empirical process takes value collection bounded functions endowed sup norm forms metric space section explore conditions class joint distribution triangular array delivers stable convergence empirical process stable convergence complete separable metric spaces defined weak convergence markov kernels see luschgy however definition extend case empirical processes taking values endowed sup norm due weak convergence empirical process gaussian process often established three steps first show class functions totally bounded respect certain second show finite dimensional projection empirical process converges distribution multivariate normal random vector third establish asymptotic empirical process let given totally bounded metric space define uniformly following theorem shows take similar strategy proving stable convergence empirical process markov kernel structure proof theorem adapted theorem pollard theorem suppose stochastic process given totally bounded metric space suppose following conditions hold finite set markov kernel asymptotically exists sup denotes outer probability exists markov kernel following properties satisfied finite dimensional projections given markov kernels conversely converges markov kernel satisfied worth noting stable convergence empirical processes conditions asymptotic equicontinuity totally boundedness function class respect standard literature weak convergence empirical processes difference convergence finite dimensional distributions replaced stable convergence finite dimensional projections maximal inequality subsection presents maximal inequality terms bracketing entropy bounds maximal inequality useful primarily establishing asymptotic empirical process also many purposes first begin tail bound sum cnd random variables janson established exponential bound sum random variables dependency graph following exponential tail bound crucial maximal inequality result obtained slightly modifying proof theorem janson lemma suppose triangular array random variables take values cnd respect let defined exp dmx dmx furthermore condition holds condition replaced replaced following holds exp dmx bound one obtained janson case dependency graphs following form maximal inequality finite set immediately follows lemma van der vaart corollary suppose triangular array random variables cnd respect let defined exists absolute constant max dmx log log max finite subset constant let elevate inequality maximal inequality function class recall allow random variables idiosyncratically distributed across define following denote number respect smallest number brackets following lemma establishes maximal inequality terms bracketing entropy bound lemma maximal inequality suppose triangular array random variables cnd respect suppose class functions envelope exists absolute constant log sup dmx bracketing entropy bound lemma involves maximum degree dmx hence bound useful neighborhood system maximum degree increasing stable central limit theorem first let say stochastic process gaussian process finite collection distribution random vector conditional multivariate normal distribution also call markov kernel gaussian markov kernel associated given gaussian process conditional distribution given given finite dimensional projection let summarize conditions neighborhood system follows assumption exists dmx log positive semidefinite dmx log dmx dav exists satisfies whenever well following result gives stable convergence empirical processes theorem suppose triangular array random variables cnd respect satisfying assumption suppose exists envelope converges gaussian process furthermore gaussian markov kernel associated fourth moment condition used ensure convergence finite dimensional distributions using theorem worth noting assumption essentially requires maximum degree dmx bounded interesting condition required clt theorem stronger condition maximum degree used establish asymptotic equicontinuity process neighborhood system generated according model stochastic graph formation condition violated many existing models graph formation used example social network modeling next section utilize approach conditioning vertices weaken condition conditional neighborhood dependence conditional vertices mentioned assumption requires dmx bounded following idea conditioning high degree vertices theorem let explore stable convergence theorem relaxes requirement prior theorem choose given subset let defined first write note sup max since cnd respect defined apply previous results gives following extension maximal inequality lemma since maximal inequality often independent interest let state formally lemma maximal inequality suppose triangular array random variables cnd respect suppose class functions envelope exists absolute constant sup log max sup second term bound dmx similarly derived thus bound improvement lemma whenever dmx let turn stable convergence empirical process modify assumption follows assumption exists log dmx positive semidefinite zero sub defined log exists satisfies whenever well condition essentially requires bounded condition allows dmx increase infinity condition zero requires thus number high degree vertices selected set bounded combination implies makes suffice focus stable limit theorem obtain following extended version theorem theorem suppose triangular array random variables cnd respect satisfying assumption suppose exists envelope converges gaussian process furthermore gaussian markov kernel associated take identical theorem reduced theorem however theorem shows approximation distribution empirical process mixture gaussian process possible even dmx appendix mathematical proofs simplify notation follow van der vaart write sequence numbers whenever cbn absolute constant absolute constant differ across different instances positive integer nnk triangular array random variables define following lemma useful proofs various results lemma suppose neighborhood system triangular array random variables cnd respect furthermore given positive integer let nnk two partitioning subvectors entries entries suppose condition holds proof choice see second statement note whenever must exist since find first statement lemma write second equality follows third equality follows outside conditionally independent given cnd property fourth equality follows fifth equality uses fact outside cnd property let turn second statement lemma assume condition holds write second equality follows outside cnd property third equality follows condition conditionally independent given let present proof theorem recall notation theorem define let define exp itw note uniformly continuous almost surely since twice continuously differentiable almost surely see yuan mei lemma suppose conditions theorem hold dmx dav ndmx dav proof first proof theorem jenish prucha decompose exp itw itw ite itw eitw let consider definition define note lemma let write sum either sum implies second equality third equality leading sum bounded dav number terms sum bounded dav constant write using let focus since hence dav let turn using series expansion exp see tikhomirov bound using mean inequality bound last term ndmx dav finally let turn write exp exp last conditional expectation equal exp first equality follows cnd second equality follows hence follows since exp itw collecting results obtain desired result proof theorem exp itw say taking integral sides obtain following expression exp exp exp exp exp note exp exp exp exp applying lemma last term bounded tan exp cndmx dav dav absolute constant hence log therefore esseen inequality see theorem ibragimov linnik obtain following bound log log taking proof lemma either let set number either number either number either number either thus hard see completing proof dav proof corollary similarly proof lemma decompose exp itw treatment proof lemma difference lies treatment using condition lemma note following argument proof lemma find bounded dav hence proof lemma need following proofs lemma theorem rest deal terms obtain desired result lemma suppose random variables cdf density function sup proof first note probability inside absolute value note also observe hence max sup sup sup taking obtain desired result proof theorem write lemma term bounded denotes density lemma cnd respect apply theorem leading term obtain bound dmx dav log dmx dav dav constant let seek desired bound second term right hand side write say choose write hence last term bounded delivering desired result lemma suppose given metric space random variables proof first note event closed set shown following proof theorem iii van der vaart wellner using following arguments proof lemma van der vaart wellner deduce lemma proof lemma since metric defined furthermore note desired result follows lemma note event open set denotes inner probability using following arguments proof theorem kallenberg continuous mapping theorem weak convergence complete proof proof theorem use following lemma lemma bounded continuous compact every exists proof theorem first let suppose hold see marginal tight borel law note stable finite dimensional convergence implies convergence finite dimensional distributions combining asymptotic using theorems van der vaart wellner obtain tight borel law fact concentrated follows theorem pollard let show convergence follow arguments proof theorem wellner let random element whose distribution since totally bounded every exists finite set points open ball center radius thus choose define convergence finite dimensional projection bounded continuous functional furthermore uniform continuity sample paths implies lim sup bounded continuous functional last two absolute values vanish combined dominated convergence theorem use asymptotic lemma fact tight law follow standard arguments show leading difference vanishes see proof theorem wellner details since convergence implies finite dimensional distributions weak convergence converse shown using standard arguments see proof theorem wellner proof lemma proof follows theorems janson particular follows theorem however need modify proof theorem necessarily hence equations page necessarily follow first without loss generality set following proof theorem janson see obtain exp cxi exp let disjoint subsets partition fix using lemma following argument janson exp exp exp last term bounded exp log exp note log exp let take rewrite exp hence exp take log let log last bound becomes exp exp exp first inequality follows inequality last inequality follows proof theorem janson rest proof proceeded taking minimal fractional proper cover proof lemma adapt proof theorem van der vaart accommodate cnd property fix hqi construct nested sequence partitions sup every definition bracketing entropy taken satisfy log log choose fixed element hqi hqi set hqi suph whenever hqi defines minimal measurable cover dudley run set functions runs define fixed following numbers indicator functions log partitions nested constant partitioning sets hqi level decompose say write say analyze empirical process control let bound sup sup since bound last expression log log due choice satisfying control functions functions since partitions nested function bounded applying corollary max dmx log log max law iterated conditional expectations jensen inequality sup dmx log max log dmx log last inequality used max max control proof parts theorem van der vaart except use instead kgn log kgn iikh log collecting results log sup dmx dmx log giving required result proof theorem prove conditions theorem let first consider convergence finite dimensional distributions without loss generality consider clt assumption together moment condition envelope implies dmx apply theorem obtain convergence finite dimensional distributions cnd property assumption condition theorem satisfied let prove asymptotic define lemma sup dmx log noting contained assumption last bound vanishes thus asymptotic follows condition proof theorem assumption max sup hence desired result follows applying theorem references ldous agleson mixing stability limit theorems annals probability ndrews empirical process methods econometrics handbook econometrics elsevier aldi inott normal approximations distributions terms dependency graphs annals probability arab lbert emergence scaling random networks science erti ratelli igo limit theorems empirical processes based dependent data electronic journal probability hou estimating sparse precision matrix optimal rates convergence adaptive estimation annals statistics anen chwartz ong estimating local interactions among many agents observe neighbors working paper hen hao normal approximation local dependence annals probability awid conditional independence statistical theory journal royal statistical society udley extended wichura theorem definitions donsker class weighted empirical distributions probability banach spaces beck dudley hahn kuelbs marcus springer ahn kuersteiner azzocco central limit theory combined time series eyde martingale limit theory application academic press new york usa uschgy stable convergence stable limit theorems springer media new york usa bragimov innik independent stationary sequences random variables groningen janson normal convergence higher semiinvariants applications sums dependent random variables random graphs annals probability large deviations sums partly dependent random variables random structures algorithms enish rucha central limit theorems uniform laws large numbers arrays random fields journal econometrics kallenberg foundations modern probability springer new york kuersteiner rucha limit theory panel data models cross sectional dependence sequential exogeneity journal econometrics auritzen graphical models springer new york auritzen awid arsen eimer independence properties directed markov fields networks eung treatment spillover effects network interference working paper einshausen graphs variable selection lasso annals statistics enrose random geometric graphs oxford university press oxford ollard empirical processes theorey applications regional conference series probability statistics volume institute mathematical statistics hayward usa inott otar multivariate clt local dependence log rate applications multivariate graph related statistics journal multivariate analysis ong measuring graph concordance locally dependent observations tein bound error normal approximation distribution sum dependent random variables proceedings sixth berkeley symposium mathematical statistics probability ikhomirov convergence rate central limit theorem weakly dependent random variables theory probability applications van der vaart new donsker classes annals statistics van der vaart ellner weak convergence empirical processes springer new york usa ellner empirical processes theory applications special topics course notes delft technical university nonlinear system theory another look dependence proceedings national academy science usa uan results following conditional characteristic functions communications statistics theory methods
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tight lower bounds planted clique sos program mar prasad raghavendra tselil abstract give lower bound sdp relaxation planted clique problem specifically show graph high probability feasible point sos relaxation clique problem objective value program distinguish random graph random graph planted clique size bound tight build works deshpande montanari meka give lower bounds respectively improve results making perturbation sdp solution proposed work showing perturbation remains psd objective value approaches independent work hopkins kothari potechin obtained similar lower bound sos relaxation berkeley prasad supported nsf career award nsf alfred sloan fellowship berkeley tschramm supported nsf graduate research fellowship nsf award introduction maximum clique problem input consists graph goal find largest subset vertices connected maximum clique problem approximate within karp suggested average case version maximum clique problem random graphs drawn distribution heuristic argument shows graph clique size log high probability given graph choose random vertex choose one neighbors choose vertex adjacent continue process vertex adjacent clique log steps probability another vertex added log steps process terminates heuristic argument made precise one show greedy algorithm find clique size log instance polynomial time indeed work shown largest clique instance actually size log high probability clique size log easily found polynomial time using heuristic previous paragraph efficient algorithm finding clique size log much elusive seminal paper probabilistic analysis combinatorial algorithms karp asked whether exists algorithm finding clique size log fixed constant despite extensive efforts algorithmic progress question since planted clique problem natural variant problem wherein input promised either graph drawn graph clique size planted within vertices goal algorithm distinguish two distributions log simple time algorithm distinguishes two distributions algorithm simply tries subsets log vertices looking clique random graph cliques size log one planted distribution clearly planted clique problem becomes easier planted clique size increases yet algorithms known problem result alon uses random matrix theory argue looking spectrum adjacency matrix suffices solve decision problem works show one able efficiently calculate injective tensor norm certain random tensor extending spectral algorithm one would algorithm however known algorithm efficiently computes injective tensor norm tensor fact computing inective tensor norm hard approximate general case algorithmic progress slow success proving strong lower bounds planted clique problem within specific algorithmic frameworks first bound given jerrum showed class markov chain monte carlo algorithms require superpolynomial number steps find clique size log fixed instance feige krauthgamer showed sdp hierarchy needed find hidden clique size feldman show planted bipartite clique problem statistical algorithm distinguish polynomial number queries random planted cases recently effort replicate results sos hierarchy powerful sdp hierarchy recent work achieves bound sos hierarchy demonstrating feasible solution sdp relaxation large enough objective value random case work achieves sharper lower bound sdp solution rounds counterexample kelner may found demonstrates analysis tight integrality gap instance within logarithmic factors line work brings fore question sos relaxation solve planted clique problem lower bounds known sdp relaxations planted clique sos relaxations general much powerful relaxations example instances unique games hard poly log log sdp hierarchy recent work shown instances solved sos hierarchy moreover even sos relaxation proves surprisingly powerful applications first work barak shows degree sos relaxation certify hypercontractivity low degree polynomials hypercube argument reason hard instances sdp hierarchies constructed via noisy hypercube gadgets easily refuted sos hierarchy second sos relaxation certify norm random subspace dimension bounded constant high probability choice subspace problem superficial similarities planted clique problem work make modest progress towards lower bound sos relaxations planted clique obtaining nearly tight lower bound sos relaxation corresponding two rounds precisely main result following theorem suppose probability exists feasible solution degree objective value polylog note work result tight logarithmic factors independent work hopkins kothari potechin obtained similar result work builds heavily previous work meka potechin wigderson deshpande montanari since sdp solution constructed works infeasible introduce modified sdp solution objective value prove random graph solution feasible high probability parameter setting objective value becomes sdp solutions violate psdness constraint equivalently exists set test vectors feasible sdp solution perturbation add spectral mass solution along vectors set enforce linear constraints sdp program notation use symbol denote psd ordering matrices saying psd wish hide constant factors clarity use denote constant made effort optimize logarithmic factors work delicate analysis required logarithmic factors certainly possible vector denote vector def def def def normalized version vector use denote drop subscript clear context notation times closely follow notation paper builds results recycle many bounds convenience use shorthand log abuse notation using refer binomial coefficient set also use notation refer union sets give vector identify entries unordered pairs elements throughout paper unless otherwise stated work fixed instance denote centered ith row adjacency matrix jth entry equal edge equal edge equal use aij denote jth index organization section give background material sos relaxation problem describe integrality gap deshpande montanari planted clique problem explain obstacle face reach integrality gap value describe integrality gap instance motivating construction using obstacle witness give overview proof integrality gap instance feasible section prove witness psd completing proof feasibility section contains concentration bounds random matrices arise within proofs proof reuse several bounds proved deshpande montanari far possible restate claims used convenience appendix list claims deshpande montanari use paper preliminaries proof overview section describe sos relaxation sdp give background witness describe modified witness give overview proof witness feasible difficult part showing witness psd full proof feasibility deferred section sos relaxation max clique degree sos relaxation maximum clique problem semidefinite program whose variables subset variable indicates whether contained maximum clique graph vertices program described follows maximize whenever subject clique instructive think variable pseudoexpectation product indicator variables pseudomoment clique intuitively constraints sdp force solution behave somewhat like moments probability distribution integral solutions although correspond moments true distribution hence term pseudomoment background see pseudmoment interpretation sdp solution motivates choice witness prior work example may notice objective function view simply pseudoexpectation size planted clique clique sdpval denotes optimum value sdp relaxation graph clearly sdpval least size maximum clique order prove lower bound degree sos relaxation sufficient argue overwhelming probability sdpval significantly larger maximum clique random graph amounts exhibiting feasible sdp solution large objective value overwhelming fraction graphs sampled formally show following theorem formal version theorem exists absolute constant sdpval log obtain theorem constructing point witness proving point feasible high probability defer description witness definition definition spend section section motivating construction however curious reader may skip ahead definition require knowledge additional notation witness henceforth fix graph sampled work meka potechin wigderson deshpande montanari construct essentially sdp solution sos relaxation sdp solution assigns clique size value depends size case essence solution takes advantage independence instance motivating observation variable thought pseudoexpectation indicator subclique planted clique idea make pseudoexpectation indicator consistent true expectation distribution clique size planted uniformly random within instance thus every vertex clique uniform probability principle applied edges traingles planted clique clique planted clique general idea sdp solution formally sdp solution specified four parameters set vertices indicator subgraph induced clique parameters determine value objective function feasibility solution convention define easy check solution satisfies linear constraints sos program since assigns values cliques key difficulty showing matrix psd appropriate choice parameters order show sufficient show gij gij indicator presence edge words matrix entry proportional indicator whether clique indicator whether subgraph bipartite clique bipartitions easy see matrix obtained dropping rows columns corresponding hence notice random matrix whose entries depend edges random graph risk approach previous works broadly summarized follows expectation show expected matrix sufficiently large positive eigenvalues concentration show high probability choice noise matrix bounded eigenvalues ensure sketch key details argument matrix decomposed blocks nab deshpande montanari use schur complements reduce problem proving facts blocks nab specifically show following lemma lemma let matrix defined let submatrix corresponding monomials psd significant challenge argue holds high probability fact inequality holds sdp solution high probability parameters objective value expected matrix expected matrix symmetric respect permutations vertices forms association scheme see virtue eigenvalues eigenspaces well understood particular following proposition immediate consequence theory association schemes proposition proposition three eigenspaces projections spaces respectively eigenvalues given def def def eigenspaces given span span used denote space vectors real numbers indexed subsets size deviation expectation given lower bound eigenvalues expected matrix next step would bound spectral norm noise however since eigenspaces stratified given one large eigenvalue several much smaller eigenvalues standard matrix concentration suffice give tight bounds along eigenspaces overcome deshpande montanari split precisely let split includes multilinear entries includes entries entries formally otherwise spectral norm matrix eigenspaces carefully bounded lemma proposition probability least following bounds hold kkk proposition lemma sufficient conclude parameter choices correspond planted clique size precisely argue high probability sufficient argue deshpande montanari fix parameter using proposition lemma matrix inequality becomes shown hold eventually necessary show stronger achieved showing bounds spectra refer reader details arguments problematic subspace sdp solution described ceases psd corresponds objective value specific obstruction arises precisely bottom principal minor yields constraint forcing clear problematic vectors precisely large aligns subspace fact identify specific subspace problematic solution describe subspace let fix notation define random variable aij otherwise follow convention aii def lemma let vectors defined aik let def span probability least proof immediate observation various matrix norm bounds specifically lemma lemma observation defer detailed proof appendix since lemma implies vectors large singular values within subspace furthermore show following lemma clearly articulates sole obstruction identified matrices matrices matrices corresponding diagonalizing according three eigenspaces expectation analagous decomposing quadratic form lemma suppose satisfies min probability proof fix recall write matrix sufficient show using proposition condidition holds using proposition lemma lemma write psd given bounds conditions see one shows principal minors psd hand write appeal fact quadratic since condition easily seen quadratic form always implying diagonalize according subspaces immediate corollary proof lemma following corollary hypothesis lemma probability corollary consequence fact corrected witness suppose unconstrained matrix wish modify little possible ensure given test vector natural update make take wwt forpa suitably chosen would suggest creating new sdp solution setting unfortunately sos sdp relaxation certain hard constraints namely entries fixed zero moreover entry must depend setting sdp solution matrix ati would almost certainly violate constraints thus natural consider multiplicative updates entries matrix clearly preserve zero entries matrix specifically idea would consider update form diagonal matrix entries given vector matrix significantly large eigenvalue along matrix multiplicative update similar effect additive update wwt norm final error term relatively small recall setting deshpande montanari sdp solution matrix large eigenvalue along formally describe sdp solution first matrix according intuition given set pseudomoments definition corrected sdp witness matrix view let defined define diagonal matrix diagonal define restriction entries also let log sdp witness matrix defined projection zeros rows columns corresponding pairs definition corrected sdp witness pseudomoments view let log let set parameters fixed later subset let graph induced subset vertices define edges clique clique otherwise factor chosen depending choice set later ensure final moments matrix psd proposition log probability least solution violate linear constraints planted clique sdp proof first whenever entries satisfy constraints sdp given aij clique notice clique depends moreover aij sum iid mean random variables therefore satisfies aij log simple union bound subsets probability least shows remains verify verifying schur complement conditions analogous submatrix one consider corresponding expression follows submatrix def matrix diagonal matrix corresponding nonmultilinear entries entries corresponding monomials like matrix matrices unchanged must simply verify concludes proof overview section verify schur complement conditions prove main result section give random matrix concentration results upon rely throughout proof proof main result allow section demonstrate conclude solution matrix psd therefore feasible point sos relaxation parameters proceed convenient parametrize choice particular useful fix def def def def two parameters finally fix setting parameters eigenvalues proposition bounded def def def convenient also use shorthand def proving make first step towards verifying schur complement conditions lemma specifically show following stronger claim showing theorem least log following holds probability proof fix definition apply lemma term corollary dropping usingdi define appeal matrix concentration bounds show section first probability vectors nearly orthogonal therefore form wellconditioned basis subspace see lemma also vectors negligible projection eigenspaces implies overwhelming probability see lemma finally dimensional space singular values moreover multiplying left right acts random linear random change basis intuitively suggests eigenvalues roughly fact probability see lemma substituting bounds get log lemma probability least kqk substituting bound kqk along finishes proof choice parameters details presented completeness log log clearly log log bounding singular values deshpande montanari observe towards bounding eigenvalues following properties regards spaces lemma consequence propositions let orthogonal projector space spanned let matrix sufficiently large probability unfortunately bound insufficient purposes require bound deviation expectation fact outside problematic subspace show much better behaved proposition let let projector space proban bility least following holds every kxk proof lemma thus may work exclusively difference expectation convenience let fix inspection entry given polynomial aac abc aac abc thus columns latter two terms eliminated span col lemma bound logn kxt applied inequality conclusion follows noting kxt kxt kxt lemma bounding singluar values splitting matrix according bound bounds eigenspaces theorem let defined choice probability proof define matrix uab verify choice conditions lemma met conclude probability let order simplify computations use following observation observation given vectors xti axi proof set inequality reduces kyi immediate consequence triangle inequality inequality using observation xta xtb xtc yut zut simplify calculations make dominant terms apparent let fix wherein log first observe setting parameters terms xta xtb xtc write kqn zut finally yut conclusion follows grouping projections taking dominating terms grows proof theorem choice given probability proof recall theorem probability least choice parameters conclusion theorem implies desired details calculation spelled sake completeness verify coefficient projector space finally concludes proof proof main theorem finally components needed prove theorem proof theorem first recall independent choice sdp solution defined definition violate linear constraints shown proposition meet program constraints remains show choice given solution psd solution matrix principal submatrix implies prove satisfies schur complement conditions lemma high probability observing apply theorem states choice term apply lower bound eigenvalues given lemma state long probability choice may conclude therefore union bound conditions lemma satisfied probability solution satisfies psdness constraint remains objective value objective value simply concluding proof concentration projected matrices section give bounds spectra random matrices part correction term though able recycle many spectral bounds deshpande montanari modification witness introduce new matrices also require description norm bounds obtain bounds employing trace power method trace power method uses fact symmetric matrix even powers bounding sufficiently large essentially obtain bounds infinity norm vector eigenvalues bound spectral norm matrix formal statement follows proof given appendix completeness lemma suppose random matrix satisfies even integer constants log log concentration proofs consist matrix question obtaining bound expression sum products along closed paths length entries case entries random matrix low degree polynomials random variables aij aij centered random variable indicates whether edge part random graph thus written polynomial random variables aij since random variables aij centered aij almost terms vanish zero nonzero terms precisely monomials every variable appears even multiplicity purpose moment calculations borrow much terminology work deshpande montanari every monomial random variables aij corresponds labelled graph consists graph labelling maps vertices labelling contributes nonzero expectation every pair appears even number times label edge problem bounding reduces counting number number contributing labeled graphs example given matrix may bound number vertices edges term function case may use following proposition allows bound number graphs every variable aij corresponding edge vertices appears least twice proposition let multigraph let labelling pair appears even number times label edge connected components proof form new graph identifying nodes label thus number nodes number labels collapse parallel edges form graph since labelled edge appears least twice number edges half number nodes thus labels number edges plus number connected components tight forest thus number distinct labels number components apply lemma well simple inductive arguments bound number contributing terms matrices question allows bound norms give concentration proofs following subsection proofs concentration let instance preceeding sections define vector preceeding sections define setting coordinates corresponding unordered pairs aic aid continue use notation notation diag begin lemma shows close orthogonal basis def lemma ati probability least proof definition vectors form basis subspace let matrix whose ith row use matrix concentration analyze eigenvalues rrt identical nonzero eigenvalues entry rrt hai hai precisely idn matrix diagonal equal hai entry let idn use trace power method prove entry given aip aiq ajp ajq aip aiq ajp ajq hai expression sum monomial products variables aip monomial product corresponds labelling graph entry mij corresponds sum links link cycle length vertices opposite ends cycle necessarily distinct vertices opposite ends cycle refer center vertices peripheral vertices link edge link weighted auv since aii every contributing labelling never case one monomial product summation corresponds labelling graph cycle links edges total vertices quantity equal sum labellings taking expectation terms contain variable auv multiplicity expectation thus equal number labellings every edge appears even number times prove contributing labelling unique vertex labels proceed induction length cycle base case cycle two links inspection cycle labels base case holds consider cycle length every label appears twice done since vertices thus must vertex appears peripheral vertex whose label repeat since two center vertices neighboring single peripheral vertex label contributing term exists center vertex whose label repeat must matching neighbors every vertex matched vertex label identify vertices remove two neighbors graph leaving cycle length removed one label graph induction hypothesis applies total labels desired thus unique labels contributing term may thus conclude applying lemma probability least therefore idn may conclude eigenvalues rrt implies since range finally probability desired following lemma allows approximate projector matrix easy describe use matrix approximation projector later proofs lemma let projection vector space let matrix defined follows proof write basis take summation outer products argue summation approximates vectors basis otherwise let matrix whose ith column given two hvi notice eigenvalues vit equal eigenvalues idn therefore matrices column row spaces vit let conclusion follows explicitly calculate entries require fact lies mostly outside prove following lemma lemma probability least log log proof call apply trace power method lemma lemma may exchange letting cyclic property trace consider expression let chain consists set quadruples identify mod let denote set chains size depending whether one common following link chain quantity consists cycles links link star outer vertices center vertex vertices link must least one vertex common next link link edges cycle connected graph see figure illustration dashed lines indicate vertex equality edges term product factor thus due scaling entries contributing terms correspond every edge variable product even multiplicity contributing term connected graph edges vertices every labeled edge appears twice may apply proposition conclude labels cycle thus applying lemma conclude desired probability log combine lemmas bound norm one final matrix arises computations theorem lemma probability idn proof begin replacing lemma replaced convenient form vector obtain second line applied obtain third line used fact obtain final line used fact kdi kxk second term overwhelming probability lemma remains bound first term end apply lemma replace let entry form aia aib aic aid aja ajb ajc ajb matrix thus see matrix whose entries form aia aib aja ajb matrix actually equal matrix lemma kbk probability thus conclude probability kbk gives desired result acknowledgements thank satish rao many helpful conversations also greatfully acknowledge comments anonymous reviewers helping improve manuscript references noga alon michael krivelevich benny sudakov finding large hidden clique random graph random struct algorithms boaz barak sum squares upper bounds lower bounds open questions lecture notes fall boaz barak fernando aram wettroth harrow jonathan kelner david steurer yuan zhou hypercontractivity proofs applications proceedings symposium theory computing conference stoc new york usa may bella bollobas paul cliques random graphs mathematical proceedings cambridge philosophical society charles brubaker santosh vempala random tensors planted cliques approximation randomization combinatorial optimization algorithms techniques international workshop approx international workshop random berkeley usa august proceedings irit dinur klaus jansen joseph naor rolim eds lecture notes computer science vol springer yash deshpande andrea montanari improved lower bounds hidden clique hidden submatrix problems proceedings conference learning theory colt paris france july peter elad hazan satyen kale eds jmlr proceedings vol improved lower bounds hidden clique hidden submatrix problems corr vitaly feldman elena grigorescu lev reyzin santosh vempala ying xiao statistical algorithms lower bound planted clique electronic colloquium computational complexity eccc uriel feige robert krauthgamer finding certifying large hidden clique semirandom graph random struct algorithms probable value relaxations maximum independent set siam journal computing alan frieze ravi kannan new approach planted clique problem iarcs annual conference foundations software technology theoretical computer science fsttcs december bangalore india geoffrey grimmett colin mcdiarmid colouring random graphs mathematical proceedings cambridge philosophical society johan clique hard approximate within annual symposium foundations computer science focs burlington vermont usa october samuel hopkins pravesh kothari aaron potechin sos planted clique tight analysis mpw moments degrees optimal lower bound degree four aram wettroth harrow ashley montanaro testing product states quantum merlinarthur games tensor optimization acm mark jerrum large cliques elude metropolis random struct algorithms richard karp probabilistic analysis combinatorial search algorithms algorithms complexity new directions recent results subhash khot improved inaproximability results maxclique chromatic number approximate graph coloring annual symposium foundations computer science focs october las vegas nevada usa subhash khot rishi saket sdp integrality gaps local ell annual ieee symposium foundations computer science focs october atlanta georgia usa david matula largest clique size random graph tech report southern methodist university dallas raghu meka aaron potechin avi wigderson lower bounds planted clique proceedings annual acm symposium theory computing stoc portland usa june prasad raghavendra david steurer integrality gaps strong sdp relaxations unique games annual ieee symposium foundations computer science focs october atlanta georgia usa matrix norm bounds deshpande montanari appendix give completeness list bounds proven deshpande montanari included body space expository considerations definition let disjoint define matrices follows abd aad abd aad abc aac aad abc abd aac abc abd aad abc abd letting matrix projector otherwise def define finally define deshpande montanari notice since defined aii since terms never considered however terms cleaner bound spectral norm subspace deshpande montanari provide trace power method bounds difference norm lemma lemma probability least deshpande montanari use trace power method bound norm bounding norms individually matrices behavior lemma lemmas probability select larger eigenvalues lemma lemmas probability also give short proof observation deshpande montanari states vanish projected observation lemmas let projector always similarly proof proof follows noting sums consider def look entry indexed vector let disjoint pair definition characterization proposition vector conclusion follows similar proof holds matrix finally use bound norm matrix difference entries lemma lemma let restriction entries indexed sets size probability least kkk also require bounds matrices used schur complement steps bounds deshpande montanari suffice since modify moments order less lemma consequence proposition define orthogonal projection space spanned suppose satisfies probability least additional proofs prove lemma follows almost immediately bounds proof lemma using matrices definition observation used fact columns lie thus bounds lemma bounds lemma projection conclusion follows prove trace power method works completeness proof lemma proof follows application markov inequality even applied stirling approximation last step choosing log log completes proof
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microwave tomographic imaging cerebrovascular accidents using computing jul tourniera aliferisc bonazzolid buhane darbasf doleand hechta joliveth kanfoudc migliaccioc natafa pichotc semenovi laboratoire lions umr cnrs sorbonne upmc paris france epc alpines paris france laboratoire leat umr cnrs nice sophia antipolis sophia antipolis france laboratoire umr cnrs nice sophia antipolis nice france umr cnrs sorbonne paris france lamfa umr cnrs picardie jules verne amiens france dept maths stats university strathclyde glasgow irit umr cnrs toulouse france emtensor gmbh techgate vienna austria abstract motivation work detection cerebrovascular accidents microwave tomographic imaging requires solution inverse problem relying minimization algorithm example successive iterations consist repeated solutions direct problem reconstruction algorithm extremely computationally intensive makes use efficient parallel algorithms computing feasibility type imaging conditioned one hand accurate reconstruction material properties propagation medium hand considerable reduction simulation time fulfilling two requirements enable rapid accurate diagnosis mathematical numerical point view means solving maxwell equations regime appropriate domain decomposition methods naturally adapted parallel architectures keywords inverse problem scalable preconditioners maxwell equations microwave imaging preprint submitted parallel computing july introduction stroke also known cerebrovascular accident disturbance blood supply brain caused blocked burst blood vessel consequence cerebral tissues deprived oxygen nutrients results rapid loss brain functions often death strokes classified two major categories ischemic strokes hemorrhagic strokes acute ischemic stroke blood supply part brain interrupted thrombosis formation blood clot blood vessel embolism elsewhere body hemorrhagic stroke occurs blood vessel bursts inside brain increasing pressure brain injuring brain cells two types strokes result opposite variations dielectric properties affected tissues quickly one detect characterize stroke fundamental importance survival patient quicker treatment reversible damage better chances recovery moreover treatment ischemic stroke consists thinning blood anticoagulants fatal stroke hemorrhagic therefore vital make clear distinction two types strokes treating patient moreover ideally one would want monitor continuously effect treatment evolution stroke hospitalization two used imaging techniques strokes diagnosis mri magnetic resonance imaging scan computerized tomography scan one downsides travel time patient home hospital lost moreover cost lack portability mri harmful character scan make unsuitable continuous monitoring hospital treatment motivated study additional technique microwave tomography measurement system lightweight thus transportable acquisition data harmless faster mri hence imaging modality could used emergency unit monitoring hospital frequencies order ghz tissues well differentiated imaged basis dielectric properties first works microwave imaging lin clarke works followed almost always synthetic simplified models new devices currently designed studied emtensor gmbh figure left operating principle diagnosis apparatus middle imaging chamber prototype emtensor courtesy emtensor company right corresponding simulation domain vienna austria purpose work solve parallel inverse problem associated maxwell equations model electromagnetic waves propagation dielectric properties brain tissues patient yield image could used rapid diagnosis brain strokes simulation results presented work obtained imaging system prototype developed emtensor gmbh see figure composed rings rectangular waveguides around metallic cylindrical chamber diameter total height head patient inserted chamber shown figure left imaging chamber filled matching solution membrane used isolate head antenna successively transmits signal fixed frequency typically ghz electromagnetic wave propagates inside chamber object imaged according electromagnetic properties retrieved data consist scattering parameters measured receiving antennas used input inverse problem raw data wirelessly transferred remote computing center hpc machine compute images patient brain formed images quickly transmitted computing center hospital see figure paper organized follows section direct problem maxwell equations form suitable boundary conditions introduced section briefly describe discretization method edge finite elements section devoted introduction domain decomposition preconditioner section explain compute scattering coefficients also compare measurement data figure design concept diagnosis technology courtesy emtensor company obtained emtensor coefficients computed simulation introduce inverse problem section section dedicated numerical results first perform strong scaling analysis show effectiveness domain decomposition method present results obtained solving inverse problem realistic configuration noisy synthetic data generated using numerical brain model simulated hemorrhagic stroke finally conclude paper section give directions future research direct problem let domain represent imaging chamber see figure right consider heterogeneous dissipative linear isotropic dielectric medium dielectric permittivity electrical conductivity transmitting antenna emitting time periodic signal angular frequency complex amplitude associated electric field solution following second order maxwell equation permeability free space note coefficient equation written next sections consider relative complex permittivity given relation permittivity free space let unit outward normal equation equipped perfectly conducting boundary conditions metallic walls impedance boundary conditions outer section transmitting waveguide receiving waveguides see propagation wavenumber along waveguide corresponding propagation fundamental mode equation imposes incident wave corresponds excitation fundamental mode waveguide hand equation corresponds first order absorbing boundary condition approximating transparent boundary condition outer section receiving waveguides bottom chamber metallic impose impedance boundary condition top chamber end following boundary value problem transmitting antenna find let curl curl space square integrable functions whose curl also square integrable transmitting antenna variational form problem reads find edge finite elements edge elements finite elements particularly suited approximation electromagnetic fields indeed given tetrahedral mesh computational domain finite dimensional subspace generated basis functions included curl since tangential component across faces shared adjacent tetrahedra continuous thus match continuity properties electric field elements called edge elements basis functions associated edges mesh precisely tetrahedron local basis functions associated oriented edges follows barycentric coordinates point respect node note polynomial degree since barycentric coordinates polynomials degree gradients constant basis functions vector functions need one set unknowns approximate components field three sets unknowns one component field instead required usual nodal scalar finite elements finite element discretization variational problem obtained taking test functions edge finite element space mesh looking solution space find locally tetrahedron write discretized field linear combination coefficients basis functions associated edges coefficients unknowns resulting linear system edge finite elements degree coefficients interpreted circulations along edges tetrahedra tangent vector edge length length consequence fact basis functions duality degrees freedom given circulations along edges domain decomposition preconditioning finite element discretization variational problem produces linear systems auj transmitting antenna however matrix combined fact underlying pde indefinite highlights need robust efficient preconditioner employ domain decomposition preconditioners extensively described naturally suited parallel computing domain decomposition preconditioner presented following let mesh computational domain first partitioned meshes using standard graph partitioners scotch metis positive integer overlapping decomposition defined recursively follows obtained including tetrahedra plus adjacent tetrahedra note number layers overlap let edge finite element space defined local edge finite element spaces defined consider restrictions local partition unity rit algebraically speaking global number unknowns numbers unknowns local finite element space boolean matrix size diagonal matrix size note rit transpose matrix gives extension using matrices one define following preconditioner called optimized restricted additive schwarz preconditioner oras moras rit local operators corresponding subproblems impedance boundary conditions ikn wavenumber boundary conditions first used transmission conditions interfaces subdomains local matrices oras preconditioner make use efficient transmission boundary conditions submatrices arit original restricted additive schwarz ras preconditioner important note direct solver used compute action multiple vectors done single forward elimination backward substitution details solution linear systems multiple sides given section preconditioner moras naturally parallel since assembly requires concurrent factorization typically stored locally different processes distributed computing context likewise applying distributed vector requires communications neighboring subdomains local forward elimination backward substitution see chapter detailed analysis partition unity construction partition unity intricate especially edge finite elements starting point construction partition unity functions classical linear nodal finite element whose degrees freedom values nodes mesh first define function continuous piecewise linear function support contained nodes nodes function defined continuous piecewise linear function support contained discrete value degree freedom evaluated thus discrete continuous level remark function also derivative equal zero border essential good convergence robin boundary conditions chosen transmission conditions interfaces subdomains indeed property satisfied continuous version oras algorithm equivalent lions algorithm see note practical implementation functions constructed locally relevant contribution removes dependency global mesh could otherwise problematic large scales degrees freedom finite elements associated edges mesh finite elements build geometric partition unity based support degrees freedom edges mesh entries diagonal matrices obtained degree freedom interpolating piecewise linear function midpoint corresponding edge partition unity property pns satisfied since software stack operators related domain decomposition method easily generated using finite element languages dsl use http since already proven enable simulations using overlapping schwarz methods used combination library hpddm unified framework domain decomposition methods https hpddm implements several domain decomposition methods ras oras feti bnn uses multiple levels parallelism communication subdomains based message passing interface mpi computations subdomains executed several threads calling optimized blas libraries intel mkl direct solvers like pardiso domain decomposition methods naturally offer good parallel properties distributed architectures computational domain decomposed subdomains concurrent computations performed coupling subdomains requires communications computing nodes via messages strong scalability oras preconditioner implemented hpddm direct problem presented section assessed section computing scattering parameters order compute numerical counterparts reflection transmission coefficients obtained measurement apparatus imaging chamber shown figure use following formula appropriate case waveguides sij fundamental mode receiving waveguide solution problem waveguide transmits signal denotes complex conjugate sij transmission coefficients sjj reflection coefficients gathered scattering matrix also called compare coefficients computed simulation set measurements obtained emtensor test case imaging chamber filled homogeneous matching solution electric permittivity matching solution chosen emtensor order minimize contrasts waveguides different brain tissues choice conductivity matching solution compromise minimization reflection artifacts metallic boundaries desire best possible ratio relative complex permittivity matching solution frequency ghz relative complex permittivity inside waveguides set experimental data hand given emtensor consists transmission coefficients transmitting antennas second ring top figure shows normalized magnitude phase degree complex coefficients sij corresponding transmitting antenna second ring top receiving antennas middle ring note measured coefficients available receiving antennas magnitude calculated normalization done dividing every transmission coefficient transmission coefficient corresponding receiving antenna directly opposite transmitting antenna thus set since normalize respect coefficient lowest expected magnitude magnitude simulation measurements magnitude receiver number simulation measurements phase degree receiver number figure normalized magnitude top phase bottom transmission coefficients computed simulation measured experimentally transmission coefficients displayed figure larger see transmission coefficients computed simulation good agreement measurements inverse problem inverse problem consider consists finding unknown dielectric permittivity conductivity solutions problem lead corresponding scattering parameters sij coincide measured scattering parameters sijmes following present inverse problem continuous setting clarity let unknown complex parameter inverse problem let denote solution direct problem dielectric permittivity conductivity corresponding scattering parameters denoted sij sij misfit parameter data defined following functional sij sijmes sijmes classical way solving inverse problem consists minimizing functional respect parameter computing differential given arbitrary direction yields mes sij sij solution following linearized problem use adjoint approach order simplify expression allow compute gradient efficiently discretization number computations independent size parameter space considering variational formulation problem test function integrating parts get introducing solution following adjoint problem mes sij sij get sij sijmes finally differential computed compute gradient use local optimization algorithm numerical results presented section obtained using algorithm note every evaluation requires solution state problem computation gradient requires solution well solution adjoint problem moreover state adjoint problems use operator therefore computation gradient needs assembly one matrix associated domain decomposition preconditioner numerical results reconstruction hemorrhagic stroke synthetic data presented next section functional considered numerical results slightly different add normalization term pair well tikhonov regularizing term sij sijmes sijempty sijempty refers coefficients computed simulation empty chamber chamber filled homogeneous matching solution described previous section object inside way contribution pair misfit functional normalized depend amplitude coefficient vary greatly pairs displayed figure tikhonov regularizing term aims reducing effects noise data regularization parameter chosen empirically obtain visually good compromise reducing effects noise keeping reconstructed image pertinent calculations carried section accommodated straightforward manner definition functional usually case medical imaging techniques reconstruction done layer layer imaging chamber emtensor study paper one layer corresponds one five rings antennas allows exhibit another level parallelism solving inverse problem independently five rings parallel precisely inverse problems solved domain truncated around corresponding ring antennas containing two rings one ring one ring impose absorbing boundary conditions artificial boundaries truncated computational domain inverse problem coefficients sij transmitting antennas corresponding ring taken account consider antennas transmitters antennas receivers time solution seconds setup solve linear speedup subdomains figure strong scaling experiment colors indicate fraction total time spent setup solution phases number gmres iterations reported parentheses numerical results results paper obtained curie system composed nodes made two intel sandy bridge processors clocked ghz interconnect infiniband qdr full fat tree mpi implementation used bullxmpi version intel compilers math kernel library version used binaries shared libraries linear algebra backend dense computations preconditioners assembled hpddm require use sparse direct solver following experiments using either pardiso intel mkl mumps linear systems resulting edge finite elements discretization solved gmres oras implemented hpddm gmres algorithm stopped unpreconditioned relative residual lower first perform strong scaling analysis order assess efficiency preconditioner solve inverse problem realistic configuration noisy synthetic data generated using numerical brain model simulated hemorrhagic stroke scaling analysis using domain decomposition preconditioner solve direct problem corresponding setting section chamber filled setup solve iterations speedup figure strong scaling experiment timings seconds setup solution phases homogeneous matching solution consider side corresponding transmitting antenna second ring top given fine mesh domain composed million tetrahedra increase number mpi processes solve linear system million doubleprecision complex unknowns yielded discretization maxwell equation using edge elements global unstructured mesh partitioned using scotch local solver pardiso intel mkl use one subdomain two openmp threads per mpi process results reported figure illustrated figure plot time solution including setup solution phases subdomains setup time corresponds maximum time spent factorization local subproblem matrix subdomains solution time corresponds time needed solve linear system gmres able obtain good speedups cores subdomains curie superlinear speedup subdomains reconstruction hemorrhagic stroke synthetic data subsection assess feasibility microwave imaging technique presented paper stroke detection monitoring numerical example realistic configuration use synthetic data corresponding numerical model virtual human head simulated hemorrhagic stroke input inverse problem numerical model virtual head comes mri tomographic images consists complex permittivity map data points figure left shows sagittal section head simulation head immersed imaging chamber shown figure right order simulate hemorrhagic stroke synthetic stroke added form ellipsoid value complex permittivity increased figure left sagittal section brain right numerical head immersed imaging chamber simulated hemorrhagic stroke test case value permittivity ellipsoid taken mean value relative permittivity original healthy brain relative permittivity blood frequency ghz imaging chamber filled matching solution relative permittivity matching solution chosen emtensor explained section equal frequency ghz real setting special membrane fitting shape head used order isolate head matching medium take membrane account synthetic test case synthetic data obtained solving direct problem mesh composed million tetrahedra corresponding approximately points per wavelength consist transmission reflection coefficients sij calculated simulated electric field subsequently add noise real imaginary parts coefficients sij additive gaussian white noise different values real imaginary parts noisy data used input inverse problem furthermore assume priori knowledge input data set initial guess inverse problem homogeneous matching solution everywhere inside chamber use piecewise linear approximation unknown parameter defined mesh used solve state adjoint problems purpose parallel computations partitioning introduced domain decomposition method also used compute store locally subdomain every entity involved inverse problem parameter gradient figure top row imaginary part exact permittivity used produce noisy data input inverse problem time evolution simulated hemorrhagic stroke indicated black arrow size ellipsoid middle right column respectively bottom row corresponding reconstructions obtained taking account first ring transmitting antennas figure shows imaginary part exact reconstructed permittivity three steps evolution hemorrhagic stroke healthy brain left column large stroke right column increasing size ellipsoid value permittivity raised simulates evolution stroke three reconstructions figure corresponds solution inverse problem truncated domain containing first two rings antennas top coefficients sij corresponding transmitting antennas first ring taken account reconstruction starts initial guess consisting homogeneous matching solution obtained reaching convergence criterion value cost functional takes around iterations algorithm figure gathers results strong scaling experiment consists time minutes linear speedup mpi processes figure strong scaling experiment total time needed obtain third reconstructed image shown figure solving inverse problem corresponding third reconstructed image figure increasing number mpi processes report total computing time needed obtain reconstructed image experiment use one subdomain one openmp thread per mpi process mesh computational domain composed tetrahedra corresponding approximately points per wavelength note evaluating functional gradient requires solution linear system sides one side per transmitter introduces trivial level parallelism since solution corresponding side computed independently however considering finite number available processors tradeoff parallelism induced multiple sides parallelism induced domain decomposition method additionally solve multiple sides simultaneously using method implemented inside gmres consists fusing multiple arithmetic operations corresponding side products dot products resulting higher arithmetic intensity scaling behavior algorithm respect number sides nonlinear scaling behavior domain decomposition method respect number subdomains thus given number processors find optimal tradeoff parallelizing respect number subdomains sides trial error example best computing time mpi processes achieved using domain decomposition communicators concurrent direct solves subdomains treating sides figure shows generate image total computing time less minutes seconds using cores curie preliminary results encouraging already able achieve satisfactory reconstruction time perspective using imaging technique monitoring allows clinicians obtain almost instantaneous images demand although reconstructed images feature complex heterogeneities brain accordance expect microwave imaging methods allow characterization stroke monitoring conclusion developed tool reconstructs microwave tomographic image brain less minutes using cores computational time corresponds clinician acceptance rapid diagnosis medical monitoring hospital images obtained noisy synthetic data accurate model brain knowledge first time realistic study operational acquisition device highly accurate synthetic data noise shows feasibility microwave imaging study made possible use massively parallel computers facilitated hpddm tools developed next step validation results clinical data regarding numerical aspects work accelerate solution series direct problems accounts elapsed time explain three main avenues research present oras solver maxwell equations one level algorithm scale well thousands subdomains introduction preconditioner adequate coarse space would allow good speedups even decompositions large number subdomains recycling information obtained convergence optimization algorithm also enable improve performance method see iterative block methods allow simultaneous solutions linear systems fully investigated arithmetic intensity would increased since block methods may converge smaller number iterations exploiting modern computer architectures effectively acknowledgments work granted access hpc resources tgcc cea allocations made genci work supported part anr project medimax lin clarke microwave imaging cerebral edema proceedings ieee semenov corfield microwave tomography brain imaging feasibility assessment stroke detection international journal antennas propagation semenov seiser stoegmann auff electromagnetic tomography brain imaging virtual human brain ieee conference antenna measurements applications cama beck hiptmair multilevel solution maxwell equations based edge elements international journal numerical methods engineering mixed finite elements numerische mathematik dolean jolivet nataf introduction domain decomposition methods algorithms theory parallel implementation siam pellegrini roman scotch software package static mapping dual recursive bipartitioning process architecture graphs computing networking springer karypis kumar fast high quality multilevel scheme partitioning irregular graphs siam journal scientific computing gander thomas optimized multiplicative additive restricted additive schwarz preconditioning siam journal scientific computing joly roberts domain decomposition method harmonic maxwell equations iterative methods linear algebra brussels amsterdam cai sarkis restricted additive schwarz preconditioner general sparse linear systems siam journal scientific computing hecht new development journal numerical mathematics jolivet dolean hecht nataf prud homme spillane domain decomposition methods massively parallel architectures journal numerical mathematics jolivet hecht nataf prud homme scalable domain decomposition preconditioners heterogeneous elliptic problems proc int conference high performance computing networking storage analysis ieee schenk solving unsymmetric sparse systems linear equations pardiso future generation computer systems amestoy duff excellent koster fully asynchronous multifrontal solver using distributed dynamic scheduling siam journal matrix analysis applications parks sturler mackey johnson maiti recycling krylov subspaces sequences linear systems siam journal scientific computing
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neural processing letters manuscript inserted editor ensemble classification algorithm based information entropy data streams aug junhong wang shuliang bingqian duan caifeng liu jiye liang received aug accepted date abstract data stream mining problem caused widely concerns area machine learning data mining recent studies ensemble classification widely used concept drift detection however regard classification accuracy criterion judging whether concept drift happening information entropy important effective method measuring uncertainty based information entropy theory new algorithm using information entropy evaluate classification result developed uses ensemble classification techniques weight classifier decided entropy result produced ensemble classifiers system concept data streams changing classifiers weight threshold value abandoned adapt new concept one time experimental analysis section six databases four proposed algorithms executed results show proposed method handle concept drift effectively also better classification accuracy time performance contrastive algorithms wang school computer information technology shanxi university taiyuan china key laboratory computational intelligence chinese information processing ministry education taiyuan china tel wjhwjh school computer information technology shanxi university taiyuan china xushulianghao duan school computer information technology shanxi university taiyuan china liu school computer science technology faculty electronic information electrical engineering dalian university technology dalian china liang key laboratory computational intelligence chinese information processing ministry education taiyuan china ljy junhong wang keywords data streams data mining concept drift information entropy ensemble classification introduction development information society many fields produced large amount data streams network monitoring telecommunication stock trading etc data fields different conventional static data due achieving fast unlimited number concept drift data streams makes technologies traditional data mining face big challenge since data streams mining problem proposed received many attentions especially widely used ensemble classification method put forward proposed data stream classification algorithm based random decision tree model algorithm creates number random decision trees randomly chooses split attribute determines split value information gain addition algorithm effectively distinguish noise data concept drift elwell proposed incremental learning algorithm periodic concept drift called preserves historical classification models historical classifier classify data correctly classifier improved weight classifying correctly many times weight historical classifier reach activation threshold predetermined user classifier activated sleeping state joins system participate deciding labels unlabeled data aiming problem unbalanced data stream classification rushing proposed incremental algorithm based overlaying mechanism called cbea algorithm training classifier training examples also saved number classifiers reaches threshold two similar coverage sets selected oldest classifier deleted final classification results determined knn algorithm gomes proposed adaptive algorithm based social network called sae algorithm introduces related concepts social network sub classifier seeing node network network consists multiple classifiers sae algorithm sets series parameters measure performances classifiers updates classifiers adapt new concept brzezinski proposed data stream classification algorithm based online data called oaue algorithm uses mean square error determine weights classification models period time detection coming replacement strategy used deal concept drift farid proposed ensemble classification algorithm based instances weighting mechanism order detect outliers algorithm combines clustering algorithm data point belong existed cluster system class data thought new concept algorithm counts data information leaf nodes confirm result liang proposed online sequential learning algorithm called oselm development extreme learning machine elm algorithm new data block coming uses new data incrementally update structures single hidden feedforward neural networks mean online sequential learning mechanism effectively deal concept drift data stream environment ensemble classification algorithms detect concept drift based accuracy classification result traditional data stream classification algorithm accuracy important indicator used detect concept drift using accuracy characterize current performance reflect amount information contained ensemble classification algorithm based information entropy data streams information system information entropy used measure degree uncertainty system proved effective method describe information content information systems information entropy powerful tool deal uncertainty problem applied many fields aiming problem data stream classification paper extended information entropy measure uncertainty concept data stream ensemble classification algorithm based information entropy ecbe proposed new algorithm based ensemble classification technique weighted voting rule adopted weights classifiers determined according change entropy values classification using hoeffding bound ecbe algorithm estimate whether concept drift appears concept drift appearing algorithm automatically adjusts classifiers according weights comparing existed algorithms ecbe algorithm effectively detect concept drift also get better classification results rest paper organized follows section describe related backgrounds section introduces ensemble classification algorithm based entropy data stream section experimental process data analysis finally section gives conclusions summary backgrounds data stream concept drift assume data stream generated system instance generating moment every instance mark features vector label order illustrate related notions following definitions definition certain number instances organized data set according time sequence call data set data block denoted size data block data stream classification face massive data storage space data far beyond capacity computer memory order make algorithm handle massive data sliding window mechanism widely used words time window allowed one data block coming system data blocks current window processed completely next data blocks acquired definition moment data sliding window used train classifier target concept get time later use new data train another classifier get target concept say concept drift happened data stream according different concept drift divided two types short time concept drift called gradual concept drift long time concept drift called abrupt concept drift concept drift appearing distribution data sliding window changed time performance classifiers decline corresponding measures taken error rate classification results continuously rise therefore many applications property used detect concept drift http junhong wang ensemble classification techniques data streams ensemble classification important classification method data streams uses number weak classifiers combine strong classifier method effectively deal problem concept drift process classification ensemble classification gives different classifiers different weights adjusting weights sub classifiers update classifiers system adapt new concept data stream classification result data eventually decided voting mechanism data stream environment classification performance ensemble classifiers better single classifier many advantages ensemble classification method widely used data stream data mining ensemble classification based information entropy data streams concept drift detection based information entropy entropy used describe disordered state thermodynamic system physics entropy used indicate disorder system shannon extended concept entropy information system utilized information entropy represent uncertainty source entropy variable greater indicates uncertainty variable larger needs information changing state variable uncertain certain classification data streams data blocks sliding window moment assumes data block sliding window yin feature matrix ith data block label set ith data block yin time entropy data sliding window calculated equation log number labels probability label instances calculated equation yij equation yim yij else yim yij utilizing classifiers classify data block classification result obtained weighted voting mechanism time use equations calculate entropy values classification result real result data block marked respectively entropy deviation data block calculated lemma hoeffding bound independently observing random variable times range random variable observed average confidence level true value least ensemble classification algorithm based information entropy data streams theorem concept data stream distribution data stable deviation two adjacent data blocks calculated according equation known proof concept data stream stable data distributions two adjacent data blocks consistent difference observed entropy values small let real entropy data block according equation hoeffding bound known plus two inequalities result follow get therefore use measure detect concept drift data streams deviation two adjacent data blocks thought concept drift appeared obvious special situation occurring using work well two adjacent data blocks represent different concepts classification accuracies two adjacent data blocks low almost equal calculating still less equal algorithm think change happening fact concept drift appeared time order solve problem correct decision criterion concept drift say concept drift appeared equation known two adjacent data blocks occuring two concept drifts less equal however still greater concept drift detected sub classifier ensemble according classification results current data block entropy classification result real result marked calculated using equation change entropy classification weights classifiers updated according equation ensemble ensemble equation ensemble new weight ensemble weight updating function showed equation constant junhong wang theorem concept data stream stable weights ensemble classifiers updated basis equation confidence level sub classifier weight must satisfy following condition proof concept data stream stable sub classifier system adaptive current concept therefore difference classifier weights classifying small assume weights ensemble classifiers according normal distribution mean variance weights calculated inequality known level confidence use approximately replace standard deviation normal distribution according principle normal distribution conclude theorem see concept drift appearing algorithm obtain statistic weights classifiers time last time last concept drift happening lower limit updating weight sub classifier calculated follow weight according classification result sub classifiers new data weights updated using equation classifiers adapt current concept weights classifiers sharply decreases weight finally system delete classifiers satisfy equation next process classification classifiers low performances continue participate ensemble classification algorithm based information entropy data streams execution ecbe algorithm knowledge know implementation steps ecbe follows algorithm ecbe input ensemble classifier data stream number sub classifier size data block winsize array preserving weights classifiers period two adjacent concept drifts output trained classifiers ensemble read winsize instances organize data block size ensemble use train new classifier ensemble ensemble else use equation calculate entropies ensemble ensemble weights classifiers calculate entropy result ensemble classifying data block update weights classifiers according equation num num ensemble end concept drift delete classifier minimum weight calculate weight weight according equation end delete classifier weights less weight use train new classifier cnew size ensemble delete classifier minimum weight end ensemble ensemble cnew use train classifier system end end ecbe algorithm uses entropy detect concept drift data block appear concept drift classifiers system good performance classification results actual labels nearly consistent difference two entropies small concept drift appearing error rate classification result increase difference entropy also increase increment promote concept drift data stream detected concept drift detected classifiers system unable adapt current concept defunct classifiers eliminated decreases accuracy classification results also gives false alarm concept drift order solve problem ecbe algorithm preserves weights classifiers concept junhong wang data stable period two adjacent concept drift concept drift happens saving statistical weights lower bound weight concept stable computed meanwhile concept changing weights classifiers dramatically reduced classifiers whose weights lower bound deleted measure ensures classifier system adapt new concept fast speed experimental analysis order validate performance ecbe algorithm paper chosen sea addexp awe dco weap comparison algorithms cart naive bayesian used base classifiers algorithms tested three artificial datasets waveform hyperplane led produced moa platform three practical datasets sensor reading shuttle parameters experimental environment follows operating system intel dual core cpu memory algorithm implemented matlab descriptions datasets datasets using experiment cited uci datasets noly give brief explain datasets detail see website waveform dataset artificial dataset data set instances instance attributes first attributes numeric different labels dataset led dataset dataset instances contains attributes values first attributes different labels dataset data contains noise hyperplane dataset hyperplane satisfies following mathepd sample dimensional matical expression dataset contains instances dimensions values first dimensions label instance marked positive otherwise label instance marked negative addition dataset contains noise sensor reading dataset dataset contains instances properties first properties real different labels dataset shuttle dataset dataset contains instances attributes values first attributes continuous different labels dataset dataset dataset contains instances attributes first attributes attributes values continuous discrete different labels dataset experimental results order verify performance ecbe algorithm sea addexp awe dco weap toselm ecbe run three artificial datasets waveform hyperplane led http ensemble classification algorithm based information entropy data streams algorithms parameters addexp follows parameters ecbe follows test results showed tables table average accuracy artificial datasets sea addexp awe dco weap toselm ecbe waveform hyperplane led table time overhead artificial datasets unit sec sea addexp awe dco weap toselm ecbe waveform hyperplane led tables known three artificial datasets ecbe accuracy highest time overhead far less compared algorithms datasets except toselm comprehensive performance ecbe best data table comparing algorithms addexp lowest accuracy time consuming reason result addexp updates classifiers based single data three datasets data contains gradual concept drift concept data constantly slowly changing makes accuracy old classifier decreasing instance classified wrongly classifier algorithm would eliminate classifier deal concept drift however classifiers addexp algorithm get better performance trained many times due training new classifiers inadequate error rate classification result higher lead classifier replaced classifiers filling system trained adequately addexp algorithm higher error rate three datasets table led dataset accuracies algorithms low except ecbe algorithm led dataset attributes relating concept drift others redundant attributes concept drift appearing speed concept changing led dataset faster waveform hyperplane datasets three algorithms sea addexp awe delete classifiers low performance basis single classifier take long training time completely eliminate weak classifiers speed junhong wang concept drift lower elimination speed performance classifiers always low level ecbe algorithm concept drift detected weights classifier greatly reduced however ecbe delete weak classifiers according weights time ecbe adapt new concept fast speed experiment shows performance ecbe best order verify effectiveness algorithm deal practical datasets sea addexp awe dco weap toselm ecbe run sensor reading shuttle datasets algorithms maximum number classifiers sensor reading datasets size data block others datasets parameters addexp follows parameters ecbe follows obtained results shown tables table average accuracies practical datasets sea addexp awe dco weap toselm ecbe sensor reading shuttle table average time overhead real datasets unit sec sea addexp awe dco weap toselm ecbe sensor reading shuttle tables known algorithms get good performances practical datasets accuracy ecbe prominent shuttle dataset every accuracy algorithm accuracies ecbe datasets close best results addition time overhead ecbe far less comparison algorithms advantage ecbe obvious data tables performances sea addexp awe dco weap toselm testing three practical datasets close achieved good results sensor reading shuttle datasets data whose labels highly concentrated cases data distribution two adjacent data blocks consistent classification model trained data better adaptability new data performances eight algorithms good practical datasets ensemble classification algorithm based information entropy data streams analyzing tables sea addexp awe dco effective datasets data labels highly concentrated dealing datasets containing gradual concept drift performance algorithms satisfactory analyze results ecbe obvious ecbe deal two types concept drift ecbe algorithm use equation calculate threshold judges change entropy data stream threshold important influence performance algorithm algorithm equation winsize explore impact winsize values algorithm select waveform dataset experimental data parameters ecbe follows run ecbe algorithm results showed table table average accuracy different winsize values winsize hyperplan led shuttle waveform winsize page sensor fig test result different winsize values hyperplane table situation different winsize values seen test result individual data blocks entire test results obvious linear rule winsize values performance ecbe ecbe algorithm small winsize value lead large threshold make ecbe sensitive change data distribution small data block cause classifiers trained data junhong wang fig test result different winsize values led fig test result different winsize values shuttle fitting performance classifiers system remarkably improved winsize value large lead small threshold makes algorithm sensitive change data distribution classifiers give false alarm concept drift affect performance ecbe therefore practical application value winsize needs consider data distribution speed concept changing trying many times select optimal winsize value series candidate values order study anti noise performance ecbe select waveform dataset experimental data add noise parameters ecbe set follows run ecbe algorithm datasets test results showed table known accuracy ecbe also gradually reduces increase noise table see comparing data containing noise ensemble classification algorithm based information entropy data streams fig test result different winsize values waveform fig test result different winsize values page table average accuracy algorithm different noise noise average accuracy noise increasing accuracy ecbe decreases respectively magnitude change small thus known ecbe good noise immunity good noise immunity ecbe comes elimination strategy noise data stream increasing difference entropies classification results actual results increases weights classifiers decay fast speed weights specified threshold junhong wang fig test result different winsize values hyperplane sensor reading fig test result different noise level classifiers trained noise data deleted weak classifiers affect classification new data order test performance ecbe compare ecbe representative neural network algorithm called datasets test different activation functions experiment choose radbas sigmoid sine hardlim functions activation functions parameters ecbe follows test results showed tables table seen accuracy ecbe better datasets waveform hyperplane datasets activation function sigmoid oselm better ecbe fact advantage weak calculating know accuracies ecbe aspect time overhead ecbe better datasets better ecbe ensemble classification algorithm based information entropy data streams table average accuracy testing datasets ecbe waveform hyperplane led sensor reading shuttle radbas sigmoid sine hardlim winsize table time overhead datasets unit sec radbas sigmoid sine hardlim ecbe waveform hyperplane led sensor reading shuttle table number concept drift detected ecbe dataset hyperplane sea waveform rbf number accuracy winsize datasets synthetically consider time overhead accuracy two algorithms sensor reading although ecbe costs time accuracy ecbe far higher ecbe still better analyses obvious ecbe best order verify ability ecbe detecting concept drift select hyperplane sea waveform rbf experimental datasets produced moa consideration concept drift datasets unknown reprocess hyperplane sea waveform make three dataset contain concept drift label changes form reverse direction changing size four datasets respectively results shown table see concept drift appearing accuracy ecbe degrade rapidly ecbe delete classifiers weak performance accuracy restored previous high level experiment ecbe detects concept drift times table know hyperplane waveform datasets algorithm detected concept drifts producing error alert sea dataset error alert produced real changing omitted rbf gradual concept drift dataset generated computer number concept drift datset unknown use rbf test effectiveness ecbe handling gradual concept drift result rbf obvious ecbe detect gradual concept drift summary include although deviations concept junhong wang fig test result different datasets ecbe drift detection algorithm experimental data indicates mechanism ecbe detecting concept drift effective ecbe cope different types concept drift conclusion paper solve classification problem data streams ensemble classification algorithm based information entropy proposed algorithm basis ensemble classification method utilizes change entropies classification detect concept drift experimental results show ecbe effective obvious ecbe algorithm suitable single labeled data apply algorithm classification data focus future research acknowledgments work supported national natural science fund china national key basic research development program china authors grateful editor anonymous reviewers constructive comments helped improve quality presentation paper references gama sebasti rodrigues issues evaluation stream learning algorithms proceedings acm sigkdd international conference knowledge discovery data mining acm paris france minku yao ddd new ensemble approach dealing concept drift ieee transactions knowledge data engineering bifet holmes pfahringer kirkby gavald new ensemble methods evolving data streams acm sigkdd international conference knowledge discovery data mining acm ouyang zhou mining noisy data streams using ensemble classifiers artificial intelligence computational intelligence aici international conference vol ieee ensemble classification algorithm based information entropy data streams learning concept drifting data streams unlabeled data neurocomputing liu jiang online sequential learning algorithm dealing concept drift neurocomputing 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jan multiplicative structure hochschild cohomology complete intersection buchweitz collin roberts dedicated foxby admiration abstract determine product structure hochschild cohomology commutative algebras low degrees obtaining answer degrees complete intersection algebras applications consider cyclic extension algebras well hochschild ordinary cohomology finite abelian groups contents introduction statement results tate models multiplication map algebra diagonal approximation graded commutativity products low degree proof theorem hochschild cohomology homological complete intersections case one variable cohomology finite abelian groups references introduction associative algebra commutative ring hochschild cohomology ring cup yoneda product graded commutative gerstenhaber fundamental result however necessarily strictly graded commutative squaring operation hochschild cohomology odd degrees necessarily hochschild cohomology twice degree might nontrivial indeed already simplest example hochschild cohomology ring isomorphic polynomial date january mathematics subject classification key words phrases hochschild cohomology complete intersection tate model diagonal approximation cup yoneda product strictly graded commutative group cohomology authors partly supported nserc grant material based upon work supported national science foundation grant first author residence mathematical science research institute msri berkeley california semester rings multiplicative identity ring homomorphisms unital buchweitz roberts ring one variable cohomological degree zero variable degree one representing class derk characteristic ring indeed graded commutative certainly strictly square returning element purpose clarify structure squaring map simplest case commutative deduce entire ring structure homological complete intersection crucial point square derivation respect cup product element determined hessian quadratic forms associated defining equations key technical tool use tate model multiplication map aev commutative show postnikov tower model carries family compatible diagonal approximations give explicit description approximation low degrees obtain desired results acknowledgement present paper relies large extent second author thesis authors wish thank solberg originally asking determine squaring map hochschild cohomology complete intersection luchezar avramov alex martsinkovsky valuable help references lisa townsley sharing synopsis thesis also thank referee paper careful reading valuable suggestions regarding presentation aside pointing numerous typos hopefully taken care statement results treatment requires simple facts differential forms calculus goes along cover facts use reader may gain full picture consulting polynomial ring commutative ring set variables module differentials free based differentials polynomial may consider taylor expansion symp concrete terms polynomial depend finitely many variables say coefficient dxa corresponding divided partial derivative usual partial derivative satisfies linear part respect expansion viewed total differential wit dxi multiplicative structure hochschild cohomology defines jacobian given polynomial derk jac derivations quadratic part respect taylor expansion dxi dxj hessian form defined definition element defines form sends derk set module derk depends solely chosen coordinate system take family polynomials associated linear jacobian respectively quadratic hessian map given jac jac derk symbols simply represent corresponding product free commutative module differentials free presentation adx basis element mapped dfb map factors surjection onto conormal module sends mod followed map induced universal sending mod follows jacobian hessian descend jacf jacfb hfb derk map jacf taking values homq normal module respect viewed submodule free kernel jacobian satisfies ker jacf derk note last isomorphism holds associative aev denotes enveloping algebra higher degrees one natural maps hhi extiaev isomorphisms projective see also buchweitz roberts formulate first result theorem let presentation commutative quotient polynomial ring commutative ring modulo ideal hessian map induces well defined map derk homa module derivations normal module explicitly mod mod variables actually occur presentation linearly ordered way canonical map homa whose image first tangent cohomology map derk sends derivation class square yoneda product equals takes image remark preceding result projective natural bijection graded commutative moreover identifies classes symmetric hochschild second result let presented assume moreover transversal tork tori ideal regular sense quillen projective canonical morphism graded commutative algebras torp isomorphism quillen points noetherian regular generated locally sequence using definition algebra structure source target one obtains isomorphism opposite algebra yoneda however already gerstenhaber pointed original paper sending element degree provides algebra isomorphism graded commutative algebra opposite algebra multiplicative structure hochschild cohomology say short homological complete intersection algebra conditions satisfied alternative terminology would say kernel multiplication map aev free exterior koszul homology notion introduced studied blanco majadas soto rodicio garcia series articles following call multiplication map intersection homomorphism key properties homological complete intersection algebras corresponding tate model small associated cotangent complex cor concentrated homological degrees precisely shifted shifted tangent complex homa complex derk jacf concentrated cohomological degrees one two link tangent hochschild cohomology established quillen fundamental paper sect essence says complex returning hochschild homology realized hopf algebra whose primitive part represented cotangent complex qualification generally spectral sequence reflecting interpretation generally degenerates characteristic zero dual side hochschild cohomology enveloping algebra tangent complex algebra structures cohomology match however complete intersection algebras things well controlled cotangent complex short detail complement picture follows hessian map interpreted defining structure graded super lie algebra follow axiomatization notion jacobian viewed lie algebra differential enveloping algebra lie see readily identified clifford algebra cliff quadratic map jacobian induces algebra differential particular cohomology algebra inherits algebra structure way theorem let homological complete intersection algebra induce hence assume free cohomology algebra cliff yoneda self extensions bimodule detail cliff degree degree differential determined mod buchweitz roberts respect multiplicative structure central mod mod projective natural homomorphism graded commutative algebras isomorphism thus hochschild cohomology given cohomology clifford algebra remark concrete terms expanding witt theorem enveloping algebras super lie algebras shows differential graded coalgebras particular complexes graded one cliff isomorphic koszul complex syma sequence mod bigrading koszul complex placing variables derk bidegree bidegree one regains hodge decomposition hochschild cohomology homological complete intersection hhp homa however preceding theorem says generally decomposition compatible multiplicative structure squaring sends homa summand homa homa words obvious multiplication hodge decomposition needs twisted quantized exterior multiplicative structure koszul complex homogeneous clifford algebra phenomenon analogous see discussion therein situation smooth schemes characteristic zero similarly arbitrary commutative rings field characteristic zero relation hodge decomposition multiplicative structure hochschild cohomology studied combinatorial point view hasten point affine homological complete intersections considered multiplicative twist make difference unit multiplicative structure hochschild cohomology end paper presenting simple examples highlight treatment encompasses unified manner various results scattered throughout literature tate models multiplication map algebra let homomorphism commutative rings seminal paper john tate presented essence following construction free resolution see also details particular details differential graded algebras divided powers special case multiplication map commutative finite type mainly interested also detailed exists factorization given ring homomorphism strictly graded commutative form syms odd even divided powers even degrees graded concentrated degree free concentratedvin degree sym denotes symmetric algebra functor exterior algebra functor divided power algebra functor differential compatible divided powers thus uniquely determined restriction vanishing necessarily degree reasons iii morphism quasiisomorphism algebras one endows concentrated degree zero possible zero differential factorization tate model usual slight abuse also simply call tate model remaining data understood given module generates subalgebra divided powers constitutes associated postnikov tower follows description given inclusion induces isomorphisms interpreted attaching map kills remark viewed complex tate model constitutes particular free resolution thus used calculate tors exts noetherian surjective one may choose free finite rank turn free finite rank degree finite type one may choose finite free finite free syms consider particular case commutative commutative ring aev multiplication map homomorphism enveloping algebra kernel multiplication map denote buchweitz roberts note ideal aev represents functor taking derivations arbitrary contrast differential represents functor taking derivations forms symmetric use convention universal derivation sends assume presented symk free ideal polynomial ring symk free module following result holds arbitrary presentation neither necessarily commutative lemma exists exact sequence aev aev induced restricting universal derivation obtained inclusion taking account aev returning commutative situation choose surjection free preceding lemma following immediate consequence corollary one may construct tate model multiplication map proof multiplication map surjective one may choose construction tate model image restriction universal derivation shows induced map generates ideal surjective thus choice resulting algebra aev nothing koszul complex composition aev whose first homology isomorphic exactness sequence term applying sides tensor product surjection yields aev surjection first onto via onto homology aev one find lifting surjection lifting extends differential whence claim concerning follows even specific lemma proof corollary indicate describe possible differentials let basis free universal derivation sends algebra generators dxa symk multiplicative structure hochschild cohomology abbreviate use notation respective residue classes aevp representing polynomial xak finite sum ordered monomials finitely many basis elements definition product rule derivation yield slightly abusing notation may base free aev construction symbols dxa define aev similarly polynomials whose classes generate may base free aev symbols dfb choose dfb differential basis elements right hand side results chosen presentation time right hand side depend chosen presentation choices easily controlled one may add right hand side change value dfb also choices collect coefficients terms dxa write dfb dxa coefficients satisfy irrespective representation chose diagonal approximation graded commutativity tate model multiplication map aev commutative note naturally complex form tensor product tensor product carries structure differential strictly graded commutative algebra divided powers denote algebra homomorphisms respectively complex underlying one free free modules make claim homology complex except buchweitz roberts note algebra homomorphism natural map onto zeroth homology note however flat aev via constitutes flat resolution via either thus homology trivial equal degree zero however even without flatness following lemma notation introduced algebra homomorphisms homotopic morphisms complexes ker vanishing homology including degree particular flat subcomplex ker ker zero homology proof difference takes values acyclic complex ker source complex projective even free claim follows concerning consider following commutative diagram complexes exact rows columns ker ker ker ker ker ker construction quasi isomorphism whence ker acyclic assumption quasiisomorphisms degrees including may take flat long exact homology sequence obtained short exact sequence say bottom yields claims definition let complex aev comes augmentation diagonal approximation morphism aev following diagram commutes multiplicative structure hochschild cohomology identified existence diagonal approximation tate model multiplication map implies graded commutativity yoneda product extaev see example definition given diagonal approximation tate model aev define cup product cochains proposition let commutative tate model aev admits diagonal approximation corresponding cup product induces graded commutative product product coincides usual yoneda composition product proof given cocycles lift morphisms complexes homotopy category complexes agree lemma following diagram complexes aev modules commutes homotopy category definition cocycle representing yoneda product composition across top yields thus two products coincide extaev endaev endaev lifts homogeneous cocycles one whence combining diagram corresponding yields graded commutativity yoneda cup product cohomology remark preceding proof shows partial diagonal approximation enough express yoneda product two cohomology classes whose degrees add cup product moreover classes commute graded sense respect product particular case flat one indeed construct diagonal approximation furthermore homomorphism algebras divided powers buchweitz roberts proposition flat exists diagonal approximation tate model aev induces approximation algebra postnikov tower proof construct desired diagonal approximation inductively along postnikov tower basis postnikov tower aev defines homomorphism algebras via view aev assume diagonal approximation defined free aev note satisfies induction hypothesis idt morphism complexes follows vanishes morphism complexes whence follows maps cycles ker latter complex acyclic flat lemma projective even free find ker ker algebra homomorphism given extended first algebra homomorphism satisfies words found extension diagonal approximation completing inductive argument products low degree sharpen proposition bit low degrees need flatness proceding proof proposition define extend homomorphism strictly commutative graded algebras divided powers indeed approximation homomorphism algebras multiplicative structure hochschild cohomology homomorphism dxa dxa dxa basis element dxa put differently diagonal approximation chosen way dxa aev dxa shortened notation dxa dxa dxa also use algebra homomorphism aev identify noteworthy induced cup product always strictly graded commutative indeed recall nothing koszul complex aev dualizing induced differential becomes zero quasi isomorphism homaev homp graded module alternating polyvector fields values homaev cocycle odd degree fact sweedler notation form suitable homogeneous whose degrees add explicit form shows zero summand summands odd one commutative demonstrated construction require flat show explicit construction one find also without flatness assumption diagonal approximation extends constructed recall free aev based elements dfb correspond polynomials whose classes generate aev module attaching map killed second homology obtained representation combination monomials algebra generators terms extending shorthand notation dfb dfb dfb determine diagonal approximation degree buchweitz roberts proposition algebra homomorphism extends means dfb finite presentation xak given polynomial defines diagonal approximation proof inspection reveals immediately idt thus remains verify compatible differentials equivalently dfb straightforward verification first exhibit use xak next apply differential double sum definition dfb obtain xak first equality uses skew derivation degree one second equality applies monomials respectively xak final equality uses form recalled reordering terms obtain dfb required note following special property diagonal approximation constructed corollary diagonal approximation cocommutative dfb automorphism graded algebras multiplicative structure hochschild cohomology ignoring differentials exchanges leaves remaining variables unchanged satisfies determined one explicit diagonal approximation choices involved provided exact low degrees easily describe diagonal approximations algebra homomorphisms algebras divided powers theorem let tate model aev exhibited diagonal approximation modified algebra homomorphism respects divided powers agrees degree zero given algebra generators degrees dxa dfb dfb aev degree degree ker ker exact degrees possible diagonal approximations homomorphisms algebras divided powers agree degree zero moreover case already chosen ker ker proof diagonal approximation along easily verified directly conversely diagonal approximation dxa cycle boundary thus form suitable degree two moreover necessarily cycle ker ker diagonal approximations lemma shows complex vanishes thus chosen ker ker also dfb dfb boundaries thus form form degree belongs necessarily ker complex vanishes argument whence may replace necessary element ker ker image differential illustrate determine cup product two assume tate model aev commutative let aev cochains degree cup product cochain degree values explicitly given buchweitz roberts dfb dfb xak xak remains find palatable form coefficients expression theorem given commutative represented given polynomial depends variables one may choose diagonal approximation way mod dxi dxj dfb cochains degree one proof let presentation polynomial variables rewrite occurring monomials xenn corresponding choice dfb proposition term contributes specializing index sorting reduces sum xenn xenn coefficient expression indeed corresponding divided second derivative monomial whence putting terms together follows choice presentation dfb cup product takes form claimed multiplicative structure hochschild cohomology proof theorem let presentation commutative quotient polynomial ring commutative ring modulo ideal need show hessian quadratic map hfb derk defined sends derk derk homa element normal element represents module every relation finitely many elements nonzero sum zero let finite subset variables involved finitely many polynomials occur nonzero coefficient given relation derk given set lifts one mod mod mod involved relation next note second divided power satisfies polynomials apply relation obtain mod middle equation term mod derivation therefore indeed element normal module proves part theorem part theorem restates fact give another deduction using tate model multiplication map aev establish along way well part theorem end recall koszul complex aev dual homaev carries cup product induced respect cohomology homaev homa strictly graded commutative algebra diagonal approximation extends diagonal approximation surjective restriction map resulting inclusion compatible buchweitz roberts cup products source target thus induces homomorphism cohomology kernel restriction map complexes long exact cohomology sequence contains derk jac homa derk algebra homomorphism strictly graded commutative algebra thus image map image known first cohomology moreover explicit description cup product homaev shows mapped via hochschild cohomology homological complete intersections consider case homological complete intersection free proposition see assume commutative admits presentation symk symmetric free ideal tork constructed tori lary already tate model free natural homomorphism strictly graded commutative algebras tor isomorphism surjection chosen induce bijection proof corollary formed koszul complex aev runs homology identified torp chap flatness together tork yields isomorphism tor tor free canonical map torp provided isomorphism algebras already tate showed map sends main result yields converse remark proposition applies particular defines complete intersection sense form regular sequence case condition tork equivalent exactness koszul complex complex simply tensor product koszul complex free resolution resolution complete intersection ring enveloping algebra tate model long history multiplicative structure hochschild cohomology wolffhardt first exhibit apparently unaware tate result wrote resolution case field characteristic zero case divided powers replaced symmetric ones expense introducing denominators explicit form resolution also described reviewers mathscinet note authors recover calculation due originally wolffhardt trans amer math soc recently since host authors including see condition tork void even special situation contrary implied one may take say see also next section detailed discussion point remark wolffhardt treats analytic case quotient noetherian smooth analytic algebra broad sense supplement leave readers retrace foregoing results convince everything works well situation provided one replaces occurring tensor products analytic counterparts turn proof theorem light theorem know explicitly form cup product homaev thus homaev case hand complex set homaev aev koszul complex polynomial ring form dual dfj employ dual dxi koszul complex compact description syma differential cohomological grading recovered putting degree degree indicated remains determine multiplicative structure identify dual cup product resulting diagonal approximation theorem explicitly given moreover elements central diagonal approximation cocommutative dfj pointed corollary explicit calculations show cup product theorem buchweitz roberts satisfies relations putting together shows cliff graded algebras differential readily identifies cliff completes proof theorem case one variable consider cyclic extensions commutative ring polynomial set abbreviate let content ideal thm polynomial annihilator content ideal zero equivalently homk provide context quote following results literature refer references cited unexplained terminology proposition let polynomial set free ideal generated monic polynomial necessarily free finite rank finitely generated projective ideal generated polynomial projective ideal generated almost polynomial flat content ideal generated idempotent cases cases content ideal generated idempotent polynomial proof parts found also terminology almost introduced part contained thm examples polynomial polynomial almost ring flat module projective cases considered proposition decomposition multiplicative structure hochschild cohomology polynomial ring moreover algebras extaop extaop extaev second factor satisfying extaop obvious multiplicative structure therefore concentrate case given polynomial natural map mod isomorphism avfree resolution tork torp trivially isomorphism words regular ideal missing piece homological complete intersection thus vanishing tork vanishing equivalent turn equivalent forming regular sequence light exercise sequence regular extik depth content ideal least example let polynomial ring commutative ring set polynomial sxd content ideal depth sxd content ideal depth thus defines homological complete intersection constructing beginning tate resolution multiplication map aev corollary one obtains complex augmented versus degree aev aev aev residue class unique polynomial satisfies summarizing discussion far following characterizations refer terminology properties exact zero divisors theorem following equivalent homological complete intersection resolves aev pair exact aev content ideal grade least homk buchweitz roberts moreover flat content ideal unit ideal proof equivalence special case proposition tautology definition pair exact aev ideals annihilators aev complex displayed exact equivalence discussed final claim follows proposition flatness means generated idempotent idempotent assume remainder section satisfy equivalent conditions theorem determine algebra call ramification algebra introduce well annihilator given note whence naturally fact homa dualizing exact sequence shows sequence read identify derk first tangent cohomology theorem assume polynomial whose content ideal depth least two yoneda aev graded commutative satisfies abf degree zero degree central degree denotes residue class second divided derivative particular even yoneda exteven aev extaev fibre product ring polynomial ring one variable degree exteven proof note algebra displayed indeed cohomology algebra clifford algebra aht relations differential apply theorem remark know graded commutativity algebra classes abf must better classes already see observe ordinary second derivative use suitable polynomials multiplicative structure hochschild cohomology one obtains product rule using product rule using mod corollary polynomial content ideal depth least two strictly graded commutative example inspired foregoing corollary fails one variable consider monic degree free rank particular projective euler identity yields whence jacobi ideal nonzero one verifies already using theorem one finds mod whence normal module mod vanish fact squaring map sends one may verify derz surjectively onto ext easily finish section looking classical situations separable unramified case recall polynomial separable derivative unit modulo equivalently case content ideal necessarily equals whence flat theorem applies retrieve higher extension groups vanish ext aev generically unramified case polynomial generically unramified derivative modulo assuming furthermore flat equivalently content unit ideal theorem applies shows generically unramified equivalently odd extension groups aev vanish case yoneda thus reduced even part fibre product polynomial ring generated cohomological degree buchweitz roberts room squaring map strictly graded commutative totally ramified case polynomial totally ramified derivative zero modulo equivalently assuming content unit ideal theorem shows yoneda algebra clifford algebra map happens example remark yoneda algebra tensor product exterior algebra generated degree symmetric algebra generated degree contrast unit whence necessarily characteristic remark yoneda algebra tensor algebra single generator degree covers case char mentioned introduction polynomials field last least let consider case coefficient ring field simply means zero polynomial trivially projective thus yoneda agrees hochschild cohomology principal ideal domain obtain explicit descriptions set gcd thus principal ideal generated isomorphic therefore obtain following presentation hochschild cohomology ring quotient graded polynomial ring degree class degree degree char necessarily mod remark description simplifies accordingly remark last example encompasses theorem lemma lemma theorem theorem completes characteristic cases handled covers well theorem cohomology finite abelian groups final example consider hochschild ordinary cohomology finite abelian groups group multiplicatively written cyclic group order may assume orders factors satisfy elementary divisors invariant factors group group algebra commutative ring given xnr homological complete intersection thus theorem applies however variables become units one simplify result case recover well fact established isomorphism graded commutative algebras multiplicative structure hochschild cohomology considered via trivial representation theorem let finite abelian group hochschild cohomology cohomology tensor product differential graded clifford algebras odd even cohomological degree cohomology second factor tensor product group cohomology trivial one retrieves isomorphism graded commutative algebras proof note accordingly multiplication map aev tensor product multiplication maps follows tate model obtainable tensor product corresponding tate models cyclic groups involved thus remains discuss tate model single cyclic group say natural number group algebra cyclic extension free therefore apply obtain tate model complex augmented versus degree residue class unique polynomial satisfies written algebra hdx homological degree degree differential diagonal approximation proposition determined buchweitz roberts earlier notation next adapt diagonal approximation using theorem setting mod form degree kernel augmentation maps satisfies mod unit adjust adding odd even obtain adapted diagonal approximation agrees degrees degree given mod odd mod even adaptation multiplicative structure case cyclic group already occurs chap xii stage cup product original differential clifford algebra odd even differential satisfies final touch set note coordinate change indeed invertible unit terms obtain finally taking tensor product tate models cyclic groups involved adapted diagonal approximations fit together endow tensor product diagonal approximation dualizing kgev results clifford algebra statement theorem establish relation group cohomology note first flat secondly observe applying tate model kgev taking cohomology homkg yields homomorphism graded algebras multiplicative structure hochschild cohomology factors thus one obtains homomorphism graded algebras one may check directly isomorphism noting one may construct tate model augmentation returns argument source homomorphism least isomorphism one initially ignores multiplicative structure alternatively one may use observe structure map induces algebra homomorphism extkg extkg provides splitting homomorphism remark one obstacle even transparent proof isomorphism diagonal approximation tate model kgev obviously induce approximation applying obstacle overcome exhibiting directly diagonal approximation tate model augmentation done agree one exhibited explicit diagonal approximation tate model augmentation also described refer reference explicit information structure augmenting reproducing results references homologie des commutatives die grundlehren der mathematischen wissenschaften band york avramov local algebra rational homotopy algebraic homotopy local algebra luminy soc math france paris avramov infinite free resolutions six lectures commutative algebra bellaterra progr basel avramov bonacho dos anjos henriques intersection homomorphisms preprint pages avramov hochschild homology criteria smoothness internat math res notices bergeron wolfgang decomposition hochschild cohomology gerstenhaber operations pure appl algebra blanco majadas rodicio acyclicity tate complex pure appl algebra bonacho dos anjos henriques free resolutions short gorenstein local rings math brewer montgomery finiteness proc amer math soc kunz divided powers hochschild homology complete intersections appendix reinhold math ann buchweitz flenner global hochschild homology singular spaces adv math buenos aires cyclic homology group hochschild cyclic homology hypersurfaces adv math buchweitz roberts calaque van den bergh hochschild cohomology atiyah classes adv math cartan lane homotopie henri cartan available electronic form numdam cartan eilenberg homological algebra princeton university press chapman cohomology ring finite abelian group proc london math soc cibils solotar hochschild cohomology algebra abelian groups arch math basel cuntz quillen algebra extensions nonsingularity amer math soc dumitrescu ionescu examples regular rings preprint pages gerstenhaber cohomology structure associative ring ann math grothendieck locale des des morphismes inst hautes sci publ math guccione guccione hochschild homology complete intersections pure appl algebra holm hochschild cohomology rings algebras algebra geom cohomology finitely generated abelian groups enseign math iyengar acyclicity tate constructions pure appl algebra majadas rodicio hochschild homology hypersurfaces comm algebra northcott finite free resolutions cambridge tracts mathematics cambridge university press ohm rush finiteness flat trans amer math soc quillen homology commutative rings applications categorical algebra proc sympos pure vol xvii new york amer math providence roberts cohomology ring finite abelian group thesis university waterloo pages roberts examples smooth regular rings canad math bull sanada hochschild cohomology crossed products comm algebra siegel witherspoon hochschild cohomology ring group algebra proc london math soc hopf algebras derivations algebra argument cup products proc amer math soc applications spectral sequence computation hochschild cohomology preprint pages tate homology noetherian rings local rings illinois math townsley kulich investigations integral cohomology ring finite group thesis university wolffhardt hochschild homology complete intersections trans amer math soc multiplicative structure hochschild cohomology dept computer mathematical sciences university toronto scarborough military trail toronto canada address ragnar dept pure mathematics university waterloo university avenue west waterloo canada address
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jmlr workshop conference proceedings unsupervised model compression multilayer bootstrap networks zhang mar department computer science engineering ohio state university columbus usa abstract recently multilayer bootstrap network mbn demonstrated promising performance unsupervised dimensionality reduction learn compact representations standard data sets mnist however bootstrap method prediction complexity mbn high paper propose unsupervised model compression framework general problem unsupervised bootstrap methods framework compresses large unsupervised bootstrap model small model taking bootstrap model application together black box learning mapping function input bootstrap model output application supervised learner specialize framework propose new technique named compressive mbn takes mbn unsupervised bootstrap model deep neural network dnn supervised learner initial result mnist showed compressive mbn maintains high prediction accuracy mbn also thousands times faster mbn prediction stage result suggests new technique integrates effectiveness mbn unsupervised learning effectiveness efficiency dnn supervised learning together effectiveness efficiency compressive mbn unsupervised learning keywords model compression multilayer bootstrap networks unsupervised learning introduction dimensionality reduction core problem machine learning classification clustering regarded special cases reduce high dimensional data discrete points paper focus unsupervised learning traditionally dimensionality reduction categorized kernel methods neural networks probabilistic models sparse coding kernel methods costly problems although neural networks scalable large scale data double computational complexity bottleneck structure take input output bottleneck structure training stage slow moreover learn data distribution globally effective learning local structures multilayer bootstrap network mbn recently proposed bootstrap method unsupervised ensemble method multiple nonlinear layers layer ensemble clusterings centers clustering randomly sampled data points called bootstrap sample input mbn easily implemented trained scales well problems neural networks training stage moreover mbn learns data distribution locally learn effective zhang zhang tions data easily however mbn contains hundreds clusterings difficult used prediction motivated aforementioned problem recent progress compressing ensemble classifiers single small classifier supervised learning bucilu hinton abstract paper propose unsupervised model compression framework framework uses supervised model approximate mapping function input unsupervised bootstrap method output application unsupervised bootstrap method specify framework taking mbn unsupervised bootstrap method dnn supervised model proposed method named compressive mbn best knowledge first work model compression bootstrap methods unsupervised learning methods compressive mbn follows first step trains mbn give training set outputs low dimensional representation training data points step driven applications second step applies low dimensional representation given application unsupervised learning outputs prediction result training data third step trains dnn training set input prediction result target finally dnn model used prediction algorithm basic framework easily extend compressive mbn techniques simply using unsupervised bootstrap techniques replace mbn first step potentially better performance also use many supervised learners replace dnn third step knowledge dnn currently already good choice may also design lot new algorithms simply specifying second step different applications examples follows compressive mbn used visualization may omit second step simply take input output mbn input output dnn respectively compressive mbn used unsupervised prediction may run hard clustering algorithm training set get predicted indicator vector training data point example data point assigned second cluster predicted indicator vector may also get probabilistic output clustering experiments conducted initial experiment mnist showed technique helpful reducing high computational cost mbn unsupervised prediction problems mnist data normalized dividing entries unsupervised model compression multilayer bootstrap networks figure mbn prediction time images seconds figure compressive mbn prediction time images seconds experiment visualizing small subsets mnist subsection consider generalization ability mbn compressive mbn instead studied visualization ability data set contained unlabeled images randomly selected training set mnist used training test mbn training trained mbn similarly second experiment zhang specifically number clusterings layer set parameters layer layer set respectively see mbn large sparse model parameter random feature selection set parameter random zhang reconstruction set getting sparse representation mbn top hidden layer mapped two dimensional space expectationmaximization principle component analysis roweis training compressive mbn omitted second step used input output mbn input output dnn model respectively parameter settings follows trained dnn dropout rate set rectified linear unit used hidden unit linear function used output unit number training epoches set batch size set learning rate set two dimensional visualizations produced mbn compressive mbn shown fig fig respectively two figures found visualizations mbn compressive mbn dnn equivalently good used features clustering nmis methods around amazingly prediction time compressive mbn images seconds accelerated prediction time mbn around times experiment full mnist used training images unsupervised model training test images test discussed unsupervised generalization ability compressive mbn test images compressive mbn without random reconstruction parameter mbn training trained mbn similarly third experiment zhang specifically number clusterings layer set parameters layer layer set respectively see mbn large sparse model parameter random feature selection set parameter random reconstruction set getting sparse representation mbn top hidden layer mapped dimensional space roweis encoded representations indicator vectors clustering specialization second step compressive mbn training compressive mbn took raw feature training set input dnn took predicted indicator vectors training target dnn parameter settings dnn follows trained dnn dropout rate set rectified linear unit used hidden unit sigmoid function used output unit number training epoches set batch size set learning rate set clustering suffers local minima ran aforementioned methods times recorded average results experimental comparison mbn mbn enable parallel computing unsupervised model compression multilayer bootstrap networks mbn compressive mbn without random reconstruction normalized mutual information training accuracy mbn training accuracy compressive mbn test accuracy mbn test accuracy compressive mbn number layers figure comparison generalization ability mbn compressive mbn clustering random reconstruction mbn used clustering accuracy evaluated normalized mutual information prediction time mbn test images seconds prediction time compressive mbn test images seconds mbn compressive mbn random reconstruction normalized mutual information training accuracy mbn training accuracy compressive mbn test accuracy mbn test accuracy compressive mbn number layers figure comparison generalization ability mbn compressive mbn clustering random reconstruction mbn used prediction time mbn test images seconds prediction time compressive mbn test images seconds compressive mbn summarized fig figure observed curves training accuracy mbn compressive mbn completely coincident zhang moreover curve prediction compressive mbn even slightly better mbn highest prediction accuracy compressive mbn reached terms nmi advanced property compressive mbn needed seconds predict images mbn needed seconds predict images prediction time accelerated around times compressive mbn random reconstruction parameter shown zhang data small scale training size similar largest parameter random reconstruction operation quite helpful however still unclear whether random reconstruction helpful data large scale training size much larger largest parameter since largest third experiment zhang experimental results mbn without random reconstruction exciting subsection enlarged section experimental settings mbn compressive mbn section except set mapped sparse features dimensional space experimental results summarized fig figure observed experimental conclusions section could also summarized except random reconstruction used performance mbn compressive mbn good without random reconstruction conclusions paper proposed general framework unsupervised model compression framework takes mbn zhang case study specialized technique named compressive mbn uses dnn auxiliary model modeling mapping function input mbn prediction result given application takes low dimension output mbn input new technique aims solve problem although mbn simple effective robust time consuming prediction initial experimental result mnist showed compressive mbn inherited generalization ability mbn even slightly better mbn also accelerated prediction efficiency mbn thousands times compressive mbn concatenates effectiveness mbn unsupervised learning effectiveness efficiency dnn supervised learning together effectiveness efficiency unsupervised learning moreover easily extend compressive mbn unsupervised model compression techniques acknowledgements author thanks yuxuan wang providing dnn toolbox prof deliang wang providing computing resources ohio supercomputing center unsupervised model compression multilayer bootstrap networks references cristian bucilu rich caruana alexandru model compression proc int conf knowl data pages acm geoffrey hinton oriol vinyals jeff dean distilling knowledge neural network arxiv preprint sam roweis algorithms pca spca advances neural information processing systems pages denver alexander strehl joydeep ghosh cluster knowledge reuse framework combining multiple partitions mach learn zhang nonlinear dimensionality reduction data multilayer bootstrap networks arxiv preprint pages
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approximating largest root applications interlacing families apr nima shayan oveis amin nikhil stanford university anari saberi university washington shayan university california berkeley nikhil april abstract study problem approximating largest root polynomial degree using top coefficients give nearly matching upper lower bounds present algorithms running time polynomial use top coefficients approximate maximum root within factor logk log log respectively also prove corresponding lower bounds log show strong lower bounds noisy version problem one given access approximate coefficients problem applications context method interlacing families polynomials used proving existence ramanujan graphs degrees solution problem bounding integrality gap asymmetric traveling salesman problem involve computing maximum root certain polynomials top coefficients subexponential time results yield algorithm running time contents introduction motivation applications preliminaries approximation largest root proof theorem algorithm approximating largest root proofs theorems matching lower bounds lower bound given approximate coefficients applications interlacing families oracle oracle atsp appendix proofs preliminary facts appendix constructions graphs large girth introduction vector let denote unique monic polynomial roots suppose know rather know top coefficients concrete terms suppose know tell roots particular maxi problem given top coefficients real rooted polynomial degree well approximate largest root problem may seem completely impossible significantly smaller example consider two polynomials largest roots two polynomials differ arbitrarily absolute value even knowing top coefficients approximate absolute value largest key approach problem exploit real rootedness one approach construct polynomial given coefficients study roots unfortunately even assuming roots original polynomial real adding exponentially small amount noise bottom coefficients lead constant sized perturbations roots complex plane famous example wilkinson polynomial instead use given coefficients compute polynomial roots moment roots use polynomial estimate largest root contributions towards problem follows efficient algorithm approximating largest root theorem present upper bound showing use top coefficients real rooted polynomial nonnegative roots efficiently obtain approximation largest root log log log moreover approximation done poly time implies exact access log coefficients sufficient largest root nearly matching lower bounds main nontrivial part work informationtheoretic matching lower bounds theorem show constant algorithms approximation factor better chebyshev polynomials critical construction well also use known constructions graphs proof variant conjecture bounds made slightly sharper assuming girth conjecture motivation applications many important polynomials easy compute top coefficients exactly whereas provably hard compute one example matching polynomial graph whose coefficients encode number matchings various sizes polynomial computing constant term number perfect matchings whereas small one compute number matchings size exactly time simply enumerating possibilities roots matching polynomial particular largest root arise number important applications natural ask well largest root approximated top coefficients another example polynomial whose top coefficients easy compute independence polynomial graph real rooted graphs whose roots connections local lemma see subexponential time algorithms method interlacing polynomials main motivation work method interlacing families polynomials essential tool development several recent results including construction ramanujan graphs via lifts solution kadisonsinger problem improved integrality gaps asymmetric traveling salesman problem unfortunately results existence expanders matrix pavings thin trees drawback nonconstructive sense give polynomial time algorithms finding desired objects notable exception situation somewhat similar local lemma able guarantee existence certain rare objects nonconstructively algorithmic proofs found section use theorem give time algorithm rounding interlacing family depth improving previously known running time leads algorithms running time problems mentioned lower bounds given approximate coefficients context efficient algorithms one might imagine computing much larger number coefficients approximately might provide better estimate largest root particular consider following noisy version problem problem given real numbers promised first coefficients polynomial well approximate largest root important extension information theoretic lower bounds problem extremely sensitive noise proposition prove even knowing coefficient exactly knowing one error better knowing first coefficients exactly exhibiting two polynomials agree coefficients except kth differ slightly nonetheless different largest roots example relevant context interlacing families polynomials lower bound common interlacing characteristic polynomials base graph means could actually arise proofs appreciate broadly one consider following taxonomy increasingly structured polynomials complex polynomials polynomials mixed characteristic polynomials characteristic polynomials lifts graphs example complements standard numerical analysis wisdom wilkinson example complex polynomial roots general terribly functions coefficients shows fact remains true even structured setting interlacing families proposition relevant quest efficient algorithms interlacing families following reason coefficients matching polynomial bipartite graph approximated poly error polynomial time fixed polynomial using markov chain monte carlo techniques one might imagine extension techniques could used approximate coefficients general expected characteristic polynomials appear applications interlacing families fact families interlacing polynomials namely mixed characteristic polynomials design markov chain monte carlo techniques approximate top half coefficients within poly error information theoretic lower bounds rule method way approximate largest root least full generality since knowing coefficients polynomial poly error poly better knowing first log coefficients exactly worst case even promise given polynomials common interlacing words even mcmc oracle gives poly approximation coefficients would generically allow one round interlacing family depth greater logarithmic since error accumulated step would connections poisson binomial distributions finally probabilistic view problem assume sum independent bernoulli random variables poisson binomial parameters problem becomes following given first moments well approximate maxi view paper related shown pair poisson binomial random variables first moments total variation distance however bound total variation distance directly imply bound maximum discussion besides conducting precise study dependence largest root realrooted polynomial coefficients results paper shed light truly efficient algorithm interlacing families might look like one hand running time shows problem eth hard unnatural enough suggest faster algorithm instance quasipolynomial may exist hand lower bounds show polynomials arise method general hard compute rather robust sense namely obtaining inverse polynomial error approximation largest roots requires knowing many coefficients exactly implies order obtain efficient algorithm even approximately simulating interlacing families proof technique one exploit finer properties polynomials hand find global proof able reason error sophisticated amortized manner perhaps track quantity place largest root computed using fewer coefficients still satisfies approximate interlacing property preliminaries let denote set use notation denote family subsets let denote set permutations set bijections use bold letters denote vectors vector denote coordinates let denote maxi mini respectively symmetric matrix denote vector eigenvalues roots det similarly denote largest smallest eigenvalues slightly abuse notation polynomial write denote vector roots also write denote largest root graph let degmax denote maximum degree vertices degavg denote average degree vertices follows mostly standard facts proofs fact fact fact fact included appendix completness facts linear algebra matrix characteristic polynomial defined det letting sum principal minors det several algorithms matrix calculate det time polynomial identity use algorithm efficiently obtain following proposition proved using formula proposition let vit det vit symmetric polynomials make heavy use elementary symmetric polynomials relate roots polynomial coefficients definition let denote elementary symmetric polynomial defined fact consider monic univariate polynomial suppose roots every means knowing top coefficients polynomial equivalent knowing first elementary symmetric polynomials roots also implies following fact shifting scaling affect elementary symmetric polynomials fact let use following relationship elementary symmetric polynomials power sum polynomials theorem newton identities polynomial written furthermore computed point time poly one immediate corollaries following corollary let univariate polynomial deg written furthermore computed point time poly theorem shows computed reverse also true second set identities also known newton identities imply following theorem newton identities written polynomial computed time poly corollary theorem following corollary two vectors chebyshev polynomials chebyshev polynomials first kind simply call chebyshev polynomials defined follows definition let polynomials recursively defined call chebyshev polynomial notice coefficients computed poly time recurrence example chebyshev polynomials many useful properties mention information see fact cos cos cosh cosh fact chebyshev polynomial degree fact fact integer monotonically increasing furthermore approximate lower bound use following connection chebyshev polynomials graphs due godsil gutman fact adjacency matrix cycle vertices det graphs large girth order prove impossibility results use existence extremal graphs small cycles definition undirected graph denote length shortest cycle girth forest girth following conjecture characterizes extremal graphs small cycles conjecture girth conjecture every integer sufficiently large exist graphs vertices girth edges words satisfy degavg conjecture proven use following general construction graphs somewhat lower girth theorem prime power odd bipartite graph vertices girth signed adjacency matrices lower bounds also utilize facts signings graphs definition graph define signing function define signed adjacency matrix associated signing follows note definition symmetric zeros diagonal following fact immediate fact signed adjacency matrix graph eigenvalues roots det real bipartite eigenvalues symmetric origin counting multiplicities signed adjacency matrices used prove existence bipartite ramanujan graphs degrees state one main results theorem every graph exists signing degmax fact following immediate corollary corollary every bipartite graph exists signing eigenvalues absolute value degmax note trivially signing every edge often far achieving bound witnessed following fact fact let adjacency matrix graph signed adjacency matrix sign every edge maximum eigenvalue least degavg approximation largest root section give answer problem witnessed fact knowing top coefficients polynomial knowing therefore without loss generality conveniently state results terms knowing theorem algorithm receives unknown input outputs approximation guarantee approximation factor logk log log furthermore algorithm runs time poly note change behavior approximation factor two regimes log log log expression logk much worse bound compared logk near threshold log logk close order complement result showing lower bounds theorem every two vectors log log log shows algorithm approximate factor better using note constant bounded away zero constant possible give constant multiplicative bound assuming girth conjecture theorem assume fixed girth conjecture conjecture true graphs girth large enough two vectors proof theorem algorithm approximating largest root consider two cases log return estimate maximum root hard see case gives approximation maximum root see claim log still use estimate maximum root guarantees logk approximation show using machinery chebyshev polynomials obtain better bound pseudocode algorithm see algorithm algorithm algorithm approximating maximum root top coefficients input output approximation log compute using newton identities theorem return else loop compute using corollary return end log end loop end prove following claims show correctness algorithm let start case log claim proof observe max taking root sides proves claim rest section handles case log first claim shows long algorithm keeps decreasing multiplicative factor log since beginning claim proof every fact finish proof correctness enough show log done next claim shows soon gets lower within one iteration loop algorithm terminates claim log log fact proof log log log log exp used inequality used fact fact conclude every claim also gives bound number iterations algorithm terminates start loop loop terminates within one iteration soon therefore number iterations log log log log log proofs theorems matching lower bounds machinery chebyshev polynomials used prove theorem show machinery also used prove weaker version theorem theorem every proof first let prove let set roots set roots two polynomials except constant term follows use following lemma prove lemma roots cos counting multiplicities cos proof cos cos cos almost roots distinct since degree follows roots roots collide perturb use fact roots continuous functions polynomial coefficients prove statement using lemma get proves moreover cos finishes proof let prove statement general applying proof get construct adding zeros make total count hard see using corollary moreover finishes proof note lower bound lower bound theorem however prove theorem theorem need tools crucial idea use get stronger theorem theorem following observation signed adjacency matrices graphs large girth lemma let graph arbitrary diagonal matrix girth top coefficients polynomial det independent signing words two signings apply lemma graphs large girth constructed based conjecture theorem order prove theorem theorem regime log order prove theorem regime log marry constructions chebyshev polynomials prove lemma end section proving theorem theorem first let prove theorem proof theorem apply lemma following graph construction proof defer appendix claim let fixed assume conjecture true graphs girth sufficiently large exist bipartite graphs girth degmax degavg let graph claim let trivial signing assigns every edge signing guaranteed corollary let square eigenvalues square eigenvalues lemma corollary another application corollary hand corollary degmax fact degavg therefore let prove theorem log proof theorem log apply lemma following graph construction proof defer appendix claim let prime sufficiently large exist bipartite graphs girth log degmax degavg fix specific prime later similar proof theorem let trivial signing signing guaranteed corollary let log even integer girth exists log constant take means log long happens sufficiently large remains show every girth lemma gives finishes proof corollary method unfortunately seem directly extend regime log since large getting girth requires many vertices degree instead use machinery chebyshev polynomials boost graph constructions proof theorem log since theorem proves desired bound theorem may without loss generality assume least large enough constant let largest integer log large constant fix later easy see log already proved theorem small regime using proof log find log log without loss generality simple scaling may assume let unique monic polynomials whose roots respectively construction top log coefficients polynomials boost number equal coefficients composing chebyshev polynomials let unless degrees case actually reprove theorem omit details note deg deg let roots together additional zeros make counts first show top coefficients show real rooted real vectors finally lower bound note degree monic leading terms besides leading terms claim top log coefficients follows fact degree polynomial expanding either terms degree log produce monomials degree log means top log coefficients shows first log elementary symmetric polynomials follows first elementary symmetric polynomials follows log log first inequality used second inequality assumed large enough cancels hidden constants obvious even real crucially use properties chebyshev polynomial note roots therefore one written cos lemma equation cos real roots counting multiplicities simply cos root roots means roots real similar argument roots real largest root since hand largest root cos cos cos means cos cos cos cos arguments satisfy almost desired properties except could negative however know using fact easily make nonnegative simply replace finally cos log used last equality finished proofs theorem theorem ready prove lemma proof lemma corollary enough prove girth proving matrix following identity mvk ease notation let identify apply formula sides sequence interpreted sequence vertices graph term inside sum vanishes therefore restrict sum terms either connected borrow abuse notation markov chains let call sequence lazy closed walk length order prove enough prove lazy closed walk get term let one lazy closed walk consider particle time resides step particle either move moves neighboring vertex time returns starting position step particle move get one entries corresponding current vertex factor clearly independent signing particle moves however get sign edge moved factor show particle must cross edge even number times therefore signs edge cancel get result consider induced subgraph girth subgraph cycles therefore must tree lazy closed walk crosses cut graph even number times edge tree constitutes cut therefore lazy closed walk must cross edge tree even number times lower bound given approximate coefficients lower bounds proved previous section show knowing small number coefficients polynomial exactly insufficient obtain good estimate largest root section generalize construction theorem provide satisfying lower bound problem proposition every integer degree polynomials coefficients except exactly equal coefficients within multiplicative factor largest root least largest root common interlacing characteristic polynomials graph laplacians laplacians correspond common base graph proof let since polynomials differ constant terms moreover fact thus agree first coefficients differ multiplicative factor coefficient establishing see observe largest root cos whereas largest root cos since difference numbers conclude ratio least see observe roots cos cos cos multiplicity roots cos multiplicity whence common interlacing according definition first apply fact interpret characteristic polynomials cycles let denote cycle length let denote union two cycles det ack det ack det whence det ack det ack det det characteristic polynomials graph laplacians weighted graphs self loops note graphs since multiplying constants change properties interested ignore considering yields examples desired dimension note multiplying simply corresponds adding isolated vertices corresponding graphs applications interlacing families section use theorem give time algorithm rounding interlacing family depth let start defining interlacing family definition interlacing say real rooted polynomial interqn laces real rooted polynomial say polynomials common interlacing polynomial interlaces following key lemma proved lemma let polynomials degree real rooted positive leading coefficients common interlacing definition interlacing family let finite sets let nonempty let real rooted polynomial degree positive leading coefficient let note define say polynomials form interlacing family following holds polynomials identically zero common interlacing definition say depth interlacing families follows repeated applications lemma interlacing family polynomial largest root largest root thm rounding algorithm interlacing family algorithm returns polynomial next design rounding algorithm given oracle computes first coefficients polynomials interlacing family theorem let finite sets let interlacing family degree polynomials suppose given oracle returns top coefficients time algorithm returns polynomial largest root times largest root time log max poly proof let log theorem polynomial find approximation largest root time poly round interlacing family many steps step round coordinates make sure step incurs multiplicative approximation cumulative approximation error desired let describe algorithm inductively suppose selected brute force polynomials identically zero note maxi many polynomials polynomial identically zero compute approximation largest root using top coefficients let argmin follows algorithm runs time poly maxi interlacing family polynomial whose largest root largest root step algorithm therefore therefore induction desired remark without use chebyshev polynomials theorem one obtain somewhat worse running time applying trick rounding vectors groups rather one time use theorem need construct aforementioned oracle application interlacing families next construct oracle several examples oracle start interlacing families corresponding weaver problem equivalent formulation problem marcus spielman srivastava proved following theorem theorem given vectors isotropic position vit maxi kvi partitioning vit give algorithm finds partitioning runs time follows proof enough design rounding algorithm following interlacing family let independent random vectors support let det rti marcus showed interlacing family next design algorithm returns first coefficients polynomial family time maxi theorem given independent random vectors support algorithm returns top coefficients time maxi poly proof fix sufficient show compute coefficient time maxi poly first observe det proposition coefficient equal rti note summation exactly terms compute rti time maxi poly need brute force vectors domain average corresponding sums rank matrices follows theorem theorem given set vectors isotropic position squared norm find two partitioning time oracle atsp next construct oracle interlacing families related asymmetric traveling salesman problem say multivariate polynomial real stable roots complex plane whenever generating polynomial defined say strongly rayleigh probability distribution real stable say homogeneous sets support size following theorem proved first second authors theorem let homogeneous strongly rayleigh probability distribution subsets let isotropic position kvi set support vit give algorithm finds set promised theorem assuming oracle returns follows proof enough design rounding algorithm following interlacing family let support let otherwise add define det follows interlacing family next design algorithm returns top coefficients polynomial family time poly theorem given strongly rayleigh distribution subsets set vectors suppose given oracle returns algorithm returns top coefficients time poly proof fix first note element nothing prove assume sufficient show compute coefficient time poly firstly observe since summing characteristic polynomials consistent work conditional distribution note since efficiently compute also compute conditioned need differentiate respect conditioned let also note instead differentiating let large number divide resulting polynomial note determinantal distribution case applications asymmetric traveling salesman problem differentiation computed exactly efficiently cases taken exponentially large tolerate exponentially small error write det proposition coefficient equal compute abovep quantity enough brute force sets compute vit time poly addition efficiently compute using oracle enough differentiate respect therefore algorithm runs time poly follows theorem theorem homogeneous strongly rayleigh distribution marginal probabilities oracle computes vectors isotropic position squared norm find time set support enough get polyloglog approximation algorithm asymmetric traveling man problem graph vertices runs time references nima anari shayan oveis gharan flows spectrally thin trees asymmetric tsp pages focs nima anari shayan oveis gharan problem strongly rayleigh measures applications asymmetric tsp beck algorithmic approach local lemma random structures algorithms yonatan bilu nathan linial lifts discrepancy nearly optimal spectral gap combinatorica michael cohen ramanujan graphs polynomial time maria chudnovsky paul seymour roots independence polynomial clawfree graph combin theory ser constantinos daskalakis christos papadimitriou sparse covers sums indicators probab theory related fields extremal problems graph theory theory graphs applications proc sympos smolenice pages publ house czechoslovak acad prague arxiv preprint shmuel friedland daniel levy approximation algorithm number bipartite graphs arxiv preprint ole heilmann elliott lieb theory systems comm math chris hall doron puder william sawin ramanujan coverings graphs arxiv preprint nicholas harvey piyush srivastava jan computing independence polynomial shearer region lll arxiv preprint mark jerrum alistair sinclair eric vigoda approximation algorithm permanent matrix nonnegative entries acm july felix lazebnik vasiliy ustimenko explicit construction graphs arbitrary large girth large size discrete appl aridam vii new brunswick adam marcus daniel spielman nikhil srivastava interlacing families bipartite ramanujan graphs degrees focs pages adam marcus daniel spielman nikhil srivastava interlacing families mixed characteristic polynomials problem adam marcus nikhil srivastava daniel spielman interlacing families bipartite ramanujan graphs sizes appear focs robin moser tardos constructive proof general local lemma acm art alistair sinclair mark jerrum approximate counting uniform generation rapidly mixing markov chains inf july alistair sinclair piyush srivastava theorems complexity computing averages stoc acm symposium theory computing pages acm new york gabor szeg orthogonal polynomials volume american mathematical valiant complexity computing permanent theoret comput wenger extremal graphs combin theory ser james wilkinson perfidious polynomial studies numerical analysis volume maa stud pages math assoc america washington appendix proofs preliminary facts proof fact conclusion trivial otherwise let vector roots similarly vector roots easy see top coefficients expansion top coefficients well since linear proof corollary statement corollary becomes theorem polynomial write theorem follows polynomial computed time poly since computed time poly proof fact cosh inverse cosh looked function cosh monotonically increasing appropriate ranges therefore monotonically increasing exp exp cosh cosh first inequality used fact cosh last inequality used fact exp proof fact maximum eigenvalue characterized called rayleigh quotient max vector degavg appendix constructions graphs large girth section prove claim claim proof claim assume fixed conjecture true graphs girth graphs promised conjecture already average degree need make bipartite make sure maximum degree making graph bipartite easy given graph following procedure makes bipartite replace vertex replace edge two edges procedure doubles number edges number vertices easy see decrease girth trim maximum degree following procedure vertex deg introduce new vertex take arbitrary subset edges incident change endpoint repeating procedure graph eventually maximum degree bounded procedure increases number vertices number new vertices easily bounded new vertex degree end number edges change procedures change number vertices get graphs arbitrary size use following trick start graph promised conjecture vertices large constant make sure edges removing edges necessary increase girth make bipartite graph aforementioned procedure make using second procedure makes sure degmax number new vertices added step end total number vertices make large enough make sure less add isolated vertices total number vertices becomes get procedure changes degavg constant factor change degmax end proof claim build graphs using theorem let largest odd number large enough exists log theorem exists bipartite graph vertices girth log construct put many copies possible side side making sure number vertices grow end add isolated vertices make total number vertices get easy see number isolated vertices added end vertices degree rest degree therefore degavg degmax
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reasoning systems elements randomly switch characteristics subhash kak abstract examine issue stability probability reasoning complex systems uncertainty structure normally propositions viewed probability functions abstract random graph implicitly assumed nodes graph stable properties nodes change characteristics situation covered abstractions either static dynamic sets changes take place regular intervals propose use sets elements change modular forms proposed account one type change expression dependence mean probability switching elements determined system also analyzed perspective decision different hypotheses sets likely use complex system queries analysis surveys introduction reasoning complex systems one often uses static data structures designed situations change constructed static sets allow query operations elements checking whether given value set enumerating values arbitrary order dynamic sets hand one insert delete elements elements stay number yet properties change randomly consider basic model set objects viewed consisting objects type remaining type properties two types distinct observation process consists choosing random element examining replacing box repeated interaction system probability easily computed let outcomes mapped random variable values respectively pick threshold determine probability greater assume objects actually types numbers nodes special sense query interaction project properties random fashion one could also view situation consisting objects type one could view class random set whose elements change type example polling surveys assume three political parties two candidates belong parties members third party must pick candidate choice varies day day based news public opinion simplicity assume members two parties vote party candidates reality groups well situation represents unstable sets subset floating elements fact considered determining system behavior elements switch characteristics randomly probability stabilize system queried say different days value fall range nodes may seen creating noise measurement indeed real systems behave like property needs taken consideration impact reasoning evaluated purpose illustration assume transitions occur end day sampling different systems change estimate elements class figure figure changing count elements example importance sets consider medicine patient might respond different active ingredients different ways due certain element interchangeability structure changeability complicate analysis simpson related paradoxes simpson paradox arises fact situations range associated numerator values get complicated picture related ranges probability inversion occurs seen table table probability inversion week week lisa bart total lisa bart assigned task editing articles week work gets edited differently supervisors performance bart superior lisa weeks excepting lisa done bart done work aggregated lisa better consideration interswitching may make possible indirect accounting interaction networks exist body complicating influence information regulatory networks examine placebo effect effect counterintuitive inert substance turns effective treatment patients arises information flows expectations associated treatment cases interaction elements may rightly seen lens quantum decision theory especially evidence superpositional behavior nodes one major consequence switching nodes quantum context probability measurements commutative thus result even apparently independent interaction disturbs system causes variables change order influence expectations associated system ideas also apply field belief vectors represent relations states underlying reasoning process specific belief may numerical value boolean variable representing choice alternatives basic form may seen representing class type choice might fluctuate two different possibilities may viewed altered random influence projected class dynamics projection may zero fixed point paper consider problem reasoning states system change dynamically interaction specifically consider model internal states change nonlinear mapping proposed possible mechanism results statistical characteristics observations obtained system also analyzed perspective noise theory dynamics system model since components system interact system analyzed based classical probability since nature interactions potentially endless must constrain analysis means specific assumptions assume system nodes type nodes switch back forth like population predators prey isolated habitat complex system dynamics characterized fixed points orbits different mechanisms may work different systems rather speak full system focus elements type let probability element projected let interaction system change proportion nodes assume dynamics determined function set positive integers taken even number interaction count system various modular schemes dynamic change probability considered let number type nodes observed class nodes ith measurement let value run scheme mod make probability change manner scheme mod prime nodes cycle cycle full primitive element prime example total nodes type changes number type nodes inspection follows first row represents number first encounter case fixed point cycle initial number nodes probability switch values initial number probability goes zero scheme consider general modular mapping even odd particular consider transformation specific case mod even odd finite version mapping associated names collatz mathematicians value determines largest orbit associated probability evolution initial count nodes type taken even population type nodes never zero excepting mod example consider mapping mod numbers mapped one step follows values figure evolution initial states observation fixed point orbit odd type nodes either fixed point part orbit example one gets sequence words orbit trajectory elements path length maximum value generated let initial distribution get orbit trajectory elements path length maximum value sequence takes steps climbing descending assumption modular mappings system classical although highly nonlinear however situations true consideration quantum model appears justified takes beyond classical sets different manner statistical analysis given number elements belonging sets respectively begin basic probability associated events figure probability sets let variable representing average give random switching elements takes values range assume uniformly distributed random variable shown figure distribution expected value mean square value may easily computed variance thus variance mean square proportion switching elements variance independent actual values estimate variance measure size may important determining many individuals poll belong separate class choices firm class noise also present noise theory perspective problem assume sake symmetry classes number elements represent two different hypotheses number measurements taken mapped decision taken threshold logic choose hypothesis otherwise hypothesis since assume changes membership occurs low rate surveys specific day membership stable large one pleases one might also assume elements also change figure source probabilities expressed subscript observation probabilities expressed subscript figure observation process bayes theorem task estimate size bayes estimate may used however random nature elements set implies probabilities fixed certain conditions probabilities may taken fixed constant general nature nonlinear process underlying change characteristics well environment determine probabilities query could ascertain probability various subsets associated system means determining structure information subsets comes sampling strategy changed reflect estimates obtained bayes testing process updated continuously conclusions considered sets elements switch characteristics random manner time situation covered abstractions either static dynamic sets changes take place regular intervals propose use sets elements change modular forms proposed account one kind change expression dependence mean probability switching elements determined system also analyzed perspective decision different hypotheses sets likely use complex system queries analysis surveys references pearl understanding simpson paradox american statistician trial design testing treatment treatment preference effects statistics medicine kaptchuk miller placebo effect medicine engl med kak three languages brain quantum reorganizational associative learning pribram king editors lawrence erlbaum associates mahwah lambert quantum biology nature physics kak communication languages agents biological systems biocommunication interactions cells organisms seckbach gordon editors london world scientific publishing lagarias problem generalization amer math monthly wirsching dynamical system generated function lecture notes mathematics kak active agents intelligence quantum computing information sciences kak initialization problem quantum computing foundations physics kak quantum information entropy int journal theoretical physics aerts quantum structure cognition journal mathematical psychology kumar varaiya stochastic systems estimation identification adaptive control siam scott multivariate density estimation theory practice visualization john wiley tuzlukov signal detection theory springer science
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local approach information transfer jan mujicab facultad ciencias aplicadas universidad del norte ibarra ecuador centro computacional laboratorio lineales escuela facultad ciencias universidad central venezuela caracas venezuela abstract work strategy estimate information transfer elements complex system time series associated evolution elements presented using nearest neighbors state local approaches deterministic dynamical rule generating data probability density functions marginals conditionals necessaries calculate measures information transfer estimated performance method using numerically simulated data real signals exposed introduction estimation directionality coupling electrocardiographic respiratory signals may help decide system exerts control allowing efficient medication determination causal relations shares stock select one monitor order make decisions buy sell detection elements control evolution others complex network allows estimate robustness simplifies control problem reducing number sites take control actions situations permanent address laboratorio sistemas complejos departamento aplicada facultad universidad central venezuela caracas venezuela useful estimate subsystem sends receives information another complex system general perspective estimation extent component contributes production information system rate shared rest components provide important information structure reason detecting transfer information directionality subject great interest due variety practical applications areas ranging physics marketing since wiener proposed improvement prediction future time series incorporation information past second time series seen indicator causal interaction operational way implement ideas proposed granger formalization strategy schreiber problem estimation relation elements complex system addressed many different points views deterministic probabilistic modeling schemes proposed problem named detection causality directionality coupling estimation detection direct links among others work propose estimate transmission information elements complex systems using formal probabilistic scheme proposed schreiber approximating probability density functions pdf involved using neighbors approach case neighborhood point used deterministic nonlinear local model estimate conditional pdf density estimator approximate marginals pdf associated data strategy although also based determination nearest neighbors conceptually simpler strategy presented advantage estimate made using nearest neighbor principle makes computationally less expensive use deterministic models determination information transfer expected offer interesting relationship strategies based probabilistic deterministic models well parametric nonparametric schemes order show usefulness scheme work organized follows section devoted present idea behind information transfer section gives methodology estimate conditional pdf involved information transfer using local approaches section combine ideas approach schreiber estimate information transfer case numerically simulated real signals finally section give concluding remarks information transfer many tools estimate information transfer time series see example references therein nevertheless attempt classify categories require titanic effort among reasons unclear relations however popular time series tools appear fall one two broad classes strategies strategies strategies representative first class approach called granger causality class represented scheme based information theory entropy transfer proposed schreiber granger method causality could tested measuring ability predict using linear model future values time series using prior values another time series using scheme case nonlinear subsystems leads erroneous conclusions problem addressed nonlinear model performed using radial basis functions despite scheme still weakness static measure case methods usual strategy quantify superimposing information contained interacting subsystems mutual information unfortunately symmetrical static measure give sense direction temporal evolution information system strategy overcomes two previously mentioned weaknesses proposed estimate transmission information based observations time series associated system elements information theory schreiber proposes measure transfer entropy represents rigorous derivation wiener causal measure shares desired properties represent transfer information mutual information takes account dynamics system measure minimal assumptions dynamics system nature coupling able quantify exchange information two systems follows may thought conditional mutual information joint conditional probabilities occurrence state system determining flow information given requires calculation probability densities associated transitions states subsystem case estimation requires coarse graining time series associated subsystems represents large computational cost increases considerably probabilities calculated pairs elements large extended system probability densities local modeling according wiener idea improvement prediction associated reduction uncertainty natural measure causality represented terms information theory concepts however determination causal relationships information transfer may also represented terms deterministic models show feasibility idea start representing joint probabilities entropy transfer definition terms conditional marginal probability densities using later propose strategy estimate conditional densities probabilities marginal densities probabilities using local nonlinear modeling dynamical system nonparametric method estimate marginals densities functions respectively inspired propose estimate conditional probabilities densities associated appearance states given predecessors states subsystems using quality predictions made deterministic model specifically base approach estimation nearest neighbors marginal probability density functions approached using strategy order start constructing cumulative distribution function derive necessary probability densities function definition strictly increasing function random variable usually represented side probability random variable takes value less equal probability lies interval therefore cumulative density function continuous random variable expressed integral probability density function follows similarly case cumulative conditional distribution function fxy pxy thus conditional probability density cpd function obtained pxy chose cumulative conditional distribution function sigmoid function model system constructed data distinguish two extremes distribution one strictly deterministic without errors given heaviside function probabilistic approach parameter sense measure randomness process clear equations approximation dynamic rule governs evolution system possible construct adequate cumulative distribution function allows estimate probability densities necessary calculate transfer information case propose chose parameter sigmoid function proportional standard deviation modeling errors since slope transition sigmoid function determines width distribution associated point summarize presentation two main ideas schreiber information transfer estimators written terms cpd cpd estimated using deterministic models rule generating data remainder section intended implement models show incorporated estimator information transfer computational cost similar determination neighbors let suppose series values states dynamical system obtained either measuring components system state regular time intervals reconstructing state space using takens theorem partial information states approach dynamical rule estimated using local approximation approach presented algorithmically given data value determine set nearest states xvi state euj clidean metric sorted ascendant order distance vji label neighbor data value integer interval iii approximation obtained taylor expanding first order around nearest neighbor xvi obtain xvi xvi xvi xvi jacobian matrix finally zero order evolution system approximated evolution nearest neighbor xvi first order approximation necessary calculate xvi marginal probabilities densities functions estimated using nearest neighbor density estimator xvi illustrate graphically methodology estimate probability densities figure shows conditional marginal probability density functions data points skew tent map thus zero order approach dynamical rule using equations cpd function given xvi xvi figure illustrates mentioned ideas gives graphical insight proposed strategy associates modeling errors cpd results order show performance strategy estimate information transfer case signals numerical simulation real data figure left cpd function doted line shows example probability occurrence values given occurrence value right marginal probability density function solid line represents local estimation doted one represents histogram data pdfs estimated using data points skew tent map parameter numerical simulations data associated numerical simulations correspond two coupled skew tent maps chaotic regime connected nonlinear bidirectional coupling function chuas system first case provides controlled experiment directionality transfer information known allows performance testing strategy second example allow compare performance strategy using numerically simulated data results system case real data coupled chaotic maps two chaotic tent maps parameter coupled according parameters define intensity coupling transmission information case associated synchronization trajectories systems understood synchronization coincidence states subsystems sufficient time elapsed figure shows synchronization error information transfer coupled maps function coupling parameters worth note flat zone information transfer surface coincide synchronization zone suggested bollt figure left synchronization error function coupling parameters right information transfer elements systems estimated using data points chua system chua system chaotic system representing rlc circuit one nonlinear elements easy construction circuit made ubiquitous example chaotic system dil ril usual linear function representing chua diode parameters chosen let rename elements circuits indexing define net flow information associated component circuit difference among sum information transfer rest elements sum information transfer rest elements figure information transfer pairs formed signals data obtained numerical integration system using order chose parameter two times standard deviation modeling errors dimension three evaluation quantities see figure produces consistent results control chua circuit authors prove chua system controlled using linear feedback controller say numerical simulations show simplest control achieved perturbing real data chua circuit case experimental implementation chua circuit done using operational amplifiers simulate chua diode inductor current inductor obtained indirect measure data sampled rate shown left side figure right side figure shows information transfer estimated explained pairs formed signals case consistent results obtained previously numerical simulation system figure left experimental data points chua circuit right information transfer pairs formed signals chose parameter two times standard deviation modeling errors dimension three number neighbors numerical experiments suggests results strongly dependent choice physiological signals experimental data correspond time series taken massachusetts general foundation waveform database https series corresponding electrocardiographic blood pressure signals measured simultaneously patients intensive care selected estimate information transfer systems data points pair series used cases greater transfer information arterial pressure system system obtained final remarks three aspects highlighted regarding performance proposed strategy first important referred low computational cost associated determination first nearest neighbors compared least cost estimate cpd using coarse graining calculating correlation integrals second must note face results using experimental data estimation transfer information based nearest neighbors appears robust presence moderate noise given consistency results references although detailed study aspects needed data set figure information transfer pairs formed electrocardiographic blood pressure signals measured simultaneously case represent information transfer system cardiac system conclusive scope present work subject current research finally emphasize although one advantages schreiber estimator independent models proposal become dependentmodel scheme nevertheless believe establishment relationship useful probabilistic definition information transfer deterministic modeling may practical idea development methodologies determine causal relationship subsystems complex systems contribute design control schemes extended systems references schultz adochier edu schroeder costin bar voss philos trans math phys eng sci liu slotine barabasi nature wiener theory prediction modern mathematics engineer granger econometrica schreiber phys rev lett bezruchko ponomarenko rosenblum pikovsky chaos aihara chen scientific reports ishiguro otsu lungarella kuniyoshi phys rev cisneros cosenza parravano phys rev rosenblum pikovsky phys rev palus stefanovska phys rev vejmelka palus phys rev rubido grebogi baptista masoller new journal physics zhu bellanger shu bouquin entropy vicente wibral efficient estimation information transfer berlin ancona marinazzo stramaglia phys rev shannon bell system technical journal palus phys rev farmer sidorowich phys rev lett marcano moleiro phys rev lett merlitti european physical journal special topics loftgaarden quesenberry annals mathematical statistics husmeier perspectives neural computing takens dynamical systems turbulence springer berlin hasler maistrenko ieee trans circuits syst matsumoto ieee transactions circuits systems bollt international journal bifurcation chaos chen dong journal circuits systems computers chen chaos solitons fractals chen dong proc ieee inr symp circ chicago gopakumar premlet gopchandran international journal electronics implementation circuit already undergraduate exercise case done using educational laboratory virtual instrumentation suite national instruments liebert phys lett welch ford teplick rubsamen waveform database cisneros revista mexicana fsica
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arxiv dec theory practice logic programming may distributed www programming using prolog pillow daniel cabeza manuel hermenegildo clip group http http facultad universidad madrid upm del monte madrid spain dcabeza herme abstract discuss practical point view number issues involved writing distributed internet www applications using systems describe pillow publicdomain internet www programming library systems designed order simplify process writing applications pillow provides facilities accessing documents code www parsing manipulating generating html xml structured documents data producing html forms writing form handlers processing templates important contribution pillow model code thus content www pages terms pillow library developed context ciao prolog system adapted number popular systems supporting functionality also describe use concurrency highlevel model interaction ciao prolog active modules context www programming propose solution downloading execution prolog code using generic browsers finally also provide overview related work topic keywords www html xml cgi http distributed execution constraint logic programming introduction wide diffusion internet popularity world wide web protocols effectively providing novel platform facilitates development new classes portable distributed applications good support network connectivity protocols communication architectures novel platform obviously requirements programming tool useful arena however alone may enough seems natural significant parts paper expanded improved version cabeza hermenegildo cabeza hermenegildo cabeza hermenegildo cabeza cabeza hermenegildo network applications require symbolic numeric capabilities necessarily related distribution important capabilities example symbolic information processing dealing combinatorial problems natural language processing general logic programming kowalski colmerauer constraint logic programming clp systems jaffar lassez van hentenryck colmerauer dincbas van hentenryck ecrc shown particularly successful tackling issues see example proceedings recent conferences practical applications prolog practical applications constraint technology seems natural study technology fares developing applications operate internet fact prolog concurrent constraint based extensions logic programming languages general many characteristics appear set particularly well placed making impact development practical networked applications ranging simple quite sophisticated notably systems share many characteristics recently proposed network programming tools java including dynamic memory management structure pointer manipulation robustness compilation bytecode furthermore unlike scripting application languages currently proposed shell scripts perl java etc systems offer quite unique set additional features including dynamic databases search facilities grammars sophisticated well understood semantics addition systems also already offer kind low level support remote communication using internet protocols generally involves providing sockets ports interface whereby possible make remote data connections via internet native protocol systems support higherlevel communication layers top interface including blackboards sicstus prolog carlsson ciao carro hermenegildo cabeza hermenegildo hermenegildo clip group hermenegildo hermenegildo hermenegildo bueno tarau bosschere etc shared communication chikayama akl janson haridi smolka ciao hermenegildo cabeza hermenegildo cases functionality provided via libraries building top basic primitives case example sicstus ciao distributed interfaces fact shown previous work sharedvariable based communication also implemented conventional systems via library predicates using attributed variables hermenegildo cabeza hermenegildo addition communication primitives several systems offer concurrency even abstractions distributed objects mobile code useful developing distributed applications concrete interest www applications applications generally distributed www programming using prolog pillow lib use specific protocols http ftp data formats html xml application architectures cgi interface different protocols typically used types distributed applications paper study good support protocols data formats architectures provided systems building widely available interfaces basic protocols aim discuss practical point view number new issues involved writing www applications using systems well architecture typical solutions process describe pillow programming logic languages web public domain programming library systems argue significantly simplifies process writing applications pillow provides facilities generating structured documents handling herbrand terms producing html forms writing form handlers processing templates accessing parsing www documents etc also describe architecture relatively sophisticated application classes using model clientserver interaction active modules cabeza hermenegildo finally describe architecture automatic code downloading local execution using library generic browsers apart tutorial value paper present number technical contributions include idea representing html xml code structured text general prolog terms use logical variable terms leading model html template pair comprising term free variables dictionary associating names variables notion active logic modules application solving efficiency issues cgi interaction simple way idea prolog scripts application cgis identification number features added existing systems order facilitate programming www applications mainly concurrency argument throughout paper small limitations functionality disappear concurrency added systems akl ciao prolog possible add extremely useful programming layer system without making significant changes implementation argue layer simplify generation applications systems including active www pages search tools content analyzers indexers software demonstrators collaborative work systems muds moos code distributors etc purpose paper also serve tutorial containing sufficient information developing relatively complex www applications prolog clp languages using pillow library pillow library developed context ciao prolog system adapted number popular systems supporting functionality ciao prolog system pillow library freely downloaded http http cabeza hermenegildo http server www browser std doc request output fig cgi interface writing basic applications simplest way writing www applications use common gateway interface cgi cgi executable standard executable file http server program responds http requests machine serves www site tell fact contains program run rather document text sent client browser usual file distinguished belonging special directory commonly named special filename ending normally set configuration http server basic idea behind cgi interface illustrated figure user selects address cgi executable document http perhaps http browser issues standard document request http server recognizing cgi executable rather document starts executable execution stores output executable buffer upon termination executable contents buffer format browser handle html returned browser normal page content accessed following example simple executable written language source might main write write html write hello write actual executable could generated usual example ciao note examples presented order shorten html code may slightly simplified result may completely however examples used popular browsers distributed www programming using prolog pillow lib system using standalone compiler writing unix shell ciaoc executable placed appropriate place accessible via http address browser right permissions executed server example systems means executable user nobody systems make executables saved states usually disadvantage generally large size system prompt one could create executable writing something like compile save main scripts cgi applications cgi executables often programs perform relatively simple tasks added slow speed network connection comparison executing program makes program execution speed less important made scripting languages shell scripts perl popular writing programs popularity due fact compilation necessary extensive string handling capabilities also play important role case perl thus changes updates program imply editing source file logic languages priori excellent candidates used scripting however relative complication making executables needing systems start compile consult file make saved state often large size resulting executables may deter cgi application programmers appears convenient provide means programs executable scripts even reduced performance generally relatively easy support scripts functionality systems ciao program also adapted sicstus hermenegildo accomplishes task first loading file given first argument skipping first lines avoiding loading messages starting execution argument provides list command line options example unix system following program run directly script without need compilation main write write html often convenient use options ciaoc generate standalone executable independent libraries example grammars databases greatly simplify many typical applications cabeza hermenegildo form http www browser http server form data form reply fig forms interface write hello write note unix versions either program must included listing first line replaced two exec execution prolog scripts may optimized systems example ciao first time script run also compiled bytecode saved file subsequent times script changed object code retrieved file avoiding compilation interpretation overhead form handling http far shown cgi executables produce output output function input coming request obviously limited interest cgi executables become useful combined html forms html forms html documents parts html documents include special fields text areas menus radio buttons etc allow providing input cgi executables steps involved handling input contained form illustrated figure document containing form accessed via browser mosaic netscape lynx etc browser displays input fields buttons menus etc indicated document locally allows user perform input modifying fields however input ultimately handled browser instead sent handler cgi program anywhere net whose address must given form forms generally submit button pressed input distributed www programming using prolog pillow lib provided menus text areas etc sent browser http server corresponding handler two methods sending input exist get post meantime sending browser waits response program come form new html document handler program invoked much way application except information form supplied handler different ways depending system method invocation content type information encoded predefined format relates piece information corresponding field form means keyword associated field handler identifies information corresponding field original form processes responds writing html document standard output forwarded server waiting browser handler terminates important point noted simple applications handler started terminate transaction reader referred example grobe naseer complete introduction cgi scripts html forms writing form handlers llow complication writing form handlers compared writing simple cgi applications need capture parse form data said data provided several ways depending system method used invoke form encoded escape sequences relatively easy write prolog program parse input using example definite clause grammars pillow library provides predicates simplify whole task hiding protocol behind principal predicates provided include dic translates input form either post get methods even dictionary dic pairs translates empty values indicate presence attribute atom empty values one line text areas files list lines strings rest atoms numbers using implemented using dcg parsers dic var val gets value val attribute var dictionary dic fail value found simplifies merging form producers form handlers see later useful check value text area empty filters spaces newlines linefeeds val default newval useful form partially filled also first invocation combined form see section value val empty else url returns url uniform resource locator www address cgi executable method returns method method invocation form handler get post cabeza hermenegildo example suppose want make handler implements database telephone numbers queried form including single entry field name handler might coded follows include library pillow main input input name write write html title telephone database write img write telephone database name write name name write provide phone name phone write telephone number write name write write phone write telephone number available write name write phone daniel phone manuel phone sacha code quite simple hand interspersion throughout text calls write html markup inside makes code somewhat inelegant also separation computation normally desirable would much preferable encoding html code prolog terms could manipulated easily elegant way predicate translate terms html output functionality provided pillow library presented next section handling html prolog terms since systems perform symbolic processing using herbrand terms seems natural able handle html code directly terms structures distributed www programming using prolog pillow lib need translated appropriate predicates html code need output general relationship html code prolog terms allows viewing www page herbrand term predicates provide functionality pillow accepts html term list html terms sends standard output text rendering term html format chars terms also relates list html resp xml terms list ascii characters rendering terms html format predicate reversible normalizes reverse direction later uses predicate transform html terms characters implemented via dcg parsing html term certain atoms structures represent special functionality html level html term recursively list html terms following legal html terms hello hello world html term converting html terms characters translates special structures corresponding format html applying recursively arguments strings always left unchanged html terms may contain logic variables provided instantiated term translated output allows creating documents piecemeal references documents etc following sections list meaning principal prolog structures represent special functionality html level special atoms translated rest assumed normal text passed html document general structures basically html two kinds components html elements html environments html element form name attributes name name element attributes possibly empty sequence attributes either attribute name attribute assignment value html environment form name attributes text name name environment attributes form general prolog structures represent two html constructions name atts defined infix binary operator represents html element name name attributes atts atts possibly empty list cabeza hermenegildo attributes either atom structure name example term img map ismap translated html source img map ismap note html use atoms name text term functor argument text represents html environment name name included text text example term address clip translated html source address clip name atts text term functor arguments atts text represents html environment name name attributes atts included text text example term http clip home represents html source http clip home env name atts text equivalent name atts text begin name atts translates start html environment name name attributes atts exists also begin name structure useful conjunction next structure including document output generated existing piece code name pre use otherwise discouraged end name translates end html environment name name rewrite previous example follows note use logic variable response allows injecting result call output term using unification include library pillow main input input name response name response html title telephone database img telephone database distributed www programming using prolog pillow lib response using logic variable response name response name response provide phone name phone response telephone number name phone response telephone number available name phone daniel phone manuel phone sacha html construction represented structures except comments declarations could included atoms strings pillow library provides additional specific structures simplify html creation specific structures section list special structures html pillow understands many cases using general structures native html names probably good practice using specific structures sometimes convenient also structures special functionality predicate provided allows defining new structures tables layers etc specific structures include reader referred pillow manual full listing start used beginning document translates html end used end document translates produces horizontal rule translates produces line break translates produces paragraph break translates comment comment used insert html comment translates comment declare decl used insert html declaration seldom used translates decl image addr used include image address url addr translates img element image addr atts list attributes atts ref addr text produces hypertext link addr url referenced resource text text reference translates addr text label label text labels text target destination label label translates label text heading text produces heading level text text used heading useful one wants heading level relative another heading translates environment cabeza hermenegildo itemize items produces list bulleted items items list corresponding html terms translates environment enumerate items produces list numbered items items list corresponding html terms translates environment description defs produces list defined items defs list whose elements definitions prolog sequence composed operators last element sequence definition defined terms translates environment img items produces list bulleted items using image img bullet predicate provides colored bullet preformatted text used include preformatted text text list html terms element list line resulting document translates pre environment verbatim text used include text verbatim special html characters translated quoted html equivalent term includes prolog term term represented functional notation variables output used include newline html source improve human readability entity name includes entity name name special character html rather cgi protocol requires content descriptor used cgi executables including form handlers replying translates includes page graphical logo message developed using pillow web programming library points manual library source additional structures rewrite previous example follows note example use equally suitable include library pillow main input input name response name response start title telephone database image heading telephone database response distributed www programming using prolog pillow lib end response name response name response provide phone name phone response telephone number name phone response telephone number available name phone daniel phone manuel phone sacha included specific structures creating forms included explained following section specific structures forms section explain structures represent various elements related forms addr atts specifies beginning form addr address url program handle form atts attributes form method used invoke atts present method defaults post translates form addr atts specifies beginning form without assigning address handler form handler executable producing form specifies end form translates checkbox name state specifies input type checkbox name name checkbox initially checked translates input element radio name value selected specifies input type radio name name several radio buttons interlocked must share name value value returned button button initially checked translates input element input type atts specifies input type type list attributes atts possible values type text hidden submit reset translates input element textinput name atts text specifies input text area name name text provides default text shown area atts list attributes translates textarea environment option name val options specifies simple option selector name name options list available options val initial selected option val options first item selected translates select environment cabeza hermenegildo menu name atts items specifies menu name name list attributes atts list options items elements list items marked prefix operator indicate selected translates select environment example order generate form suitable sending input previously described phone database handler one could execute following goal start title telephone database heading telephone database http click enter name clip member press return input text end course one could also simply written directly resulting html document html title telephone database telephone database form post http click enter name clip member press return input text merging form producer handler interesting practice producing html forms handlers merge operation form producer handler program idea produce generalized handler receives form input parses computes answer produces new document contains answer input well new form special case must made first invocation input would empty form generated following example merges producer handler phones notice one text field exists form form submitted simply pressing return inside text field distributed www programming using prolog pillow lib include library pillow main input input name response name response start title telephone database image heading telephone database response click enter name clip member press return input text end response name response name response phone name phone response telephone number name phone response telephone number available name phone daniel phone manuel phone sacha combination form producer handler allows producing applications give impression interactive even step involves starting running handler completion note forms contain fields displayed passed input next invocation handler allows passing state one invocation handler next one finally note testing debugging cgi scripts unfortunately straightforward could useful techniques include carefully checking permissions looking data logs server replacing predicates versions print really received etc cabeza hermenegildo templates problem previous programs layout output page easily configurable source changed modifying program something normal user even expert programmer size program large may want order address pillow provides facility reading html templates also xml templates converting term format natural manipulate html template file contains standard html code slots defined given identifier means special tag slots represent parts html code html code inserted html template read pillow slots appear free logic variables corresponding pillow terms way user define layout html editor choice taking care marking left parts given names parts filled appropriately program functionality associated parsing terms encapsulated following predicate chars terms dict parses string chars contents html template unifies terms list html terms comprised template substituting occurrences special tag name prolog variables dict instantiated dictionary substitutions list pairs following example template file called assumed hold formatting output page defining html variable called response substituted response cgi program note predicate defined ciao library reads file returns second argument contents file list character codes note also calling third argument instantiated response response effect instantiating slot contents response makes use fact one slot template normally call used locate appropriate pair include library pillow library main input input name response name response contents contents response response distributed www programming using prolog pillow lib response name response name response phone name phone response telephone number name phone response telephone number available name phone daniel phone manuel phone sacha example contents template file could html head title telephone database body img telephone database response form post click enter name clip member press return input text accessing www documents facilities presented previous sections allow generating html documents including forms handling input coming forms many applications search tools content analyzers also desirable able access documents internet access generally accomplished protocols ftp http built top systems connectivity interface required protocols easily coded source language using facilities dcg parsers present http protocol supported pillow html code library uses internal representation uniform resource locators urls able manipulate easily provides predicates translate internal representation textual form facilities provided pillow accessing www documents include following predicates url info translates url url internal structure info details various components urls make predicate fail http info gives info http url http gives url http string cabeza hermenegildo url baseinfo info translates relative url url appears html page referred baseinfo given structure complete structure info absolute urls translated previous predicate http info gives info http http info gives info http dic args translates list pairs dic form dictionary returned string args appending url pointing form handler url request response fetches document internet url uniform resource locator document given structure request list options specify parameters request response list includes parameters response request parameters available include head specify interested header timeout time time specifies maximum period time seconds wait response predicate fails timeout date get document newer date example structure represents date date tuesday january name provide field authorization scheme params provides authentication field accessing restricted sites name param functor translates field name user machine parameters returned response list include see definition information content content returns content actual document text list characters status type code phrase gives status response type informational success redirection code status code phrase textual explanation status pragma data miscellaneous data date time message sent location url document moved url server identifies server responding allow methods list methods allowed server date sender believes resource last modified distributed www programming using prolog pillow lib expires date entity considered stale type subtype params returns mime document type encoding document length length size document bytes authenticate challenges request authentication chars terms already explained predicate transforms html terms html format used way around parse html code example retrieved resulting list html terms terms normalized contains structures example simple fetch document done follows http member content note error occurs document exist moved example simply fail following call retrieves document modified since october http wednesday october last one retrieves header document timeout seconds get last modified date http head timeout member date following simple application illustrating use example defines url badlinks predicate fetches html document pointed url scours check links produce errors followed list badlinks contains bad links found stored compound terms form badlink link error link problematic link error error explanation given server url badlinks url urlinfo urlinfo response member text html response member content content response content terms terms urlinfo badlinks baseurl cabeza hermenegildo baseurl baseurl env anchoratts baseurl member anchoratts url baseurl env baseurl baseurl url baseurl url baseurl urlinfo urlinfo status phrase status success name phrase name url badlink url status phrase url head timeout response member status status phrase response timeout timeout providing code www facility easily built top primitives presented far remote www modules program modules reside net particular http address way normal program modules reside particular location local file system allows example always fetching recent version given library pillow program compiled example form handler section rewritten http main input input name would load current version library time executed generalized module declaration syntactic sugar using document distributed www programming using prolog pillow lib form http http server www browser form data form reply active module predicate argn fig forms interface using active modules fetch using followed standard declaration obviously interesting combine facility caching strategies interesting straightforward implement additional feature fetch remote generally done available possible two systems use normally checked easily bytecode also may interesting combine type code downloading www document accesses code downloaded automatically particular document fetched issue addressed section finally obvious security issues related downloading code general addressed standard techniques security signatures model interaction active modules despite power interface also shortcomings serious perhaps fact handler started expected terminate interaction two disadvantages first state preserved one query next however mentioned fixed passing state form using hidden fields saving temporary file server side using cookies etc second importantly starting stopping application may inefficient example idea query large database natural language understanding system may take long time start stop system order avoid propose alternative architecture applications similar idea although based idea active modules proposed independently ken bowen bowen basic idea illustrated figure operation identical standard form handlers illustrated figure step step handler started application rather interface actual application running continuously thus contains state thus cabeza hermenegildo interface started stopped every transaction interface simply passes form input received server running application forwards output application server terminating application continues running interface application written using predicates presented interface simple script application typically compiled interesting issue communication interface application course done sockets however cleaner much simpler alternative concept active modules cabeza hermenegildo used advantage application active module active object modularity implemented via objects ordinary module computational resources attached example process unix machine resides given socket address compiling active module produces executable running acts server number relations predicates exported module relations exported active module accessed program network simply loading module thus importing remote idea process loading active module involve transferring code rather setting things calls local module executed remote procedure calls active module possibly network except compiling special way active module identical programmer point view ordinary module also program using active module imports uses way module except uses rather see also active module address network address must known order use address announced active module started via file name server would another active module fixed address present constructs related active modules ciao module predicates declaration used import predicates list predicates active module module point code written standard declaration used declaration needs following predicate accessible module module address predicate must return address address module active module imported code number standard libraries defining versions predicate address predicate define way publish address used active modules name active module taken name current executable number standard libraries defining versions predicate correspondence libraries define versions previous predicate also possible provide active modules via www address however find straightforward simply use socket addresses case generally hidden inside access method thus made transparent user distributed www programming using prolog pillow lib modulefile publishmodule makes active module executable module residing modulefile using address publish module name publishmodule executable run example operating system level module socket created hook predicate mentioned supposed defined publishmodule called order export active module address required standard driver run attend network requests module exported predicates note code modulefile need written special way scheme flexible allowing completely configure way active modules located accomplished writing pair libraries one defining way active module address published second defining way address given active module found example ciao standard libraries include example implementation libraries uses directory accessible involved machines via nfs store addresses active modules predicate examines directory find required data solutions provided examples include posting address www address implementation name server another active module one known fixed address records addresses active modules supplies data modules import serving contact agency servers clients implementation point view active modules essentially daemons prolog executables started independent processes operating system level ciao system library communication active modules implemented using sockets thus address active module unix socket machine requests execute goals module sent socket remote programs request arrives process running active module takes executes returning socket computed results results taken remote processes thus compiler finds declaration defines imported predicates remote calls active module example predicate imported active module predicate would defined compiling following code active module writing ciao toplevel using standalone compiler executing ciaoc creates executable started process example typing unix shell prompt saves address socket file waits queries module imports module also provides predicate dynamically add information database module cabeza hermenegildo response name response name response provide phone name phone response telephone number name phone response telephone number available name name phone assert phone name phone dynamic phone daniel phone manuel phone sacha following simple script used executable active module interface previous active module started process form input issue call automatically handled active module produce new form terminating locate address active module via predicate defined library library include library pillow main input input name response name response start title telephone database image heading telephone database response click enter name clip member press return distributed www programming using prolog pillow lib input text end many enhancements simple schema brevity sketched one add concurrency active module whatever means handling interaction used order handle queries different clients concurrently easy systems support concurrency natively ciao akl feel ciao offer advantages area offers compatibility prolog clp systems time efficiently supporting concurrent execution clause goals via local distributed threads carro hermenegildo goals communicate different levels abstraction shared fact database similarly blackboard shared variables also supports threads somewhat different communication mechanisms tarau bosschere finally shown szeredi also possible exploit concurrency present prolog systems aurora implementing multitasking server also interesting set things single active module handle different forms done even dynamically capabilities active module augmented fly able handle new form designating directory code loaded active module would put active module consulting directory periodically increase functionalities finally another important issue addressed providing security ensuring allowed clients connect active module case remote code downloading standard forms authentication based codes used automatic code downloading local execution section describe architecture using facilities presented previous sections allows downloading local execution prolog code accessing www address without requiring special browser complementary approach giving www access active module sense provides code executed client machine java concretely functionality desire simply clicking www pointer transparently user remote prolog code automatically downloaded way queried via forms processing done locally allow http server server machine configured give specific example files hold prolog code example special suffix like side browser configured start helper application receiving data type cabeza hermenegildo form http www browser http form data form reply loaded formreply active module code formdata answerform formdata formreply loadcode fig automatic code downloading architecture application interface prolog engine execute www downloaded code acting active module sketch procedure see figure form used query downloaded code assume already loaded browser contains link points prolog code file clicking link produces download explained note browsers handle mime types modern browsers form code file could alternatively combined document however brevity describe case separate handler form specified local executable server file tells browser page type browser starts passes file example saving file temporal directory passing name process checks whether prolog engine currently running browser necessary starts one prolog engine configured active module call predicate active module loadcode file handler asks active module read code active module reads code compiles distributed www programming using prolog pillow lib waits active module complete compilation writes done message browser browser receives done message submit button form pressed following standard procedure forms browser starts process sending form data process gets form data translates dictionary formdata passes active module call exported predicate answerform formdata formreply active module processes request returns formreply www page term contains answer possibly new form process translates formreply raw html gives back browser dying afterwards subsequent queries active module accomplished either going back previous page using back button present many browsers answer page contains new query form using case procedure continues net effect approach simply clicking www pointer remote prolog code automatically downloaded local prolog engine queries posed via form answered locally prolog engine obvious security issues need taken care architecture standard authentication techniques used however since source code passed around comparatively easy verify dangerous predicates example perhaps access files executed note also possible download bytecode since supported current systems using similar approach related work previous general purpose work www programming using computational logic systems includes best knowledge publicly available library cabeza hermenegildo manual logicweb system loke davison pillow library also described previously cabeza library built cabeza hermenegildo using input naish forms code hermenegildo bueno experiments building www interface warren pereira program released publicly available www library systems announced among places internet newsgroup cabeza hermenegildo library since ported large number systems adapted several prolog vendors well used different programmers various institutions particular ken bowen ported library als prolog extended provide group processing forms alternative use active modules bowen present work essentially significant extension library main previous body work related interfacing cabeza hermenegildo logic programming www knowledge logicweb loke davison system loke davison aim logicweb use logic programming extend concept www pages incorporating programmable behavior state shares goals java also offers rich primitives accessing code remote pages module structuring aims logicweb different logicweb presented system implementation done tight integration mosaic browser making use special features browser contrast general purpose library meant used general computational logic systems offers wide range functionalities syntax conversion html logic terms access predicates www pages predicates handling forms generally somewhat lower level abstraction logicweb believe using pillow ideas sketched paper possible add quite interesting functionality offered logicweb standard clp systems shown examples including access passive remote code modules ftp http address programs automatic remote code access querying using standard browsers forms addition discussed active remote code functionality rather code exported recently larger body work topic presented workshop held topic logic programming internet joint international conference symposium logic programming also previous version paper presented work presented loke sterling based logicweb aims provide distributed lightweight databases www basic logicweb system believe pillow library used implement systems interesting ideas proposed therein briefly mentioned work szeredi proposes architecture similar active modules order handle form requests solution handling multiple requests performed using feel ciao threads natural modeling kind concurrency ideas proposed quite interesting eclipse bonnet thomsen aimed implementing internet agents offers functionality part similar ciao libraries including facilities similar active modules approach different however several respects eclipse library implements special http servers clients contrast pillow uses standard http servers interfaces using special purpose servers may interesting approach possibly allows greater functionality hand approach general requires either substitution standard server given machine setting special server different socket address standard one eclipse library also contains functionality related active modules although interface provided lower level finally papers describing interesting www applications presented regularly underline suitability distributed www programming using prolog pillow lib computational logic systems task believe ciao pillow library contribute making even easier develop applications future additional work topic logic programming internet found proceedings workshop sponsored compulognet research network reader referred tutorials papers presented two workshops information number applications libraries topics interfacing compilation computational logic systems java examples prolog systems interfaced java binprolog see http ciao bueno others calejo experimental prolog java compilers built academia see example jprolog http commercially see example prolog tools http approach quite attractive although results compete performance conventional prolog compilers open research whether improvements java performance improved compilation technology bridge gap commercial work topic interfacing prolog www addition done als system mentioned include amzi prolog webls system http lpa prologweb system http recent work using pillow includes web integrator davulcu webbase system integrates data various web sources allows users query web sources single webdb cabeza hermenegildo database management interface also within radioweb project partners developed collaboration group codish ben gurion university language describing www page layout style rules engine interpreting rules generate www sites dynamically adapt parameters user characteristics cederberg clip group additional applications developed pillow library accessed pillow www site see later page pointers proceedings previously mentioned workshops well information including technical reports tutorial regarding topic logic programming constraint programming internet maintained http conclusions future work discussed practical point view number issues involved writing internet www applications using systems described pillow programming library systems pillow provides facilities generating structured documents producing html forms writing form handlers processing templates accessing parsing www documents accessing code posted http addresses also described architecture application cabeza hermenegildo classes including automatic code downloading using model clientserver interaction active modules finally also described architecture automatic code downloading local execution using generic browsers believe ciao pillow library ease substantially process developing www applications using computational logic systems recently developed several extensions library example setting getting cookies sample applications make extensive use concurrency systems support overlap network requests also developed complementary library interfacing prolog virtual reality modeling language vrml addition included part ciao system pillow library provided standard standalone public domain library sicstus prolog prolog clp systems supporting functionality please contact authors consult www site http pillow page http download details online version pillow manual ciao prolog system also freely available http http acknowledgments authors would like thank lee naish mats carlsson tony beaumont ken bowen michael codish markus fromherz paul tarau andrew davison koen bosschere useful feedback previous versions document pillow code first versions ciao system library developed partial support acclaim esprit project subsequent development occurred context mcyt projects ella edipia mcyt esprit project radioweb collaboration eccosic fulbright references cailliau luotonen nielsen secret web communications acm bowen march personal communication available http bueno cabeza carro hermenegildo puebla august ciao prolog system reference manual ciao system documentation school computer science technical university madrid upm cabeza hermenegildo distributed concurrent constraint execution ciao system proc workshop parallelism implementation technologies utrecht utrecht madrid available http cabeza hermenegildo march html package systems spain available http distributed www programming using prolog pillow lib cabeza hermenegildo february html www interface publicly available posting available http cabeza hermenegildo april www programming using computational logic systems library proceedings workshop logic programming www cabeza hermenegildo june www database management interface prolog technical report school computer science technical university madrid upm facultad upm del monte cabeza hermenegildo varma september library programming using computational logic systems proceedings workshop logic programming tools internet applications available http calejo land opportunities pages proceedings first international conference practical application constraint technologies logic programming practical application company also available http carlsson february sicstus prolog user manual box spanga sweden carro hermenegildo concurrency prolog using threads shared database pages international conference logic programming mit press cambridge cederberg per clip group june flexible layout styling last language technical report clip radioweb project chikayama fujise sekita portable efficient implementation tick evan proc workshop parallel concurrent programming oregon colmerauer les gramaire metamorphose tech rept univ groupe colmerauer introduction prolog iii communications acm davulcu hasan freire juliana kifer michael ramakrishnan june layered architecture querying dynamic web content acm sigmod international conference management data url http bosschere another approach parallelizing prolog pages proceedings parallel computing elsevier north holland ecrc eclipse user guide european computer research center grobe naseer hasan july instantaneous introduction cgi scripts html forms available http hermenegildo april writing shell scripts sicstus prolog posting available http hermenegildo clip group methodological issues design ciao generic parallel concurrent constraint system pages principles practice constraint programming lncs hermenegildo bueno banda puebla december ciao compiler system experimentation workbench cabeza hermenegildo future systems proceedings ilps workshop visions future logic programming available http hermenegildo cabeza carro using attributed variables implementation concurrent parallel logic programming systems pages proc twelfth international conference logic programming mit press hermenegildo bueno cabeza carro banda puebla ciao compiler system experimentation workbench future systems pages parallelism implementation logic constraint logic programming commack usa nova science hermenegildo puebla bueno using global analysis partial specifications extensible assertion language program validation debugging pages apt marek truszczynski warren eds logic programming paradigm perspective jaffar joxan lassez constraint logic programming pages acm symposium principles programming languages acm janson haridi programming paradigms andorra kernel language pages international logic programming symposium mit press kowalski predicate logic programming language pages proceedings ifips loke davison logic programming web pages acm conference hypertext acm press available http dincbas simonis van hentenryck solving large combinatorial problems logic programming journal logic programming partners radioweb project july radioweb automatic generation web sites radio brodcasting industry project description technical annex technical report radioweb project bonnet bressan leth thomsen september towards eclipse agents internet proceedings workshop logic programming tools internet applications available http carro hermenegildo interfacing prolog vrml application constraint visualization pages practical application constraint technologies logic programming practical application company smolka november definition kernel dfki documentation series german research center artificial intelligence dfki loke davison sterling september lightweight deductive databases web proceedings workshop logic programming tools internet applications available http szeredi katalin scott rob september serving multiple html clients prolog application proceedings workshop logic programming tools internet applications available http tarau april binprolog posting available http van hentenryck constraint satisfaction logic programming mit press warren pereira efficient easily adaptable system distributed www programming using prolog pillow lib interpreting natural language queries american journal computational linguistics
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neededness delia alejandro jan irif cnrs univ universidad buenos aires conicet universidad buenos aires abstract show observationally equivalent needed reduction proof result uses semantical argument based intersection type system called interestingly system also allows syntactically identify needed redexes term introduction one fundamental notions underlying paper one needed reduction used understand lazy evaluation functional programs key notions reducible programs former programs represented containing nonevaluated subprograms called reducible expressions redexes whereas latter seen definitive results computations called normal forms turns every reducible program contains special kind redex known needed words every normal form contains needed redex redex said needed contracted evaluated sooner later reducing normal form informally said way avoiding reach normal form needed strategy always contracts needed redex normalising term reduced way normal form contraction needed redexes necessarily terminates excellent starting point design evaluation strategy unfortunately neededness redex decidable consequence real implementations functional languages directly based notion goal however establish clear connection semantical notion neededness different implementations lazy functional languages miranda haskell implementations based calculi pioneered wadsworth extensively studied indeed calculi fill gap operational semantics actual implementations lazy functional languages argument time used operation quite seen memoized version value argument stored first time work partially founded lia infinis evaluated subsequent uses example duplicates argument lazy languages first reduce value uses argument need evaluate notion needed reduction defined respect full strong normal forms calculi evaluate programs special values called normal forms either abstractions arbitrary applications headed variable terms form arbitrary terms overcome shortfall first adapt notion needed redex terms going fully reduced normal forms weakhead normal forms thus informally redex needed term contracted sooner later reducing normal form derived notion strategy called needed strategy always contracts needed redex paper introduces two independent results neededness obtained means intersection types survey found consider particular typing system show allows identify needed redexes normalising term done adapting classical notion principal type proving redex normalising term needed iff typed principally typed derivation second goal show observational equivalence needed reduction two terms observationally equivalent empirically testable computations identical means term evaluated normal form using machinery needed reduction normalises means system mentioned far use technique reason observational equivalence flexible general easy verify even certify indeed system provides semantic argument first showing term typable system iff normalising needed strategy whnd resorting results showing system complete term typable system iff normalising name normalising iff normalising need thus completing following chain equivalences whnd typable name need leads observational equivalence needed reduction structure paper sec introduces preliminary concepts sec defines different notions needed reduction type system studied sec sec extends derivation trees show sec system identifies needed redexes sec gives characterisation normalisation needed reduction sec devoted define finally sec presents observational equivalence result preliminaries section introduces standard definitions notions concerning reduction strategies studied paper head weakhead reduction neededness later notion based theory residuals given countable infinite set variables consider following grammar terms values contexts name contexts set denoted use denote terms respectively use chti resp ehti term obtained replacing hole resp sets free bound variables term written respectively defined usual work standard notion renaming bound variables abstractions thus example term form called redex clear context called anchor redex onestep reduction relation resp given closure contexts resp rewriting rule denotes standard substitution thus forbids reduction inside arguments neither holds write resp reflexivetransitive closure resp head leftmost reductions order introduce different notions reduction start formalising general mechanism reduction consists contracting redex specific occurrence occurrences finite words alphabet use denote empty word notation concatenations letter alphabet set occurrences given term def def defined induction follows def given two occurrences use notation mean prefix denote subterm occurrence defined expected thus example def set redex occurrences defined roc use notation mean roc reduces contracting redex occurrence notion extended reduction sequences expected noted list redex occurrences contracted along reduction sequence use nil denote empty reduction sequence nil holds every term term exactly one following forms latter case say head redex former case head redex moreover say redex terms occurrences head redex minimal redex occurrence form particular satisfies abstraction every redex reduction sequence contracting step head redex resp redex corresponding term called head reduction resp reduction given two redex occurrences roc say anchor left anchor thus example redex occurrence term alternatively relation understood dictionary order redex occurrences either proper prefix share common prefix side application side notice case implies since notion defines total order redexes every term normal form unique leftmost redex term leftmost reduces reduces reduction step contracts leftmost redex example leftmost reduces leftmost reduces notion extends reduction sequences expected towards neededness needed reduction based two fundamental notions residual describes given redex traced along reduction sequence normal form gives form expected result reduction sequence section extends standard notion needed reduction head needed reductions residuals given term roc descendants written set occurrences defined follows rkq instance given roc notice furthermore occurrence redex roc roc position called residual reducing notion extended sets redex def occurrences indeed residuals particular given roc residuals def def sequence stability relation makes use notion residual lemma given term let roc roc every proof first notice implies immediate see roc result holds immediately otherwise implies every definition may distinguish two cases either cases imply share common prefix say resp case know proper prefix thus proper prefix either hence every notice result implies leftmost redex preserved reduction redexes also residual leftmost redex occurs exactly occurrence original one corollary given term roc leftmost redex reduction contracts neither residuals roc leftmost redex proof induction length using lem notions normal form expected result evaluating program specified means appropriate notion normal form given relation term said form iff term iff exists thus given term define set forms def nfr particular turns term form whnf form arbitrary terms redex set def forms whnf similarly term head form hnf turns form head redex set head forms given def hnf last term form form form set singleton may use either set unique element worth noticing hnf whnf indeed inclusions strict instance head form head form notions needed reduction different notions normal form considered sec suggest different notions needed reduction besides standard one literature indeed consider roc say used reduction sequence iff reduces residual needed every reduction sequence form uses head needed every reduction sequence head form uses needed every reduction sequence form uses notice particular resp implies every redex needed resp head needed needed reduction needed resp head needed noted resp contracted redex needed resp head needed reduction sequence needed resp head needed noted resp every reduction step sequence needed resp head needed instance consider reduction sequence needed head needed since redex might contracted reach head normal form moreover second reduction sequence head needed needed since redex needed get normal form notice following equalities hold hnd hnf whnd whnf leftmost redexes reduction sequences indeed needed lemma leftmost redex term normal form resp head normal form needed resp head needed proof since existing proof extend normal forms give alternative argument cover standard case needed reduction also new ones head reductions let term normal form resp head normal form normal form let consider normal form resp head normal form normal form assume towards contradiction leftmost redex used cor occurrence still leftmost redex leads contradiction normal form resp head normal form normal form particular leftmost redex term head normal form resp normal form necessarily head redex resp redex theorem let roc leftmost reduction resp head reduction reduction starting resp needed resp head needed iff used proof immediate definition needed resp head needed let hypothesis leftmost redex corresponding term lem needed resp head needed notice given redex needed follows definition residual needed either therefore needed resp head needed implies needed resp head needed well notice reduction prefix head reduction turn prefix leftmost reduction normal form consequence immediate see every needed redex particular head needed every head needed redex needed well example consider needed redex head needed needed however needed head needed needed redex term needed type system section recall intersection type system extension used characterise normalising terms strategy precisely show typable system normalising needed redexes contracted characterisation used sec conclude needed strategy observationally equivalent calculus introduced sec given constant type denotes answers countable infinite set base type variables define following sets types types multiset types finite set empty multiset denoted remark types strict sides functional types never multisets thus general form type constant type base type variable typing contexts contexts written functions variables multiset types assigning empty multiset finite set def variables domain given dom union def contexts written defined denotes multiset union example notion extended several contexts expected denotes finite union contexts notation understood empty context write context type judgements form typing context term type intersection type system given fig val fig intersection type system constant type rule val used type values axiom relevant weakening rule multiplicative note argument application typed times premises rule particular case subterm occurring typed term turns untyped type derivation tree obtained applying inductive typing rules system notation means derivation judgement system term typable system iff subject derivation iff use capital greek letters name type derivations writing example short usually denote derivation subject type context size derivation denoted defined number nodes corresponding derivation tree write rule access last rule applied derivation likewise prem multiset proper maximal subderivations instance given rule prem also use functions ctxt subj type access context subject type judgement root derivation tree respectively short also use notation denote type associated variable def typing environment conclusion ctxt intersection type systems usually seen models typing stable convertibility typable typable property splits two different statements known subject reduction subject expansion respectively first one giving stability typing reduction second one expansion particular case types subject reduction refines weighted stating typability stable reduction also size type derivations decreasing moreover decrease strict reduction performed special occurrences redexes called typed occurrences introduce concepts given type derivation set toc typed occurrences subset subj defined induction last rule rule val toc def rule subj prem def toc toc rule subj andsprem def toc toc toc remark two kind untyped occurrences inside untyped arguments applications inside untyped bodies abstractions instance consider following type derivations val toc remark redex typed term always typed occurrence convenience introduce alternative way denote type derivations refer result applying subject type def denote val result applying val abstracting term def val val refer abs result applying premise abstracting variable def abs ctxt type likewise write app result applying premises argument argument untyped note application valid provided type type def app ctxt ctxt given toc multiset subderivations occurrence inductively defined follows def rule subj prem def rule subj prem def recall denotes multiset union given type derivations position subj replacing subderivations occurrence written type derivation inductively defined follows assuming subj subj every call unique subject derivations def either rule subj prem def abs rule subj prem def app rule subj prem def app unique subject derivations denotes replacement subterm variable capture allowed remark decomposition sets thus replacement turns operation state two main properties system whose proofs found sec theorem weighted subject reduction let exists moreover toc toc theorem subject expansion let exists note weighted subject reduction implies reduction typed redex occurrences turns normalising substitution reduction derivations order relate typed redex occurrences convertible terms extend notion derivation trees making use natural basic concept typed substitution contrast substitution terms operations derivation trees see discussions examples given variable type derivations typed substitution written making abuse notation type derivation inductively defined type def val val def capture variable abs def abs capture variable app type type def app remark decomposition sets thus substitution derivation trees turns operation intuitively typed substitution replaces typed occurrences corresponding derivation matching type matching chosen way moreover also substitutes untyped occurrences untyped operation completely deterministic thus example consider following substitution defined sec following lemma relates typed occurrences trees composing substitution substituted tree lemma let derivations defined toc iff toc toc iff exists toc toc proof induction based previous notion substitutions derivations able introduce reduction derivation trees reduction relation derivation trees defined first considering following basic rewriting rules typed untyped occurrences case reduction closed usual term contexts need close previous relation derivation tree contexts however reduction given subterm causes many reductions corresponding derivation tree recall defined multiset informally given redex occurrence type derivation multiset minimal subderivations containing written apply reduction rules elements thus obtaining multiset recompose type derivation reduct formalise idea given type derivation occurrence subj define maximal typed prefix written unique prefix satisfying toc toc notice multiset subderivations position corresponds multiset minimal subderivations containing instance consider type derivation presented sec val toc indeed minimal subderivation containing occurrence also given terms type derivations respectively say reduces written iff exists roc denotes lifting multisets basic rewriting rules introduced applying rule elements multiset gives reduction relation trees reduction sequence derivation trees contracting redexes typed positions dubbed typed reduction sequence note typed reductions normalising thm yielding special kind derivation indeed given type derivation say normal iff toc roc reduction trees induces reduction terms subj subj abuse notation may denote sequences letter neededness typed occurrences section presents one main results establishes connection needed redexes typed redex occurrences precisely first show sec every needed redex occurrence turns typed occurrence whatever type derivation converse however hold show sec typed occurrence special kind typed derivation call principal corresponds needed redex occurrence start technical lemma lemma let toc iff exists toc proof result holds since toc every possible definition assume proceed induction roc toc moreover two possibilities definition toc iff toc moreover lem toc iff toc notice thus conclude definition toc iff toc lem toc iff exists toc toc thus conclude analyse form moreover two possibilities definition toc iff toc since implies inductive hypothesis toc iff exists toc thus definition toc iff exists toc thus toc iff toc iff toc definition rule val result immediate since toc toc definition toc iff toc inductive hypothesis toc iff exists toc thus conclude previous case case symmetric one presented may conclude definition otherwise toc iff toc thus result follows definition inductive hypothesis needed redexes typed order show every needed redex occurrence corresponds typed occurrence type derivation start proving typed occurrences come untyped ones lemma let exists toc toc proof straightforward induction using lem theorem let needed redex let type derivation toc proof thm used reduction whnf rem reduction contracts typed redexes thus residuals typed occurrence corresponding derivation tree finally conclude lem toc principally typed redexes needed mentioned converse thm hold typed occurrences correspond needed redex occurrence illustrated following examples recall defined sec indeed occurrence resp term resp typed needed since terms already normal form fortunately typing relates redex occurrences restrict type derivations principal ones given term form derivation normal principally typed times type variable none typed given normalising term say principally typed implies normal principally typed note particular previous definition depend chosen normal form suppose another normal form convertible terms property normal principally typed iff thm lemma let type derivation subject roc toc let normal used proof straightforward induction using lem notions leftmost needed reductions untyped terms naturally extends typed reductions tree derivations thus lemma let normalising term principally typed leftmost typed reduction sequence starting needed proof induction leftmost typed sequence called empty result immediate show typed needed redex leftmost definition conclude inductive hypothesis indeed assume whnf definition normal principally typed thus typed redexes contradicts hence redex whnf moreover typed rem needed lem thus conclude theorem let normalising term principally typed roc toc needed redex proof let leftmost typed reduction sequence normal note exists definition principally typed lem needed reduction sequence moreover lem used hence needed redex direct consequence thm given normalising term typed redex occurrences principally typed derivation always exists correspond needed redexes hence system allows identify needed redexes normalising term characterising needed normalisation section presents one main pieces contributing observational equivalence result indeed relate typing neededness showing typable term system normalising needed reduction characterisation highlights power intersection types start technical lemma lemma let normal implies whnf proof induction analysing last rule applied whnf val whnf whnf normal normal thus inductive hypothesis whnf moreover since toc conclude whnf let say reduction sequence iff every left every corresponding subsequence words every every residual redex left relative given reduction subsequence reductions define particular standard strategies give canonical ways construct reduction sequences one term another theorem exists reduction theorem let iff whnd proof thm know strategy reducing typed redex occurrences normalising exist normal lem whnf thm exists reduction let write whnf whnf claim reduction steps leftmost assume towards contradiction exists leftmost redex written since reduction residual contracted step thus reduction sequence whnf whnf used leads contradiction needed lem consequence leftmost reduction sequence moreover lem whnf whnd thus whnd consider reduction let normal principally typed derivation defined sec finally conclude induction using thm section describes syntax operational semantics introduced concise previous specifications operationally equivalent results could also presented using alternative specifications given countable infinite set variables define different syntactic categories terms values list contexts answers need contexts terms values list contexts answers need contexts nhhxii denote set terms terms form closures called explicit substitution set without set terms notions free bound def variables defined expected particular def def def extend standard notion expected use special notation nhhuii lhhuii free variables captured context abstractions explicit substitutions context binds free variables thus example given nhyi nhhyii nhxi written nhhxii notice use special notation last case needed contexts example case calculus introduced given set terms reduction relation union respectively closure need contexts following rewriting rules lht nhhxii lhnhhvii rules avoid capture free variables example sequence following redex step underlined clearness reduction preserves free variables implies notice reduction also weak answers observational equivalence results sec used prove soundness completeness neededness second main result precisely interpreter stops value needed reduction stops value means observationally equivalent formally given reduction relation term language associated notion context define observationally equivalent written iff chti chui every context order show final result resort following theorem theorem let iff name terms iff observations allows conclude theorem terms iff proof thm sufficient show iff proof proceeds follows iff definition chti name chui name iff thm chti typable chui typable iff thm chti whnd chui whnd iff definition conclusion establish clear connection semantical standard notion neededness syntactical concept use types powerful technique able characterise different operational provides simple natural tool show observational equivalence two notions refer reader proof techniques based intersection types used connect semantical notions neededness syntactical notions lazy evaluation interesting difficult extension result sec reduction defined explicit substitutions contracts needed redexes appropriate natural notion needed redex explicit substitutions technical tool obtain result would type system straightforward adaptation system syntax given recent formulation strong describing deterministic strategy normal form instead normal form would natural extend technique obtain observational equivalence result standard notion needed reduction full normal forms strong strategy remains future work references beniamino accattoli pablo barenbaum damiano mazza distilling abstract machines johan jeuring manuel chakravarty editors proceedings acm sigplan international conference functional programming gothenburg sweden september pages acm zena ariola matthias felleisen lambda calculus funct zena ariola matthias felleisen john maraist martin odersky philip wadler lambda calculus ron cytron peter lee editors conference record popl acm symposium principles programming languages san francisco california usa january pages acm press franz baader tobias nipkow term rewriting cambridge university press thibaut balabonski pleine paresse une certaine thesis thibaut balabonski pablo barenbaum eduardo bonelli delia kesner foundations strong call need pacmpl icfp hendrik barendregt lambda calculus syntax semantics volume north holland revised edition hendrik barendregt richard kennaway jan willem klop ronan sleep needed reduction spine strategies lambda calculus inf antonio bucciarelli delia kesner daniel ventura intersection types logic journal igpl stephen chang matthias felleisen lambda calculus revisited helmut seidl editor programming languages systems european symposium programming esop held part european joint conferences theory practice software etaps tallinn estonia march april proceedings volume lecture notes computer science pages springer mario coppo mariangiola extension basic functionality theory notre dame journal formal logic daniel carvalho logique temps calcul phd thesis philippa gardner discovering needed reductions using type theory masami hagiya john mitchell editors theoretical aspects computer software international conference tacs sendai japan april proceedings volume lecture notes computer science pages springer delia kesner reasoning means types bart jacobs christof editors foundations software science computation structures international conference fossacs held part european joint conferences theory practice software etaps eindhoven netherlands april proceedings volume lecture notes computer science pages springer john maraist martin odersky philip wadler lambda calculus funct simona ronchi della rocca principal type scheme unification intersection type discipline theor comput steffen van bakel complete restrictions intersection type discipline theor comput pierre vial intersection types beyond phd thesis christopher wadsworth semantics pragmatics lambda calculus phd thesis oxford university
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differential complex cat cubical spaces oct introduction pierre julg alain valette also tadeusz pytlik ryszard szwarc constructed studied certain fredholm operator associated simplicial tree operator defined least two ways combinatorial flow tree similar flows forman discrete morse theory theory unitary cocycles applications theory surrounding operator theory completely bounded representations groups act trees selberg principle representation theory groups crucial property fredholm operator introduced julg valette initial operator continuous family fredholm operators parametrized closed interval applications emerge properties family circumstance group acts properly underlying tree case operators family act hilbert spaces carry unitary representations roughly speaking family connects regular representation trivial representation within context calls mind kazhdan property rather negation property well haagerup property strong negation property groups act trees known haagerup property essentially due haagerup pytlikszwarc construction perhaps best viewed geometric incarnation fact immediate consequence amenability group acts property tree another strong negation property main aim paper extend constructions julg valette pytlik szwarc cat cubical spaces cat cubical supported part epsrc grants supported part grant simons foundation supported part nsf grant space thing simplicial tree secondary aim illustrate utility extended construction developing application operator ktheory giving new proof groups act properly cat spaces expect uses constructions beyond operator shall associate bounded geometry cat cubical space fredholm operator differential complex cohomology construction rather challenging general cat cubical spaces trees whereas trees less canonical notion flow towards distinguished base vertex tree higher dimensions example vertex typically connected given base vertex large number addition need consider cubes need impose condition oblige carefully consider orientations cubes way quite unnecessary trees interesting still problem defining final complex family complexes aim construct solve shall rely theory hyperplanes cat cubical spaces case tree hyperplanes simply midpoints edges general nontrivial geometry fact cat cubical spaces rights shall also introduce study related notion parallelism among cubes cat cubical space tree two vertices parallel two distinct edges parallel higher dimensions parallelism subtle instance finite tree number vertices precisely one plus number edges simple geometric fact fact essential part construction proof following generalization higher dimensions quite bit involved proposition finite cat cubical space number vertices equal number parallelism classes cubes dimensions expect parallelism aspects constructions interest value elsewhere theory cat cube complexes one last challenge comes passing cat cubical geometry fredholm complexes operator two standard paradigms operator bounded cycles unbounded cycles geometry faced forces consider hybrid two however done shall arrive application theorem second countable locally compact group admits proper action bounded geometry cat cube complex groups act properly cat cube complexes known haagerup property proved theorem advantage present approach constructions proof tied cube complex whereas authors rely auxilliary action group euclidean space rather hard understand directly brief outline paper reviewing concept hyperplane section shall study orientations define complex section shall introduce parallelism section define final complex shall call complex section family complexes connecting two constructed stages sections application operator subject sections cubes hyperplanes shall begin fixing basic notation concerning cubes hyperplanes cat cube complex shall follow exposition niblo reeves adaptations throughout paper denote cat cube complex section though everywhere necessary shall assume throughout finitedimensional bounded geometry sense number cubes intersecting one cube uniformly bounded varies cubes every contains exactly faces face disjoint precisely one shall call opposite face shall use standard terms vertex edge cubes concept midplane cube introduced section identify standard cube midplanes precisely intersections cube coordinate hyperplanes thus midplanes cube particular closed subsets contains precisely midplanes particular vertex contains midplanes niblo reeves describe equivalence relation set midplanes cube complex two midplanes hyperplane equivalent arranged first last members finite sequence midplanes intersection two consecutive midplanes midplane definition see definition hyperplane union set midplanes equivalence class midplanes hyperplane cuts cube contains midplane cube hyperplane cuts edge say edge crosses hyperplane see figure figure hyperplane union three midplanes examples tree hyperplanes precisely midpoints edges plane divided cubes integer coordinate lines hyperplanes coordinate lines hyperplanes particularly relevant context cat cube complexes previous two examples following reason lemma see theorem lemma cat cube complex every hyperplane totally geodesic subspace separates two connected components components complement hyperplane two associated hyperplane open totally geodesic subsets moreover union cubes contained given cat cube complex right totally geodesic subcomplex later helpful approximate infinite complex finite complexes follows lemma every bounded geometry cat cube complex increasing union finite totally geodesic cat subcomplexes whose hyperplanes precisely nonempty intersections hyperplanes proof fix base point integer form set hyperplanes whose distance base point greater form intersection hyperplanes contain base point denote union cubes included intersection totally geodesic subset cat cube complex moreover intersection hyperplane connected union since set hyperplanes distance less base point finite finite subcomplex definition hyperplane vertex adjacent vertex included edge crosses hyperplane lemma hyperplanes cat cube complex intersect pairwise intersect within proof see theorem lemma assume distinct hyperplanes cat cube complex intersection adjacent vertex intersect contains vertex proof see lemma proposition lemma two hyperplanes cat cube complex disjoint one contained one proof see lemma complex let bounded geometry cat cube complex dimension aim section define differential complex generalizes complex introduced julg valette case tree motivate subsequent discussion recall construction let tree vertex set edge set fix base vertex differential defined mapping vertex first edge unique geodesic path mapped zero adjoint differential maps edge furtherest vertex composite identity whereas natural projection onto subspace spanned base vertex follows easily cohomology complex degree zero otherwise construction shall need concept orientation cubes begin definition presentation cube consists vertex cube together linear ordering hyperplanes cut cube two presentations equivalent distance two vertices parity permutation two orderings orientation cube positive dimension choice equivalence class presentations orientation vertex choice sign remark every cube precisely two orientations oriented cube shall write underlying unoriented cube equiped opposite orientation definition space oriented vector space comprising functions set oriented function every oriented cube remark space subspace vector space finitely supported functions shall call full space formula defines involution full space shall write dirac function oriented cube way belongs full space shall write oriented difference dirac functions two possible meanings symbol agree next introduce geometric ideas allow define differential higher dimensions first following generalization notion adjacency introduced definition definition adjacent hyperplane disjoint exists containing face cut lemma adjacent hyperplane cut vertices adjacent proof clearly cube adjacent vertices converse assume vertices adjacent lemma suffices show every hyperplane cuts must also cross let vertices separated denote qop vertices separated respectively four vertices belong four distinct intersections associated hyperplanes lemma hyperplanes intersect shall fix base vertex complex definition let hyperplane define operator follows let oriented put adjacent addition put adjacent lies base point adjacent separated base point define unique cube containing face cut orientations positive dimension oriented vertex listing hyperplanes orient vertex separated hyperplane alone listing hyperplanes vertex orientation oriented orientation receives opposite orientation remark linear operator previous definition initially defined full space specifying values oriented form basis space omit elementary check oriented allows restrict operator spaces oriented shall employ similar conventions consistently throughout linear operators defined initially full space cochains restricted space oriented cochains formulas hold restricted operators shall point instances definition differential linear map given formula sum taken hyperplanes note finitely many terms sum nonzero example case tree vertex distinct base point first edge geodesic operator agrees one defined julg valette base point chosen every edge cat cube complex canonically oriented selecting vertex nearest base point vertices canonically oriented orientation thus original construction julg valette involves vertices edges assumes base point orientations appear explicitly lemma two hyperplanes oriented cube nonzero distinct adjacent separate remark means proof item follows lemmas prove note first result left hand side nonzero right hand side nonzero case underlying unoriented namely unique cube containing face cut orientation suppose presented ordering vertex cube presented ordering vertex vertex immediately opposite cube presented ordering vertex argument applies vertex orientation remaining case follows identity lemma differential regarded operator space oriented cochains satisfies proof let consequence lemma sum vanishes important work larger full space result true see remarks definition let hyperplane let define operator follows let oriented cut cut define face lies entirely separated base point orientations presented ordered list vertex separated base point presented ordered list vertex separated alone edge presented vertex separated base point vertex opposite orientation presented vertex separated base point orientation remark convenience shall define operator vertices zero example let consider tree selected base vertex edge zero unless cuts case vertex farthest away choose orientation oriented vertex orientation otherwise definition let define operator definition oriented vector space basis full space equip space inner product orthogonal basis oriented length subspace oriented inherits inner product otherwise thus selecting unoriented one possible orientations gives collection oriented corresponding form orthonormal basis space oriented basis canonical signs coming relations proposition operators definitions formally adjoint bounded respect inner products definition proof fact operators bounded follows assumption complex bounded geometry fact adjoint follows following assertion hyperplane oriented oriented see definitions conclude section let compute cohomology complex form laplacian operators defined space oriented cochains larger full space cochains available formula hence also proposition oriented number hyperplanes adjacent separate particular form orthonormal basis eigenvectors invertible orthogonal complement also space oriented proof shall show oriented eigenvector acting full space eigenvalue statement vertex dimension reasons irrespective choice orientation higher dimensions oriented similarly adding separating sum terms terms obtain follows lemma term second sum zero understand first sum observe hyperplane oriented cube cuts otherwise also adjacent separated otherwise proposition follows lemma distinct hyperplanes every oriented cube proof vertex sides formula zero generally one following two conditions fails sides formula zero adjacent separates base point cuts crosses assume conditions suppose may presented listing hyperplanes vertex separates base point edge always possible shall leave exceptional case edge oriented vertex closest base point reader let vertex separated alone let qop vertices directly opposite respectively cube presented listing together vertex hence also listing vertex qop follows presented listing vertex right hand side presented listing vertex presented listing vertex follows presented listing vertex required corollary cohomology complex degree zero otherwise proof degree kernel one dimensional spanned degrees proceed follows follows also calculation shows oriented also oriented conclude section slight generalization needed later definition weight function function set hyperplanes weighted differential linear map given formula addition adjoint operator defined remark mainly interested following examples small variations minimal distance base point vertex adjacent calculations section easily repeated weighted context operators formally adjoint although unbounded case unbounded weight function example differentials restricted spaces oriented cochains cohomology either complex degree zero otherwise record formula weighted laplacian compare proposition proposition oriented sum squares weights hyperplanes cut sum squares weights hyperplanes adjacent separate base vertex parallelism classes cubes remaining aspects generalization theory cat cube complexes rest following geometric concept definition two cubes cat cube complex parallel dimension every hyperplane cuts also cuts figure darker edges form parallelism class determined hyperplane see definition every parallelism class determined determines set pairwise intersecting hyperplanes namely hyperplanes cut cubes parallelism class call determining hyperplanes parallelism class proposition intersection determining hyperplanes associated parallelism class carries structure cat cube complex intersections space cut every determining hyperplane proof case assertion cat cube complex case assertion hyperplane cat cube complex cat cube complex manner described proved sageev thm general result proceed inductively follows suppose given distinct hyperplanes intersection cat cube complex described statement result follow another application thm verify hyperplane cubes also midplanes exactly intersections cubes midplanes must show two midplanes belonging hyperplane intersect intersections hyperplane equivalent follows fact totally geodesic subspace proposition let cat cube complex let vertex parallelism class unique cube closest measured distance closest point cube metric beginning proof recall distance two vertices equal number hyperplanes separating vertices see example theorem addition let make note following simple fact lemma hyperplane separates two vertices distinct cubes parallelism class must intersect every determining hyperplane proof obvious hyperplane one determining hyperplanes otherwise hyperplane must fact separate two cubes parallelism class must separate two midplanes determining hyperplane since hyperplanes connected result follows proof proposition choose vertex among cubes parallelism class every vertex shall prove addition formula certainly prove uniqueness addition formula consequence following hyperplane property satisfying every hyperplane separates parallel intersect least one determining hyperplane indeed follows lemma hyperplane property hyperplane separate follows characterization edge path distance given remains prove hyperplane property satisfying shall use notion normal cube path section exists normal cube path vertices means every pair consecutive diagonally opposite cube called normal cube whose hyperplanes separate every separating hyperplane cuts exactly one normal cube also means every hyperplane separating parallel least one hyperplanes separating normal cube turn large possible note hyperplane contained completely contains hyperplane separating intersect every determining hyperplane would follow lemma determining hyperplanes would intersect vertex vertex separated alone would belong cube parallelism class would strictly closer consider second normal cube opposite vertices hyperplane separating parallel hyperplane separating turn parallel determining hyperplane contained completely contains determining hyperplane contained completely contains meet determining hyperplane continuing fashion successive normal cubes find every hyperplane separates indeed parallel determining hyperplane required verify formula mentioned introduction proposition finite cat cubical space number vertices equal number parallelism classes cubes dimensions proof fix base vertex associate vertex first cube normal cube path correspondence induces bijection vertices parallelism classes cubes indeed follows hyperplane property nearest cube within parallelism class vertex furthest first cube normal cube path map surjective hand follows addition formula nearest within equivalence class vertex furthest hyperplane separates nearest cube also separates choosing hyperplane adjacent cut find first cube normal cube path map injective proposition let cat cube complex let vertices separated single hyperplane nearest within parallelism class either opposite faces separated cut proof denote nearest vertices respectively among vertices cubes equivalence class suppose hyperplane separates must separate addition formula applied nearest point also separate addition formula applied nearest point must separate hence must either hyperplane separating case course nearest cubes opposite across determining hyperplane vertices parallelism class determining hyperplane belong opposite one another across required complex described introduction ultimate goal involves deforming julgvalette complex call complex complex cohomology equivariant case group acting cat cube complex short section describe algebraic complex motivation follows consider compare orientations parallel cubes key observation vertex uniquely determined position relative cutting hyperplanes thus natural isometry vertex sets two parallel shall say parallel positive dimension compatibly oriented orientations presented vertices correspond isometry common listing cutting hyperplanes vertices compatibly oriented oriented choice sign shall generalize considerations pairs comprising cube one faces definition cube pair pair cube containing face two cube pairs parallel cubes parallel cubes parallel shall call pair always keeping mind notation codimension may describe parallelism class pair grouping determining hyperplanes parallelism class symbol determine parallelism class hyperplanes cut complementary hyperplanes cube pair parallelism class orientation cube pair orientation face order compare orientations parallel cube pairs compare orientations faces parallel cubes must also take account position faces within ambient cubes introduce following notion definition two parallel cube pairs parity number complementary hyperplanes separate even otherwise opposite parity definition let parallel cube pairs orientation orientations aligned one following conditions holds parity compatibly oriented opposite parity compatibly oriented symbol describing parallelism class cube pair hyperplanes ordered relevant data left right vertical bar cube pair oriented symbol receives additional structure coming orientation group determining hyperplanes include vertex new symbol case hyperplanes form ordered list together vertex presentation oriented cube case list empty replace sign representing orientation vertex obtaining symbol form either case hyperplanes remain unordered set conversely formal expression symbol oriented pair precisely hyperplanes distinct nonempty pairwise intersection vertex adjacent following definition captures notion alignment orientations terms associated symbols definition symbols form equivalent sets equal permutation number hyperplanes among separating parity permutation case symbols form omit replace number hyperplanes among separating even orientation signs agree odd otherwise oriented equivalence class symbols shall denote equivalence class symbol simply confusion arise use similar notation case symbols form shall denote set oriented symbols hqp disjoint union proposition oriented symbols associated oriented pairs agree precisely orientations cube pairs aligned generalization complex differential complex designed capture combinatorics oriented aligned cube pairs definition space oriented type complex space finitely supported functions hqp function used involution hqp defined reversing orientation symbol shall denote space hqp space oriented defined similarly using oriented symbols type splits direct sum remark cochains space oriented type subspace full space type vector space finitely supported functions set hqp shall follow conventions similar section write dirac function oriented symbol symbol hqp difference dirac functions linear operators defined full space cochains specifying values basis dirac functions oriented symbols shall typically omit elementary check operator commutes involution restricts operator spaces oriented cochains define differential complex definition differential linear map oriented symbols type satisfies oriented vertex separated alone usual hat means entry removed formula used symbols form together requirement commute involution determines symbols form since maps oriented symbol type linear combination oriented symbols type cases splits direct sum linear maps hqp lemma differential regarded operator space oriented cochains satisfies example let tree complex form identifying space finitely supported functions set edges identity identifications note identified set oriented edges involution acts reversing orientation space functions identifies space finitely supported functions set edges goal remainder section analyze complex emphasizing similarities complex begin providing formula formal adjoint differential definition let linear map oriented symbols type satisfies oriented symbols type hat means entry removed since maps oriented symbol type linear combination oriented symbols type splits direct sum linear maps hqp definition define inner product full space qcochains declaring elements orthogonal length subspace oriented inherits inner product otherwise lemma operators definitions formally adjoint bounded respect inner products definition proposition laplacian acts summand hqp scalar multiplication proof prove statement operator defined full space cochains operator equals restricted subspace oriented cochains proof direct calculation result applying oriented symbol type sum whereas result applying added second summands cancel first summands combine give corollary cohomology complex dimension zero otherwise continuous fields hilbert spaces objective next several sections construct family complexes continuously interpolates complex complex shall construct interpolation within hilbert space context using concept continuous field hilbert spaces refer reader chapter comprehensive treatment continuous fields hilbert spaces brief continuous field hilbert spaces topological space consists family hilbert spaces parametrized points together distinguished family sections satisfies several axioms important pointwise inner product two sections continuous function see definition following theorem gives convenient means constructing continuous fields theorem let topological space let family hilbert spaces parametrized points let family sections satisfies following conditions pointwise inner product two sections continuous function every linear span dense unique enlargement gives structure continuous field hilbert spaces proof enlargement consists sections every every section linear span neighborhood see proposition definition shall call family statement theorem generating family sections associated continuous field hilbert spaces ultimately shall use parameter space section shall concentrate open subspace extend next section section next shall deal construction continuous fields hilbert spaces shall construct differentials acting fields section begin completing various cochain spaces section natural way obtain hilbert spaces definition denote hilbert space completion oriented cochain space inner product definition basis comprised oriented cochains orthonormal remark case section shall also consider full cochain space comprised functions set oriented completion full space inner product definition contains space previous definition subspace functions shall construct every families hilbert spaces parametrized topological space completions spaces full oriented respect family pairwise distinct inner products considering oriented cochains obtain family hilbert spaces completion corresponding hilbert space space defined definition parallel compatible orientations denote number hyperplanes disjoint separate parallel incompatible orientations set compatibly oriented vertices distance higher dimensions parallel may identified vertices cat cube comples intersection determining hyperplanes parallelism class addition compatibly oriented distance complex compare theorem definition let every two oriented define exp course set exp extend linearity form full space note formula definition makes sense exp particular form one underlying definition used define theorem form positive proof consideration oriented opposed unoriented cubes merely gives two orthogonal copies space functions aside result proved technical lemma case see also prop case reduces case using theorem definition denote hilbert space completion oriented cochain space inner product remark hilbert spaces previous definition completions quotient elements zero norm shall soon see every nonzero linear combination oriented nonzero every natural maps injective next define generating family sections using either one following lemmas basis theorem easy check continuous fields arising lemmas one lemma let set sections form indexed generating family sections continuous field lemma set sections form continuous scalar function oriented generating family sections continuous field continuous fields constructed particularly interesting continuous fields fact isomorphic constant fields become much interesting structure taken account shall later paper sequel important fix particular isomorphism conclude section required unitary isomorphism defined using certain cocycle operators analogues studied valette case trees case cocycle operators general cat cube complexes constructed case involves minor elaboration case shall refer details follows definition adjacent hyperplane define dop opposite face unique cut contains cube exists lemma case oriented orient dop compatibly either case shall refer pair dop adjacent across definition let adjacent across hyperplane previous definition oriented adjacent define dop separated separated addition define adjacent extend linearity linear operator spaces full oriented example indeed identity operator generally restricted space spanned ordered basis dop adjacent separated operator acts unitary matrix particular extends unitary operator completed cochain spaces definition subsequent remark let assume two parallel necessarily adjacent across hyperplane follows theorem exists path consecutive pair consists parallel adjacent let define notation omits mention path justified following result proposition unitary operator independent path proof let two cube paths connecting cubes cubes parallel theorem thought vertices cat cube complex created parallelism class paths give rise vertex paths cat cube complex common beginning end vertices way reduce general case proposition zero dimensional case proved lemma follows shall use base vertex selected construction complex definition let every oriented let cube nearest base vertex parallelism class see proposition extend linearity linear operator spaces full oriented particular oriented cochains lemma linear operator vector space isomorphism proof consider increasing filtration cochain space indexed natural numbers nth space spanned cubes whose nearest vertex metric distance less operator preserves filtration fact simple direct calculation see lemma shows constant linear combination cubes closer formula shows induced map associated graded spaces isomorphism isomorphism lemma two oriented inner product left hand side remark lemma implies form positive definite since isomorphism proof lemma assume parallel compatibly oriented since otherwise sides formula zero let denote parallelism class nearest base vertex unitarity proposition give elaboration proposition multiples oriented cubes hence conclude required following results immediate consequences theorem map extends unitary isomorphism theorem unitary operators determine unitary isomorphism continuous field generated sections lemmas constant field fiber extension continuous field section shall extend continuous fields defined section adding following fibers definition shall denote completion space oriented inner product definition subspace functions hilbert space functions set oriented symbols following two definitions focus particular continuous sections shall extend definition let let oriented pair associated basic type linear combination full cochain space sum parallel given orientation compatible orientation associated basic oriented cochain belonging space oriented example basic type single oriented basic type difference vertices across edge finally dim basic type since definition basic section type continuous field continuous section form oriented pair shall extend basic sections sections assigning value hilbert space namely symbol associated cube pair section shall write compare definition remark shall prove following result theorem let pointwise inner product two basic sections possibly different types extends continuous function value continuous function equal inner product example suppose tree basic sections type functions oriented edge theorem easily checked case basic sections type easily handled also basic sections type form adjacent vertices tree one calculates converges agreement theorem addition second distinct basic cochain vertices arranged sequence along path tree short calculation reveals distance particular inner product converges agreement theorem definition extended basic section type continuous field hilbert spaces section form oriented pair basic sections form generating family sections continuous field course symbols span follows theorem extended basic sections form generating family sections continuous field fibers whose restriction continuous field previous section shall prove theorem carrying sequence smaller calculations following formula common also use section subsequently shall write finite sum oriented times coefficient functions bounded constant times lemma oriented pair dop dop separated complementary hyperplanes pair compatible orientation proof shall prove lemma induction case clear case let hyperplane cuts aim apply induction hypothesis faces separated denote faces face containing denote face directly across finally denote face separated complementary hyperplanes pair except particular dop expression left hand side depends cube pair course proof shall denote compute summand corresponding face belongs using path see used coefficient face equals coefficient face directly across induction hypothesis since first term expression second term induction follows dop used putting things together lemma proved previous section defined unitary isomorphisms defined using specific choice base point within parallelism class choice important far unitarity concerned shall exploit making judicious choices base point calculate inner products theorem lemma let oriented pair let associated basic type pointwise inner product converges proof choose base point defining unitary isomorphisms exactly expression previous lemma follows lemma dop dop result follows lemma let parallel pairs parity faces compatibly oriented pointwise inner product converges proof may assume lies side complementary hyperplanes parallelism class indeed replacing face necessary change corresponding basic cochain choose base point defining unitary isomorphisms lemma also using identity faces hyperplanes separating precisely separating terms orthogonal putting everything together get result follows lemma let oriented cube pairs types respectively let associated basic parallel compatibly oriented pointwise inner product converges particular case proof fail parallel incompatible orientations orthogonal full cochain space lemma proved assume parallel compatibly oriented therefore parallel reindexing necessary hyperplane passes neither choose base point unitary face parallel compatibly oriented side cube also dop dop face separated complementary hyperplanes pair particular dop opposite sides follows definition basic properties cocycle cubes appearing support side lemma inner product proof theorem possible values inner product positive value occurs oriented cube pairs parallel aligned negative value occurs parallel aligned occurs parallel result follows lemmas already pointed theorem allows extend continuous field sequel convenient work following generating family continuous bounded sections definition necessarily continuous section continuous field geometrically bounded finite set supported proposition space geometrically bounded continuous sections continuous field spanned extended basic continuous sections proof every basic continuous section certainly geometrically bounded finite complex converse true since fiber dimension continuous field finite constant case continuous field vector bundle basic continuous sections span fiber bundle general case regard geometrically bounded continuous section section continuous field associated suitable finite subcomplex lemma express combination basic continuous sections differentials continuous field purpose section construct differentials continuously interpolate differentials differentials later purposes important use weighted versions differentials definition first shall proceed without weights indicate end section weights incorporated recall operators definition proved isomorphisms lemma definition define dut differential definition addition define differential definition aim prove following continuity statement concerning operators theorem continuous geometrically bounded section continuous field pointwise differential continuous geometrically bounded section according proposition space continuous geometrically bounded sections generated module extended basic sections suffices prove theorem section shall following two preliminary lemmas lemma let oriented pair assume complementary hyperplanes pair separate base point associated basic type satisfies closest base point among cubes parallel face parallel separated complementary hyperplanes proof according definitions applied lemma result follows lemma let previous lemma let nearest cube parallelism class let face parallel separated base point complementary hyperplanes let face parallel separated complementary hyperplanes nonzero complementary hyperplane case proof consider first case case vertex characterized following hyperplane property proof proposition every hyperplane separating parallel least one determining hyperplane parallelism class nonzero exactly adjacent separates hyperplanes cutting certainly satisfy condition conversely hyperplane satisfying condition must intersect determining hyperplanes lemma separate must cut determining hyperplane parallelism class separates follows easily hyperplane separates separates argument applies cocycle property evaluate observe hyperplane appearing along geodesic path must cross every determining hyperplane follows commutes last equality follows elaboration proposition finally hyperplane separating adjacent putting things together result follows reduce general case case using proposition according set parallel vertex set cat cube complex way correspond complex key observation complex corresponding cube statement lemma closest vertex corresponding proof theorem let oriented pair associated extended basic shall show section linear combination extended basic cochains plus term geometrically bounded possibly changing sign assume furthest base point among faces parallel words assume complementary hyperplanes pair separate base point therefore face shall show equality suffices show equivalently left hand side applying lemmas statement lemma complete verification suffices check follows lemmas applied pair although little care must taken since base cube nearest within parallelism class replaced analogous base cube parallelism class consider adjoint operators together adjoint differential theorem continuous geometrically bounded section continuous field continuous geometrically bounded section continuous field proof could approached computations similar used prove theorem shortcut continuous geometrically bounded section viewed associated finite subcomplex lemma case finite complex differentials constitute map vector bundles pointwise adjoints automatically give map vector bundles finally return issue weights important next section work context kasparov theory let function hyperplanes defined formula dist dist next section shall work weighted operators dwt isomorphism definition dwt weighted differential described definition operator extend bounded operator since pointwise values geometrically bounded section lie theorem makes sense weighted case without extending domains operators beyond moreover theorem remains true weighted family operators proof reduces immediately unweighted case weighted unweighted differentials applied continuous geometrically bounded section differ term applies theorem equivariant fredholm complexes shall assume second countable locally compact hausdorff topological acts cat cube complex preserving cubical structure shall assume fixes base point goal section place complexes within context equivariant fredholm complexes need begin definitions definition fredholm complex hilbert spaces bounded complex hilbert spaces bounded operators identity morphism complex chain homotopic chain homotopy consisting bounded operators morphism consisting compact hilbert space operators words fredholm complex hilbert spaces complex form topological restrictions group really necessary allow easily fit concept equivariant fredholm complex context kasparov next section hilbert space differential bounded operator moreover exist bounded operators operator compact perturbation identity operator fredholm condition implies cohomology groups fredholm complex main reason definition interested following concept equivariant fredholm complex cohomology groups relevant definition let second countable hausdorff locally compact topological group fredholm complex hilbert spaces bounded complex separable hilbert spaces bounded operators hilbert space carries continuous unitary representation differentials necessarily equivariant differences gdg compact functions identity morphism complex chain homotopic chain homotopy consisting bounded operators morphism consisting compact hilbert space operators operators chain homotopy necessarily equivariant differences ghg compact normcontinuous functions remark differentials necessarily equivariant cohomology groups equivariant fredholm complex hilbert spaces necessarily carry actions direct interest far concerned nevertheless definition due kasparov minor variant form played important role number mathematical areas notably study novikov conjecture manifold topology see survey topics going manufacture equivariant fredholm complexes complexes complex difficult two understand disregarding group action differentials definition extend bounded operators hilbert space completions cochain spaces associated inner products resulting complex hilbert spaces bounded operators fredholm definition moreover group certainly acts unitarily differentials typically fail since defined using choice base point complex need fixed means technical items definition need considered carefully fact handle technical items necessary finally make use weight functions introduced definition following computation starting point assemble together cochain spaces form single space dim form hilbert space completion dim lemma weight function operator viewed operator domain essentially proof operator formally sense essential consequence fact range operator dense turn consequence fact laplacian diagonal operator indicated proposition since essentially operator study resolvent operators extend initial domains definition namely ranges bounded operators lemma weight function proper sense every set finite resolvent operators compact hilbert space operators proof two resolvent operators adjoint one another suffices show product compact compactness clear proposition let examine dependence operator initial choice base point lemma weight function sense sup every difference linear operator extends bounded linear operator proof suffices prove estimate place since adjoint one another denote operators definition associated two indicated choices base points since uniformly bounded replace second sum change overall expression term defines bounded operator suffices show pair base points expression defines bounded operator expression parentheses separates finitely many hyperplanes lemma follows fact hyperplane formula defines bounded operator long cube complex bounded geometry shall assume complex weighted using proper weight function fact next section shall work specific weight function let even though yet necessary since weighted differential bounded shall need make adjustment fit weighted complex framework fredholm complexes hilbert spaces bounded operators forming normalized differentials strictly speaking formula mean closure sense unbounded operator theory normalized complex complex indeed complex commute one another fredholm complex adjoints constitute chain homotopy identity compact cochain map fact compact lemma shall use following computation functional calculus show normalized complex equivariant fredholm complex hilbert spaces lemma compare positive hilbert space operator bounded positive constant integral converges norm topology theorem normalized complex defined using proper weight function equivariant fredholm complex proof suffices show normalized operator property compact function use lemma formula write difference linear combination two integrals integrand compact operator valued function whose norm integrals converge compact operators required let examine complex inner products pytlikszwarc cochain spaces given definition differentials given definition bounded story much simpler theorem complex equivariant fredholm complex proof follows proposition formula hqp set defines exactly bounded chain homotopy identity compact cochain map namely orthogonal projection onto degree zero zero operator higher degrees conclude section introduce following notion topological opposed chain homotopy two equivariant fredholm complexes next section shall construct homotopy equivariant fredholm complexes constructed using continuous field complexes constructed section definition two equivariant complexes hilbert spaces homotopic bounded complex continuous fields hilbert spaces adjointable families bounded differentials continuous field carries continuous unitary representation differentials necessarily equivariant differences gdg compact functions identity morphism complex chain homotopic chain homotopy consisting adjointable families bounded operators morphism consisting compact operators continuous fields operators homotopy necessarily equivariant differences ghg compact normcontinuous functions restrictions complex points complexes need supply definitions concepts mentioned usually formulated language hilbert modules example consistency rest paper shall continue use language continuous fields hilbert spaces definition adjointable family operators soon shall contract adjointable operator continuous fields compact space family bounded operators carries continuous sections continuous sections whose adjoint family also carries continuous sections continuous sections adjointable operator unitary unitary definition representation unitary adjointable operators continuous field continuous action map continuous sections continuous sections continuous place space continuous sections topology associated norm max definition adjointable operator continuous fields hilbert spaces compact base space compact norm limit banach space operator continuous sections continuous sections sequence linear combinations operators form continuous sections domain range continuous fields respectively compact operators form closed ideal adjointable operators theorem shall prove next section theorem equivariant fredholm complexes obtained complexes theorems homotopic sense definition purpose section prove theorem giving proof shall explain relevance theorem shall need use language kasparov equivariant emphasize proof theorem involve definitions last section work ealier paper shall assume familiarity kasparov theory complex hilbert spaces definition determines class kasparov equivariant representation ring kkg way homotopic complexes definition determine element complex whose differentials exactly determines class complex cohomology groups unitary representations zero differentials complex trivial representation degree zero cochain spaces determines multiplicative identity element definition see definition second countable locally compact hausdorff topological group multiplicative identity element representable equivariant fredholm complex hilbert spaces cochain space viewed unitary representation weakly contained regular representation theorem see corollary natural homomorphism cmax cred induces isomorphism groups cmax cred remarks homomorphism theorem isomorphism group amenable explains term every group example infinite group kazhdan property certainly homomorphism certainly isomorphism quickly surveyed background information state main result section theorem second countable locally compact group admits proper action bounded geometry cat cube complex theorem proved julg valette case cube complex tree used complex called tree showed continuous field complexes constructed paper homotopy connecting complexes shall general case construction homotopy proves theorem view following simple result whose proof shall omit lemma assume second countable locally compact group acts proper action cat cube complex hilbert spaces complex weakly contained regular representation remark theorem new proved higson kasparov theorem using different argument far general applies much broader class groups far less geometric prove theorem therefore suffices prove theorem shall shall construct homotopy theorem requires modifying constructions section less way modified complex construct complex shall therefore applying functional calculus family operators dwt dwt differential associated weight function course adjoint differential apply functional calculus shall need know family resolvent operators carries continuous sections continuous sections consequence following result proposition let nonzero real number family operators carries space continuous geometrically bounded sections dense subspace space continuous geometrically bounded sections norm actually shall need small variation proposition definition denote operator fiber orthogonal projection onto span single basic type course basic cochain follows formula laplacian proposition operators essentially bounded form resolvent operators including proposition let real number possibly zero family operators carries space continuous geometrically bounded sections dense subspace space continuous geometrically bounded sections propositions proved examining action laplacians dwt continuous geometrically controlled sections field proof propositions family operators maps space continuous geometrically bounded sections consider compositions suffices show families operators map space continuous geometrically bounded sections dense subspace let basic type lemmas tell property precisely hyperplanes adjacent separate base point according formula laplacian proposition dwt applying sides get similarly max ranges families contain perturbations every basic section propositions follow form bounded operators lemma family maps continuous sections continuous sections consider bounded complex continuous fields hilbert spaces bounded adjointable operators differential component mapping indicated continuous fields proposition disregarding complex homotopy fredholm complexes proof set compact operator continuous field remains show equivariant homotopy resolvent families compact would able follow route taken previous section prove equivariance fredholm complex associated complex compactness fails need bit careful following two propositions substitute lemmas used handle complex previous section proposition every every restricted family operators compact operator continuous field moreover proposition every operators uniformly bounded sup moreover taking granted moment result calculation theorem complex homotopy equivariant fredholm complexes sense definition proof need check families differentials complex modulo compact operators also varies normcontinuously let discuss first sufficiently close identity fixes base point actually locally constant function proof equivariance modulo compact operators small variation proof theorem suffices show family operators compact since compact operator suffices prove operator equivariant modulo compact operators applying lemma find difference sum two integrals integrands written terms compact operator valued functions first virtue proposition second compact moreover norms integrals converge compact operators required remains prove propositions first easy deal immediately proof proposition want show family operators compact since compact operators form closed ideal algebra adjointable families operators suffices show family compact compare proposition conjugating unitaries suffices prove family dwt dwt constant field hilbert spaces fiber compact one things restricting makes possible final assertion simple consequence explicit formula laplacian proposition together fact weight functions uniformly proper sense every finitely many hyperplanes satisfy norm estimate proposition holds operator bounded elementary let turn proposition complicating factor fails preserve differential also fails preserve unitary operators appear definitions differentials proposition correct two failures certain extent cancel one another definition let vertices define unitary operator cocycle operator follows define respects decomposition definition higher cubes according parallelism classes summand determined given class cubes equivalence class set nearest immediate definition unitary operator definition definition find dwt write let use abbreviation dwt dwt side rearranged dwt dwt norm expression dwt dwt suffices show operators dwt dwt satisfy conclusions proposition second operator adjoint first fact suffices prove conclusions proposition first operator alone shall proceed let adjust notation bit follows given vertex shall denote differential defined using base vertex weight function whose definition also use base vertex rather new notation drop mention group proposition consequence following assertion proposition operator bounded moreover recall differential defined using operation hyperplanes cubes since operation depends choice base vertex shall write indicate choice earlier prove proposition suffices consider case distance one another separated unique hyperplane shall make assumption lemma hyperplane fails separate oriented proof first fails separate operators equal one another shall drop subscripts rest proof next cuts cuts cubes parallel therefore cuts sides equation lemma cubes make zero assume disjoint let hyperplane separates according proposition nearest parallelism class either equal either opposite faces across cut combination another cube opposite face cut see fails separate equivalently fails separate also fails separate terms accordingly sides equation lemma zero assume separate suppose fails adjacent either cuts vertex adjacent side equation either zero case equation obviously holds also fails adjacent case side equation simple assume adjacent let separated alone since fails separate separates see lemma intersect lemma adjacent either cut contains faces case sides equation lemma finally adjacent neither sides equation zero lemma separates smooth bounded functions vanish proof fails adjacent sides displayed formula zero suppose adjacent case according definitions opposite across sign separated find separated say separated write addition finally obtain required proof proposition shall use previous lemmas formula differential get let separate sum part indexed hyperplanes separate followed single term indexed hyperplane separate according lemma first part inserting definition weight function obtain dist dist moreover dist part indexed keeping mind dist dist obtain lemma following formula bounded vanish required estimates follow terms uniformly bounded number supported uniformly close uniformly bounded size vanish references baum connes higson classifying space proper actions group san antonio volume contemp pages amer math providence bekka harpe valette kazhdan property volume new mathematical monographs cambridge university press cambridge baaj julg bivariante kasparov non dans les hilbertiens acad sci paris cherix cowling jolissaint julg valette groups haagerup property modern classics basel gromov paperback reprint edition cuntz amenability discrete groups reine angew dixmier publishing amsterdam translated french francis jellett mathematical library vol forman morse theory cell complexes adv guentner higson weak amenability cat groups geom dedicata haagerup example nonnuclear metric approximation property invent higson kasparov groups act properly isometrically hilbert space invent julg valette pour les groupes sur les arbres acad sci paris julg valette amenability action associated tree funct julg valette twisted coboundary operator tree selberg principle operator theory julg valette tordu sur arbre principe selberg operator theory kasparov equivariant novikov conjecture invent connection dual space group structure closed subgroups funkcional anal lance hilbert toolkit operator algebraists volume london mathematical society lecture note series cambridge university press cambridge niblo reeves geometry cube complexes complexity fundamental groups topology niblo roller groups acting cubes kazhdan property proc amer math pedersen automorphism groups volume london mathematical society monographs academic press harcourt brace jovanovich publishers york pimsner cocycles trees operator theory pytlik szwarc analytic family uniformly bounded representations free groups acta sageev ends group pairs curved cube complexes proc london math soc valette cocycles arbres acad sci paris department mathematical sciences university southampton southampton address department mathematical sciences university hawaii manoa honolulu usa address erik department mathematics penn state university university park usa address higson
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nov arash akbarinia contrast variant pooling mechanism colour constancy contrast variant pooling mechanism arash centre per computador universitat barcelona barcelona spain department information communication technologies universitat pompeu fabra barcelona spain raquel gil http alejandro parraga abstract pooling ubiquitous operation image processing algorithms allows processes collect relevant features region interest currently one commonly used operators computational literature however lack robustness outliers due fact relies merely peak function pooling mechanisms also present primate visual cortex neurons higher cortical areas pool signals lower ones receptive fields neurons shown vary according contrast aggregating signals larger region presence low contrast stimuli hypothesise mechanism address shortcomings maxpooling modelled contrast variation histogram clipping percentage pooled signal inversely proportional local contrast image tested hypothesis applying phenomenon colour constancy number popular algorithms utilise step methods investigated consequences replacing original proposed experiments three colour constancy benchmark datasets suggest previous results significantly improve adopting mechanism introduction many computer vision frameworks contain pooling stage combines local responses different spatial locations operation often mechanism implemented wide range algorithms feature descriptors sift hog convolutional neural networks choosing correct pooling operator make great difference performance method current standard pooling mechanisms lack desired generalisation find equilibrium rare informative descriptors therefore many computer vision applications benefit dynamic pooling solution takes account content pooled signals copyright document resides authors may distributed unchanged freely print electronic forms arash akbarinia contrast variant pooling mechanism pooling operators commonly used modelling phenomenon colour constancy visual effect makes perceived colour surfaces remain approximately constant changes illumination biological computational solutions despite decades research colour constancy still remains open question solutions phenomenon important practical point view camera manufacturers need produce images objects appear actual objects order satisfy customers motivated article propose mechanism investigate feasibility context computational colour constancy computational models colour constancy mathematically recovery spectral reflectance scene illuminated light unknown spectral irradiance problem infinite possible solutions simplest popular solution impose arbitrary assumptions regarding scene illuminant chromatic content broadly speaking colour constancy algorithms divided two categories driven reduce problem solving set mathematical equations train machine learning techniques relevant image features approaches may offer highest performance results however major setbacks make unsuitable certain conditions rely heavily training data easy obtain possible situations likely slow unsuitable deployment inside limited hardware large portion driven models summarised using general minkowski expression kec represents illuminant colour channel image pixel value spatial coordinate minkowski norm multiplicative constant chosen illuminant colour unit vector distinct values minkowski norm results different pooling mechanisms setting reproduces well known algorithm assumed colours scene average grey setting replicates algorithm assumes brightest patch image corresponds scene illuminant general challenging automatically tune every image dataset time inaccurate values may corrupt results noticeably minkowski framework generalised replacing derivatives solutions analogous mechanisms visual cortex typically modelled dog operators narrower positive gaussian plays role centre broader negative gaussian plays role surround recently models colour constancy grounded dog offered promising results however efficiency largely depends finding optimum pooling strategy higher cortical areas short pooling crucial component many colour constancy models driven features even solutions primate visual systems size receptive field varies according local arash akbarinia contrast variant pooling mechanism contrast light falling presenting dynamic solution dependent region interest models mentioned related early stages visual processing primary visual cortex area likely involved colour constancy physiological measures suggest although receptive fields triple size area area basic circuity respect surround modulation similar keeping size dependency contrast properties found consistent large body physiological psychophysical literature highlighting significance contrast visual cortex computer vision models also shown encouraging results various applications visual attention tone mapping boundary detection name hypothesise convenience explanatory value various pooling strategies proposed previous colour constancy methods rest work explore advantages replacing different pooling configurations successful colour constancy models visual system described current neurophysiological literature aim dual one hand want explore technological possibilities creating efficient algorithm hand would like test idea might play important role colour constancy summary contributions figure flowchart proposed cvp mechanism context colour constancy implemented cvp given input image feature map value computed according inverse local contrast channel estimate scene illuminant average value pooled pixels right side depicted dashed lines present article propose generic cvp mechanism replace standard operators wide range computer vision applications figure illustrates flowchart cvp based local contrast therefore offers dynamic solution adapts pooling mechanism arash akbarinia contrast variant pooling mechanism ing content region interest tested feasibility cvp context colour constancy substituting pooling operation four algorithms whitepatch doubleopponency three benchmark datasets results experiments show quantitative qualitative benefits cvp model colour constancy one earliest computational models colour constancy grounded assumption brightest pixel image corresponds bright spot specular reflection containing necessary information scene illuminant mathematically equivalent operation intensity pixels arg max represents estimated illuminant chromatic channel original image spatial coordinates image domain one important flaw simple approach single bright pixel misrepresent whole illuminant furthermore algorithm may fail presence noise clipped pixels image due limitations operator one approach address issues account larger set white points pooling small percentage brightest pixels top operation referred manner pooling mechanism collectively computed considering group pixels rather single one small variant might crucial factor estimation scene illuminant similar mechanism also deployed successfully applications shadow removal practice given chosen implemented clipping mechanism let histogram input image let represents number pixels colour channel intensity histogram domain scene illuminant computed nkc nkc total number pixels within intensity range values determined chosen nkc total number pixels image chosen percentage colour channel within formulation cumbersome define universal optimallyfixed percentage pixels pooled consequently free variable requires specific tuning image dataset individually naturally limitation restricts usability operator following sections show automatically compute percentage based local contrast image arash akbarinia contrast variant pooling mechanism pooling mechanisms visual cortex know physiology cerebral cortex neurons higher cortical areas pool information lower areas increasingly larger image regions although exact pooling mechanism yet discovered sparse coding kurtotical behaviour common many groups neurons visual cortex conceivable mechanism analogous might present within cortical layers indeed behaviour discovered across population cells cat visual cortex activation level cells behaviour reported vary depending contrast visual stimuli results reported suggest inverse relationship contrast stimulus percentage signal pooled pooling neurons exposed low contrast stimuli responses shifted slightly away pure selecting highest activation response within region towards integrating larger number highly activated neurons language computer vision regarded assumes smaller value high contrast larger value low contrast interestingly pooling neurons remained always much closer linear integration neurons mathematically interpreted small value come great surprise pooling mechanism visual cortex depends stimulus contrast large body physiological studies showing receptive fields neurons contrast variant comprehensive review refer quantitative results suggest rfs visual area one awake macaques double size measured low contrast similar expansions also reported rfs neurons extrastriate areas implies typical neuron higher cortical areas normally pool responses preceding areas three neighbouring spatial locations access substantially larger region pool presence low contrast stimuli line reported pooling mechanism cat visual cortex contrast variant pooling order model mechanism first computed local contrast input image every pixel location means local standard deviation defined indexes colour channel spatial coordinates pixel isotropic average kernel size represents neighbourhood centred pixel radius simulate inverse relation stimulus contrast percentage signal pooled operator determined percentage average inverse local contrast signals computed spatial image domain fashion instead defining fix percentage signal pixels pooled chose adaptive arash akbarinia contrast variant pooling mechanism percentage according contrast image terms colour constancy effectively relates number pooled pixels compute scene illuminant average contrast image illustrated mechanism central panel figure red green blue signals correspond histogram chromatic channel pixels right side dashed lines pooled example contrast higher red signal therefore smaller percentage cells pooled red channel bearing mind contrast fraction range characterising absolutely uniform area representing points highest contrast edges predict percentage small number natural images homogeneous regions likely form majority scene consequently operator always pool small percentage agreement observations indicate pooling mechanism always much closer generalisation colour constancy models number colour constancy models literature driven features require pooling mechanism top computed feature maps order estimate scene illuminant algorithm feature map computed convolving representation image dog kernel followed operation model derivatives image calculated convolution derivative gaussian kernel complemented minkowski summation oscillates depending norm similar algorithm pooling mechanism models also replaced operator computed according local contrast image explained difference instead pooling intensity image case algorithm pool respective feature maps means receives feature map instead intensity image input experiments results order investigate efficiency model applied proposed cvp mechanism four different colour constancy algorithms whose source code publicly available greyedge simply replaced operator proposed pooling mechanism evaluate method used recovery angular error defined arccos kee kket represents dot product estimated illuminant ground truth stands euclidean norm vector worth mentioning error measure might correspond precisely observers preferences however commonly used comparative measure literature also computed reproduction arash akbarinia contrast variant pooling mechanism angular error experiments due lack space results reported readers encouraged check accompanying supplementary materials conducted experiments three benchmark sfu lab images colour checker images iii grey ball images table reported best median trimean angular errors considered methods metrics proposed respectively evaluate colour constancy algorithms since robust outliers mean angular errors reported supplementary materials sfu lab colour checker grey ball median trimean median trimean median trimean cvp cvp cvp cvp table recovery angular errors four colour constancy methods cvp three benchmark datasets lower figures indicate better performance method figure illustrates three exemplary results obtained proposed cvp operator two colour constancy models qualitatively observe cvp better job estimating scene illuminant also confirmed quantitatively angular errors shown right bottom side computed output original ground truth cvp cvp figure qualitative results cvp angular errors indicated bottom right corner panel images sfu lab colour checker grey ball dataset respectively source code materials available supplementary submission arash akbarinia contrast variant pooling mechanism influence free parameters free variable tested models compared performance free variable therefore exempted analysis figure reported impact different receptive filed size algorithm best worst results obtained free variable dataset results available supplementary material observe almost cases outperforms improvement tangible colour checker grey ball datasets low sfu lab colour checker grey ball figure best worst results obtained free variables algorithm figure illustrates impact different gaussian size secondorder algorithm observe similar patterns outperforms practically cases improvement significant low colour checker dataset derivative must noted objective article merely study performance cvp top algorithm however angular errors cvp happen par best results reported obtained using optimum minkowski norm dataset important caveat cvp extra variables tuned whereas minkowski norm optimisation value must dataset sfu lab colour checker grey ball figure comparison free variable greyedge algorithm derivatives figures observe greatest improvement occurs colour checker dataset speculate one reasons larger range intensity values colour checker dataset comparison two datasets contain images therefore inaccurate severely penalised arash akbarinia contrast variant pooling mechanism discussion would like emphasise objective article improve colour constancy show cvp almost always produces improvements surprisingly results obtained even competitive instance sfu lab dataset lowest reported angular error obtained gamut algorithm means cvp angular error outperforms dataset colour checker grey ball datasets models method obtain lower angular errors comparison cvp nevertheless results comparable physiological evidence besides better performance cvp explained intuitively fact relies merely peak function region interest whereas model pooling defined collectively based number elements near maximum consequently peaks outliers likely caused noise get normalised pooled elements rationale within model pool larger percentage low contrast since conditions peaks informative whereas high contrast peaks likely informative irrelevant details must removed therefore smaller percentage pooled although importance choosing appropriate pooling type demonstrated experimentally theoretically current standard pooling mechanisms lack desired generalisation believe offer dynamic general solution article evaluated cvp colour constancy phenomenon however formulation cvp generic based local contrast principle applied wider range computer vision algorithms pooling decisive factor implementation cvp approximated local contrast local standard deviation see least two factors require profound analysis incorporating sophisticated models contrast perception accounting extrema human contrast sensitivity analysing role kernel size computation local contrast conclusion article presented novel cvp mechanism grounded physiology visual cortex main contribution summarised linking percentage pooled signal local contrast stimuli pooling larger percentage low contrast smaller percentage high contrast cvp operator remains always closer rather since natural images generally contain homogeneous areas abrupt discontinuities tested efficiency cvp model context colour constancy replacing operator four algorithms proposed pooling conducted experiments three benchmark datasets results show outperforms commonly used operator nearly cases explained fact model allows informative peaks pooled suppressing less informative peaks outliers argued proposed cvp generic operator thus application arash akbarinia contrast variant pooling mechanism extended wider range computer vision algorithms offering dynamic automatic framework based local contrast image pixel opens multitude possibilities future lines research remains open question whether model reproduce excellent results domains well therefore certainly interesting investigate whether cvp improve convolutional neural networks acknowledgements work funded spanish secretary research innovation references vivek agarwal andrei gribok mongi abidi machine learning approach color constancy neural networks arash akbarinia alejandro parraga edge detection surround modulation proceedings british machine vision conference pages arash akbarinia alejandro parraga feedback surround modulated boundary detection international journal computer vision pages alessandra angelucci shushruth beyond classical receptive field surround modulation primary visual cortex new visual neurosciences pages kobus barnard improvements gamut mapping colour constancy algorithms computer pages springer kobus barnard lindsay martin brian funt adam coath data set color research color research application jonathan barron convolutional color constancy proceedings ieee international conference computer vision pages boureau jean ponce yann lecun theoretical analysis feature pooling visual recognition proceedings international 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data discovery anomaly detection using atypicality theory sep anders fellow ieee elyas sabeti member ieee chad walton abstract central question era big data enormous amount information one possibility characterize statistics averages classify using machine learning order understand general structure overall data perspective paper opposite namely value information applications parts deviate average unusual atypical define mean atypical axiomatic way data encoded fewer bits rather using code typical data show definition good theoretical properties develop implementation based universal source coding apply number real world data sets index terms big data atypicality minimum description length data discovery anomaly ntroduction one characteristic information age exponential growth information ready availability information networks including internet big question enormous amount information one possibility characterize statistics think averages perspective paper opposite namely value information parts deviate average unusual atypical rest background noise take art truly valuable paintings rare atypical could true scientific research entrepreneurship take online collections photos photos rather pedestrian snapshots interest wider audience photos interest unique flickr collection photos rated interestingness one notice photos indeed different typical photos atypical aim approach extract rare interesting data big data sets central question interesting means first thought focus rare part interesting data something unlikely based prior knowledge typical data examples typical data training way sabeti department electrical engineering university hawaii manoa honolulu ahm sabeti walton department surgery university hawaii honolulu email cwalton work supported part nsf grants ccf paper presented part ieee information theory workshop seville september draft outlier usually defined unlikeliness could measured terms likelihood terms codelength called surprise according distance measure also common principle anomaly detection however perhaps unlikely sufficient something many cases outliers junk eliminated contaminate typical data makes something interesting maybe new unusual structure quite different structure data already seen return example paintings make masterworks interesting different paintings structure intriguing take another example many scientific discoveries like theory relativity quantum mechanics began experiments fit prevailing theories experiments outliers anomalies made truly interesting possible find new theory explain data relativity quantum mechanics principle pursue finding data better alternative explanations fit typical data something unlikely even necessary data suppose typical data iid uniform sequence bits equally likely therefore sequence consisting purely way yet catch interest look new interesting data characteristic know looking looking unknown unknowns instead looking specific statistics data need use universal approach provided information theory idea finding alternative explanations data rather measuring kind difference typical data separates method usual approaches outlier detection anomaly detection far determine reading hundreds papers approach explored previously obviously information theory coding used anomaly detection data mining knowledge discovery discuss compares approach later methodology also connections tests randomness run length test aim different applications atypicality relevant large number various applications list applications ecg electrocardiogram ecg recordings patterns heart rate variability known indicate possible heart disease modern technology possible individual wear unobtrusive heart rate monitor atypical patterns occur could indicative disease individual doctor could notified perhaps important application medical research one analyze large collection ecg recordings look individuals atypical patterns potentially used develop new diagnostic tools genomics another example application interpretation large collections genomics data given mammals essentially set genes must exist significant differences distinguish obvious distinct attributes species well subtle differences within species although genome mined exhaustive studies applying panoply approaches regions thought uninteresting recently come increased study potential role defined morphological september draft physiological differences individuals applying atypical evaluation tool genomic data individuals known irregularities may provide valuable insight genetic mechanisms underlying condition ocean monitoring passive acoustic monitoring pam oceans one hydrophones towed behind ship deployed fixed suspended array order record vocalizations marine mammals one major focus detect perhaps count rare endangered species would highly interesting scan data unusual patterns examined researcher plant monitoring example nuclear plants atypical monitoring data may indicative something wrong computer networks atypical network traffic could indicative cyberattack already used anomaly detection however abstract atypicality approach used find subtle attacks unknown unknowns airport security already software used flag suspicious flyers likely based past attacks atypical detection could used find innovative attackers stock market atypicality could used detect insider trading could also used investors find unusual stocks invest promising outstanding returns ruin astronomy atypicality used scan huge databases new kinds cosmological phenomena credit card fraud unusual spending patterns could indicative fraud already used credit card companies obviously simple annoying way anyone credit card blocked overseas trip testify gambling casinos constantly fighting fraudsters game cat mouse fraudsters constantly find new ways trick casinos one inventor shannon therefore abstract atypicality approach may best solution catch new ways fraud notation use denote sequence general need make length explicit denotes single sample sequence use capital letters denote random variables rather specific outcomes finally denotes subsequence logarithms base unless otherwise indicated atypicality starting point theory randomness developed kolmogorov kolmogorov divides infinite sequences typical typical sequences call random satisfy laws probability characterized kolmogorov complexity sequence bits random iid uniform kolmogorov complexity sequence satisfies constant sequence incompressible finite sequence algorithmically random terms coding iid random sequence also incompressible put another way best coder identity september draft function let assume draw sequences iid uniform distribution optimum coder identity function code length suppose one sequences find universal coder code length less directly equivalent one could state interpretation kolmogorov terms would typical sequence special sequence instead call sequences considering general distributions general finite alphabets instead iid uniform distributions state following general principle definition sequence atypical described coded fewer bits rather using optimum code typical sequences definition central approach atypicality problem definition optimum code typical sequences quite specific following principles example assume prefix free codes within class coding could done using huffman codes shannon codes codes arithmetic coding etc care code length among variation length within bits code length typical encoding quite accurately calculated hand described coded fewer bits less precise principle one could use kolmogorov complexity kolmogorov complexity calculable given except constant comparison code length therefore comparison rather type universal source coder used given quite precise meaning class finite state machine sources following work strongly related minimum description length mdl essential adhere strict decodability decoder decoder sees stream bits able accurately reconstruct source sequence example sequence atypical must type header telling decoder use universal decoder rather typical decoder atypical sequences encoded multiple ways decoder must informed sequence bits encoder used one could argue things irrelevant example anomaly detection since actually encoding sequences problem terms omitted far easy encode sequence like choosing complex model fit data without accounting model complexity exactly mdl sets solve although also case actual encoding done therefore try account factors needed describe data believe one key strengths approach major difference atypical data anomalous data atypicality axiomatic property data defined definition based randomness hand far know anomaly something strictly defined usually think anomaly something caused outside phenomenon intruder computer network heart failure gambler playing tricks influences think performance detector fails give indication anomaly miss type error gives indication things happening false alarm type error september draft atypicality hand purely property data ideally therefore misses false alarms data atypical mean anomaly expresses observed data must mean structure data theory source coder would discover exploit structure reduce code length thus data atypical means simply way detect anomaly observations theory therefore really call miss hand suppose casino gambler long sequence wins could due fraud could also simply due randomness casino security would interested either case scrutiny thus reason atypicality really matter atypicality matters still distinguish two cases call sequence intrinsically atypical atypical according definition generated typical probability model extrinsically atypical fact generated probability law definition two parts work concert write simplified typical codelength atypical codelength typical code length simply expression likelihood seeing particular sequence large means given sequence unlikely happen detecting sequences would catch many outliers extreme example sequence impossible according typical distribution would always caught would work universally started typical sequences iid uniform sequence equally likely would catch sequences case test sequence structure possible sequences would caught atypicality thus calculating essential calculating also essential suppose instead use code length used encode typical sequences average essentially entropy rate catch sequences test sequence less structure typical sequences omit obvious examples test sequence use typical sequence swapped hand impossible sequences would caught absolute certainty declare something outlier find coder sufficient large sequence unlikely happen however always use trivial coder transmits data uncoded sequence unlikely happen according typical distribution likely length thus seen two parts work concert catch sequences part might catch sequences catch anomalies parts used another point view following suppose typical model binary uniform iid look collection sequences want find atypical sequences interesting sequences without specification interesting seems reasonable choose sequences structure reasonably measured much sequence compressed rissanen calls useful information need take account typical model uniform iid example typical sequences much structure sequence little structure might interesting therefore end reasonable measure interesting sequences might september draft alternative approaches argued introduction outlined aiming anomaly detection traditional sense still many similarities certainly information theory universal source coding used previously anomaly detection approaches mostly heuristic fundamental systematic approach information distance defined without able claim applies perhaps hundreds papers think various approaches summarized using universal source coding type distance measure whether satisfies strict mathematical metric properties heuristic hand methodology definition classified distance measure traditional sense instead trying find alternative explanations data comment approach contrasts approaches similarity distance developed directly applicable problem consider extent adapt useful contrast similarity distance min max instead given typical distribution imagine given long typical sequence used case within certain approximation suppose starting point typical distribution binary iid uniform also binary iid uniform within constant drawn distribution help describing either still makes sense two completely random sequences similar whether distribution thus similarity distance used anomaly detection defined looking special sequences words kolmogorov problem similarity metric exactly designed really deterministic similarity sequences appropriate classification reason similarity distance still gives results anomaly detection actually universal source coders approximate kolmogorov complexity poorly heuristic methods using anomaly detection using universal source coding mostly based comparing code length let code length encode sequence universal source coder let training string test sequence compare could seen measure entropy rate compare detect change issue many completely dissimilar sources entropy rate example let data binary iid original source new source optimum code original source optimum code new source length hand atypicality immediately distinguish sequences september draft iii inary iid ase order clarify ideas first consider simple model typical model iid binary alternative model class also binary iid unknown want decide given sequence typical atypical stated hypothesis test problem problem ump universal powerful test however common approach solving type problem glrt generalized likelihood ratio test let sequence length number glrt log log log log log log relative entropy threshold glrt heuristic principle satisfies optimality properties case equal invariant ump test considered optimum solution certain constraints thus reasonably take optimum solution problem need appeal kolmogorov information theory solve problem complications start consider sequences variable length test depends sequence length need choose threshold function result false alarm probability detection probability obvious argument choose hypothesis testing point view could choose independent another arbitrary choice consider problem context definition order need model problem coding point view assume infinite sequence sequences variable length need encoded need encode bit also encode whenever new sequence starts september draft typical encoding bits use shannon code huffman code arithmetic coding etc code length sequence length log log log log also need encode sequence ends except small constant factor new one starts simplicity let assume lengths geometrically distributed model problem one three source symbols iid distribution assume small expression still valid content part sequence added constant log encode separators decide sequence atypical according definition use universal source coder source encodes first number ones enumerates sequences ones transmits index given sequence analysis important simple expression code length therefore use log approximation good reasonably large also reaches lower bound also needs inform decoder following atypical sequence knows use atypical decoder rather typical encoder ends former use indicate start atypical sequence rather typical sequences probability sequence atypical code length log log mark end atypical sequence could insert code either based distribution lengths typical sequences assume known whereas would knowledge length atypical sequences instead seems reasonable encode length specific atypical sequence argued done log constant log log log log log log sum continues long argument log positive summarize log log log log log log log log log log log criterion sequence atypical easily seen equivalent log lengths fixed reduces lengths variable provides threshold function term september log ensures seems reasonable instead used draft easy see except property term log might seem arbitrary based solid theory seen later several important theoretical properties examine criterion detail inequality gives two thresholds impossible find explicit expressions clear therefore large replace series expansion end explicit criterion log log following use considerably simpler analyze also write pql term would central limit type statement probability sequence classified intrinsically atypical would independent main interest exactly dependency given following theorem theorem consider iid let probability sequence length classified intrinsically atypical according bounded lim strengthened bounds tight sense lim proof chernoff bound states inf september draft usual lpq log moment generating function bernoulli random variable inf exp pes minimizing gives used equation directly leads hoeffding inequality gives bound exp exp log tighter lower bound use moderate deviations define define rewrite satisfies lal using theorem gives lim inf lim inf september draft together upper bound gives figure compares upper bound simulations upper bound simulation fig simulated upper bound also bound miss probability extrinsically atypical sequences follows theorem suppose typical sequence iid let test sequence iid probability test sequence missed according criterion upper bounded ppa lim proof may assume similarly proof theorem chernoff bound inf exp minimizing gives september lqa draft lqa using series expansions hypothesis testing interpretation solution may seem arbitrary nice interpretation terms hypothesis testing return solution solution gives test given however problem reconcile tests different one way solve issue consider random variable introducing prior distribution bayesian sense let prior distribution equation becomes log log log log log log log hypothesis test log log course problem know still compare without approximations log term corresponds distribution integers namely except term log equations identical use prior distribution rissanen argues distribution reasonable distribution integers really prior knowledge mainly coding point view therefore seems reasonable distribution term log model hypothesis one unknown parameter complex null hypothesis account additional complexity goal find explanation atypical sequences among large class explanations distribution zeros ones penalty finding complex explanation data explained data atypical occam razor penalty one unknown parameter argued rissanen exactly log therefore following explanation september draft fact criterion understood hypothesis test prior distribution penalty log unknown parameter seen light theorem surprising replaced implicitly corresponds prior distribution log log exactly distribution seen atypical subsequences one problem believe approach excels finding atypical subsequences long sequences difficulty find atypical subsequences may short subsequences deviate much typical model long subsequences deviate little choose among definition gives precise answer formal problem statement consider sequence finite alphabet section sequence generated according probability law known sequence embedded infrequent finite subsequences finite alphabet generated alternative probability law probability law unknown might known certain class probability distributions example parametrized parameter subsequence may drawn different probability law problem consider isolate subsequences call atypical subsequences section assume binary iid solution similar one variable length sequences atypical subsequences encoded universal source coder code length log start sequence encoded extra symbol code length log length encoded bits conclusion end exactly criterion repeated log difference slight different meaning subsequence problem central question probability given sample part intrinsically atypical subsequence notice infinitely many subsequences contain probability atypical given theorem obtain upper bound follows let say determined part atypical sequence clear sequence must also atypical according therefore upper bound probability part atypical sequence probability event using approximate criterion september pql draft rewrite pql lpq log could upper bound union bound using theorem however quickly seen converge problem events union bound highly dependent need slightly refined approach results following theorem theorem consider case probability given sample part atypical subsequence upper bounded constants proof without loss generality assume let set subsequences containing length let length subinterval theorem know therefore constant argument work allow arbitrarily long subsequences sum divergent however write probability atypical subsequence least length proof bound define following events lpq log lpq log rewrite lpq log log ease notation define september log draft using union bound write excluded length one sequence consisting consider think simple random use upper bound probability probability interpreted probability random walk passes given times since random walk increase one since threshold increasing means time must furthermore easy see probability upper bounded probability given random walk times thus denominator interpreted probability maximum random walk stays theorem expressed september draft sufficiently large constant since discussed start proof assume choose large enough satisfied furthermore since increasing choose independent long sufficiently large next upper bound numerator probability path stayed steps step hits count paths divide two groups count separately first group paths start zero hit first time steps second group easily described reverse time paths start step stay time hit finally hit time according section count paths number length paths need upper bound probability path starting hits steps use section get bound power exponent follows thus use bound probability set paths second term bound september draft sum looked reverse time interpreted probability path starting hits zero time write see section use proof theorem specifically bound exp exp next bound probability paths first term log september draft thus constant first evaluate sum term decreasing sufficiently large evaluate sum separately convergence depends latter tail threshold increasing example put proportional therefore write erfc september draft constants used verified three sums inserted convergent using bound second sum exp ignore small constants write exp dldt remaining integral clearly convergent decreasing therefore two important implications theorem first sufficiently large fact made arbitrarily small large enough important theoretical validation definition resulting criterion theory resulted everything would atypical atypicality would meaningless trivially satisfied shown proposition proposition says equation instead atypical log log everything would log corresponds forgetting length atypical sequence also needs encoded resulting sequence decodable thus strict adherence decodability september draft lead meaningful criterion although decodability first seems unrelated detection turns crucial importance similarly first term log may seen arbitrary however within margin sufficient ensure everything becomes atypical second important implication theorem validates meaning way introduced number bits needed encode fact atypical sequence starts therefore put log atypical sequence starts theorem confirms desired meaning purely random sequences reasons trivial chosen probability atypical sequence theorem gives probability sample atypical proposition consider case suppose instead use criterion giving probability given sample part atypical subsequence proof assume continue random walk framework proof theorem define event log log log namely declare atypical endpoint atypical sequence clearly could start midpoint atypical sequence rather loose lower bound write september acl acl ack ack draft consider probability ack way conditional event happen last inequality true sufficiently large september draft example sufficiently large divergent proving theorem states convergence proposition shows divergence gap values hard fill theoretically therefore tested numerically see fig course testing convergence numerically quite still figure indicates phase transitions divergence convergence happens right around probability atypicality samle different values middle probability fig transition divergence convergence function recursive coding instead using definition directly could approach problem follows first sequence encoded typical code distribution sequence agreement typical code results sequence iid binary bits purely random sequence sequence encoded try encode sequence universal code categorize sequence atypical let length sequence typical coding typical atypical codelengths therefore log september log log draft estimated encoded sequence log log log log log log log log argument follows without detailed calculations encode sequence wrong code later correct code induced statistic result originally encoding correct code thus criterion approximately equivalent state follows proposition definition applied encoded sequences instead original data course ignores integer constraints block boundaries etc importance statement sometimes easier operate partially encoded sequences simply amount data already reduced problem standardized need know typical codebook even model typical data since everything typical model reduced stream iid binary digits atypicality algorithms therefore applied data streams without knowledge original data also means theoretical results theorem assume typical data iid uniform general applicability however first encoding sequence atypicality detection also disadvantages practical finite length setting atypical subsequences become embedded typical sequences unpredictable ways example could difficult determine exactly atypical sequence starts ends practical implementation therefore uses definition directly eneral ase return problem considered start section iii given sequence fixed length need determine atypical iid case simple hypothesis test problem solution given general case would like find alternative explanations large abstract class models issue often possible fit alternative model well data allow complex enough models well known occam razor problem rissanen mdl solution problem therefore general case even fixed length sequences problem straightforward hypothesis test problem resort information theory september draft finite state machines possible class models general case class finite state machines fsm rissanen defines complexity sequence class fsm sources min log sequence state machines used emphasize probability estimated rissanen uses laplace estimator could also used except integer constraints valid descriptive length therefore used definition natural extension iid case considered section iii opposed kolmogorov complexity complexity could actually calculated although high complexity complexity mostly useful theoretical considerations one result following generalization theorem theorem assume typical distribution iid uniform atypical descriptive length given maximum number states independent probability intrinsically atypical sequence satisfies lim proof since consider state machines number states certain maximum must also include state machine single state equivalent iid model section iii therefore get lower bound proof upper bound probability section iii use log bits indicate beginning end atypical sequences probability sequence atypical therefore log log log log log prove log log constants number states state machine since slowest decay dominates get upper bound fixed state machine code length according log log denotes number occurrences state number times next symbols state log log september log draft want upper bound probability event log write log log log log let let remaining small terms dependent log log upper bound notice log chernoff bound exp log log log log exp order get valid bound need show independent easy see exp show exp show true state machines class finite state machines states done showing max exp fsm states turns easier prove expand class take maximum clearly expanding class decrease maximum fsm states function satisfies bit fsm state steps next state transition dependent next bit got state extend class dispensing requirement describe program run follows based choose state without knowledge except independent uniformly september draft distributed assumption typical distribution think slightly differently program puts bucket updates order maximize exp based past data opposed state machine setup choice way restricts choices states buckets since program knowledge program optimize based values rather sufficient look easy see worst case obtained bits distributed evenly states thus worst case independent thus problem reduced case single state showing exp exp lnt used sum actually decreasing function seems hard prove instead upper bound sum september lnt draft upper bound lnt ldx ldx constants using gaussian moments proves probability intrinsically atypical sequence probability upper bound length fig probability intrinsically atypical sequence typical distribution iid uniform detection atypical sequences ctw algorithm used section theorem typical model iid uniform outlined section principle also applies general sources since first encode look atypical sequences theorem shows looking complex explanations data essentially increase probability intrinsically atypical sequences fig compare fig confirms experimentally atypical detection based ctw explained section good approximation fsm modeling hand one fsm models fact fit data chance detecting sequence greatly increased although hard quantify think intrinsically atypical sequences false alarms shows power methodology since fsm sources iid case seems reasonable conjecture theorem still valid sufficiently large clearly essential theoretical property atypicality september draft however theorem follow directly theorem verify conjecture requires formal proof present atypical encoding terms coding definition stated following form code length encoded optimum coder according typical law encoded argued section iii need put header atypical sequences inform encoder atypical encoder used therefore write number bits header number bits used encoding data encoding data obvious solution use universal source coder many approaches universal source coding transform partial predictive mapping ppm anyone could applied problem considered paper idea atypicality linked particular coding strategy fact coding strategy need decided could try several source coders choose one giving shortest code length could even combined however control complexity choose single source coder popular simplest approach source coding perhaps issue convergence slow according optimum sense lim log thus poor short sequences exactly var interested atypicality therefore chosen use context tree weighing ctw algorithm ctw approach advantages setup natural extension simple example considered section iii allows estimation code length without actually encoding flexibility estimate probabilities importantly seen practical implementation fsm based descriptive length used section typical encoding training definition example section iii assumed typical model data exactly known case typical encoding straightforward using example arithmetic coding notice need codelength calculated arithmetic coding without actually encoding however many cases typical model known exactly simplest case typical model small class parametrized parameters binary iid unknown case finding typical model simple parameter estimation problem discuss focus case typical model given specific model case seems obvious also use universal source coding typical data however straightforward want stay faithful idea definition argue september draft let typical model estimated typical model specific model given principle could given estimate various joint probability mass functions however useful approach estimate conditional probabilities called context source finite memory probabilities characterize source otherwise could give good approximate model issue possible contexts even moderately large amount training data required even observe every context large get good estimates every context even larger realistically therefore every probability estimated issue universal source coding designed deal therefore turn universal source coding let assume given single long sequence training rather model based need encode sequence let denote coder understand means realize encoded according known coding probabilities fixed affected discussed section important part definition reacts outliers data fit typical model coding probabilities allowed depend extreme case would always issue universal source coders often easily adapts new types data desirable property good universal source coder problematic light discussion therefore need freeze source coder example updating dictionary however training data likely incomplete discussed freezing hard resulting encoder universal source coder rather training based fixed source coder implementation quite different universal source coder requiring careful consideration suggest one algorithm based principle ctw algorithm naturally complements using ctw algorithm atypical encoding could also used atypical encoders following discussion requires good knowledge ctw algorithm easily obtained algorithm based estimating contexts estimate given done number respectively seen context training data unaffected test sequence discussed complication every context might seen particular long contexts rarely seen estimates might accurate shorter contexts solve weighting idea ctw algorithm weights thought prior distribution different models summarize follows every context subsequence associated could either memoryless could memory call former model latter ctw algorithm uses prior distribution weights models basic idea weigh instead algorithm best described example trained algorithm resulting context tree seen fig suppose want find coding distribution actual data begin september draft fig example context tree root calculate empty context model data memoryless find look context tree calculate similarly model data iid september draft context tree seen context seen context seen either needed calculation completed algorithm implemented follows run standard ctw algorithm training data freeze node corresponding context data needs stored algorithm described root implementation simpler recursion top root often source coder might trained many separate sequences rather one long sequence issue care taken startup sequence original ctw paper assumes context length least available prior start sequence true practice paper solves introducing indeterminate context context may start indeterminate context multiple training sequences could happen better approach therefore use start training sequence purely context code wastes training bits sequences long loss minor different case would need find short atypical sequences rather subsequences case careful treatment start sequences would needed freezing encoder essential implementing atypicality simulation confirming shown fig ctw algorithm trained markov chain transition probability generating according markov chain mostly generates following pattern test sequence generated another markov chain transition probability generating according generating mostly pattern algorithm code length difference typical atypical encoding small easily missed although difference patterns raw data clearly visible naked eye reason algorithm work quickly learns new pattern good source coder would including advantageous source coding case means missing obvious atypical pattern large amount data used training complexity become high mainly terms memory namely contexts might observed context tree completely filled example suppose typical data actually iid means every string seen equal probability ctw algorithm means every node context tree filled number nodes depth dictionary based algorithms means dictionary size becomes huge needed algorithm estimates unknown parameters also model iid one way could trim context tree dictionary looked detail atypical subsequences finding atypical subsequences long sequences basic setup previous sections used let subsequence want test september draft effect freezing typical coder learn atypical sequences typical code length frozen typical code length atypical code length fig importance freezing source coding testing atypicality atypicality section start sequence needs encoded well length additionally code length minimized maximum depth context tree atypical code length given min log except denotes probability root context tree depth since also interested finding short sequences encoding initialized importance atypical coding therefore use algorithm section typical coding use either known fixed model shannon codes algorithm section model known encode use context assume equivalently encode total sequence algorithm section let codelength sequence put need test every subsequence every length need test subsequences every value atypical coding means new ctw algorithm needs started every sample time maximum sequence length separate ctw trees need maintained time completely independent run parallel processors result every bit data calculate min state atypicality criterion advantage stating like need choose prior running algorithm sort according first examine data smallest atypical parts data thus algorithm really parameter free september draft one advantage approach many anomaly detection algorithms multiple parameters need adjusted implementation clearly quite complex still feasible implement medium sized data sets due speed source coder order process larger data sets faster approximate search algorithm needed working algorithms leave topic later papers xperimental esults order verify performance algorithm used three different experiments first evaluated randomness sources accordance starting point randomness second looked infection human dna third looked arrhythmia ecg word presentation results outcome method plot given time would like illustrate raw data source cases stream bits convert plot call random walk representation experience allows one quickly assess obvious pattern data data random results look like typical random walk small fluctuations large fluctuations experimental data software used available http coin tosses experiment typical data iid binary random source typical data used experimental coin tosses data consists tosses two berkeley undergraduates fair coin result heads therefore consider real binary iid experiment indeed example pure random data experiments data examine randomness types data one type data one might think purely random word length changes text first experiment generate binary data using consecutive word length comparison part synge shores great sea following manner next word longer current word assigned binary data otherwise case two consecutive words length random generated good random number generator insert data coin toss data since assume coin tosses data iid need train ctw code length iid case used typical coding fig illustrates result algorithm mixed data thus word length changes iid random perhaps word length bounded limits long runs possible second experiment generated random data infamous randu random number generator random number generator widely used discovered clear deviation randomness randu generates random numbers interval number needs bits binary representation instead using bits number sum bits compare september draft mixture coin tosses consecutive word length comparison data shores great sea synge random walk mixed data bits cummulative bits text word length comparison samples fig random walk mixed coin tosses consecutive word comparison mixture coin tosses randu data random walk mixed data bits cummulative bits randu samples fig random walk mixed coin tosses randu generate either data inserted part coin tosses data fig shows atypical segment inserted data randu random number generator third experiment generate binary data using consecutive heart rate comparison part normal sinus rhythm downloaded bih database way text fig represents result algorithm mixed data seen atypicality measure shows huge difference iid randomness randomness consecutive heart beats know case september draft mixture coin tosses consecutive normal heart rate comparison normal heart rate comparison cummulative bits random walk mixed data bits samples fig random walk mixed coin tosses hrv dna collection experiments detect viral bacterial insertion human genomic dna dna foreign species inserted human genome either natural processes typically though viral infections bacterial infections genetic engineering inverse also occurs genetic engineering experiments creation transgenic organisms insertion human dna bacteria yeast worms mice experiments show focused former case train ctw algorithm pure human genomic dna data used comprised kilobases human genomic dna sequence different chromosome either bacterial viral random dna sequences kilobases per insertion inserted since software slow find atypical sequences length hundreds removed middle insertions notice actually makes detection harder used human dna training test sequences first experiment tried detect short sequences streptococcus pneumoniae bacterial infection high fatality rate frequent cause death elderly randomly inserted larger segments human genomic dna fig illustrates result experiment based figure inserted streptococcus pneumoniae dna fragment detected algorithms second experiment tried detect hiv inserted human genomic dna mimic viral infection realistic experiment since viruses typically insert dna host genome every time human obtains viral infection fig illustrates result experiment seen infected viral fragment detected algorithms september draft primarily human genomic dna infected streptococcus pneumoniae fatal bacterial infection elderly cummulative bits streptococcus pneumoniae infection random walk infected human dna bits dna samples fig random walk human dna bactrial infection primarily human genomic dna infected hiv viral infection random walk infected human dna bits cummulative bits hiv infection dna samples fig random walk human dna viral infection hrv hrv heart rate variability powerful indicator arrhythmia common issue known exactly look data aim application use atypicality localize signs subtle complex arrhythmia based modest goal localizing simple type known arrhythmia managed find premature beats using hrv signal attempted detect subtle arrhythmia hrv signals used downloaded bih database used normal sinus rhythm database nsrdb supraventricular arrhythmia database svdb encoding hrv signals september draft done manner text word length comparison subsection experiment ctw trained hrv normal sinus rhythm applied hrv signal supraventricular rhythms fig shows result simulation algorithm able localize segment suffers abnormal rhythms hrv random walk atypicality measure supraventricular arrhythmia cummulative bits random walk hrv bits hrv samples fig random walk hrv onclusion paper developed criterion finding atypical sub sequences large datasets criterion based solid theoretical foundation shown criterion amenable theoretical analysis particular shown probability sample intrinsically atypical less important theoretical requirement trivially satisfied also shows examples data method able find known atypical subsequences aim set introduction ambitiously find interesting data large datasets context purpose current paper introduce methodology show theoretical properties order analyze really big datasets need much faster probably approximate algorithms much efficient software say python faster computers working future work eferences akoglu tong vreeken faloutsos fast reliable 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international conference speidel note estimation string complexity short strings information communications signal processing icics international conference speidel eimann brownlee detecting network events via information communications signal processing international conference speidel gulliver analytic upper bound information theory proceedings isit ieee international symposium july yang speidel string similarity detection information theory workshop ieee volf willems switching two universal source coding algorithms data compression conference dcc proceedings jacquet szpankowski limiting distribution lempel ziv redundancy information theory proceedings isit ieee international symposium willems shtarkov tjalkens reflections context tree weighting method basic properties newsletter ieee information theory society vol willems weighting method extensions information theory ieee transactions vol mar department statistics berkeley http physiobank atm http september draft britten dna 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nov spikes regularizers anders department computer science university copenhagen copenhagen soegaard abstract present learning algorithm ral spike regularized adaptive learning relying encoding activation spikes adaptively update weight vector relying confidence estimates activation offsets relative previous activity regularize updates proportionally confidence support loosely inspired observation neurophysiology high spike rates sometimes accompanied low temporal precision experiments suggest new learning algorithm piral robust less prone overfitting averaged perceptron row learning linear classifiers perceptron rosenblatt conceptually simple widely used discriminative linear classification algorithm originally motivated observations signals passed neurons brain return perceptron model neural computation technical point view main weakness perceptron linear classifier prone overfitting one particular type overfitting likely happen perceptron learning feature swamping sutton frequent features may prevent features updated leading catastrophic performance frequent features absent less frequent test time words perceptron well learning crammer parameters updated features occur rare features therefore often receive inaccurate values several ways approach overfitting capping model supremum norm focus specific line research learning linear classifiers learning explicitly estimates confidence induction often maintaining gaussian distributions parameter vectors words model parameter interpreted mean augmented covariance estimate learning cwl dredze first learning algorithm crammer later introduced adaptive regularization weight vectors row simpler effective alternative row passes data item item computing margin dot product weight vector item updating covariance matrix standard additive fashion cwl weights interpreted means covariance matrix form gaussian distribution weight vectors specifically confidence add smoothing constant compute learning rate adaptively research funded erc starting grant lowlands well danish research council conference neural information processing systems nips barcelona spain max update proportionally update covariance matrix follows cwl row shown robust averaged perceptron several studies crammer johannsen show replacing binary activations samples spikes lead better regularized robust models spikes regularizers neurophysiological motivation neurons fire synchronously constant rate neural signals onset increase signal followed spike decrease signal inhibition neuron returning equilibrium simplify picture bit assuming spikes gaussians learning algorithm piral propose motivated observation spike rate speed neuron fires increases neuron fires kawai sterling keller takahashi futhermore keller takahashi show increased activity may lead spiking higher rates lower temporal precision means active neurons less successful passing signals leading neuron return stable firing rate words brain performs implicit regularization exhibiting low temporal precision high spike rates prevents highly active neurons swamping less active neurons hypothesise implementing similar mechanism learning algorithms prevent feature swamping similar fashion finally blanco show periods increased spike rate lead smaller standard deviation synaptic weights loosely inspired implement temporal imprecision high spike rates decreasing weight standard deviation algorithm single layer feedforward model perceptron sampling gaussian spikes effect input therefore implement regularizer noise injection bishop variance relative confidence model input item parameters means parameter values multiply input inverse sample reflecting intuition highly active neurons less precise likely drop clip sample give pseudocode algorithm following conventions crammer experiments main experiments extract binary classification problems mnist training odd data points testing even ones since algorithm explicit parameter tuning implementation piral experiment first ten problems left upper corner test robustness piral relatively perceptron row randomly corrupt input test time removing features inspired globerson roweis plots figure presents number features kept deleted observe two tendencies results piral outperforms perceptron consistently features sometimes large margin except cases perceptron figure sampling activations gaussian spikes row acc acc acc acc acc perceptron perceptron arow noise arow noise noise perceptron arow noise perceptron arow noise perceptron arow acc acc acc acc acc perceptron perceptron arow arow noise perceptron perceptron arow noise perceptron arow noise arow noise noise piral acc acc acc acc perceptron perceptron spikes acc perceptron spikes noise noise perceptron spikes noise perceptron spikes spikes noise noise acc acc acc acc acc perceptron noise perceptron spikes perceptron spikes noise noise perceptron spikes perceptron spikes noise spikes noise figure performance noise levels percentage features kept first two lines compare perceptron blue row green third fourth compare perceptron blue piral green algorithm piral except lines identical row end sampling activations gaussian spikes clipping values outside window vvt max end end return better features contrast row less stable improves significantly perceptron noise levels cases perceptron almost always superior full set features since relatively simple learning problem overfitting unlikely unless noise injected test time practical rademacher complexity compute piral practical rademacher complexity ability piral fit random data randomly label dataset ten times compute average error reduction random baseline perceptron achieves error reduction random baseline average overfitting quite bit random labelling data contrast piral reduces errors random baseline average suggesting almost resilient overfitting dataset conclusion presented simple single layer learning algorithm ral uses sampling gaussian spikes regularizer loosely inspired recent findings neurophysiology piral outperforms perceptron row large margin noise injected test time lower rademacher complexity algorithms acknowledgments references cristopher bishop training noise equivalent tikhonov regularization neural computation wilfredo blanco catia pereira vinicius cota annie souza cesar sharlene santos gabriella dias ana guerreiro adriano tort adriao neto sidarta ribeiro synaptic homeostasis restructuring across cycle plos computational biology koby crammer ofer dekel joseph keshet shai yoram singer online algorithms journal machine learning research koby crammer kulesza mark dredze adaptive regularization weighted vectors nips koby crammer mark dredze fernando pereira linear classification text categorization journal machine learning research mark dredze koby crammer fernando pereira linear classification icml amir globerson sam roweis nightmare test time robust learning feature deletion icml fusao kawai peter sterling cgmp modulates spike responses retinal ganglion cells via current visual neuroscience clifford keller terry takahashi spike timing precision changes spike rate adaptation owl auditory space map journal neurophysiology frank rosenblatt perceptron probabilistic model information storage organization brain psychological review anders anders johannsen robust learning random subspaces equipping nlp oov effects coling charles sutton michael sindelar andrew mccallum reducing weight undertraining structured discriminative learning naacl
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mar computers create art aaron hertzmann adobe march abstract essay discusses whether computers using artificial intelligence could create art first part concerns tools assisting art making history technologies automated aspects art covered including photography animation case see initial fears denial technology followed blossoming new creative professional opportunities artists hype reality artificial intelligence tools art making discussed together predictions tools used second part speculates whether could ever happen systems could conceive artwork credited authorship artwork theorized art something created social agents computers credited authorship art current understanding ways could change also hypothesized introduction artificial intelligence research made staggering advances recently including many developments web search image recognition conversational agents robotics developments stoked fear artificial intelligence effect many aspects society context art news media hype presents new image video creation algorithms automating creation art perhaps empower everyday users putting artists work rob humanity beyond hype confusion technology influences art pervades serious discussions professional artists often concerned computers might put jobs essay expresses opinions employer manuscript submitted journal arts special issue machine artist century http invitation please send comments hertzman however promise respond feedback concern heard decades practitioners present algorithms potentially artists journalists recently contacted prominent social psychologist inspired recent results neural networks wished conduct experiments assess whether ordinary people might willing buy artwork made computer assumed computers already happily making artwork hand whenever informally asked friends colleagues question whether computers create art answer usually decisive art requires human intent inspiration desire express something thus definition thing art created computer would anyone worry concepts art inspiration often spoken mystical terms something special primal beyond realm science technology humans create art humans surely scientific explanation essay tackle question computers create art might seem like simple question hides substantial complications people sometimes approach question one technological capability computers smart enough creative enough analogous asking whether computers perform tasks like solving mathematical problems accurately resolving search queries argue right way look question attempt sort survey existing technology opinion whether computers create art computer science question much question philosophy art human psychology first part essay discuss history current state automation art begin historical perspective previous moments history new technologies automated image film creation particularly invention photography case see new technologies caused fears displacing artists fact new technology created new opportunities artists invigorating traditional media argue new technologies benefit art artists creating new tools modes expression new styles expression sadly art science often viewed separate even opposition competition rather foe technological development stimulates much continued vitality art new artistic technologies create new job opportunities often artists important contributors innovations artists technologists much common tinkerers experimenters explorers new technologies follow trend forseeable future new algorithms provide new tools expression transform art culture positive ways many technologies past computers create art artists using computers create art despite many decades procedural art never computer widely accepted author artwork second part discuss whether could ever change would ever agree assign authorship computer argue artistic creation primarily social act action people primarily interaction humans society means computers create art people give gifts coffeemakers marry cars however boundaries art fluid discuss scenarios algorithms could come accepted artists along dangers eagerly accepting software artists essay expresses point view topics someone developed various kinds technology art followed development new technologies affecting art culture years small instances exhibited work artworks written style meant accessible general audience since topics many kinds people care history current practice making general claims computer tools relate art begin first describing several useful historical examples including inventions photography film computer animation examples show previous situations new technology appeared poised displace artists reality provided new opportunities roles artists also discuss procedural art computer art employs automation many fields outsiders view computer technology work creativity yet cases human artist artists behind work true authors work argue throughout history technology expanded creative professional opportunities artists dramatically providing newer powerful tools artists advent new technologies often cause fears displacement among traditional artists fact new tools ultimately enable new artistic styles inject vitality art forms might otherwise grow stale new tools also make art accessible wider sections society creators consumers trends particularly visible past two centuries since industrial revolution photography became artform today painting dead paul delaroche painter demonstration daguerreotype lessons past art perhaps invention significant prior invention photography realistic images world could produced artists today world swamped images hard imagine special unique must felt see realistic painting technical skills realism inseparable creative aspects changed photography automated task producing images real world first two photographic processes invented daguerre dagguereotype william henry fox talbot process mainly presented ways produce practical records world historical information section distilled two texts connection made previously expand upon much figure traditional silhouette portrait daguerreotype portrait abraham lincoln photographic techniques like completely displaced previous portraiture techniques two daguerreotype popular several decades talbot process restricted patents improvements talbot method eventually made daguerreotype obsolete evolved modern film processes portraiture main driver early adoption today people enjoyed possessing pictures friends ancestors portrait painting available aristocrats wealthy even today portrait painting symbol great wealth status century number inexpensive alternatives developed silhouette tiny black representation individual outline figure typically handcut artisan black paper daguerreotype offered economical way create realistic portrait figure slow required locking subject head place head brace several minutes subject gripped chair tightly flutter fingers nonetheless numerous daguerreotype studios arose became commonplace technologies improved many portraitists switched new technology named henri secq said one knows photography harmed painting considerably killed portraiture especially livelihood photography largely replaced older forms portraiture silhouette one seems particularly regret loss much appreciate mystery beauty looking old etchings portraits rather use mobile phone camera pictures try paint hand another early use daguerreotype produce souvenirs tourists daguerreotypes roman ruins completely replaced etchings lithographs tourists previously purchased technology improved photography became indispensable source records engineering projects disappearing architectural ruins well figure unhappy painter theodor hosemann satirizes painter victim progress made obsolete daguerreotype documentary purposes matthew brady photographs horrors american civil war photography art question debated many decades coalescing three main positions many people believed photography could art made mechanical device rather human creativity many artists dismissive photography saw threat real second view photography could useful real artists reference considered equal drawing painting finally third group relating photography established forms like etching lithography felt photography could eventually significant art form painting photography ultimately profound unexpected effect painting painters mimetic abilities improving centuries many painters century like john everett millais neoclassicists like ingres painted depictions world dazzling realism ever seen however cameras became cheaper lighter easier use grew widespread among amateurs professionals realistic photographs became commonplace end century photorealism could reduced mechanical process artist role question drove painters away visual realism toward different forms abstraction james mcneill whistler tonalist movement created atmospheric moody scenes wrote imitator poor kind creature man paints tree flower surface sees artist king artists would photographer artist something beyond impressionists sought capture perceptions scenes likely influenced perceptual imperfections photographs contrast symbolists artists moved away perceptual realism altogether edvard munch wrote fear photography long used heaven hell going paint people breathe feel love vincent van gogh describing artistic breakthroughs around wrote brother must boldly exaggerate effects either harmony discord colors produce thing drawing accurate drawing accurate color perhaps essential thing aim reflection reality mirror could caught color would picture photography continued influence modern art century one infer significant influence marey photography futurism cubism duchamp nude descending staircase one argue fact photography one major catalysts modern art movement influence led decades vitality world painting artists inspired photographic images pushed beyond realism meanwhile pictorialist movement begun around attempt firmly establish photography art form pictorialists introduced much artistic control photographs often using subjects classical painting manipulating images darkroom many works hazy atmospheric work similar tonalism deliberately softened realism photography seemed deliberately mimicking qualities fine art painting time today seem rather affected pursued various strategies toward legitimization work art form organization photographic societies periodicals juried photography exhibitions works achievements made harder harder deny artistic contributions photography culminating buffalo show organized alfred stieglitz albright gallery buffalo first photography exhibition american art museum photography firmly established art free move beyond pretensions pictorialism story provides several lessons directly relevant artistic tool first photography like seen many mechanical process saw photography threat argued legitimacy photography displace old technologies fulfilled functions portraiture social function artists enthusiastically embraced new technology began explore potential technology improved became widespread nearly century artists learned better control express new technology real controversy status photography new technology made imagemaking much accessible hobbyists today everyone experiment photography furthermore new technology breathed new life old art form provoking toward greater abstraction wherever controversy artistic tool predict trajectory eventually new tools fully recognized artists tools tools may stimulate traditional media well new aesthetic technology cinema story filmmaking technology important lessons artists technologists work together pushing early photographers necessity artists technologists experimenting new techniques driven art inspire art film animation interaction much central art form figure interplay painting early photography century western painters achieved dazzling levels realism early cameras took though evocative pictures daguerreotype took ten minutes expose however camera technology steadily improved capturing greater greater realism much faster exposures challenged painters create works depiction whistler tonalist nocturne pictorialist photographers attempted establish photography art form mimicking styles abstraction painting works ophelia john everett millais boulevard temple daguerre portrait sarah bernhardt nadar nocturne blue gold old battersea bridge james macneill whistler morning clarence white figure technological developments art filmmaking first captured film workers leaving lumiere brothers factory george trip moon filmed like stage play fantastical special effects citizen kane used numerous experimental camera lens effects tell story history film filled well teams artists technologists brothers created first film simple recording workers leaving factory also experimented wide range camera technologies color processing artistic ways use stage magician george filmed fantastical stories like trip moon employing wide range clever tricks techniques create delightfully inventive beguiling films walt disney employed pushed new technologies sound color recording drove innovations along way multiplane camera many orson welles innovative film techniques made possible new camera lenses employed cinematographer gregg toland introduction portable camera audio equipment enabled experiments french new wave turn influenced young american directors like francis ford coppola george lucas george lucas team star wars early developer many new visual effects shoestring budget think ben burtt hitting telephone create blaster sound effect well early innovation digital film editing compositing digital computer graphics technology obviously revolutionized film storytelling since directors like michel gondry james cameron pushing technology unforeseen directions case see technologies rapidly adopted directors create new storytelling techniques styles transforming medium computer animation collaboration computer animation artform pioneered pixar animation studios success due close collaboration artists engineers began catmull animation enthusiast received phd computer science thesis invented several core techniques every major computer graphics system uses today time graduate school quietly set goal make world first film consequently founded lucasfilm computer division hired team brilliant engineers invent computer systems used however none group could animate bring character life movement hence catmull recruited john lasseter animator trained deeply disney tradition tight collaboration lasseter technical staff able invent new technologies discover together computer animation could start become art form group spun pixar following years invented numerous technical innovations aimed answering needs set pixar artists turn artists inspired new tools pushed new extremes one mantras art challenges technology technology challenges art pixar design treats artists engineers crucial company success minimizes barriers groups worked sabbatical despite technical role many energizing conversations different kinds artists attended many lectures art storytelling sketched open life drawing session watched performance employee improv troupe participated many social educational events deliberately mixed people different parts company culture though still flaws address achieved many years technical creative innovation ultimately commercial artistic success computer animation another technology scared traditional artists early prepixar days catmull group made many attempts interest disney animators work alvy ray smith later said animators frightened computer felt going take jobs away spent lot time telling people creativity misconception everywhere common misconception computer animation amounts computer solving everything programmer presses button characters move reality computer animation extraordinarily requiring skills talented artists especially animators almost every little detail character animation art form extreme skill talent requiring laborious effort using fundamental skills performance bringing character life pure movement conventional animation traditional cel animation jobs last disney various reasons disney feature animation underwent renaissance early starting little mermaid following changes management disney animation began slow sad decline releasing duds like brother bear home range management shut traditional animation disney converted studios entirely computer animation many conventional animators retrained animation disney first animation chicken little still dud following disney acquisition pixar several years later revived disney beloved animation productions result charming enjoyable film called princess frog performed box office moreover animators creative energy focused newer art form today traditional animation disney today computer animation thriving industry thrives many places cel animation ever many different film studios visual effects films video games television studios web startups independent web animators many types opportunities animators ever traditional animation styles still vital countries like japan france unlike america believe animation kids even visual styles evolved considerably due computer technology figure procedural artworks fine art world galleries art museums left electric sheep scott draves evolves dazzling procedural abstract animations based thousands votes right top grossing film time jason salavon shows average color frame movie titanic story destruction jobs evolution growth art form technology another story contradicts popular notion art technology operating conflict fact opposite usually true procedural artwork art world long tradition procedural artwork jean arp created artworks governed laws chance claimed beginning john cage used random rules compose music term generative art appears originated sol lewitt wall drawings provided lists precise instructions people still drawing new versions paintings death starting painter harold cohen began exhibiting paintings generated program wrote called aaron since many current artists karl sims scott snibbe golan levin scott draves jason salavon produce abstract artworks writing computer programs produce either static images create interactive artistic experiences installation works figure sims draves work artwork evolves according audience input popularity processing computer language artists speaks growth procedural art cases despite presence procedural emergent crowdsourced elements human behind credited author artwork would seem perverse suggest otherwise human done creative around visual style designing framework process testing evaluating alternative algorithms state art computer science research recent developments computational artistic image synthesis quite spectacular mistaken artists figure painterly rendering algorithms process input photograph using rules algorithms paul haeberli interactive painting system automatic painting system processes photograph without user input aside selecting parameter settings rendering npr subfield computer graphics research worked many years npr research develops new algorithms artistic tools creating images inspired look conventional media painting drawing paul haeberli groundbreaking paper introduced paint program began photograph whenever user clicked canvas initially blank system placed brush stroke color orientation based photograph way user could quickly create simple painting without particular technical skill figure paper pete litwinowcz automated process entirely placing brush strokes grid first research paper arose experimenting modifications algorithm method came places long curved strokes beginning large strokes refined small details figure algorithm inspired experience real painting way artists often start rough sketch refine type artistic algorithm design reflects majority computer graphics research area algorithms automated explain complete detail algorithm works intuitions artistic process embodies mathematical modeling artistic representation continues tradition begun renaissance filippo brunelleschi invention linear perspective viewpoint written elsewhere point found difficult embody richer intuitions artistic process figure image analogies algorithm stylizes photograph style given artwork case van gogh starry night figure neural style transfer algorithm stylizes photograph style given artwork case van gogh starry night algorithm led numerous new apps research stylization source code instead inspired recent results computer vision began develop method working examples collaborators published method calling image analogies presented work learning artistic style example learning quite shallow amounted rearranging pixels source artwork clever way generalizing radically new scenes style figure since researchers improved method substantially making much robust leon gatys colleagues published new breakthrough space neural style transfer based recent advances neural networks method transfers certain neural network correlation statistics painting photograph thus producing new painting input photograph figure method still shallow sense understanding photograph artwork method seems robust original image analogies algorithm paper led flurry excitement new applications including popular prisma app facebook live video stylization well many new research papers improving upon ideas work ongoing today another development received considerable attention invention deepdreams alexander mordvintsev developing visualization tool neural networks discovered simple activation excitation procedure produced striking figure images generated using neural network trained different subsets database artistic imagery one meant typify single stylistic category medium though biases convolutional neural network architecture also evident cinatory imagery type never seen many current projects particularly around generative adversarial networks project magenta google also show promise new artistic tools example figure shows images generated visualizing trends learned neural network large collection artistic images different styles variety related images produced creative adversarial networks visual style seems familiar familiar probably driven part biases convolutional neural network representation cases artworks produced procedure human author imagery case artificial intelligence intelligent unfortunately considerable amount media hype around techniques news media algorithms often anthropomorphized consciousness humans sometimes described artists fact really know consciousness despite many theories would mean embody algorithm today machine learning algorithms best thought glorified procedures algorithms basically like fitting curve set datapoints except sophisticated ways fit curves millions datapoints researchers speak training algorithm algorithm learns easy misinterpret thing human learning words mean quite different things two contexts general training model learn task involves careful human effort formulate problem acquire appropriate data test different formulations laborious requires considerable expertise experimentation new task needs solved human starts compared human intelligence algorithms brittle bespoke example image recognition algorithms undergone breathtaking breakthroughs past decade widely used consumer products yet often fail inputs suggest bizarre misunderstandings existence robustness adversarial examples images demonstrates algorithms really learned anything like understanding algorithm learned general purpose understanding world like tourist foreign country repeat combine phrases phrasebook lacks true understanding language culture visiting systems autonomy except within narrow scope trained typically failsafes must put place well google photos longer classifies anything gorilla one failure indeed fascinating parallels human learning machine learning seem likely humans way machines optimization evolutionary principles going analogies actual machine intelligence problem solution even horizon technology changes art based history make several specific claims technology changes art far replacing artists new technologies become new tools artists invigorating changing art culture claims apply equally current developments previous developments like photography animation algorithms artists tools every technology currently employ whether photography film software algorithm technologies algorithms use basic tools like brushes true new algorithms appearing always predictable results often surprising delightful could said way watercolor flows page plausible sense current systems reflect true artificial intelligence always human behind artwork applying standard current research neural networks neural style transfer would seem equally perverse assign authorship outputs software deepdream software authored human another human selected view reflects think conventional wisdom computer graphics research field research background field always close ties certain artistic communities especially computer animation visual effects based experience field often resistant attempts automate creative tasks contrast artificial intelligence researchers use terminology much aspirationally historically using words like intelligence learning expert systems ways far simpler human versions things input image experimented many parameter settings running software obtaining good results indeed recent art exhibition meant promote methods exploration human artists credited individual works process selection tools inputs adjusting settings even modifying code iterating desirable output produced occurs current computer artworks cases algorithms presented artists potential artists reasons think claim misunderstands nature procedural art short present understanding art algorithms including methods based machine learning tools artists artists technology helps art stay vital rather afraid new technologies enthusiastic new artworks enable artists produce think art external influences normally think social political influences ignore effect new tools contrast argue especially centuries technological developments played pivotal role advancing art keeping vital injecting fresh ideas stories gave photography cinema include many examples however effect far widespread one important breakthroughs history western art invention oil paint flemish painters jan van eyck century previously painting done primarily tempera lack subtle coloration fresco cumbersome paint oil paint existed form centuries van eyck others found new techniques compounds gave practical new medium fast drying allowed create richer colors sharp edges hard surfaces much wider gamut colors rich light color associate northern renaissance italian renaissance due technology figure decade since many works used technology invented within previous ten years example technology used today feature films exist ten years ago widespread use digital cameras facial performance capture goes artworks using smartphones crowdsourcing artworks involving white leds arduino controllers djs performing stage behind laptops even romantic comedies frequently involve recent digital video editing digital backdrops conversely artistic styles fail change become stale lose cultural relevance adoption exploration new technology one main ways art stays vibrant example introduction synthesizer music pop music created new sound exciting modern sound diversified tools improved grunge became popular made synthpop sound seem superficial nowadays recent revival instruments bands like daft punk lcd soundsystem seems exciting times creating new types music using old instruments figure development oil painting technology changed painting art form fresco painting michaelangelo sistine chapel fresco process difficult achieved limited tonal range today fresco defunct medium oil painting jan van eyck ghent alterpiece much richer colors lighting possible oil paint contrast swing music revival never went anywhere bands like big bad voodoo daddy squirrel nut zippers opinion bands aped classic styles classic instruments without inventing anything particular original era radical technological innovations met artists enthusiasm rejection example moog synthesizer became popular adopted bigname bands like emerson lake palmer bands felt twisting knobs make music cheating queen album covers proudly state band use synthesizers robert moog described one new york musician said instrument end world seems silly imagine people might ever categorically objected synthesized music scratching sampling djs seems silly people rejected waltzing impressionists rite spring invalid immoral addition stimulating professional artists new tools make art accessible larger portions society photography accessible determined early adopters continually become easier faster compact point nearly everyone carries mobile phone camera pocket purse goes tools cinematography heavy cameras steadicams handycams iphones modern computers give nearly everyone access digital equivalents darkrooms mixing studios painting studios formerly technologies requiring laborious effort figure les paul inventor electric guitar robert moog pioneer electronic synthesizer respectively technologies transformed popular music ways could foreseen general story jobs technology concerns technology displaces jobs around since least century luddite protesters destroyed mechanical weaving machines folk songs john henry competed steam drilling machine fears real understandable yet despite many technological disruptions live world massive unemployment caused technology old roles erased many arise stead fears keep recurring given time easy imagine losing specific jobs requires superhuman imagination forecast new opportunities created transformative new technologies nowadays jobs would hard even explain detail worker real workforce concerns technology whether economic system shares benefits new productivity fairly across society versus concentrating wealth among richest whether machine learning systems misused magnify existing forms inequality displacements due new technology occur effects eased social safety nets better educational foundations employment flexibility retraining conversely society fails distribute wealth economic gains fairly much bigger problems impact fears new technology seem human nature suspect many people view normal state things came age view significant change scary yet nearly familiar modern technologies viewed threatening previous generation fear life long time notably scientists discovered electricity searched understand discovered effect galvanism muscles dead frogs could stimulated electrical currents secret life discovered inspired mary shelley novel frankenstein modern prometheus student uses modern science create new life today story vivid evocative intellectually recognize preposterous fear essentially irrational fear skynet frankenstein monster neural networks promethean spark instead present terminator autonomous slightly plausible ability travel backwards time future tools artists general trends around evolution technology art seem quite robust discussed current algorithms autonomous creators foreseeable future still tools ready artists explore exploit new developments tools enthusiastically adopted artists leading exciting new forms styles currently foresee possible tasks performed human artists gradually fade generally mechanical tasks require much creativity fill societal functions artistic expression traditional arts may fade simply due seeming nature art nothing fresh forever blamed technology conversely new technologies enable new styles aesthetics job descriptions novices access new simplified tools expression artistic technology imagination amplifier better technology allow artists see even aside general trends hard make specific predictions art future les paul invented guitar primarily performed light pop country showtunes could hardly predicted electric guitar would used say led zeppelin hard imagine daguerre predicting instagram generally making predictions technologies might transform society hard little understanding technologies might actually even science fiction writers completely failed imagine transformative power internet mobile computing computers future would still monstrosities one sit front operate predict moon colonies replicants short predict new inventions ideas artists come future predict amazing amazing make use technology new unpredictable ways artist far described computer technologies currently accepted tools artists artists computers human tasks like speak search print navigate extent drive cars several obvious reasons computers make art including tradition incentives involved relatively predictable nature existing automation still one could imagine alternate history fact frankenstein presented cautionary tale quest knowledge general victor frankenstein tells story warning learns captain walton driven obsessive quest knowledge entirely unrelated frankenstein machines computer programs already called artists believe fundamental reason explains happened unlikely happen anytime soon section theorize prerequisites artist apply discussion next section art social artist trying people bring closer something course art sharing artist want share experience david hockney create consume art argue art primarily social behavior art communication displays people example people often speak art personal expression act communication directly inspired theory going back charles darwin adaptive product biological evolution sets persuasive argument theory briefly summarize though full justice creating art served several functions pleistocene ancestors served fitness signal mating sexual selection art also used displays wealth status storytelling music dance strengthen social bonds within group storytelling additionally plays important role communicating information would otherwise hard share observe functions art social art arose forms communication displays sharing people although art takes many different forms different cultures today forms serves one basic social functions pleistocene generalize theory beyond humans hypothesize art interaction social agents social agent anything status akin personhood someone worthy empathy ethical consideration many behaviors interactions social agents gifts conversation social relationships like friendship competition romance contrast get emotionally attached computers possessions feel real empathy needs ethical duty toward possessions participate shallow versions interactions example frequently talk possessions statements like brakes bicycle complaining gave new pair gift much happier love statement indicates emotional attachment true empathy bike feelings despite anthropomorphizing language live social hierarchy possessions compete status try impress care people say care people care computers say insofar useful generally treat conversational agents like siri alexa user interfaces software like people also benefits artist example help practice skills like dexterity problem solving creating art often pleasurable meditative benefits secondary social benefits reasons evolution produced art human activity similarly one may also talk sing oneself alone talking singing still fundamentally social activities note evolutionary argument optional one discuss whether art fundamentally social without believe evolutionary view art gives additional understanding authors seen despite many technological advances current algorithms accepted artists existing examples objects created authors processes give support theory natural processes natural processes including landscapes like grand canyon huangshan mountains considered art even though may extraordinarily beautiful change one perspective immensely beautiful structures made instinctively animals honeycombs coral considered art indicates simply creating complex beautiful outputs sufficient art since creative social communication cases animals higher mammals including chimpanzees elephants dolphins trained paint many writers skeptical art typically animal owner handler steers process letting animal throw paint canvas stopping painting believe done selecting works show animals seem show interest artwork afterward animal artwork significant cultural impact popularity seems largely product media stunts people ethical treatment animals recently tried claim copyright favor monkey failed copyright law allows humans claim copyright interesting aspect discussion whether animals create art decide discussions whether artifacts art based priori rules whether animals artists instead attempts study evidence animal behavior around artwork infer whether artwork form inner expression artifact animal special appreciation words open idea animals creating art social relationships found creature satisfies criteria creating art whatever judging work instead tempting judge whether computer artist based solely merits work produces hypothesis whether computer artist judgment figure art beautiful landscapes considered art pierre brassau monkey painter part hoax satirized modern art quality work produces independent properties computer algorithm outputs continual stream diverse stimulating beautiful skillful outputs without many duds might quite tempted call algorithm artist better art becomes hear questions whether computers artists skill clearly real requirement someone something artist human make art including unskilled amateurs children conversely computers already programmed create infinite sequences dazzling realistic abstract imagery exhibiting technical proficiency way beyond typical human suspect look computer output ask work good enough call computer artist actually judging quality work per instead really looking evidence system intelligent conscious feeling traits associate social agents matter skillful surprising computer output accept artist infer sort social inside intent machine another hypothesis follows modern art world role artist supply intent idea work necessary artist execute work coordinating production humans clearly true numerous examples section well appropriation works like duchamp readymades richard prince questionable instagram reproductions artists also employ helpers crowdworkers scott draves electric sheep aaron koblin sheep market consequently computer artist simply needs supply intent easy imagine designing system creates intent even coordinates labor producing work example one could write simple procedural algorithm generate basic intents portray ominous landscape sample intents recursive neural network trained artists statements scraped web method could randomly select news item photograph historical event randomly sample attitude toward thing starting intent crowdworkers could used refine idea convert new image similar systems like soylent one could also automate steps process using gans generate entirely new starting images scratch crowdworkers could also used rate evaluate outputs system selecting best results discarding rest final steps process could also automatically hire professional designers sites like upwork system could run continuously generating new images time payment automatically made crowdworkers involved workers could group images common themes intents create separate collections around themes artist statements could generated around themes system preferences could grow adapt time data gathered external data streams photography blogs change suppose someone build system calling say intent machine exhibits work art show gallery suppose moreover convinced curator gallery owner credit intent machine authorship works created fully disclosed procedure worked would people credit artist authored works would say real artist believe general consensus would real artist really artwork probing nature art nature commercial artworld note procedure used define system fundamentally different procedural art algorithm sections even work ended quite good viewers would ask good doubtful computer contribution would judged significant beyond humans involved artists intent machines art social act answer creativity growth finally two related criteria intuitive appeal notions creativity growth important human artists perhaps artists well several authors proposed energy terms criteria creativity often attempts express idea system output somehow surpass human programmer training data example proposes judge creativity system part whether system output surprises system author believe weak criterion since many mechanical algorithmic phenomena may surprising discoverer author first example basic algorithm produces mandelbrot set images specified single yet produces dazzling animations infinite mandelbrot set surprising produces beautiful color image location proportional number steps iteration requires reach starting large constant example https unprecedented images call iteration equation creative artist current procedural art systems mandelbrot set recognizable style awhile lose novelty believe true systems specifically designed creativity objectives unlike human artists systems grow evolve time perhaps artist would need exhibit form growth example harold cohen painter began write software generate art described evolution views ten years would said look program another ten years would said fact program central issue denoting belief program potential growing autonomy whole business producing complex images high quality could forever without rewriting single line code much autonomous one get virtually impossible imagine human similar position human artist modified act making art program similarly would required merely capable assessing output modifiable worldview existing system easy think trivial ways system evolve change time subtly change color palette training data years superficial growth easy meaningful growth hard even define computer someone could design system produces sequence art meaningful people also significantly evolves time would truly remarkable seems hard imagine achieving without enormous technological advances may possible without true social form definitions art guidance nature art could also looked existing definitions art however royal society prescribes valid art instead art phenomenon results interplay cultural institutions general population analyze changes time philosophers attempted devise concise definitions art include existing types music dance painting etc styles art institutional definition originated response conceptual art states roughly art anything style broadly accepted art approach identify attributes common many different types art definitions art attempt fit data draw line things call art like theatre like spectator sports understandably definitions assume artist always human without exploring much whether create art thus provide much guidance discussion ever artist background turn speculating future seen authorship current algorithmic art assigned human author behind algorithms ever say created art ever recognize piece software author work art ever develop artificial intelligence consciousness definition would able create art since would capacity consciousness emotions social relationships discussed section scenario science fiction idea possible would achieved making meaningful predictions world true impossible little idea specifically would actually operate moreover would transform society much make unrecognizable may well speculate kind artwork made aliens ever meet pressing questions kind music like hence interesting question whether could artwork without artificial intelligence consciousness social argued creating art fundamentally social act expression communication follows granted authorship view social agent performing communication sharing art mean view social agent view deserving empathy ethical consideration way however need intelligence social relationships pets expect something say socially suggest inner consciousness feeling short true intelligence science fiction scenario think way happen shallow agents people sometimes fooled shallow classic example eliza simple psychiatrist program developed based simple repetition user types meant demonstration superficiality ais time unexpectedly many people attributed emotions machine since many anecdotes people fooled chatbots online settings including recent plague twitter bots veil lifted clear chatbots exhibit real intelligence software robots designed relationships owners including talking baby dolls tamagotchi paro therapeutic baby seals related effect people behave toward computers social agents certain ways even believe intelligent example dialogue systems like siri alexa use female voices default based many findings male female users respond better female voices perhaps many users system need truly intelligent perceived social agent like siri alexa ask make artwork day may come agents integrated daily lives forget software one easily imagine development simulates emotion affection toward user easy imagine example toy doll paints pictures owner along one many behaviors designed display companionship affection seems possible algorithms could successfully promoted artists tentative forays direction reasons given skeptical methods accepted true artists without plausible belief underlying social conscious attributes perhaps curator museum would download otherwise acquire various artifacts software artists list software systems authors would controversy discussion newspapers journals perhaps curators galleries would follow suit perhaps people would find enough value artworks also convinced human could rightly given credit works sort process happened things like abstract expressionism chimpanzee art could happen computer art suitcase words term artist could come used multiple meanings words like intelligence learning come mean something different humans algorithms software program say automatically stylizes photos could called artist way software applications like calendar mail programs replaced namesakes unfortunately use word mean different things different contexts causes endless confusion discussed section dangers continual danger new technology human users misunderstand nature call shallow artist risk seriously misleading lying people believe convince people artist also falsely attribute emotions feelings ethical weight true would argue calling ais artists unethical leads sorts dangers including overselling competence abilities misleading people nature art seems likely companies scruples example hanson robotics promoted many contexts social robot truly intelligent even though clearly nothing chatbot face yann lecun words potemkin related concern deprive designer authorship credit present credit author piece automatic software output software usually acknowledges skill effort required engineer iterate software produces good outputs artistic credit important understanding real sources something made outside science fiction see positive benefit calling computer artist see dangers conclusion believe software system current understanding could called art social activity mean warning misleading oneself others nature art course ambitious reader could take challenge laid serious objections must overcome wish create software think done anytime soon also know proving critics wrong one ways art science advance one main goals essay highlight degree technology contributes art rather antagonistic lucky alive time artists explore tools every time see artist create something wonderful new technology get little thrill feels like new art form evolving danny rosin wooden mirror jason salavon top grossing film time bob sabiston snack drink michel gondry like rolling stone kutiman thruyou amon tobin permutation ian bogost cow clicker christian marclay video installations quilez procedural renderings wesley allsbrook goro fujita virtual reality paintings examples artworks affected way years today github twitter extremely fast interplay machine learning researchers artists seems like every day see new tinkerers artists tweeting latest creative experiments rnns gans janellecshane helena christophrhesse quasimondo art maintains vitality continual innovation technology one main engines innovation occasionally avant garde tremendous cultural impact electronic music sampling domain experimental electronic musique pioneers like wendy carlos delia darbyshire likewise one time computeranimated films could seen obscure festivals today seeing many intriguing beguiling experiments techniques artists tools surely transform way think art thrilling unpredictable ways acknowledgements thanks shira katz craig kaplan dani oore valuable comments manuscript thanks everyone shared discussion encouragement online including aseem agarwala mark chen lyndie chiou michael cohen james landay nevena lazic jason salavon adrien treuille many others references rcas art age machine intelligence arts lexander optima animals princeton university press ernstein ittle iller artmannn ackermann arger rowell panovich soylent word processor crowd inside proc uist rooks seven deadly sins predictions mit technology review atmull wallace creativity inc overcoming unseen forces stand way true inspiration random house ohen exploits aaron painter stanford humanities review ohen acm siggraph awards harold cohen distinguished artist award lifetime achievement https ohen imagination amplification ieee comp graph app olton creativity versus perception creativity computational systems proc aaai spring symposium creative intelligent systems olton painting fool stories building automated painter proc computers creativity otter residence walls luscious austerity new york times danto artworld journal philosophy roix tansey irkpatrick gardner art ages harcourt brace esmond zoos make money selling paintings made animals art washington post ickie defining art american philosophical quarterly utton concept art theories art today carroll university wisconsin press utton art instinct beauty pleasure human evolution bloomsbury press dvorsky computer science students fooled artificially intelligent gizmodo dvorsky artificially intelligent robot composes performs music gizmodo fros eung texture synthesis sampling proc international conference computer vision lgammal lhoseiny azzone creative adversarial networks generating art learning styles deviating style norms proc int conf computational creativity pstein russia love scientific american mind jamri hechtman sente stylit stylization renderings acm trans graphics siggraph july atys cker ethge image style transfer using convolutional neural networks proc computer vision pattern recognition aut art cluster concept theories art today carroll university wisconsin press aut cluster account art defended british journal aesthetics ershgorn inside mechanical brain worlds first robot citizen quartz oodfellow badie irza warde zair ourville engio generative adversarial nets proc neural information processing systems riggs computer voices mostly female cnn aeberli paint numbers abstract image representations proc siggraph ertzmann painterly rendering curved brush strokes multiple sizes proc siggraph ertzmann algorithms rendering artistic styles phd thesis new york university ertzmann survey rendering ieee computer graphics applications ertzmann rendering science art proc npar ertzmann jacobs liver urless image analogies proc siggraph alesin ockney way see chronicle essler twitter bots fool thinking real people fast company koblin sheep market proc creativity cognition koerding olpert bayesian decision theory sensorimotor control trends cognitive sciences special issue probabilistic models cognition rauss writers imagine internet slate asseter principles traditional animation applied computer animation proc siggraph evinson defining art historically british journal aesthetics itwinowicz processing images video impressionist effect proc siggraph etz creating building blocks reshape music art new york times illward monkey selfie case british photographer settles animal charity royalties dispute telegraph ordvintsev lah yka inceptionism going deeper neural networks google research blog guyen yosinski lune understanding innovation engines automated creativity improved stochastic optimization via deep learning evolutionary computation icholas stunning tools change art world slate eil weapons math destruction big data increases inequality threatens democracy crown paik infinity beyond story pixar animation studios chronicle books erez microsoft new drawing bot artist techcruch rice pixar touch vintage eeves nass media equation people treat computers television new media like real people places csli rosenblum world history photography abbeville press rosin ollomosse image artistic stylisation springer rubin droidmaker george lucas digital revolution triad charf art photography penguin hein computing arts commun acm helley frankenstein modern prometheus lackington hughes harding mavor jones imonite comes gorillas google photos remains blind wired two cultures london cambridge university press nowden robert moog call music classic interview mark google doodle guardian terling essay new aesthetic wired ternberger amateur aesthete legimitization photography art america university new mexico press tiglitz price inequality norton zegedy aremba utskever runa rhan oodfel low ergus intriguing properties neural networks proc int conf learning representations enenbaum emp riffiths oodman grow mind statistics structure abstraction science yka exploring intersection art machine intelligence google research blog incent facebooks head really hates sophia robot good reason verge eizenbaum computer program study natural language communication man machine comm acm ilber fang ertzmann ollomosse longie bam behance artistic media dataset recognition beyond photography proc international conference computer vision ilson inventing languages humans understand stop fast company hang uang hang uang attngan text image generation attentional generative adversarial networks
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hypothesis testing via affine detectors anatoli juditsky arkadi nemirovski oct abstract paper develop approach originating hypothesis testing via convex programming existing results hypothesis testing aim quantify closed analytic form separation sets distributions allowing reliable decision precisely stated observation models contrast descriptive highly instructive traditional framework approach promote qualified operational testing routines risks yielded efficient computation know advance favorable circumstances specified risk test whether high low provably circumstances compensation lack explanatory power approach applicable much wider family observation schemes hypotheses tested closed form descriptive analysis possible present paper primary emphasis computation make step extending principal tool developed testing routines based affine detectors large variety testing problems price development loss blanket proposed procedures though still preserved observation schemes studied become particular cases general setting considered introduction paper considered extension following simple observation starting point numerous developments imagine want decide two composite hypotheses distribution random observation taking values observation space hypothesis stating given families probability distributions let detector let risk detector defined smallest test given observations deduces otherwise accepts true hypothesis least shown ljk grenoble alpes grenoble cedex france georgia institute technology atlanta georgia usa nemirovs first author supported project gargantua labex research second author supported nsf grants tests near optimal hypotheses decided upon test risk exists detector comparable risk note risk seems much larger especially small compensate risk deterioration passing test associated detector test based detector pusing kobservations risk test thus worse risk ideal test already quite moderate value good certain precise sense parametric families distributions primarily gaussian distributions poisson distributions parameters corresponding random variables independent entries poisson parameter discrete distributions parameter distribution vector probabilities take value optimal minimal risk thus detectors found efficiently provided convex hypotheses meaning cut family distributions question restricting distribution parameter reside convex domain closer inspection common denominator gaussian poisson discrete families distributions cases minimal risk detector pair convex hypotheses results case deciding pair convex hypotheses stemming good family distributions sum following best smallest possible risk affine detector risk efficiently computed smallest risk affine detector best terms risk detector available circumstances associated test note far practical applications approach concerned one survive without constructed detectors must paper focus families distributions obeying class turns incomparably larger defined good particular includes nonparametric families distributions staying within much broader class still able construct computationally efficient way best affine detectors pair convex certain precise sense hypotheses along valid upper bounds risks detectors general claim anymore tests associated detectors said believe investigating possibilities building tests quantifying performance computationally friendly manner value even provably guarantee tests retrospect results seen development line research initiated pioneering works chernoff kraft cam developed among many others see also references affinity detector makes sense naturally identified subset indeed case gaussian poisson distributions make case discrete distributions suffices encode basic orth thus making set basic orths encoding every real valued function becomes affine paper organized follows families distributions well suited constructing affine detectors computationally friendly fashion introduced investigated section particular develop kind fully algorithmic calculus families calculus demonstrates families probability distributions covered approach much common commodity good observation schemes defined section explain build within framework tests pairs larger tuples hypotheses quantify performance tests computationally efficient fashion aside general results type work detail case family distributions giving rise convex hypotheses tested comprised distributions section section discuss application statistical problem aggregation estimators show results extended general situation considered finally section show framework extended gaussian case include quadratic detectors streamline presentation proofs exceeding lines collected appendix setup let fix observation space let given families borel probability distributions put broadly goal given random observation decide upon hypotheses intend address goal case families simple comprised distributions functions admit explicit upper bound regular simple probability distributions let nonempty closed convex set symmetric origin closed convex set continuous function convex concave refer satisfying restrictions regular data regular data define family borel probability distributions exp say distributions satisfying regular given regular data refer regular family distributions associated data regular data define smaller family borel probability distributions exp say distributions satisfying simple given regular data refer simple family distributions associated data recall starting point study plausibly good test given two families distribution common observation space independent observations drawn distribution decides whether interest families distributions stems fact families type building test reduces solving game thus carried computationally efficient manner postpone related construction analysis section continue presenting basic examples simple regular families distributions simple calculus families basic examples simple families probability distributions distributions let closed convex subset set cone positive semidefinite matrices space symmetric matrices let case contains distributions parameters borel probability distributions admitting upper bound exp exp function parameters bound belonging particular contains gaussian distributions poisson distributions let let closed convex subset nonnegative orthant let exp family contains poisson distributions poisson vectors parameters belonging poisson distribution random vector independent entries entry poisson random variable parameter discrete distributions consider discrete random variable taking values set let think variable random variable taking values standard basic orths probability distribution variable identified point probabilistic simplex probability variable take value identifications setting specifying closed convex subset setting exp family contains distributions discrete random variables taking values probabilities comprising vector distributions bounded support consider family borel probability distributions supported closed bounded convex set let max support function following result refined section proposition every holds exp expectation right hand side function convex result setting get regular data proof see appendix calculus regular simple families probability distributions regular simple families probability distributions admit fully algorithmic calculus main calculus rules follows direct summation let regular data given let set symmetric origin closed convex set nonempty closed convex set continuous function clearly family contains distributions family contains distributions iid summation let observation space regular data space let collection reals associate outlined entities new data setting given probability distribution associate withpit new probability distribution follows distribution drawn independently immediate observation data regular whenever probability distribution belongs distribution belongs particular distribution sum independent copies belongs summation let regular data given avoid complications assume every bounded let also given assume small namely let aggregate given regular data new one setting let define function follows inf evident reasons infimum description achieved continuous addition convex concave postponing moment verification consequences form regular data claim whenever borel random variable taking values rdl marginal distributions belong families distribution belongs indeed since exists exp let set let given find applying inequality get whence see exp exp exp thus claimed remains verify function defined indeed convex concave concavity evident functions perspective transformation convex functions jointly convex thus obtained partial minimization indeed convex affine image let regular data embedding space affine mapping let set note regular data immediately seen whenever probability distribution random variable belongs belongs distribution random variable belongs respectively belongs incorporating support information consider situation follows given regular data interested family distribution known belong addition know distributions supported given closed convex set could incorporate domain information pass family containing smaller family type still containing give answer simplest case specifically denoting support function selecting somehow closed convex set containing origin let set inf continuous function participating original continuous domain definitely regular data assuming case compact set finite continuous note domain regular data claim contains provided family first family two families smaller second one verification claim immediate let properly selected holds exp besides every support whence every one exp exp since resulting inequality holds true get exp since arbitrary first part claim justified implying latter due inclusion readily given inequality illustration distributions bounded support revisited section given convex compact set support function checked data regular family contains family borel probability distributions supported moreover every family contains supported distributions expectations note describes well behaviour logarithm function small indeed far overestimates large utilizing construction replace continuous function inf still ensures inclusion easy see every distrir bution reproduces reasonably well small large indeed since small reproduces even better clearly correct description large affine detectors hypothesis testing situation assume given two collections regular data common specifically collections start construction specific test pair hypotheses building test impose regular data question following assumption regular data function saddle point min max associate saddle point following entities risk exp quantity uniquely defined saddle point value thus independent select saddle point detector affine function given simple sufficient condition existence saddle point condition sets compact function max coercive meaning along every sequence khi indeed condition theorem holds sadval inf max sup inf optimization problems opt min opt max equal optimal values condition problem clearly problem minimizing continuous coercive function closed set solvable thus opt opt real problem clearly problem maximizing compact set upper semicontinuous since continuous function taking real values perhaps value identically equal since opt real thus solvable thus solvable common optimal value therefore saddle point pairwise testing regular families distributions main observation immediate crucial observation follows proposition situation section assumption one exp proof every every exp exp properly selected thus exp exp whence also exp exp similarly every every exp exp properly selected thus exp exp whence exp exp testing pairs hypotheses repeated observation follows given let random observations generated nature exists random sequence driving factors deterministic function refer situation case repeated observation let regular data observation space associate data four hypotheses stochastic nature observations denoting conditional given distribution say distribution repeated observation obeys hypothesis hri independent belongs hsi independent belongs note last two hypotheses contrast first two require observations note also weaker hri weaker hsi hypotheses weaker respective stationary counterparts hri hsi tests considered sequel operate initial fragment prescribed length repeated observation call observation say distribution obeys one hypotheses cut repeated observation distributed according hypothesis question think hri hsi hypotheses distribution observation pairwise hypothesis testing repeated observations assume given two collections regular data common observation space common given positive integer observation want decide pair hypotheses distribution assume function associated pair regular data question saddle point let induced saddle point affine detector risk see let set consider decision rule hypotheses given observation accepts rejects accepts rejects case tie test say accepts rejects follows refer test associated detector proposition situation question obeys exp obeys exp result test accepts exactly one hypotheses risk test maximal probability accept hypothesis true observation obeys hypothesis exceed proof fact test always accepts exactly one hypotheses evident let denote expectation distribution let stand expectation conditional distribution assuming holds true invoking first inequality exp exp exp exp exp taken account deterministic function case conditional distribution belongs arrive inequality clearly implies probability reject hypothesis true since rejects assuming true invoking second inequality similar reasoning shows holds true probability reject hypothesis true exceed illustration gaussian cases let nonempty closed convex set compact convex subset interior positive semidefinite cone assume compact setting get two collections regular data know section families distributions contain families distributions parameters see well families gaussian distributions parameters expectation covariance matrix running besides pair regular data question clearly satisfies condition consequently test given construction applied collections regular data well defined allows decide hypotheses distribution observation test also used decide stricter hypotheses stating observations drawn gaussian distribution belonging goal process detail situation question refine conclusions risk test gaussian hypotheses considered situation symmetric observe first function situation consideration becomes interested solutions saddle point problem find saddle point function min max structure compactness combined fact comprised positive definite matrices immediately follows saddle points exist saddle point satisfies relations immediately follows affine detector risk given exp exp note symmetric case always exists saddle point test associated saddle point quite transparent maximum likelihood test two gaussian distributions common value bound risk test nothing hellinger affinity two gaussian distributions equivalently exp applying proposition arrive following result proposition symmetric case case saddle point problem admits saddle point form associated affine detector risk given exp result deciding via hypotheses fact even weaker hypotheses risk test associated symmetric gaussian case apply test observation order decide hypotheses risk bound improved proposition consider symmetric case let symmetric saddle point function given let affine detector given let also exp erf erf let erf exp normal error function particular deciding via single observation gaussian hypotheses stating risk test associated erf proof let prove proof completely similar erf erf since erf particular part proposition readily given applied testing multiple hypotheses repeated observations consider situation follows given observation space symmetric origin closed convex set closed convex sets rnj along continuous functions data give rise hypotheses distribution repeated observation top assume given closeness relation subset assume contain diagonal symmetric sequel interpret indexes hypotheses indexes goal given positive integer observation decide closeness hypotheses convenient thought hypotheses distribution let act let make assumption every pair function saddle point hij min max let set exp hij hij aij aij hij set set construction observe every observation obeys hypothesis exp construction present first related result proposition originate section reproduce make exposition used along furthermore whenever matrix shifted detectors satisfy scaled version specifically every observation obeys hypothesis exp exp bottom line follows proposition situation question given closeness consider following test tck deciding hypotheses distribution observation given shift matrix observation tck accepts hypotheses whenever reject hypotheses predicate take place test tck accepts perhaps none hypotheses accepted hypotheses besides tck max exp meaning every distribution obeys hypothesis hypothesis true event either true hypothesis accepted list accepted hypotheses contains hypothesis exceed infimum shifts exactly spectral norm symmetric nonnegative matrix eij infimum attained eigenvector selected positive case optimal selection selection proof given test tck accept hypotheses impossible due thus tck accept let distribution obey hypothesis invoking relation every event event exp exp using union bound event accepted exp exp construction test accepted accepted already seen test never accepts pair hypotheses thus tck indeed symmetric nonnegative matrix leading eigenvector selected nonnegative positive setting get every eij exp eij thus fact smallest possible shifts value proved nonnegative consider close symmetric matrix positive entries bij utilizing automatically strictly positive eigenvector matrix explained get shifts bij exp kek left hand side inequality eij exp right hand side close enough thus indeed make made arbitrarily close making arbitrarily close special case inferring colors assume given collections regular data common observation space common thus disposal hypotheses distribution observation let also index set partitioned nonempty subsets convenient think common color indexes colors indexes inherited hypotheses color inference problem want solve amounts decide given observation obeying one hypotheses color hypotheses note may happen distribution obeys pair hypotheses different colors case distribution obeys pair hypotheses two distinct colors call problem case speak color distribution provided distribution obeys union hypotheses color hypotheses obeyed distribution problem infer color given order process problem via machinery let define closeness follows color assuming resulting ensures validity assumption apply scheme build test tck convert test color inference follows given observation may happen tck applied accepts one among hypotheses case item proposition accepted hypotheses words color claim color distribution observation dealing tck applied accepts nothing claim color interested remains undecided let analyze described color inferring procedure let denoted observe first situation question assuming sets consecutive fragments matrix given naturally partitioned blocks comprised entries eij construction diagonal blocks zero blocks positive since clearly positive pairs different colors follows eigenvectors strictly positive implies properly selected shifts quantity proposition equal follows assume test tck utilizes exactly optimal shifts case may happen bad case case proposition says nothing meaningful quality test tck consequently say much quality contrast claim lemma assume problem well posed whenever distribution obeys one hypotheses recovers correctly color least proof immediate good case entries magnitude whence whenever see max follows nonzero entries nonnegative whence particular properly selected applying proposition role see distribution obeys hypothesis due origin hypotheses exactly say distribution test tcm accepts hypothesis obeys least follows obeys tcm accept simultaneously positive since already explained tcm never accepts two hypotheses different color simultaneously conclude color conclusion holds true whenever distribution obeys one hypotheses meaning problem well posed invoking proposition conclude distribution obeys hypothesis tck accept least preceding analysis whenever tck accepts obeys correctly infers color claimed finally remark holds implies problem well posed every desired risk level find efficiently observation time result color inferring procedure recovers color distribution provided distribution obeys hypotheses least application aggregating estimates testing let consider situation follows given triples regular data common parameter sets sharing common embedding space sake simplicity assume bounded thus nonempty convex compact sets continuous functions coercive whenever sequence satisfies kht observe realization observation drawn unknown probability distribution known belong family thus nature exists exp call parameter associated goal recover observation image given linear mapping undoubtedly parameter estimation problem fundamental problem mathematical statistics subject huge literature particular several constructions estimators based testing convex hypotheses studied connection signal reconstruction linear functionals estimation actual goal addressed modest assume given candidate estimates estimates could outputs various estimation routines applied independent observations sampled form goal select best among estimates problem aggregating estimates goal process aggregation problem via color inference procedure section aggregation procedure stressed stated aggregation problem appears could several pairs satisfying values pairs could different different pairs necessary well defined one way resolve ambiguity would assume given relation uniquely defines however prefer another setting satisfying selected nature perhaps several alternatives performance aggregating procedure develop independent nature selection processing aggregation problem assume points distinct let split space takes values voronoi cells note comprised points one among points let set convex compact sets observe solution aggregation problem closest point among belongs implying least one sets nonempty whether latter condition indeed satisfied given found efficiently via solving convex feasibility problems latter condition hold let call associated estimate redundant eliminate list estimates aggregated without affecting solution aggregation problem redefine voronoi cells reduced list estimates role original list check whether list still contains redundant estimate eliminate latter exists proceed recursively list estimates construction still contains solutions aggregation problem redundant estimates built lose nothing assuming purification carried advance already initial list estimates contain redundant ones course lose nothing assuming thus assume every least one sets nonempty note solve aggregation problem optimality exactly terms sets identify intend reduce task solving series color inference problems start presenting principal building block construction individual inference procedure individual inference procedure parameterized real given initialize algorithm follows mark red nonempty sets along elements corresponding regular data look one one sets associate sets chunks note resulting sets convex compact whenever nonempty mark blue set along elements corresponding regular data result actions get collection nonempty convex compact subsets associated regular data sets elements regular data colored red blue note sets different colors intersect since images mapping intersect point gets one color note also collection definitely contains red components individual inference procedure infers color regular data given observation drawn distribution collection contains red blue regular data exactly color inference procedure section associated collection blue regular data present always infers color red observe collection regular data built contains blue data value holds true let define risk individual inference procedure parameters follows contains blue regular data otherwise stated proposition note whenever probability distribution satisfies exp procedure well defined since assumption functions need deal continuous coercive minimization variable closed convex minimization compact convex maximization domains required saddle points exist quantity construction upper bound event applied observation drawn independently recover correctly color observe large enough values since large collection contains blue data recall parameter sets bounded besides claim nonincreasing support claim assume contains blue data let show indeed recall respective symmetric nonnegative matrices see proposition increasing reduce associated reduce entries thus reduce norm matrix aggregation procedure propose follows given tolerance every specify smaller better way given observation run procedures whenever returns color assign index vector processed get color red get color blue get color aggregation procedure returns solution whatever red vector one discovered returns say otherwise proposition situation assumptions described beginning section let sample drawn probability distribution taken along satisfies event min max least simple illustration let let nonempty convex compact subset suppose given positive definite matrix also given sample drawn distribution parameters let also aggregation problem interested reads given estimates expectation random vector parameters known krepeated sample distribution vector want select true expectation assume otherwise projecting estimates onto could provably improve quality distinct sets become empty shrink grows thus blue sets nonempty increases indeed proposition entries obtained saddle point values functions monotone transformation immediately seen grows functions domains minimization argument remain intact domains maximization argument shrink saddle point values increase instance could start large enough ensure select either last term progression first term progression negligibly small say less depending happens first situation sets functions become case thus suppress index notation note individual inference procedure deals exactly one red hypothesis hsi blue hypotheses hsi associated nonempty sets applying construction section arrive aggregation routine follows vary set max given assign vector color red otherwise assign color red vectors found output one solution aggregation problem red vectors found output say solution proposition applied situation question states whenever drawn independently distribution parameters least result aggregation routine satisfies relation min max essentially recovers classical oracle inequality theorem beyond scope affine detectors lifted detectors tests developed sections based affine detectors affine functions associated pairs composite hypotheses probability distribution observation detectors built satisfy relations exp exp small possible affinity absolutely importance constructions sections based upon availability pairwise detectors affine satisfying respective pairs composite hypotheses relations known far affinity detectors utilized building detectors satisfying via generic scheme presented section note given random observation taking values along deterministic function convert observation observation deterministic transformation remembers make statistical inferences observations exactly make observations however detectors affine nonlinear instance affine detectors exactly detectors quadratic see within framework approach passing allows consider wider family detectors thus arrive wider family tests potential bottleneck necessity bring augmented observations scope setup example distributions bounded support consider case distribution observation belongs family borel probability distributions supported given bounded set sake simplicity unit euclidean ball given continuous function goal cover family distributions induced distributions family associated regular data thus making machinery developed far applicable family distribution assuming observe distribution parameters follows point convex compact set specifying regular data ensure family contains family probability distributions useful pretty crude observation context approach depends much information captured properly selected convex compact set satisfying consider details quadratic lifting case quadratic lifting gaussian case consider situation given nonempty bounded set nonempty convex compact subset positive semidefinite cone matrix affine mapping given matrix pair specifies gaussian random vector thus specifies borel probability distribution let family probability distributions stemming fashion gaussian distributions parameters goal cover family family type already explained would allow use machinery developed far order decide pairs composite gaussian hypotheses via tests based detectors quadratic convenient represent linear form vector coefficients form argument form denote last basic orth assume holds whence spectral norm observe every satisfied thus assume always finally set desired covering given following proposition notation assumptions section let given let convex compact computationally tractable set contain matrices denoting support function max let set det frobenius norm hbt form regular data besides function coercive convex argument whenever every proof see appendix note constraint valid valid matrices induces linear constraint thus incorporated description special case situation proposition let vary convex compact set case simplest way define set let compute function setting last column direct computation yields max max imagine given two collections number rows associated bounds matrices want use proposition build affine detector capable decide hypotheses distribution observation stating observation end solve saddle point problem min max functions associated explained proposition first respectively second collection view computation boils necessity solve convex minimization problem min max optimal solution problem induces affine detector max max risk detector pair families gaussian distributions question exp hand could build affine detector families machinery section solving saddle point problem min max risk resulting affine detector exp assume elements elements selected table unrestricted risk quadratic detector case computation says minimal values identically equal functions thus equal thus case machinery proposition produces quadratic detector better terms risk affine detector yielded proposition numerical illustration get impression performance quadratic detectors compared affine ones case present results experiment singletons risks affine quadratic purely quadratic set detectors pair families gaussian distributions given table see deciding families gaussian distributions common covariance matrix expectations varying associated families convex sets passing affine detectors described proposition quadratic detectors affect risk first row table general fact results situation question affine detectors optimal terms risk among possible detectors deciding families gaussian distributions case distributions different families close expectations third row table affine detectors useless quadratic ones provided differs case first moments distribution observation bear definitive information family distribution belongs makes affine detectors useless contrast quadratic detectors able utilize information valuable stored second moments observation general second row table affine purely quadratic components quadratic detector useful suppressing one increase significantly attainable risk quadratic lifting bounded observations convenient represent quadratically lifted observation matrix seems tautology quadratic detectors affine ones best risk achievable quadratic detectors smaller best risk achievable affine detectors point however proposition guarantee building best terms risk quadratic detector deals computationally tractable approximation problem result quadratic detector constructed latter proposition principle worse affine detector yielded proposition assume distributions supported solution set system quadratic constraints properly selected consequence bounded since strongly convex quadratic form setting observe distribution induced distribution supported closed convex set support function set max max recalling example section section arrive regular data inf expectation therefore references approximation dans les spaces estimation asymptotique des tests phd thesis paris vii vitesses maximales des erreurs tests optimaux zeitschrift wahrscheinlichkeitstheorie und verwandte gebiete sur minimax son application aux tests probab math approximation dans les espaces estimation zeitschrift wahrscheinlichkeitstheorie und verwandte gebiete model selection via testing alternative penalized maximum likelihood estimators annales institut henri poincare probability statistics volume pages elsevier robust tests model selection banerjee bunea huang koltchinskii maathuis editors probability statistics back models processes festschrift honor jon wellner pages institute mathematical statistics indeed large matrix contained positive definite conclude bounded set burnashev minimax detection imperfectly known signal white noise background theory probab burnashev discrimination hypotheses gaussian measures geometric characterization gaussian distribution math notes chernoff measure asymptotic efficiency tests hypothesis based sum observations annals mathematical statistics pages donoho liu geometrizing rate convergence technical report tech report dept university california berkeley donoho liu geometrizing rates convergence annals statistics pages goldenshluger universal procedure aggregating estimators annals statistics pages goldenshluger juditsky nemirovski hypothesis testing convex optimization electronic journal statistics lemarechal convex analysis minimization algorithms fundamentals grundlehren der mathematischen wissenschaften ingster suslina nonparametric testing gaussian models volume lecture notes statistics springer kraft conditions consistency uniform consistency statistical procedures univ california publ cam convergence estimates dimensionality restrictions annals statistics pages proofs proof proposition need verify check right hand side function relation convex latter evident since convex verify let fix set expectation note definitely holds true let let denoting distribution induced distribution noting distribution supported expectation get hence exp inf max inf max inf max max function convex cosh sinh combines computation yield relation cosh sinh cosh sinh need verify indeed holds true implies recalling exactly want prove verification follows left hand side convex cosh sinh containing due range furthermore minimum left hand side cosh coth attained sinh equal sinh coth sinh need prove latter quantity nonnegative whenever coth coth coth since get proof proposition let denote collection regular data processed let random observation probability distribution index vector premise proposition let let finally let event every obeys one hypotheses hsi processed procedure procedure correctly recovers color construction due union bound least follows need verify claim proposition show relation takes place thus let fix observe first whence thus running obeys red one among hypotheses hsi processed since case gets color namely color construction aggregation procedure output either case clearly holds true another vector let denoted also assigned red color claim vector satisfies relation indeed otherwise meaning obeys hypothesis hsi processed running hypothesis blue since case implies color inferred blue desired contradiction nearly done indeed closest point among implying recalling relations tell following story points hyperplane symmetric distance immediate observation case arrive justify note shift rotation reduce situation one belong linear span first two basic orhts first two coordinates three vectors respectively hence claimed proof proposition exp exp det det bht hbt bht hbt observe implies det bht hbt det det taken account premise proposition therefore let function det domain fixed used fact due whence denoting frobenius norm matrix noting kabkf kakkbkf computation completely similar one yields besides setting det int equipping frobenius inner product properly selected conclude denoting eigenvalues noting max see satisfy therefore eigenvalues whence noting max see conclude yields matrix satisfies kdk whence consequently combines relation yield conclude det clearly continuous note combining origin see arrive exp need verification claim regular data boils checking continuous latter check recalling indeed continuous reduces verifying convex continuous recalling nonempty compact set function continuous implying continuity defined prove convexity note contained implying convex hand schur complement lemma bht implying convex since see epigraph convex since set epigraph claimed remains prove coercive let let prove looking expression immediately seen terms expression except terms coming remain bounded grows need verify goes observe uniformly bounded due implying khi denoting last basic orth last basic orth note construction let taking account satisfy positive due matrices observe hti khi kri khi result khi eet khi bbt kri khi kbw concluding quantity tends due khi
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jun relative modules majid rahro zargar abstract let commutative noetherian local ring let proper ideal paper main result shown gorenstein hia completely encoded homological properties hca particular bass numbers notice result provides generalization result hellus schenzel proved main result case introduction throughout paper commutative noetherian ring proper ideal prime ideal residue class field denoted integer let hia denotes local cohomology module respect see definition basic results gorenstein module local ring maximal module finite injective dimension concept introduced sharp studied extensively authors present paper use concept relative modules generalization concept modules kind modules studied title cohomologically complete intersections continued hellus schenzel main result showed gorenstein local ring ideal dim relative following statements equivalent hia relative respect iii extdr hca extir hca hca moreover satisfies conditions follows homr hca hca extir hca hca denotes completion mathematics subject classification primary secondary key words phrases local cohomology gorenstein module relative research part supported grant ipm zargar main result theorem generalize result gorenstein indeed shown gorenstein dim relative suppr following statements equivalent hia relative respect hdm hca hca iii extd exti hca moreover satisfies conditions follows homr homr extr denotes type main results definition say finitely generated relative cohen macaulay respect precisely one local cohomology module respect clearly case grade denotes cohomological dimension respect largest integer hia observe notion relative module connected notion cohomologically complete intersection ideal studied remark let relative module respect let view theorems easy see supp hna supp grade inf dimrp supp definition see theorem finitely generated module local ring said gorenstein following equalities hold true depth dim idr dim definition prime ideal bass number respect defined vdim extirp local finitely generated depth number called type denoted let finitely generated module local ring let ideal let minimal injective resolution welll known fact denotes injective hull let grade relative modules therefore hca ker observation provides embedding hca complexes hca considered complex concentrated homological degree zero definition see definition cokernel embedding hca called truncation complex therefore denoted short exact sequence hca complexes observe following proposition assistance proof main result proposition assume gorenstein module local ring previous notation following statements true exact sequence homr extcr hca isomorphisms extr iii suppose relative respect homr extcr hca proof first notice since gorenstein gorenstein let canonical module ring view exercise isomorphic direct sum finitely many copies hence one use theorem theorem fact exti exti see exti homr flat let minimal injective resolution since bounded complex injective applying functor homr exact sequence obtain following short exact sequence complexes homr homr homr hca zargar let generating set ideal let complex respect view theorem exists isomorphism derived category since extr see homr homr thus get following isomorphisms homr homr homr homr homr homr derived category since homr flat one use theorem isomorphisms deduce homr homr homr hand since grade homr homr one deduce extir hca therefore aid considerations induced long exact cohomology sequence exact sequence complexes provides statements claim iii assume relative respect one exact complex hom easily check complex hom also exact therefore extir hence assertion follows following lemma definition needed proof next result lemma see proposition let integer let arbitrarily local ring following conditions equivalent extir definition let local ring let proper ideal say relative suppr whenever relative respect arp arp suppr following theorem extends main result theorem theorem let ideal local ring gorenstein rmodule suppose relative suppr set dim following statements equivalent hia relative respect relative modules hdm hca hca iii extd exti moreover one statements holds extir hca hca homr homr denotes completion proof first use proposition assumption see hca since dimension hnm hca hand view theorem assumption hnm injective therefore one use corollay achieve isomorphism iii follows lemma extir hca hand since hdm hca injective one use proposition see exti extd homr hdm hca hence assertion done implications iii clear iii first view lemma hca since dimr hca dimr one use vanishing theorem see hca therefore proposition implies homr artinian hence theorem hdm hca artinian thus one use corollary see hdm hca hca hdm hca hence hdm hca injective obtain isomorphism since relative suppr one use proposition deduce prime ideals prp arp prp suppr hand since gorenstein one use theorem corollay see therefore assertion follows theorem next one use proposition iii proposition establish final assertion next provide example show even suppr relative respect arp different cohomological dimensions theorem longer true example let filed set grade also one use sequence see zargar hand easy see prime ideals arp arp hence relative respect arp prime ideals arp next since one see hence using fact gorenstein dim next single certain case theorem denotes canonical module ring corollary let local ring admits canonical let ideal dim suppose relative following statements equivalent hia hdm hca iii extd ext moreover satisfies conditions extir hca hca homr denotes completion proof first notice supp supp therefore dim dim hand one use proposition see gorenstein type homr hence assertion follows theorem following corollary immediate consequence corollary proved theorem main result corollary let local gorenstein ring let ideal set dim suppose relative following statements equivalent hia relative respect iii extdr hca extir hca hca moreover one conditions satisfied homr hca hca extir hca hca relative modules acknowledgements grateful professor hossein zakeri kind comments assistance preparation paper also author grateful referee careful reading suggesting several improvements manuscript references brodmann sharp local cohomology algebraic introduction geometric applications cambridge university press cambridge bruns herzog rings cambridge university press cambridge hellus schenzel cohomologically complete intersections algebra hellus schenzel notes local cohomology duality algebra rahro zargar zakeri injective gorenstein injective dimensions local cohomology modules algebar appear schenzel proregular sequences local cohomology completion math scand sharp gorenstein modules math majid rahro zargar school mathematics institute research fundamental sciences ipm box tehran iran address
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distributed bayesian filtering using logarithmic opinion pool dynamic sensor networks saptarshi bandyopadhyay chung dec jet propulsion laboratory california institute technology pasadena usa graduate aerospace laboratories california institute technology pasadena usa abstract distributed bayesian filtering dbf algorithm presented problem tracking target dynamic model using network heterogeneous sensing agents dbf algorithm sensing agents combine normalized likelihood functions distributed manner using logarithmic opinion pool dynamic average consensus algorithm show agent estimated likelihood function globally exponentially converges error ball centered joint likelihood function centralized bayesian filtering algorithm rigorously characterize convergence stability robustness properties dbf algorithm moreover provide explicit bound time step size dbf algorithm depends target dynamics desired convergence error bound modeling communication error bounds furthermore dbf algorithm models cast modified form kalman information filter performance robust properties dbf algorithm validated using numerical simulations key words bayesian filtering distributed estimation sensor network data fusion logarithmic opinion pool introduction network heterogeneous sensing agents could use distributed estimation algorithm estimate states target dynamics distributed manner potential applications include environment pollution monitoring analyzing communication social networks tracking mobile targets air land water space paper present new distributed estimation algorithm based logarithmic opinion pool guarantees bounded convergence probability distribution states target dynamics bandyopadhyay chung supported part afosr grant nsf grant research carried part jet propulsion laboratory california institute technology contract national aeronautics space administration california institute technology rights reserved email addresses saptarshi bandyopadhyay sjchung chung corresponding author tel preprint submitted automatica distributed estimation algorithms broadly classified three categories based representation states target dynamics algorithms first category estimate mean covariance matrix target states algorithms usually deal linearized target dynamics measurement models also neglect information captured moments estimated probability distribution target states second category aims reach agreement across sensor network discrete set hypotheses states target although algorithms use entire information estimated probability distribution target states applicable cases target states represented discrete finite set hypotheses therefore algorithms suitable estimation continuous domains third category algorithms estimates posterior probability distribution states target category forms general class distributed estimation algorithms algorithms used estimation continuous domains incorporate nonlinear target dynamics heterogeneous nonlinear december rithm heterogeneous sensor network communication graph much sparser complete graph agent estimate converges posterior probability distribution target states furthermore assume communication network topology periodically strongly connected agent communicate neighboring agents time instant ment models uncertainties algorithms also use entire information mean covariance matrix estimated probability distribution target states light advantages paper focuses development distributed estimation algorithm belongs third category algorithms agents exchange local probability distributions neighboring agents combine using fusion diffusive coupling rules estimate aggregate probability distribution schemes combining probability distributions distributed manner like linear opinion pool linop logarithmic opinion pool logop first studied statistics literature logop scheme deemed ideal purpose favorable properties externally bayesian paper present distributed bayesian filtering dbf algorithm address open question time instant agents exchange normalized likelihood functions neighboring agents combine using fusion rule fusion rule combining arbitrary probability distributions relies logop scheme dynamic average consensus algorithm show finite time instants estimated likelihood function agent converges error ball centered joint likelihood function centralized bayesian filtering algorithm also provide explicit upper bound time step size dbf algorithm depends target dynamics convergence error bound moreover analyze effect communication modeling errors dbf algorithm target dynamics models show dbf algorithm simplified modified kalman information filter finally show distributed estimation algorithms special cases dbf algorithm focus distributed estimation algorithms use logop scheme first algorithm proposed particular generates weights logop scheme combining probability distributions within exponential family probability distributions expressed exponential functions discussed distributed estimation algorithm presented well prior work agents combine local posterior probability distributions using consensus algorithm multiple consensus loops within time step executed much faster original time steps bayesian filter moreover show agent estimated probability distribution target states converges around pdf minimizes sum divergences posterior probability distributions target states similar algorithms combining local likelihood functions using consensus algorithm proposed number consensus loops within estimator time step grows fast number agents due convergence properties consensus algorithm hence algorithms feasible target dynamics comparatively fast connection target dynamics time step size distributed estimation algorithm explored literature furthermore analyzed algorithm using lineargaussian models focused probability distributions within exponential family contrast present rigorous proof technique first introduced prior work logop scheme applicable general probability distributions paper organized follows section presents preliminaries problem statement logop scheme general convergence results presented section dbf algorithm special cases presented section results numerical simulations presented section paper concluded section agents perfectly connected complete communication graph agent could communicate instantaneously every agent without loss information communication links agents exchange local likelihood functions use centralized bayesian filtering algorithm estimate posterior probability distribution target states open question design distributed estimation preliminaries problem statement let represent sets natural numbers positive integers real numbers respectively state space target states closed set rnx dimension states target let borel probability space defined complete probability measure let denote density probability distribution respect measure continuous lebesgue measure probability density function pdf therefore probability event written integral paper deal continuous case function represents pdf lebesgue measure let represent set pdfs state space distance divergence pdfs dkl rnx rnw rnx possibly nonlinear function state discretization time step size independent identically distributed process noise dimension process noise vector consider network heterogeneous sensing agents simultaneously tracking target let represent measurement taken ith agent time instant measurement model agents given log hik hik rnx rnvi rnyi possibly nonlinear function state measurement noise nyi nvi dimensions measurement measurement noise vectors respectively note measurements conditionally independent given target states assume target dynamics measurement models known paper algorithms presented discrete time let time step size two consecutive time instants time index denoted right subscript agent index denoted lowercase right superscript frequently used symbols listed following table table list frequently used symbols symbol definition adjacency matrix periodicity communication network jki lik inclusive neighbors ith agent number agents network ski prior pdf tki uki wki wkc estimated likelihood function state space true states target predicted states target updated states target measurement taken ith agent time step size bayesian filtering algorithm agent uses bayesian filtering algorithm estimate pdf states target let represent predicted updated states target time instant let pdfs ski wki denote ith agent prior posterior pdfs target states time instant normalized likelihood function normalized joint likelihood function prediction step prior pdf ski obtained previous posterior pdf using equation estimated pdf posterior pdf centralized posterior pdf probabilistic model state evolution obtained known target dynamics model assume prior pdf available start estimation process new measurement used compute posterior pdf wki update step using bayes rule wki target dynamics measurement models let represent true states target time instant dynamics target discrete time given ski ski likelihood function obtained ith agent known measurement model let pdf lik represent normalized likelihood function lik therefore equivalent wki convergence error given tki lim tki dbf algorithm shown fig algorithm achieves objective note agents exchange estimated pdfs neighboring agents time instant fusion step lik ski lik ski sensing agents hypothetically connected complete graph agents exchange likelihood functions agent use centralized bayesian filtering algorithm compute centralized posterior pdf target states wkc using bayes rule wkc prior knowledge states posterior pdf previous time instant prediction step compute pdf transmit pdf neighbors exchange pdfs neighboring agents receive pdfs neighbors fusion step compute pdfs normalized joint likelihood function measurement next time instant bayesian filtering optimal posterior pdf wkc integrates uses available information expressed probabilities moreover optimal state estimate respect criterion computed posterior pdf wkc minimum meansquare error mmse estimate maximum posteriori map estimate given wkc arg max wkc update step compute pdf fig flowchart dbf algorithm ith agent kth time instant communication network topology communication network topology sensor network denoted directed graph edge ith agent receives information agent time instant inclusive neighbors ith agent denoted jki matrix represents adjacency matrix jki potential criteria optimality maximum likelihood minimum conditional divergence minimum free energy discussed main advantage original bayesian filtering formulation approximation needed filtering process complete information dynamics uncertainties model incorporated filtering algorithm however direct implementation bayesian filtering computationally expensive practical implementation algorithms general form achieved using particle filtering bayesian programming assumption digraph adjacency matrix satisfy following properties exists positive integer directed graph strongly connected time instants matrix doubly stochastic iii matrix product defined problem statement let pdf tki denote estimated joint likelihood function ith agent time instant positive constants let aim design distributed estimation algorithm communication network topology described section agent tki converges normalized joint likelihood function exists constant element therefore digraph periodically strongly connected matrix balanced note digraph strongly connected time instants linop logop logarithmic opinion pool convergence results pklinop pki linop logop let pdf pki denote ith agent pdf time instant linop logop schemes combining pdfs pki given pklogop fig pdfs combined using linop logop note logop solution preserves modal nature original pdfs definition assumption constant pki pklogop using simple algebraic manipulation get pklogop logop pki logop pki pki weights integral denominator finite thus combined pdf obtained using linop logop gives weighted algebraic geometric averages individual pdfs respectively shown fig combined pdf obtained using logop typically preserves multimodal unimodal nature original individual pdfs compelling reason using logop scheme externally bayesian logop combination step commutes process updating pdfs multiplying commonly agreed likelihood pdf thus represented logop scheme linear equation using functions pki pklogop removed effect normalizing constants pki logop pklogop state useful convergence results using functions definition proofs given appendix therefore logop scheme ideal combining pdfs distributed estimation algorithms definition pointwise convergence pdf pki converges pointwise pdf pki due multiplicative nature logop scheme agent veto power pki agent pklogop combined pdf irrespective pdfs agents order avoid veto condition enforce following assumption used literature lemma pdfs satisfy assumption exists lemma function converges pointwise function corresponding pdf pki also converges pointwise pdf assumption nonzero probability property paper pdfs strictly positive everywhere closed set definition convergence measure defined measure induced pdf pki event similarly let denote measure induced pdf order analyze logop scheme general probability distributions satisfy assumption use following functions distance defined ktv supa measure converges measure ktv necessary conditions satisfying given dkl lik dkl state dbf algorithm whose steps shown fig let pdf uki denote estimated pdf ith agent time instant pdf tki defined section assumptions dbf algorithm given algorithm lemma pdf pki converges pointwise pdf measure converges measure moreover ktv pki another reason using logop scheme minimizes information lost combination process information loss measured using divergence algorithm distributed bayesian filtering algorithm ith agent steps time instant lemma pdf pkkl globally minimizes sum divergences pdfs pki agents given arg min dkl compute prior pdf ski using obtain local measurement compute normalized likelihood function lik using receive pdfs agents jki compute pdfs uki tki follows lik note pdf pkkl equivalent pdf pklogop obtained using logop scheme weights agents tki proof lemma given prior work note normalized joint likelihood function also given lkl lkl lkl tki ski tki ski following theorem shows dbf algorithm satisfies problem statement section theorem assumptions agents execute dbf algorithm algorithm time step size algorithm defined log convergence error pdf tki pdf bounded lim max tki furthermore convergence error pdfs tki time instants bounded max tki assumption time step size exists constant agents section present main dbf algorithm convergence robustness properties application special cases first state assumption nature pdfs lik agents directly link target dynamics measurement models time step size distributed estimation algorithm compute posterior pdf wki follows distributed bayesian filtering algorithm lik wki show dbf algorithm also estimates pdf lkl distributed manner uki uki log otherwise log log log log max max lki lki lki lkkl note lkkl lki follows shows functions uki converge towards converge function lkkl let define error vector uki ukn positive constants denotes second largest singular value matrix upper bounded uki evolution error vector given moreover periodicity communication network topology smallest positive element defined assumption defined assumption error measures induced pdfs tki bounded max ktv lim max ktv overall evolution error vector time instants given defined proof using definition define thei functions lkl lkkl lkl lki lik uki tki since functions note therefore investigate convergence along directions orthogonal follows assumption matrix irreducible therefore matrix primitive lemma denotes hthe secondi fined henceforth drop term brevity largest eigenvalue matrix let vtr orthonormal matrix eigenvectors symmetric primitive matrix spectral decomposition get vtrt vtr step first show pdf uki converges pdf lkl equation using functions uki vst used since eigenvectors orthonormal vst vst gives jki defined section since doubly stochastic satisfies conservation property max lkkl vst vst vst vst vst vst vst kvst kvst lkc lkkl max lkc using lemma select tki therefore max max moreover follows assumption lkn max tki therefore since point therefore tki max max kvst hence presence disturbance term get kvst kvst kvst kvst hence convergence error pdfs bounded follows time step size found using error term using discrete gronwall lemma obtain kvst thus error function tki function lkc bounded max tki kvst bounded therefore given sin tki uki therefore follows since matrix irreducible matrix positive matrix maximum path length two agents less equal hence measure irreducibility matrix atk lower kvst step show pdf tki converges pdf equations denotes largest singular value matrix since vst orthonormal vst also orthogonal vst matrix atk primitive denotes second largest singular value matrix therefore error vector vst globally exponentially stable absence disturbance term thus error function uki function lkkl bounded depends time instant kvst kvst vst first investigate stability system without disturbance term vst let kvst candidate lyapunov function system therefore get kvst kvst moreover defined therefore follows therefore log otherwise log computed using constraint error follows lemma note exponentialconvergence proof substantially different proof section study robustness dbf algorithm presence communication modeling errors order implement dbf algorithm agents need communicate estimated pdfs neighboring agents see line algorithm remark key advantage dbf algorithm require sensors observe target agent observe target sets normalized likelihood function uniform distribution lik agent likelihood function influence joint likelihood function estimated pdfs geometric nature fusion rule moreover dbf algorithm avoids double counting summation weights paths constant due weights adjacency matrix theorem explicitly bounds time step size distributed estimation algorithm target dynamics effectiveness dbf algorithm predicated assumption moreover upper bound time step size decreases increasing number agents remark communication pdfs information theoretic approach communicating pdfs studied particle filters used implement bayesian filter combine pdfs resampled particles represent agent estimated pdf hence communicating pdfs equivalent transmitting resampled particles another approach involves approximating pdf weighted sum gaussian pdfs transmitting approximate distribution several techniques estimating gaussian parameters discussed gaussian mixture model literature let pdf denote pdf uki corrupted communication errors similarly let pdf represent normalized likelihood function lik corrupted modeling errors first state assumptions errors state main result section following corollary provides sharper bounds special case static communication network topology corollary communication network topology time step size theorem given log log log log log robustness analysis assumption exists constants agents uki lik therefore uki lik adjacency matrix proof case written kvst kvst corollary assumptions time step size theorem given using discrete gronwall lemma obtain kvst kvst log defined assumption proof equation written hence get lkkl get respectively substituting bounds assumption gives algorithm information filtering algorithm ith agent steps time instant compute prior pdf ski evolution error vector given similar proof theorem get kvst kvst obtain local measurement receive pdfs agents jki compute pdfs uki tki follows tik uik section apply dbf algorithm special case target dynamics measurement models given linear systems additive gaussian noise special case multiple consensus loops within time instant section show proposed dbf algorithm easily extended recursively combine local likelihood functions using multiple consensus loops within time instant agent estimated likelihood function converges joint likelihood function resultant dbf algorithm equivalent bayesian consensus algorithms note multiple consensus loops within time step significantly reduces practicality algorithms let pdfs denote local pdfs agent consensus loop time instant since pdf lik updated time instant define pdfs lik lik coni sensus loop agent updates local pdfs using following fusion rule special case information filter compute posterior pdf wki follows corollary order generate satisfactory estimates using dbf algorithm bounds substantially smaller tjk get respectively iik rik rik uik hence get lkkl process noise measurement noise rik zero mean multivariate normal distributions therefore adopt information representation dbfkalman information filtering algorithm lineargaussian models given algorithm prior pdf ski posterior pdf wki estimated pdfs uki uik tki tik also multivariate normal distributions tki axis theorem assuming strongly coni nected agent pdf globally exponentially converges pointwise nloop consensus loops norm error vector bounded nloop kek nloop benchmark example axis centralized bayesian filter dbf subssection compare performance dbf algorithms centralized bayesian filtering algorithms using benchmark example studied target dynamics modeled linear nearly constant velocity model time step sec mse position mse position fig motion target position sensing agents toa sensors doa sensors agents sensors communication network topology shown numerical simulations section demonstrate properties dbf algorithm using benchmark example section complex estimation control task section proof follows theorem thus distributed estimation algorithm special case dbf algorithm motion target agent sensor toa sensor doa sensor communication network centralized kalman filter information filter time step sec fig variation mse position respect time step size shown centralized bayesian filtering algorithm dbf algorithm scenario centralized kalman filtering algorithm dbf algorithm models scenario sensors measurement models sensors given covariance matrix process noise time step size state vector denotes position velocity components along coordinate axes shown fig sensing agents distributed given region able communicate neighboring agents undirected communication network topology assumed weights used compute doubly stochastic adjacency matrix hik doa doa sensor toa sensor denotes position ith agent inverse tangent function doa sensor measurement noise doa variance toa sensor measurement noise variance jki max scenario agent executes dbf algorithm algorithm using particle filters particles comparison dbf algorithm centralized bayesian filtering algorithm varying time step sizes shown fig target motion shown fig used simulations see dbf algorithm mse position converges centralized algorithm time step size decreases mse smaller time step size sec note mse denotes degree ith agent scenario five agents equipped nonlinear position sensors measure distance target using time arrival toa sensors another five agents equipped direction arrival doa sensors measure bearing angle target remaining agents centralized algorithm change much time step size constrained measurement noise intensities shows performance dbf algorithm approaches performance centralized bayesian filter time step size reduced moreover fig shows distances estimated likelihood functions joint likelihood function bounded approach maintain distance nearest neighbors rki maintain distance dcm cos estimated center mass propagation step dbf algorithm agents use estimated positions estimate control input applied agents therefore estimation errors contribute process noise propagation step fusion step time instant ith agent communicates agent either nki nkj sec time sec tki sec simulations use sec particles execute dbf algorithm start estimation process particles selected uniform distribution state space simulation results multiple values shown fig since agents use relative measurements orientation final regular polygon global frame fixed therefore conclude agents successfully estimate relative positions using dbf algorithms achieve complex desired formations time sec fig trajectories distances estimated likelihood functions joint likelihood function ten sensing agents shown scenario ten agents doa toa sensors linear position sensors hik lin measurement noise lin rki covariance matrix rki agent executes information filtering algorithm algorithm fig shows performance information filtering algorithm approaches performance centralized kalman filtering algorithm time step size reduced agent agent agent agent polygon agent agent agent polygon relative position estimation formation control tki subsection agents estimate relative positions using range measurements reconfigure regular polygon specifically agent measure distance nearest two neighbors using toa sensor whose measurement model described agent simultaneously executes dbf algorithms estimate relative positions agents ith agent dynamics control inputs given agent agent agent agent agent polygon agent agent agent agent polygon xik uik dcm nki denotes two nearest neighbors ith agent agent estimate agent position obtained using dbf algorithms agents use artificial potential field apf based fig initial position final position trajectories agents final regular polygon shown agents conclusions ahmed schoenberg campbell fast weighted exponential product rules robust general data fusion robotics science systems viii roy newman srinivasa eds mit press paper presented novel distributed estimation algorithm namely dbf algorithm ensures agent estimated likelihood function converges error ball around joint likelihood function centralized bayesian filtering algorithm rigorously proven convergence properties algorithm shown explicit connection time step size distributed estimation algorithm timescale target dynamics also presented dbfkalman information filtering algorithm special case models properties algorithms illustrated using complex numerical examples envisage novel proof techniques presented paper also used distributed estimation algorithms rely logop scheme fraser bertuccelli choi hyperparameter consensus method agreement uncertainty automatica vol hlinka rupp distributed vol hlawatsch djuric likelihood consensus application particle filtering ieee trans signal hlinka hlawatsch djuric consensusbased distributed particle filtering distributed proposal adaptation ieee trans signal vol battistelli chisci average consensus probability densities distributed state estimation guaranteed stability automatica vol bandyopadhyay chung distributed estimation using bayesian consensus filtering proc amer control portland june references bandyopadhyay chung distributed estimation using bayesian consensus https speyer computation transmission requirements decentralized control problem ieee trans autom control vol degroot reaching consensus amer statistical vol borkar varaiya 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markov chains mixing times american mathematical
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arxiv sep analysis equality relationships imperative programs pavel institute informatics systems avenue lavrentiev novosibirsk russia emelianov march abstract article discuss analysis equality relationships imperative programs describe semantic domains general purpose operations abstract computational states term evaluation identification semantic completion widening operator etc semantic transformers corresponding program constructs summarize experiences last years concerning analysis give attention applications analysis automatically generated code among illustrating examples consider program analysis diverges without widening operator results analyzing residual programs produced automatic partial evaluator example analysis program generated evaluator given keywords abstract interpretation value numbering equality relationships program terms formal grammars semantic transformers widening operator automatically generated programs introduction semantic analysis powerful technique building effective reliable programming systems presented new kind semantic analysis designed framework abstract interpretation analysis determines approximation sets invariant term equalities called analysis equality relationships program terms hereinafter referred era traditional static analyses imperative programs interested finding equalities specific kind value analyses discussed work partly done author laboratoire informatique ecole polytechnique palaiseau france ecole normale ingenieur bourges france describing regular approximations simple mathematical descriptions machine representations sets values convex affine congruent counterparts well etc carefully designed reasonable express semantic properties effectively computed polynomial algorithms handle regular nature allow treat well programs irregular hence special interest investigations approximations based sets terms potentially arbitrary nature could powerful due irregularity effectively computed one well known example analysis case terms represent expressions computed programs enables analysis take account different aspects program behavior unified way unified treatment semantic information allows analysis improve accuracy mean era generalization value analyses except constant propagation one use different approaches semantic domains transformers extract effectively precisely limited classes semantic properties general results analyses comparable era provides interesting possibilities gathering propagating different invariant information programs unified way information used verification optimization purposes second especially interesting automatically generated programs residual obtained process partial evaluation synthesized specifications due nature automatic generation processes programs specific control flows example hierarchy nested conditional statements specific conditions case residual programs hierarchy deep degree partial evaluation increases successfully optimized base gathered invariant information besides peculiarity era mentioned let discuss common properties semantic analyses taxonomic properties analysis algorithms attribute dependence context sensitivity flow sensitivity scalability properties well known however author opinion notion interpretability semantic analysis considered adequately yet interpretability analysis means extensively properties primitive operations language arithmetical logical etc type information allowed analyzing handled analysis works one extreme point view interpretability approach accepted pure program scheme theory interpretations functional symbols type information unfortunately results obtained approach reasonably strong nevertheless must underscored era dates back sabelfeld works program scheme theory another extreme leads complete description program behavior also workable obviously closely allied flow sensitivity ignoring part semantic information example sort approximation value sets conic shapes one computational disadvantage semantic transformers involve algorithms known proposed many years ago gain ground precisely exist works program scheme theory semantic interpretations functional symbols like commutativity superposition etc considered allow treat precisely control flow constructs analyzed programs scalability attempts take account large quantity semantic properties example using theorem prover invoked semantic analyzer works deduces new properties lead combinatorial explosion abstract computational states possible interpretability highlighted enough analysis algorithms take account limited classes primitive operations type information enriched natural way example interval analysis able incorporate congruence properties natural way essentially another case era choice handle expressiveness analysis intend illustrate notion interpretability analysis importance usefulness example analysis among analyses closely related would like point following semantic analysis detection equalities program variables simple relationships among described makes list sets variables discovered equal using hopcroft partitioning algorithm automata algorithm quite efficient however precise enough techniques developed algorithms demonstrate adequacy value numbering example last case time complexity algorithm number program variables number program size another important example analysis mentioned approximating sets terms found resolving system equations formal grammars used analysis recursive data structures functional languages see example formal languages applied coding memory access paths values program variables analysis established common foundations connecting generalizing different approaches using formal languages represent semantic properties course mention techniques automatic proof theory term rewriting theory widely applied analysis stage improve accuracy stage present results user article organized follows sections describe semantic properties concrete abstract respectively considered era section discuss basic operations semantic properties used define semantic transformers next presented section section consider widening operator complexity era discussed finally section describes processing era invariants presents results experiments era appendix example analysis residual program considered course possible use sophisticated approaches combinations analyses introduces complicated problems implementations enrichment original ones properties interest concrete properties usual choice description operational semantics specification transition relationship pairs control point state program memory see example states program memory described mapping cells memory universe values variables groups cells values constants asymmetric roles another example asymmetry manipulations structured objects programs arrays records etc transparent primary ones describe operational semantics era used another approach objects program considered identical following meaning let set symbols representing variables constants last ones may following kinds scalars compositions scalars constant arrays records etc names record fields indefiniteness let set functional symbols represent primitive operations programming languages arithmetic logic type casting kinds memory addressing well let trs set terms hereinafter referred program terms represent expressions computed execution program state program memory take reflexive symmetrical transitive relationship equivalence relationship trs relationship defines set term equalities use describe operational semantics call computation state suppose following code var integer array odd mod else end example executed least twice different parities variable table static semantics five control points given present subset term equalities concerning dynamic behavior piece shall use property see table illustrate reasoning set equalities completed number consistent equalities entry odd exit odd odd odd exit odd odd table description collecting semantics example constant represents constant arrays equality relationships grammar representation functional net representation figure semantic properties representations set equalities contain trivial equalities like equalities given symmetry equality relationship formally described follows let eqs set equalities terms trs eqs trs set eqs computation state interpreted following way equality values expressions represented must equal point execution trace take set eqs set concrete semantic domain describing collecting semantics era element concrete semantic domain set computation states particular point particular program set computation states corresponds execution trace program reaches point properties considered era presented means grammars finite set nonterminals denoted capital letters initial symbol grammar finite set terminal symbols finite set grammar rules give precise description shall use quite simple machinery formal languages theory object considerations serves demonstrations could use functional nets language well rather widely used expect descriptive ways become apparent examples figure ones way state computation represented language generated grammar described form say nonterminal lang guage know terms obviously grammar representation superfluous syntactic sugaring use rules say classes equal values evidently may suppose set rules contain rules identical right parts convenient consider grammars contain useless redundant nonterminals rules rule useless produces one term language knows trivial equality like term argument terms nonterminal useless participate derivations sentential forms grammars arise result operations grammars nonterminals rules useless redundant possible remove see lemma operation called state reduction consist detecting removing set revealed incremental markup algorithms see example illustrating purposes shall use special functional nets grammars nonterminals represented ovals containing functional symbols right parts rules arcs functional symbols ovals represent argument dependencies ordered left right see figure abstract properties interesting peculiarity era abstract approximate properties nature computation states operational semantics formally approximation defined follows functions abstraction concretization given concrete property eqs abstract property eqs abstraction function eqs eqs concretization one eqs eqs defined following way eqs otherwise union eqs inclusion intersection languages eqs respectively take empty language infimum computed expressions abstract semantic properties supremum inaccessible computation state language containing possible equalities program terms also theoretical inclusion eqs lemma abstraction function monotonic proof function monotonic iff eqs imprecise element eqs soundly approximated eqs example table best approximation concrete property figure intersection computation states intersection finding intersection languages undecidable problem general case case languages term equalities algorithm exists see example fig similar constructing cartesian product automata algorithm intersection two languages term equalities input grammars output grammar description let set rules defined follows rule introduced rule introduced add rules initial nonterminal apply state reduction described algorithm intersection impractical improve choose efficient strategy generating functional symbols first topological sorting functional symbols appearing right parts rules intersect symbol sets generate next functional symbol conformity topological order arguments symbol already exist new grammar practical cases intersection done linear average time respect grammar size linear space demands quadratic time worst case operations semantic properties shall discuss basic operations semantic properties eqs eqs used define semantic transformers era notation means removing terms operations abstract computation states use certain common transformation sets term equalities consists removing subset following statement holds lemma removing subset term equalities preserves correctness approximation proof easy see states removing term equalities makes approximation rough preserve correctness example shall write single term term set removing followed state reduction operation defined term evaluation define abstract semantics evaluation term abstract computation state result term evaluation state knowing evaluated term term evaluation otherwise add new rules grammar nonterminal exist figure term evaluation otherwise functional symbol calculated exist derivations add new rules grammar nonterminal still exist improve accuracy analysis take account example commutativity primitive operations knows commutative identification terms standard semantics defines program execution values computed expressions equal incorporate information computation state also left identification terms transforms state new one incorporating information example know value term representing conditional expression coincides terms representing constants true false respectively executed identification terms along semantic completion considered provides powerful facilities take account real control flow programs identification terms replace nonterminal nonterminal let rules rules identical right side appeared certain nonterminal left sides rules example must taken nonterminals grammar must replaced figure identification values terms repeat step stabilization state containing inconsistent term result else reduction example identification given figure lemma identification values terms correct transformation resulting state unique proof let eqs concrete semantic property holds identification terms values equal concrete semantics equal abstract semantics values equal identification gives inconsistent computation state obviously includes transformation correct identification done finite steps size grammar decreases step uniqueness resulting state explained following observation two pairs terms candidates identification identification one close possibility another remove duplication functional symbols fact identification pair terms obtain new state including source one thus existing identification possibilities remain order merging term pairs important resulting state semantic completion yet considered interpretation constants functional symbols could continue developing era way result shall obtain exists wide spectrum inconsistency conditions simplest equality two different constants see section discussion figure semantic completion noninterpretational version analysis likewise analysis algorithms program scheme theory however natural use semantics primitive operations programming language interest order achieve better accuracy era provides possibilities taking account properties language constructs especially important easily handle complexity manipulations fact inclusion properties corresponds carrying finite part completion computation states consistent equalities manipulation called semantic completion completion see basic version semantic completion take computations constant equal arguments detect term specific value also possible apply identification involving dependencies among result operation arguments rue etc identification process iterative new possibilities identification appear next steps conceivable shall detect inconsistency computation state case result semantic completion version extended intelligent theorem prover inferring new reasonable equalities checking inconsistency computation states combining analysis proofs offers powerful facilities analyzer see mentioned previous works prover reusable consequent automatic processing results analysis size used completion tuned options interpretability analyzer arithmetical errors division zero type range etc appear semantic completion case analyzer tells error sets current computation state notice languages incomplete boolean evaluation admissible semantic completion boolean expressions carefully designed especially presence pointers example semantic completion presented figure turning back identification example figure consider following interpretation constants functional symbols exclusive disjunction negation constant true constant false easy see case application semantic completion gives analyzer implemented interpretational version era uses semantic completion approach definitions basic transformations mentioned changed following short shall omit interpretability prime semantic transformers section describe semantic transformers eqs eqs corresponding common statements existing imperative programming languages statement input computation state output computation state notation means assignment statement among program terms considered era pick access program terms including array elm record fld pointer val referencing playing important role determination effect assignment statement abstraction program memory manipulations storeless based notion memory access paths represented access program terms example address expression bar access term fld val elm elm bar shall assume operations memory addressing example comparisons allowed structured variables arrays records previous example neither elm val elm elm appear arguments operations elm fld respectively limitation allows simplify definition assignment statement preserve safety analysis take account memory aliasing appearing programs two access terms alias address memory location general case era inadequate handle precisely kinds aliasing use analyses next assumed access term know set access terms covering set aliases information let otherwise either accept assignment structured variables destroys equalities involving components implement strategy example copying preserving useful safe access terms roots memory access terms notice approximations alias information may cause conservative results era therefore era alias analyses used implementation sensitivity control flow assignment statement exp state evaluate exp using evaluation transformer formally defined earlier section let result evaluation let nonterminal perform remove add term unfortunately cases abstraction assignment statement fails example assignment transformer corresponding applied state rue gives trivial identity improve accuracy analysis cases consider artificial variables associated scalar variables program store previous values original ones approach first second steps assignment statement effect definition insert step let associated remove second rule add needed approach shall rue deduce also transformers program given program program var begin end variables statements define following transformer corresponding rogram represents indefinite value notice constant empty statement sequence statements read statement read notice read statement well statements set user represented form equalities program terms supplied analyzer take consideration check consistency include current computation state write statement writ conditional statement else end cycle statement cycle body cycle end composed statement possibly contains occurrences statements exitk sequence becomes stabilize cle stationary entry state exitk process become stabilize widening operator used see section halt exit return statements exit ret call function assume return results function calls implemented assignment variables names invoked functions connection call sites taken account function bodies may contain return statements well function var begin end local variables statements ion figure function call factual parameter function see note read statement intersection stationary entry states return statements exist otherwise term evaluated result function call represented widening operator convergence analysis abstract semantic domain satisfy descending chain condition therefore requires widening operator guarantee convergence abstract interpretation use dual widening operator eqs decreasing chains decreasing chains defined strictly decreasing iteration sequence widening convergent limit sound approximation fixpoint widening operator infinite chains appear corresponding languages common infinite subsets generated cyclic derivations grammars source era term identification avoid problem imposing constraint grammars must acyclic within semilattice eqs eqs subsemilattice finite generated grammars satisfies chain condition languages expressive enough solution following grammars originally restricted course abstract interpretation grammar size becomes greater parameter harmful cycles must destroyed end remove grammar rules participating cyclic derivations correctness approximation intersection follows lemma notice sets term equalities special form corresponding languages used sabelfeld develop effective algorithms recognizing equivalence classes program schemata detecting rules simpler problem mfas mfvs consider grammars directed graphs sets smallest sets arcs vertices respectively whose removal makes graph acyclic suppose feedback vertices choice natural purposes general case problem approximate algorithms solve problem polynomial even linear time consideration weighted digraphs makes possible distinguish grammar rules respect worth accuracy analysis algorithm however perspectives approach clear reasons algorithms example proposes algorithm weighted requiring time complexity matrix multiplication acyclic subgraph cyclomatic graph grammar cyclomatic graph grammar figure widening operator analysis equality relationships defined following way vertex set cyclomatic grammar set functional symbols existing arc belongs arc set contains rules transformation cyclomatic graph involves detecting fvs upper approximation minimal feedback set removing vertices fvs said example shown figure let language obtained applied grammar generating define otherwise parameter reasonable choose parameter depending number variables analyzed programs linear function small factor proportionality notice case lengths appearing chains linearly depend number variables living simultaneously divergence analysis widening operator rather complex really needed analysis equality relationships programs analyzed generate infinite chains semantic properties mentioned constructing program examples problem long time stated belief existence seems hardly probable attempts failed concentrated constructing example completely functional symbols frame pure theory program schemata already noticed widely vary interpretability analysis algorithm order construct example suffice use following rule completion rue consider following example end end program scheme sign abs end end program properties computed body entry belong infinite decreasing chain figure state describes properties valid cycle execution states graph represents cyclic derivations grammar author find another appropriate name object computer science discrete mathematics literature figure case divergence analysis describe properties entry cycle body first second iteration respectively easy see coincides except equality relationships containing terms generated therefore every time obtain next state functional element absent placed dashed box repeat times obtain program program scheme interpret functional symbols following manner sign abs would like point following interesting property execution piece code behavior determined standard semantics diverges two values time analysis algorithm execution piece code nonstandard semantics always divergent condition widening operator used assumption interpretation mentioned holds program actually reader decide would like underline one hand interpretability analysis algorithm varied wide ranges able prove formally impossibility behavior analyzer considered interpretation choose either lean analysis using acyclic grammars another one using arbitrary grammars widening operator penetrating reader may notice program considered human written really stupid slightly intellectualized interpret add automatic generators programs code seem improbable complexity analysis pointed following upper bound time algorithm era program size maximum number program variables existing time gmax maximum sizes grammars appearing course analysis due construction widening operator namely choice parameter linearly depended assume gmax bound deduced help theorem theorem states recursive strategy chaotic iteration maximum complexity maximum length increasing chains built widening operator set control program points depth control point hierarchy nested strongly connected components control flow graph containing set vertices widening operator applied analysis programs assume maximum depth nested loops depend program size bounded constant conclude number algorithm steps exceed since time complexity operations used analysis estimated obtain upper bound notice improve results analysis possible use rich semantic completion precise quadratic time complexity however experimental results show approximation fixed point heads cycle bodies usually attained two iterations time complexity analysis proportional ngmax also user turn checking threshold widening started case consciously admits chance analysis diverges believe chance big easy see space complexity equality relationship analysis ngmax essentially depended number variables estimate actual space requirements per program lines programs processing invariants experimental results usage era produces set invariants involving program terms useful different steps program development processing debugging verification invariants interesting specialization optimization automatic prover mentioned used step results analysis notice analyzer tell user useful information stage analyzing means possible exist execution traces computational states appear mark properties detected stage stage processing results analysis properties hold execution trace leading program point shall briefly list program properties extracted computation states lin lout statement input output states respectively variable indefinite value lin contains error evaluation expression division zero type ranges dereferencing etc inaccessible lin information correspond different properties program execution potentially infinite cycles recursive calls dead branches conditional statements useless procedural definitions etc assignment statement redundant lout contains variable associated unused definitions constants variables types constant propagation notice era detect expression constant constants variables expression known general expression exists expression equal original one calculated efficiently respect given criterion target computer architecture obviously list complete many properties extracted invariants example consider systems contained gathered invariants try solve derive precise ranges values expressions inconsistency computational state apart automatic mode invariants processed automatically provide interactive mode visualize results analysis hypertext system hypercode presented open tunable system visualization properties syntactic structures two cases visualization properties detected automatic mode processing visualization properties experiments show program properties interest automatically extracted computed invariants judicious consider many particular cases hardly embed system instead system facilitates specification user request friendly interface chooses program point expression obtains equality relationships valid point expression occurs program examples example program presented properties detected analyzer indicated comments program kmp lambert automaton intf ackerman length lines era average improv size bytes era average improv table comparison era var integer procedure integer integer begin parameters always equal return expression simplified end begin read read variable might uninitialized simplified else simplified end write call transformed write end div arithmetical error write inaccessible point end basis analysis program transformed following var integer begin read read write end error exception end table present results optimization based analysis residual programs generated specializer compile examples used xds compiler following programs investigated kmp matching algorithm specialized respect pattern residual program comparable knuth morris pratt algorithm efficiency see also appendix lambert program drawing lambert figure specialized respect number points automaton interpreter deterministic automaton specialized respect language intf interpreter mixlan specialized respect program computing fibonacci numbers ackerman program computing values akcerman function specialized respect first argument let comment briefly obtained results reducing length program considered reducing number operators declarations examples optimizing effect typically attained removal redundant assignments dead operators unused variables reduction operator strength exception kmp program characterized high degree polyvariance roughly speaking means presence conditional statements active usage array references constant conditions redundant range checks eliminated notice last optimizing transformation important languages checks defined language standard notable optimizing effect lambert program explained deep reduction power operations achieved optimizing techniques used compilers since automaton ackerman programs quite small optimization gives conservative results however would better ackerman program implementation era substantial optimized programs obtained less surprising since great bulk specializers take criterion optimality experiments show average reduction size residual programs case kmp program seems suppose improvement achieved practice programs increased large residual programs high degree polyvariance active usage arrays arithmetics author opinion analysis automatically generated specifications well programs promising direction application especially implementation era unfortunately experiments exhaustive enough since partial evaluation involved yet real technological process software development hence finding large resudial programs hard problem acknowledgement author wishes thank bulyonkov cousot cousot sabelfeld support useful discussions remarks references aho ullman theory parsing translation compilation volume alpern wegman zadeck detecting equality variables programs proc annual acm symposium principles programming languages pages acm press baburin bulyonkov emelianov filatkina visualization facilities program reengineering programmirovanie berman markowsky linear approximate invariants research report watson research center ibm yorktown bourdoncle efficient chaotic iteration strategies widenings proc international conference formal methods programming applications volume lecture notes computer science pages springerverlag bulyonkov kochetov practical aspects specialization programs proc international seminar partial evaluation volume lecture notes computer science pages bulyonkov kochetov visualization program properties research report institute informatics systems novosibirsk russia cortadella octahedron abstract domain proc international static analysis symposium volume lecture notes computer science pages cousot cousot abstract interpretation unified lattice model static analysis programs construction approximation fixpoints rec acm symposium principles programming languages pages acm press cousot cousot abstract interpretation application logic program journal logic programming cousot cousot abstract interpretation frameworks journal logic computation cousot cousot formal languages grammar program analysis abstract interpretation rec conference functional programming languages computer architecture pages acm press cousot cousot software analysis model checking proceedings international conference computer aided verification volume lecture notes computer science pages cousot halbwachs automatic discovery linear restraints among variables program records annual acm symposium principles programming languages pages acm acm press january deutsch interprocedural analysis pointers beyond sigplan notices proc acm sigplan conference program language design implementation emelianov baburin semantic analyzer proc international static analysis symposium volume lecture notes computer science pages emelianov sabelfeld analyzer semantic properties modulaprogramms software intellectualization quality pages institute informatics systems novosibirsk russia emelianov analysis equality relation program terms proc third international static analysis symposium volume lecture notes computer science pages even naor schieber sudan approximating minimum feedback sets proc international conference integer programming combinatorial optimization volume lecture notes computer science pages excelsior llc native development toolset gargi sparse algorithm predicated global value numbering proceedings acm sigplan conference programming language design implementation pages gordon pitts editors high order operational techniques semantics publications newton institute cambridge university press granger static analysis arithmetical congruences international journal computer mathematics granger static analysis linear congruence equalities among variables program proceedings international joint conference theory practice software developement volume lecture notes computer science pages gulwani necula discovering affine equalities using random interpretation proc acm principles programming languages pages gulwani necula algorithm global value numbering proc international static analysis symposium volume lecture notes computer science pages halbwachs proy roumanoff verification systems using linear relation analysis formal methods system design nevin heintze joxan jaffar voicu framework combining analysis verification records annual acm symposium principles programming languages pages heintze jaffar set constraint program analysis borning editor principles practice constraint programming volume lecture notes computer science pages jones muchnick editors program flow analysis theory applications jones flow analysis lazy functional programs abramsky hankin editors abstract interpretation declarative languages pages ellis horwood karr affine relationships among variables program acta informatica kochetov effective specialization programs thesis institute informatics systems novosibirsk russia masdupuis array operations abstraction using semantic analysis trapezoid congruences proceedings international conference supercomputing pages acm press masdupuis semantic analysis interval congruences proceedings international conference formal methods programming applications volume lecture notes computer science pages new numerical abstract domain based matrices proceedings second symposium programs data objects volume pages rosen robust linear algorithms cutsets journal algorithms knoop steffen detecting equalities variables combining efficiency precision proc international static analysis symposium volume lecture notes computer science pages sabelfeld polynomial upper bound complexity equivalence decision doklady akademii nauk matematika sabelfeld equivalence decidable information processing letters speckenmeyer feedback problems digraphs proceedings international workshop concepts computer science volume lecture notes computer science pages venet automatic analysis pointer aliasing untyped programs science computer programming wegman zadeck constant propagation conditional branches acm transactions programming language systems appendix analysis kmp appendix presents results application era kmp program generated specializer program specialization program implementing pattern matching match str respect pattern ababb invariants written comments program points hold conclude target string necessarily ends variable equal string length line every time element str lines used second loop value index expression exceed value variable true value variable increment statements inc lines therefore suffices check value beyond ranges determined type input target string line second cycle range checks eliminated assignment cfg counter redundant line conditions str str always false lines respectively two different constants equal code dead conditions lines false however automatic detection properties easily previous using semantic information possible build new program functionally equivalent match ababb str text program given underlined code eliminated module match fio import file open readchar writeint stdout var cfg counter cardinal str file file type type array char var str begin str file open loop str readchar str file str exit else inc end end cfg counter counter loop case cfg counter writeint stdout exit end str cfg counter else inc cfg counter end writeint stdout exit end str cfg counter else inc cfg counter end writeint stdout exit end str cfg counter else inc cfg counter end writeint stdout exit end str writeint stdout exit end str writeint stdout exit else inc writeint stdout exit end str writeint stdout exit end counter str counter counter counter counter str str str str str str line str writeint stdout exit end str cfg counter else inc writeint stdout exit end str cfg counter else inc cfg counter end end else inc cfg counter end else inc cfg counter end end else inc writeint stdout exit end str writeint stdout exit end str str str str str str str str str str str str str str str str str str str str str str str str str str str str str str str str str str str str str cfg counter else inc cfg counter end else inc cfg counter end end writeint stdout exit end str cfg counter else inc cfg counter end writeint stdout exit end str cfg counter else inc cfg counter end writeint stdout exit end str cfg counter else inc cfg counter end writeint stdout exit end str cfg counter else inc cfg counter end end end end match str str str str counter counter counter counter
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may bees algorithm vehicle routing problem aish fenton department computer science university auckland auckland new zealand colophon typeset latex using typefaces computer modern image courtesy preface msc thesis prepared aish fenton university auckland department computer science supervised michael dinneen work undertaken thesis grown research project sponsored new zealand trade enterprise nzte company vworkapp research vehicle route optimisation use within software product acknowledgements like thank partner crime anna jobsis encouragement cajoling threatening bribing dishes proof reading guilt tripping grammar policing comforting whatever took help get done also like thank mum dad encouragement always making feel like higher education within reach thanks steve taylor steve harding holding fort work disappeared thesis likewise thanks team jono rash elena bob marcus yuri robin ball always despite absence thanks brendon petrich vworkapp providing time work supportive undertaking lastly especially owe michael dinneen supervisor big thank persevering even though must doubted ever finish iii abstract thesis present new algorithm vehicle routing problem called enhanced bees algorithm adapted fairly recent algorithm bees algorithm developed continuous optimisation problems show results obtained enhanced bees algorithm competitive best available vehicle routing able achieve results within optimal solution commonly used set test instances show algorithm good runtime performance producing results within optimal solution within seconds making suitable use within real world dispatch scenarios additionally provide short history well known results literature along detailed description foundational methods developed solve vehicle routing problem contents introduction content outline background overview tsp introduction history exact methods classic heuristics constructive heuristics heuristics iterative improvement heuristics simulated annealing genetic algorithms tabu search large neighbourhood search swarm intelligence ant colony optimisation bees algorithm problem definition capacitated vehicle routing problem variants multiple depot vehicle routing problem vehicle routing time windows pickup delivery problem algorithm objectives problem representation enhanced bees algorithm bee movement search space coverage search neighbourhood destroy heuristic repair heuristic neighbourhood extent results enhanced bees algorithm experiments bees algorithm versus enhanced bees algorithm large neighbourhood search summary comparison conclusion chapter introduction thesis present new algorithm solve vehicle routing problem vehicle routing problem describes problem assigning ordering geographically distributed work pool resources aim minimise travel cost required complete work meeting specified constraints maximum shift duration often context used fleet vehicles delivering goods customers although problem equally applied across many different industries scenarios applied applications microchip layout interest vehicle routing problem increased last two decades cost transporting delivering goods become key factor developed economies even decrease percent transportation costs offer savings billions economy context new zealand home country virtually every product grown made used carried truck least lifetime success new zealand export industries inextricably linked reliability cost effectiveness road transport moreover growth national output requires increase transport services vehicle routing problem offers real benefits transport logistics companies optimisation planning distribution process modelled vehicle routing problem offer savings anywhere transportation costs accordingly vehicle routing problem focus intense research since first formal introduction fifties vehicle routing problem one studied combinatorial optimisation problems hundreds papers covering family related problems since introduction fifty years ago challenge vehicle routing problem combines two case variants combinatorially hard problems chapter introduction known membership family problems makes unlikely algorithm exists reasonable runtime performance able solve problem exactly therefore heuristic approaches must developed solve smallest sized problems many methods suggested solving vehicle routing problem thesis develop new algorithm call enhanced bees algorithm adapt another fairly recent algorithm named bees algorithm developed solving continuous optimisation problems show results obtained enhanced bees algorithm competitive best modern available vehicle routing problem additionally algorithm good runtime performance producing results within optimal solution within seconds makes enhanced bees algorithm suitable use within real world dispatch scenarios often dispatch process dynamic hence impractical dispatcher wait minutes hours optimal content outline start chapter providing short history vehicle routing problem well providing background material necessary understanding enhanced bees algorithm particular review classic methods brought bear vehicle routing problem along influential results achieved literature chapter provide formal definition vehicle routing problem briefly describe variant problems developed literature chapter provide detailed description enhanced bees algorithm operation along review objectives algorithm designed meet description algorithm internally represents vehicle routing problem chapter provide detailed breakdown results obtained enhanced bees algorithm algorithm tested well known set test instances due christofides mingozzi toth contrasted well known results literature finally chapter provide summary results achieved enhanced bees algorithm context methods available solving vehicle routing problem literature additionally offer thoughts future directions areas warrant research enhanced bees algorithm developed part new zealand trade enterprise research development grant use within dispatch software product vworkapp hence consideration given runtime performance typically afforded literature chapter background chapter provides short history background material vehicle routing problem particular review solution methods brought bear vehicle routing problem classic results reported literature chapter laid follows start section informally defining vehicle routing problem providing timeline major milestones research also review closely related problem traveling salesman problem cornerstone vehicle routing problem review section exact methods developed solve vehicle routing problem distinguished methods review provide exact solutions globally best answer produced follow section reviewing classic heuristics methods developed vehicle routing problem methods guaranteed find globally best answer rather aim produce close optimal solutions using algorithms fast running times able scale large problem instances section review methods adapted vehicle routing problem methods provide competitive results available solving vehicle routing problem considered currently lastly section review modern family called swarm intelligence inspired problem solving abilities exhibited groups animals natural processes last methods become popular area research recently starting produce competitive results many problems thesis uses swarm intelligence method solving vehicle routing problem chapter background overview vehicle routing problem commonly abbreviated vrp describes problem assigning ordering work finite number resources cost undertaking work minimised often context used fleet vehicles delivering goods set customers although problem equally applied across many different industries scenarios including scenarios microchip layout aim split deliveries vehicles specify order vehicle undertakes work distance travelled vehicles minimised constraints met classic version vrp constraints must met vehicle must start end route depot goods must delivered goods dropped single time single vehicle good requires specified amount capacity however vehicle finite amount capacity exceeded adds complexity problem necessarily influences selection deliveries assigned vehicle figure example customers assigned three different vehicle routes depot black dot centre formally vrp represented graph vertices graph represent locations visited includes customer location location depot convenience let denote vertex represents depot denote set customers next let set edges correspond valid connections customers connections vrp connections possible edge corresponding cost cij cost typically travel distance two locations overview solution given vrp instance represented family routes denoted route sequence customer visits performed single vehicle denoted customer demand maximum demand permissible route maximum capacity cost solution value aim minimise given following formula cvi formalise vrp constraints follows equation specifies customers included least one route equations ensure customer visited across routes equation ensures route starts ends depot lastly equation ensures route exceed capacity version problem come known capacitated vehicle routing problem often appreciated cvrp literature see chapter alternative formation states problem integer linear programming problem standard vrp vrp first formally introduced dantzig ramser paper truck scheduling problem vrp remained important problem logistics transport one studied combinatorial optimisation problems hundreds papers written intervening fifty years large number implementations use today clear vrp real benefits offer transport logistics companies anywhere believe formation provided chapter simpler precise understanding algorithmic methods described chapter however provide standard formation chapter chapter background savings reported vehicle routing procedure implemented vrp comes family related problems problems model constraints encountered real world applications vrp classic problems include vrp time windows vrptw introduces time window constraint customer vehicle must arrive within vrp multiple depots mdvrp vehicles dispatched multiple starting points pickup delivery problem pdp goods picked delivered course route courier would tsp introduction history vrp combination two problems combinatorial hard traveling salesman problem precisely multiple traveling salesman problem bin packing problem traveling salesman problem tsp informally defined follows given points map provide route points point used total distance travelled minimised problem name traveling salesman comes classic real world example problem salesman sent trip visit cities must select order visit cities travel least amount distance although problem sounds like might easily solvable fact best known exact algorithms solving tsp still require running time karp famous paper reducibility among combinatorial problems showed hamiltonian circuit problem implied tsp thus supplied mathematical explanation apparent difficulty finding optimal traveling salesman tours tsp history reaching back many years related another classic graph theory problem hamiltonian circuit hamiltonian circuits studied since hamilton kirkman whereas tsp informally discussed many years become actively studied menger whitney flood robinson produced much early results field robinson rand report probably first article call problem name since become known traveling salesman problem overview robinson rand report purpose note give method solving problem related traveling salesman problem one formulation find shortest route salesman starting washington visiting state capitals returning washington generally find shortest closed curve containing given points plane figure shown example city tsp tour tour one possible tours cities early result provided dantzig fulkerson johnson paper gave exact method solving city problem large number cities time algorithm used cutting plane method provide exact solution approach inspiration many subsequent approaches still bedrock algorithms attempt provide exact solution generalisation tsp multiple traveling salesman problem mtsp multiple tours constructed multiple salesman used visit cities pure mtsp trivially turned tsp constructing graph additional copies starting vertex forbidding travel directly starting vertices however pure formulation mtsp places additional constraints routes constructed real life applications mtsp typically require additional constraints limiting size duration route one salesman working hour shift another work mtsp leads naturally family problems given vrp vrp family related problems understood generalisation mtsp incorporates additional constraints constraints capacity limits introduce additional dimensions problem hard combinatorial problems chapter background exact methods first efforts providing solution vrp concerned exact methods started sharing many techniques brought bear tsp follow laporte nobert survey classify exact methods vrp three families direct tree search methods dynamic programming integer linear programming first classic direct tree search results due christolds ellison paper provided first branch bound algorithm exactly solving vrp unfortunately time memory requirements large enough able solve problems customers result later improved upon christolds using different branch model improvement allowed solve customers christofides mingozzi toth provide lower bound method sufficiently quick terms runtime performance used lower bound excluding nodes search tree using lower bound able provide solutions problems containing customers laporte mercure nobert used mtsp relaxation vrp within branch bound framework provide solutions realistically sized problems containing customers dynamic programming approach first applied vrp eilon watsongandy christofides approach allowed solve exactly problems customers since christofides made improvements algorithm solve exactly problems fifty customers set partitioning method given balinski quandt produce exact vrp solutions however problem sets used small containing customers even able produce solutions problems however taking approach starting point many authors able produce powerful methods rao zionts foster ryan desrochers desrosiers solomon extended basic set partitioning algorithm using column generation method integer programming later papers produced best exact results notwithstanding preceding discussion exact methods use advancing theoretical understanding vrp providing solutions real life routing problems mostly attributed fact real life vrp instances often involve least tens customers often hundreds involve richer constraints modelled classic vrp classic heuristics classic heuristics section review classic heuristic methods developed vrp methods guaranteed find globally best answer rather aim produce close optimal solutions using algorithms fast running times able scale large problem instances classic heuristics vrp classified three families constructive heuristics heuristics divided two subfamilies cluster first route route first cluster finally improvement methods constructive heuristics start looking constructive heuristics constructive heuristics build solution ground typically provide recipe building route total cost routes minimised trivial intuitive constructive heuristic nearest neighbour method method routes built sequentially step customer nearest last routed customer chosen continues route reaches maximum capacity point new route started practice nearest neighbour algorithm tends provide poor results rarely used figure shown example nearest neighbour method applied partially constructed route selects customer add closest last added customer early influential result given clarke wright paper paper present heuristic extending dantzig ramser earlier work since become known clarke wright savings heuristic heuristic based simple premise iteratively combining routes order pairs provide largest saving chapter background figure clark wright savings algorithm customers selected candidates merge merge results new route algorithm works follows algorithm clark write savings algorithm initialiseroutes savingsmatrix sortbysavings lij findroutes lij feasiblemerge combineroute end end algorithm starts initialising candidate solution creates route calculates matrix contains savings sij cij edges produces list enumerates cell matrix descending order savings entry list lij selects two routes contain customers tests see two routes merged merge permissible first last vertices excluding depot respective routes combined demand two routes exceed maximum allowed heuristic comes two flavours sequential parallel sequential version adds additional constraint one route constructed time case one two routes considered must route construction classic heuristics neither routes route construction list item ignored processing continues list merge permissible merge routes parallel version entire list savings enumerated resulting solution returned answer sequential version loop repeated feasible merges remain clark write savings heuristic used solve problems customers results often within optimal solution using seconds runtime parallel version clark write savings algorithm outperforms sequential version cases typically one employed heuristic proven surprisingly adaptable extended deal specialised vehicle routing problems additional objectives constraints must factored flexibility result algebraic treatment problem unlike many vrp heuristics exploit problem spatial properties many heuristics see section savings formula easily adapted take consideration objectives example solomon equally ubiquitous algorithm extends clark wright savings algorithm cater time constraints classic algorithm extended gaskell yellow paessens suggested alternatives savings formulas used clarke wright approaches typically introduce additional parameters guide algorithm towards selecting routes geometric properties likely produce better combinations altinkemer gavish provide interesting variation basic savings heuristic use matching algorithm combine multiple routes step construct graph vertex represents route edge represents feasible saving edges weights represent savings realised merge two routes algorithm proceeds solving maximum cost weighted matching graph heuristics next look heuristics start looking cluster first route second subfamily one foundational algorithms method given gillett miller provided new approach called sweep algorithm paper popularised approach although similar method suggested earlier wren book subsequently wren holliday paper approach initial clustering phase used cluster customers base set routes routes treated separate tsp instances optimised accordingly approach typically prescribe method tsp solved assumes already developed tsp methods used classic sweep algorithm uses simple geometric method cluster customers routes built sweeping ray centered chapter background depot clockwise around space enclosing problem locations sweep method surprisingly effective shown solve several benchmark vrp problems within best known solutions figure diagram shows example sweep process run ray swept clockwise around geographic area example one route already formed second start customer fisher jaikumars paper builds upon approach providing sophisticated clustering method solve general assignment problem form clusters instead limitation method amount vehicle routes must fixed front method often produces results better results produced classic sweep algorithm christofides mingozzi toth expanded upon approach proposed method uses truncated branch bound technique similar christofides exact method step builds collection candidate routes particular customer evaluates route solving tsp selects shortest tsp route petal algorithm natural extension sweep algorithm first proposed balinski quandt extended foster ryan basic process produce collection overlapping candidate routes called petals solve set partitioning problem produce feasible solution approaches assumed order customers within route solved using existing tsp heuristic petal method produced competitive results small solutions quickly becomes impractical set candidate routes must considered large lastly route first cluster second methods basic premise techniques first construct grand tsp tour customers visited second phase concerned splitting tour feasible routes route first cluster second methods generally thought less competitive methods although interestingly haimovich rinnooy kan shown customers unit demand simple shortest path algorithm solved polynomial time used produce solution tsp tour classic heuristics asymptotically optimal iterative improvement heuristics iterative improvement methods follow approach initial candidate solution iteratively improved applying operation improves candidate solution typically small way many thousands times operations employed typically simple change small part candidate solution position single customer edge within solution set solutions obtainable current candidate solution applying operator known neighbourhood typically iterative improvement heuristics new solution selected exhaustively searching entire neighbourhood best improvement possible improvement found heuristic terminates initial candidate solution starting point algorithm randomly selected produced using another heuristic constructive heuristics typically used initially seeding improvement heuristic see section information probably one best known improvement operators operator takes two edges edges traversed particular route removes candidate solution splits route two disconnected components new candidate solution produced reconnecting using vertices alternate edges figure diagram shows applied candidate solution producing new solution example edges exchanged edges rationale behind due triangle inequality edges cross unlikely optimal aims detangle route chapter background number operations suggested literature christofides eilon give one earliest iterative improvement methods paper paper make simple change increase amount edges removed two operation fittingly called found heuristic produced superior results general operations remove edges search optimal recombination components take number edges removed profitable strain research focused producing operations reduce amount recombinations must searched presents operation since come known restricted searches relocation sets consecutive vertices calls chains improvement made improvement made tries chains consecutive vertices shown produce similar results running time recently renaud boctor laptorte presented restricted version called operates similar vein running time denotes number edges spanned building chain iterative improvement heuristics often used combination heuristics case run candidate solution initial heuristic completed however used way often fine balance producing operation improves solution one sufficiently destructive enough escape local minimum interest iterative improvement heuristics grown operations developed directly applicable modern heuristics family known presented next section broad collection methods make assumptions type problem solved provide framework allows individual problems modelled plugged typically take approach candidate solution solutions initially produced iteratively refined towards optimal solution intuitively thought searching problem search space iteration searches neighbourhood current candidate solution looking new candidate solutions move closer global optimum limitation guaranteed find optimal solution even good one moreover theoretical underpinnings makes one effective another still poorly understood figure diagram shows example search space metaheuristic moves example peak centre figure globally best answer also hills valleys metaheuristic may become caught called local minima maxima within literature tend tuned specific problems validated empirically number produced vrp recent years many competitive results produced last ten years due next review well known results vrp simulated annealing simulated annealing inspired annealing process used metallurgy algorithm starts candidate solution randomly selected moves nearby solutions probability dependent quality solution global parameter reduced course algorithm classic implementations following formula used control probability move represent solution quality current solution new solution respectively analogy metallurgy process represents current temperature solution initially set high value lets algorithm free local optima may caught cooled course algorithm forcing search converge solution chapter background one first simulated annealing results vrp given robuste daganzo souleyrette define search neighbourhood solutions obtained current solution applying one two operations relocating part route another position within route exchanging customers routes tested solution large real world instances customers reported success approach test cases unique direct comparison possible osman given best known simulated annealing results vrp algorithm expands upon many areas basic simulated annealing approach method starts using clark wright algorithm produce initial position defines neighbourhood candidate solutions reached applying operator names operation works selecting two sequences chains customers two routes note chains necessarily length customers within chain exchanged turn exchange produces infeasible solution neighbourhood produced typically quite large osman restricts less suggests first move provides improvement used rather exhaustively searching entire neighbourhood figure diagram shows example applied candidate solution two sequences customers selected routes respectively customers swapped feasible osman also uses sophisticated cooling schedule main change temperature cooled improvements found improvement found resets temperature using max reset temperature temperature best solution found far although simulated annealing produced good results many cases outperforms classic heuristics compare competitive tabu search methods discussed section genetic algorithms genetic algorithms first proposed since applied many problem domains particularly well suited applications must work across number different domains fact first algorithm applied combinatorial problems basic operation genetic algorithm follows algorithm simple genetic algorithm generate initial population termination condition met evaluate fitness individual select fittest pairs mate pairs produce next generation mutate optional end classic genetic algorithm candidate solution encoded binary string chromosome individual candidate solution initially created randomly used seed population technique often employed literature initially bootstrap population making use another heuristic produce initial population however special care must taken approach ensure diversity maintained across population risk premature convergence introducing enough diversity initial population next fittest individuals selected population mated order produce next generation mating process uses special operator called crossover operator takes two parents produces offspring combining parts parent optionally mutation operation also applied introduces change exist either parent classic crossover operation takes two individuals encoded binary strings splits one two points along length string strings recombined form new binary string turn encodes new candidate solution entire process continued termination condition met often predetermined running time population converged single solution special consideration needs given problems encoded crossover mutation operators work using genetic algorithms solve discrete optimisation problems vrp example classic crossover operation works binary strings would work well tsp tour two components two tours combined way likely contain duplicates therefore common vrp tsp use direct representation use specially designed crossover operators instance vrp represented set sequences holding ordered list customers chapter background crossover operators designed take consideration constraints vrp two crossover operators commonly used combinatorial problems order crossover edge assembly crossover eax operates selecting two cut points within route substring two cut points copied second parent directly offspring likewise string outside cut points copied first parent offspring duplicates removed potentially leaves partial solution customers routed partial solutions repaired inserting unrouted customers child order appeared second parent figure diagram shows crossover operator applied two tours tsp child produced taking customers position second parent injecting position first parent removing duplicates leaves customers unrouted reinserted back child order appear first parent another common crossover operator eax eax originally designed tsp adapted vrp eax operates using following process combine two candidate solutions single graph merging solution edge sets create partition set graph cycles alternately selecting edge graph randomly select subset cycles generate incomplete child taking one parents removing edges selected subset cycles add back edges parent chosen cycles child connected route repair iteratively merging disconnected cycles connected cycles alternative interesting approach found literature instead encode set operations parameters fed another heuristic turn figure diagram shows example eax applied two parent solutions parents first merged together new graph created selecting alternate edges parent subset cycles taken applied edges removed child solution produced infeasible contains broken routes would need repaired produces candidate solution well known example approach suggested encoded ordering customers ordering fed insertion heuristic produce actual candidate solutions influential result uses genetic algorithms solve vrptw given gideon algorithm gideon uses approach inspired sweep method overview sweep method provided section builds routes sweeping ray centered depot clockwise around geographic space enclosing customer locations customers collected candidate routes based set parameters refined genetic algorithm gideon uses genetic algorithm evolve parameters used algorithm rather operate problem directly finally gideon uses local search method optimise customers within route making use operator description operator provided section generally speaking genetic algorithms competitive chapter background solving vrp however recently two promising applications genetic algorithms used solve vrp nagata adapted eax operator use vrp berger barkaoui presented hybrid genetic algorithm called adapts construction heuristic use crossover operator basic premise select set routes parent located close one another customers removed one parent inserted second using operation inspired solomon construction heuristic vrptw methods reached best known solution number classic vrp benchmark instances christofides mingozzi toth competitive best tabu search methods tabu search tabu search follows general approach shared many iteratively improves candidate solution searching improvements within current solution neighbourhood tabu search starts candidate solution may generated randomly using another heuristic unlike simulated annealing best improvement within current neighbourhood always taken next move introduces problem cycling candidate solutions overcome tabu search introduces list solutions already investigated forbidden next moves hence name first instance tabu search used vrp willard willard approach made use fact vrp instances transformed mtsp instances solved algorithm uses combination simple vertex exchange relocation operations although opening door research results competitive best classic heuristics osman gives competitive use tabu search simulated annealing method makes use operation define search neighbourhood osman provides two alternative methods control much neighbourhood searched selecting next move searches entire neighbourhood selects move optimal searches move found optimal current position osman heuristic produced competitive results outperformed many heuristics however since refined improved upon newer tabu search methods toth vigo introduced concept granular tabu search gts method makes use process removes moves neighbourhood unlikely produce good results reintroduce moves back process algorithm stuck local minimum idea follows existing idea known candidate lists toth vigo method produced many competitive results taillard provided one successful methods solving vrp tabu search method talliard tabu search uses neighbourhood structure borrows two novel concepts use sophisticated tabu mechanism duration number iterations item tabu chosen randomly diversification strategy vertices frequently moved without giving improvement penalised novel aspect taillard algorithm decomposition problem problem split regions using simple segmentation region centred depot taillard also provides alternative approach problems customers evenly distributed around depot subproblem solved individually customers exchanged neighbouring segments periodically taillard observes exchanging customers beyond geographically neighbouring segments unlikely produce improvement moves safely ignored taillard method produced currently best known results standard christofides mingozzi toth problem sets large neighbourhood search large neighbourhood search commonly abbreviated lns recently proposed heuristic shaw large neighbourhood search type heuristic belonging family heuristics known large scale neighbourhood search vlsn large scale neighbourhood search based simple premise rather searching within neighbourhood solutions obtained single typically quite granular operation might profitable consider much broader neighbourhood candidate solutions obtained applying many simultaneous changes candidate solution distinguishes heuristics others neighbourhoods consideration typically exponentially large often rendering infeasible search therefore much attention given providing methods successfully traverse neighbourhoods lns somewhat confusingly named given type vlsn competing approach chapter background large neighbourhood search uses destroy repair metaphor searches within neighbourhood basic operation follows algorithm large neighbourhood search initial solution termination condition met destroy repair better current solution end end result starts selecting starting position done randomly using another heuristic iteration algorithm new position generated destroying part candidate solution repairing new solution better current solution selected new position continues termination conditions met large neighbourhood search seen type large scale neighbourhood search iteration number neighbouring solutions exponentially large based number items removed destroyed obviously key components approach functions used destroy repair solution care must given functions constructed must pinpoint improving solution large neighbourhood candidates also providing enough degrees freedom escape local optimum empirical evidence literature shows even surprisingly simple destroy repair functions effective applications large neighbourhood search vrp pair simple operations commonly used often alongside complex ones destroy repair functions specifically solution destroyed randomly selecting removing customers repaired finding least cost reinsertion points back solution customers shaw applied large neighbourhood search vrp original paper introducing method introduced novel approach destroy repair functions destroy function removes set related customers defines related customer two customers share similar geographic location sequentially routed share number similar constraints overlapping time windows time constraints used idea removing related customers simply removing random customers related customers likely profitably stated another way unrelated customers likely reinserted back original positions shaw repair swarm intelligence tion makes use simple branch bound method finds minimum cost reinsertion points within partial solution results immediately impressive reached many best known solutions christofides mingozzi toth problems recently ropke proposed extension basic large neighbourhood search process method adds concept using collection destroy repair functions rather using single pair function use selected iteration based previous performance way algorithm adapts use effective function search neighbourhood ropke makes use several destroy functions uses simple random removal heuristic shaw removal heuristic worst removal heuristic removes costly customers terms customer contribution route overall cost likewise makes use several different insertion functions include simple greedy insertion heuristic novel insertion method calls regret heuristic informally regret heuristic reinserts customers first impacted terms increased cost inserted optimum positions specifically let set customers reinserted let xik variable gives lowest cost inserting customer partial solution let words cost difference inserting customer second best position first iteration repair function choose customer maximises max ropke presents series results show large neighbourhood search competitive solving vrp related problems vrptw pdptw darp considering large neighbourhood search proposed successful short space time attracted large amount research produced competitive results solving vrp swarm intelligence recent area research producing heuristics mimic certain aspects swarm behaviour probably well known heuristics family particle swarm optimisation pso ant colony optimisation aco real life swarm intelligence interesting combinatorial optimisation researchers demonstrates form emergent intelligence individual members limited reasoning capability simple behaviours still able arrive optimal solutions complex resource allocation problems chapter background context combinatorial optimisation behaviours mimicked exploited algorithms make use approach produce solutions simulating behaviour across number agents typically perform rudimentary operations feature class algorithms ease parallelised making easily adaptable large scale problems swarm intelligence algorithms employed solve number problems look two examples ant colony optimisation bees algorithm thesis makes use ant colony optimisation ant colony optimisation inspired ants forage food communicate promising sites back colony real life ants initially forage food randomly find food source return colony process lay pheromone trail ants stumble upon pheromone trail follow probability dependent strong therefore old pheromone trail follow find food return colony thus also strengthening pheromone trail strength pheromone trail reduces time meaning younger shorter pheromone trails take long traverse attract ants figure diagram depicts ants make use pheromone trails optimise exploitation local food sources ant colony optimisation mimics behaviour graph simulating ants marching along graph represents problem solved basic operation swarm intelligence algorithm follows algorithm ant colony optimisation data graph representing problem termination condition met positionants solution built marchants end updatepheromones end iteration algorithm ants positioned randomly within graph ants stochastically marched graph completed candidate solution case tsp would tour vertices stage march ant selects next edge based following probability formula pkij pkij probability ant traverse edge set edges traversed ant yet amount pheromone deposited edge desirability edge based priori knowledge specific problem global parameters control much influence term march complete set candidate solutions constructed ant pheromone deposited edge using following equation pheromone persistence function gives amount pheromone deposited ant function defined edge visited ant otherwise chapter background represents total distance travelled graph ant ensures shorter paths result pheromone deposited example ant colony optimisation use combinatorial problems show applied tsp build weighted graph representing city visited wij representing cost travel city step iteration ensure following constraints met city visited set equal wij ant colony optimiser starts positions ant randomly selected vertex city within graph step ant march builds tour cities ant completed tour serves candidate solution tsp initially solutions low quality use length tours ensure pheromone deposited shorter tours end iterations ants converged near optimal solution like guarantee global optimum figure shown example aco used solve tsp initially ants explore entire graph end iteration optimal tours pheromone deposited meaning next iteration ants likely pick edges constructing tour eventually ants converge solution ant colony optimisation applied vrp bullnheimer hartl strauss adapted straightforward implementation used tsp detailed preceding discussion forcing ant create new route time exceeds capacity maximum distance constraint also use modified edge selection rule takes account vehicle capacity proximity depot updated rule given swarm intelligence pkij sij sil represents proximity customers depot giving maximum capacity giving capacity already used vehicle additional load added influences ants take advantage available capacity bullnheimer implementation ant colony optimisation vrp produces good quality solutions christofides mingozzi toth problems competitive best modern recently reimann stummer doerner presented competitive implementation ant colony optimisation vrp implementation operates graph represent savings combining two routes given classic clark wright savings heuristic see section heuristic ant selects ordering merges applied implementation reported competitive best bees algorithm last decade inspired success ant colony optimisation number algorithms proposed aim exploit collective behaviour bees includes bee colony optimisation applied many combinatorial problems marriage honey bees optimization mbo used solve propositional satisfiability problems beehive used timetabling problems virtual bee algorithm vba used function optimisation problems mating optimisation hbmo used cluster analysis finally bees algorithm focus thesis see bibliography high level overview many algorithms bees algorithm first proposed inspired foraging behaviour honey bees bee colonies must search large geographic area around hive order find sites enough pollen sustain hive essential colony makes right choices sites exploited much resource expended particular site achieve sending scout bees directions hive scout bee found promising site returns hive recruits hive mates forage site bee performing waggle dance dance communicates location quality site fitness time bees successfully forage site recruited exploit aspect bee behaviour shares similarities ant foraging behaviour chapter background figure shown waggle dance performed honey bee image courtesy direction moved bee indicates angle bees must fly relative sun find food source duration dance indicates distance informally algorithm described follows bees initially sent random locations fitness site calculated proportion bees reassigned sites highest fitness values bee searches local neighbourhood site looking improve site fitness remainder bees sent scouting new sites words set new random position process repeats one sites reaches satisfactory level predetermined termination condition met formally algorithm operates follows algorithm bees algorithm settorandomposition termination condition met sortbyfitness searchneighbourhood cnep searchneighbourhood dnsp settorandomposition end set bees used explore search space initially bees set random positions function sortbyf itness sorts bees order maximum fitness proceeds taking promising sites found bees partitioning two sets first best sites represents called elite bees next promising sites searchn eighbourhood function explores neighbourhood around provided set bees site explored nep bees recruited search swarm intelligence nsp recruited search practice means nep nsp number positions explored within neighbourhoods sites respectively moves typically made stochastically possible deterministic approach used remaining bees words set random positions repeated termination condition met may running time threshold predetermined fitness level advantage promised bees algorithm ability escape local optima ability navigate search topologies rough terrain figure achieves scouting search space promising sites committing resources exploration sites produce better results figure shown search space many valleys hills search spaces provide challenge approaches many local minima maxima get caught bees algorithm ameliorates searching many different areas simultaneously bees algorithm applied manufacturing cell formation training neural networks pattern recognition scheduling jobs production machine data clustering many others areas see examples comprehensive bibliography however best knowledge bees algorithm adapted vehicle routing problem chapter background chapter problem definition chapter provide formal definition vrp briefly describe variant problems arisen literature capacitated vehicle routing problem cvrp correct name vrp distinguishes variants start section providing formal definition cvrp formulate integer linear programming problem become standard vrp literature follow section overview vrp variants commonly used capacitated vehicle routing problem formulate cvrp integer linear programming problem although possible solve cvrp using integer programming solver uncommon practice best solvers still able solve small problem sizes provide formulation become lingua franca combinatorial problems start formation specifying variables used within represent cvrp weighted graph vertices graph represent locations visited includes customer location location depot convenience let denote vertex represents depot denote set customers thus set vertices given let set edges correspond valid connections customers connections depot cvrp connections possible words set clique edge correspondingpcost cij let cost euclidian distance two locations cij represent coordinates customer location chapter problem definition use denote set vehicles used visit customers maximum number vehicles allowed define maximum capacity maximum work duration respectively allowable vehicle demand required capacity customer denoted likewise denote service time required customer use decision variable xijk denote particular edge traversed vehicle words travels customers true let xijk xijk use sequencing variable gives position customer within route vehicle visits able define problem follows minimise cij xijk subject xijk xvkd xjv xijk xijk xijk cij xijk xic xcj objective function minimises costs cij constraint ensures customer serviced single vehicle constraint enforces capacity constraint vehicle exceed maximum vehicle capacity likewise constraint enforces vehicle work duration constraint vehicle work duration sum service times customer visited vehicle travel time convention travel time taken equal variants distance traversed vehicle turn equal costs cij edges traverses constraints ensure vehicle starts depot finishes depot exactly constraint constraint flow constraints ensure number vehicles entering customer equal number vehicles leaving eliminated lastly constraint ensures integrality conditions constraint enforces maximum vehicle work duration often left traditional cvrp formation included present problem instances use benchmarks chapter variants section provide overview common variations vrp used variations arisen real world vehicle routing scenarios constraints often involved modelled cvrp multiple depot vehicle routing problem simple extension cvrp allow vehicle start different depot part problem becomes assigning customers depots hard combinatorial problem cvrp formation easily relaxed allow two variations problem one constrains vehicle finish depot starts allows vehicles start finish depot long number vehicles return depot left vehicle routing time windows vehicle routing problem time windows vrptw adds additional constraint classic vrp customer must visited within time window specified customer formally vrptw customer also corresponding time window goods must delivered vehicle permitted arrive start time however case vehicle must wait time adding time takes complete route however permitted job start time additional constraint added formation cvrp ensure time window constraints met sik decision variable sik provides time vehicle arrives customer chapter problem definition pickup delivery problem pickup delivery problem pdp generalises vrp problem goods picked delivered vehicle along route vehicle work comes two flavours pickup jobs delivery jobs additional constraints added cvrp formation ensure pickup deliver jobs completed vehicle represents sequence jobs undertaken vehicle pickup job appears corresponding delivery job sequence jobs undertaken vehicle vehicles capacity exceeded goods loaded unloaded requires use intermediate variable yik represents load vehiclepk customer adds constraints xijk yjk yik yik enforce also variation pdp adds time windows called pdptw case extra constraints vrptw problem merged given pdp much harder problem computationally cvrp extra constraints add new dimensions problem complexity pdp actively researched last decade chapter algorithm chapter provides detailed description enhanced bees algorithm algorithm developed thesis operation start reviewing objectives algorithm designed meet section section provide description algorithm internally represents vrp problem candidate solutions next section provide detailed description operation algorithm finally section describe neighbourhood structures used algorithm define search space objectives enhanced bees algorithm built use commercial setting developed part new zealand trade enterprise grant company vworkapp scheduling dispatch software accordingly different objectives aimed design runtime performance typically sought vrp literature algorithm objectives order priority follows ensure constraints met specifically route maximum duration observed good runtime performance desirable algorithm produce reasonable quality result quickly within seconds produce better result require longer processing time specifically algorithm could reach optimum value within seconds would sufficient produce good quality results notwithstanding objective results produced must close global optimum chapter algorithm design lends parallelisation able make use additional processing cores available within modern hardware problem representation enhanced bees algorithm represents problem direct straightforward manner directly manipulates candidate solution set routes route contains ordered sequence customers starting ending depot vertex figure shown example simple vrp candidate solution represented internally enhanced bees algorithm general representations sometimes used commonly seen genetic algorithms allow algorithm easily adapted combinatorial problems however often comes cost added complexity inferior algorithm designed specifically solving instances vrp direct representation chosen algorithm makes use fitness concept common many describe cost solution fitness function includes terms distance cost solution penalties breaking capacity maximum route duration constraints enhanced bees algorithm uses penalties encourage feasible solutions produced rather outright barring infeasible solutions fitness function allows algorithm wriggle room traverse way towards feasible solution occurs operators act problem representation longer exploit information specific problem domain must rely general purpose operations instead enhanced bees algorithm specifically defined follows max max function calculates cost distance given route function calculates overcapacity given route define overcapacity much larger sum route demands stated maximum allowable capacity likewise function calculates overtime given route route duration calculated sum customer service times travel time convention travel time equal distance route function returns much maximum allowable route duration duration lastly fitness function weighted sum three terms parameters used used control much influence term determining candidate solution fitness purposes benchmarking algorithm see chapter use travel cost equal euclidian two points real life problems found using manhattan often provides superior results presumably due manhattan distance better modelling road system tested auckland new zealand although strict grid still closer euclidian distance models enhanced bees algorithm algorithm based bees algorithm see section overview standard bees algorithm enhanced bees algorithm makes changes adapt bees algorithm vrp domain interesting aspect bees algorithm covers broad search area minimising risk stuck local optimum achieves randomly probing bees algorithm parlance scouting many areas search space entire run however approach well suited hard combinatorial problems newly constructed specifically use cij specifically use cij chapter algorithm solution let alone randomly generated one often far optimal instance clark wright savings heuristic still produces solutions global optimum would require many operations get close optimal adapted bees algorithm many unique characteristics like relative robustness maintained working well hard combinatorial problems vrp enhanced bees algorithm summarised high level follows algorithm enhanced bees algorithm seedsites termination condition met explore removeworstsite end end end algorithm maintains collection sites site maintains collection bees bee proxy problem domain trying solve case vrp problem representation covered section initially site seeded site contains collection bees bee corresponding vrp candidate solution candidate solution initialised seeding route randomly chosen customer filled using insertion heuristic outlined section site turn improved upon achieved iteratively exploring neighbourhood site process used explore site majority algorithm processing takes place interesting aspects algorithm come play exploration process covered detail sections number sites explored reduced run algorithm borrows idea cooling schedule used simulated annealing sites reduced using formula mod represents worst site terms fitness represents current iteration algorithm represents period iterations number sites reduced algorithm complete solution best enhanced bees algorithm overall fitness returned answer next section review detail aspect algorithm bee movement bees moved around search space look improvements collection candidate solutions maintained bee represents candidate solution valid bee move new candidate solution reached within neighbourhood see section operations neighbourhood defined neighbourhood best site position last best positions sitek sitei figure three sites shown along neighbourhoods site shows detail site maintains list last promising positions exploration feature enhanced bees algorithm two bees occupy position algorithm maintains register current positions occupied bee use current fitness quick simple representation bee current bee tries occupy position another bee share candidate solution bee trying occupy position forced explore neighbourhood find another position obviously work circumstances reasonable likelihood two candidate solutions case problem instances used thesis may need modified algorithm used general problem instances chapter algorithm enforcing constraint bee must occupy unique position two benefits forces diversification bees sites hence encouraging greater proportion search space explored increases chance local optimum escaped bee ensnared local optimum forces remainder hive explore alternative positions feature similar intent effect tabu lists used tabu search another feature enhanced bees algorithm role sites play concentrating exploration certain areas search space site maintains list last best positions taken launching point site bees explore bees recruited exploration positions explored best positions taken used launching points site next round exploration exploration method two purposes firstly allows simple type branching promising positions traversed way current position also explored secondly prevents cycling promising solutions close vicinity conversely sites interact site maintains unique list promising positions constraint two bees occupy position ensures site covers non overlapping area search space practice found sufficient encourage sites diverge explore distinct areas search space search space coverage mentioned one unique aspects bees algorithm ability produce robust results probing large area search space however work well hard combinatorial problems ascertained quickly area search space shows promise overcome limitation instead use approach inspired simulated annealing use cooling schedule bees initially divided equally site ensuring site explored equally every period iterations reduce number sites maintained site lowest fitness measure site fitness fitness best position found date process continues single site remains show experimentally chapter process improves robustness algorithm produces better results overall standard bees algorithm search neighbourhood search neighbourhood already discussed bee seeks improve upon current fitness exploring local neighbourhood solution represents enhanced bees algorithm applying large neighbourhood search lns operator candidate solution lns operator differs common vrp operators single operation applies many changes candidate solution widens neighbourhood encompass exponentially many candidate solutions lns navigates vast space spans selecting changes high likelihood improving solution lns operation comprised destroy phase repair phase lns destroy phase used vrp typically involves removing proportion customers solution enhanced bees algorithm destroy phase uses two heuristics along lines somewhat intelligent heuristic attempts remove customers likely able recombined profitable way simple random selection heuristics covered detail section second phase lns used repair partial solution enhanced bees algorithm uses simple insertion heuristic inserts customers locations lowest insertion cost heuristic covered formally section destroy heuristic enhanced bees algorithm employs two destroy heuristics first simply selects customers randomly solution removes routes second slightly complicated due shaw shaw removal heuristic stochastically selects customers higher likelihood customers related one another removed purposes define related mean two customers either geographically close one another cij small share adjacent position within route rationale removing related customers likely profitably exchanged one another conversely unrelated customers likely reinserted back positions removed repair heuristic repair heuristic used enhanced bees algorithm randomly selects one removed customers calculates cost reinserting pair chapter algorithm jobs actually reinsertion positions considered see section description positions considered reinsertion cost calculated follows cij cjk cik cost calculates cost difference terms travel distance defined route customer inserted respectively functions defined section final cost sum added travel distance two extra penalties route overcapacity maximum duration algorithm selects position lowest insertion cost reinsert customer repeated customers reinserted solution reason customers reinserted random order adds beneficial amount noise heuristic ensures healthy diversity solutions generated heuristic neighbourhood extent use two techniques adjust extent neighbourhood searched first technique use allows algorithm flexibility selecting infeasible solutions seen formulation candidate solutions fitness values see section violations problem capacity duration constraints penalised rather forbidden allows bees navigate infeasible solutions aspects solution sufficiently attractive enough outweigh penalties however feasible solutions allowed counted final solutions returned algorithm second technique use adjust number insertion positions considered part repair heuristic number insertion positions considered starts sides three closest customers increases site ages formally let customer inserted ordered sequence candidate insertion points lvi kept lists customers increasing geographic distance cij lns repair operator tests positions lvi find cheapest insertion point repair operator tests possible insertion points represented lvi tests insertion cost inserting immediately route contains search neighbourhood site also maintain counter denotes age site site age incremented iteration site improve upon currently best known solution defined solution fitness whenever site improves upon best known solution counter reset initial neighbourhood final neighbourhood figure diagram shows vertex considered reinsertion back solution starts considering insertion points geographically close position search area widened course algorithm take consideration larger set insertion points use following formula increase much lvi considered site ages min constant controls rate search area expanded process extends number insertion positions considered repair heuristic also serves extend neighbourhood solutions surrounding candidate solution way algorithm also dynamically extends size neighbourhood surrounding site becomes stuck local optimum chapter algorithm chapter results chapter provide detailed breakdown results obtained enhanced bees algorithm algorithm tested well known set test instances christofides mingozzi toth start section presenting results obtained running algorithm two standard configurations first configuration optimised produce best overall results regardless runtime performance second configuration optimised produce best results possible within second runtime threshold follow section providing contrasting results problem instances instead using standard bees algorithm lns local search aim demonstrate enhancements suggested thesis fact improve solution quality finally end section comparing ranking enhanced bees algorithm performs compared results literature enhanced bees algorithm results depicted figures show algorithm performance two configurations first configuration optimised produce best overall results minimum travel distance meets capacity duration constraints consideration made algorithm runtime performance configuration configuration denoted best following diagrams tables second configuration optimised produce best results possible within second runtime window configuration denoted fast following diagrams tables results depicted figures standard vrp problem instances due christofides mingozzi toth obtained chapter results percentage best known average time secs figure best shown results obtained standard christofides mingozzi toth problem instances algorithm optimised producing best overall results algorithm left run minutes lefthand axis shows relative percentage compared best known result problem instance bottom axis shows elapsed runtime seconds infeasible solutions shown best known result average result obtained across problem instances shown red macbook pro ghz intel core duo best result problem instance selected runs algorithm figures best configuration algorithm set start sites reduce number iterations number promising solutions remembered site set algorithm left run minutes problem terminated infeasible solutions solutions capacity duration constraints allowed traversed scored best known solution conversely figures fast configuration algorithm set start sites reduce number iteration number promising solutions remembered site set algorithm left run seconds problem experiments percentage best known average time secs figure shown results depicted figure area left axis blown terminated lns improvement heuristic set destroy mean solution step repair operator initially considers first closest customers reinsertion points increases closest customers site ages infeasible solutions solutions duration service time constraints allowed traversed scored best known solution table provides tabular summary results covered section experiments section review results obtained implementing standard bees algorithm lns local search aim experiments prove algorithmic enhancements suggested thesis fact produce better results would obtained used standard bees algorithm also demonstrate combination bees algorithm lns local search produces better results either algorithm used separately chapter results percentage best known average time secs figure fast shown results obtained standard christofides mingozzi toth problem instances algorithm optimised producing best results within second runtime limit lefthand axis shows relative percentage compared best known result problem instance bottom axis shows elapsed runtime seconds infeasible solutions shown best known result bees algorithm versus enhanced bees algorithm start figures showing results obtained using standard bees algorithm described pham problem instances section used results compared directly results depicted figures obtained macbook pro ghz intel core duo best result instance selected runs algorithm algorithm configured following parameters used sites selected best sites elite elite site bees recruited search another sites selected bees recruited search bees remaining sites left search randomly algorithm left run seconds problem terminated infeasible solutions solutions capacity duration constraints allowed traversed scored experiments percentage best known average time secs figure shown results depicted figure area left axis blown best known solution improvement heuristic used improvement phase bee see chapter overview heuristic works large neighbourhood search next show figures results obtained using standalone lns search embedded within hill climb lns search used section one employed enhanced bees algorithm noted sophisticated lns algorithms available comparatively simple one used would probably produce better results lns results presented however believe one attractive features enhanced bees algorithm uses fairly simple local search procedure moreover aim experiment demonstrate limitations simple local method offset embedded within bees algorithm results depicted figures obtained macbook pro ghz intel core duo best result problem instance selected runs chapter results table enhanced bees algorithm given results obtained running enhanced bees algorithm using two different configurations standard christofides mingozzi toth problem problem instances fast configuration optimised produce results within second runtime limit conversely best configuration optimised producing best results within much longer period minutes instance average best results run seconds run minutes reported gendreau laporte potvin algorithm algorithm initialised starting position generated simple insertion heuristic lns heuristic set destroy mean solution step repair enumerated customers deciding best reinsertion point infeasible solutions solutions capacity duration constraints allowed traversed scored best known solution summary table provides summary results obtained bees algorithm lns experiments alongside results obtained enhanced bees algorithm bees algorithm worst three surprising given bees algorithm devised solve continuous problems rather discrete problems experiments percentage best known average time secs figure standard bees algorithm shown results obtained standard christofides mingozzi toth instances algorithm optimised producing best results within second runtime limit lefthand axis shows relative percentage compared best known result problem instance bottom axis shows elapsed runtime seconds infeasible solutions shown best known result average result obtained across problem instances shown red bees algorithm used discrete problems literature adapted incorporate sophisticated local search techniques much enhanced bees algorithm also note two problem instances produce feasible solutions within second runtime threshold believe given longer running time algorithm would probably found feasible solution however one objectives enhanced bees algorithm produce robust results reliably count standard bees algorithm competitive lns improvement heuristic produced much stronger results shows lns improvement plays important part results obtained enhanced bees algorithm lns heuristic fairly new heuristic vrp research least nevertheless produced competitive results borne results obtained section however lns local search fail find feasible solution one problem instances chapter results percentage best known average time secs figure standard bees shown results depicted figure section left axis blown note average shown blow percent probably due limited runtime permitted problem instance removed results lns search average result becomes getting much closer results obtained enhanced bees algorithm comparison lastly table figure provide comparison enhanced bees algorithm along well known results literature seen table figure best results known due taillard tabu search heuristic reaches best known solutions set problems enhanced bees algorithm comparison finds best known solutions however enhanced bees algorithm still competitive runtime duration required find best known solution smaller many although direct comparison hard make many reported results run significantly older hardware additionally solutions produced enhanced bees algorithm within best known comparison percentage best known average time secs figure lns algorithm shown results obtained standard christofides mingozzi toth instances algorithm optimised producing best results within second runtime limit lefthand axis shows relative percentage compared best known result problem instance bottom axis shows elapsed runtime seconds infeasible solutions shown best known result average result obtained across problem instances shown red solutions average meaning algorithm competitive best available vrp chapter results percentage best known average time secs figure lns shown results depicted figure area left axis blown comparison table experiments given results contrast standard bees algorithm lns local search enhanced bees algorithm one another algorithm run seconds used standard christofides mingozzi toth problem instances instance bees average run seconds configured described section run seconds configured described section run seconds configured described section enhanced chapter results best table results comparison given comparison enhanced bees algorithm alongside well known results literature results standard christofides mingozzi toth problem instances commonly used vrp literature running times given parenthesis known instance clark write savings parallel algorithm implemented laporte semet sweep algorithm due gillett miller implemented christofides mingozzi toth reported generalised assignment due fisher jaikumar reported local search applied clark write savings parallel algorithm first improvement taken implemented laporte semet simulated annealing due osman runtime duration given parentheses reported seconds vax tabu search due taillard runtime duration given parentheses reported seconds sillicon graphics workstation ant colony optimisation bullnheimer hartl strauss bullnheimer provided two papers ant colony optimisation vrp better two used runtime duration given parentheses reported seconds pentium enhanced bees algorithm results shown best configuration section runtime duration given parentheses reported seconds macbook pro ghz intel core duo best known results reported gendreau laporte potvin comparison aco gen assign percent best known solution sweep eba average time seconds figure results comparison shown average results across problem instances obtained enhanced bees algorithm shown red mapped results listed table chapter results chapter conclusion thesis described new vrp called enhanced bees algorithm results obtained competitive best available vehicle routing problem additionally algorithm good runtime performance producing results within optimal solution within seconds took bees algorithm starting point bees algorithm originally developed solving continuous optimisation problems part work undertaken thesis adapt use vrp although could argued enhanced bees algorithm inspired adapted approach developed enhanced bees algorithm could equally applied combinatorial optimisation problems traveling salesman problem job shop scheduling problem cutting stock problem showed empirically quality solutions obtained enhanced bees algorithm competitive best modern available vrp additionally showed algorithm good runtime performance producing results within optimal solution within seconds runtime performance algorithm makes suitable use within real world dispatch scenarios often dispatch process fluid hence impractical optimisation take minutes hours environments acceptable trade fraction percent solution quality quicker runtime performance also gave results demonstrated algorithmic enhancements suggested thesis fact produce better results would obtained using standard bees algorithm also demonstrated combination bees algorithm lns local search heuristic produced better results algorithms used separately chapter conclusion additionally provided comprehensive survey vrp literature provided short history results foundational vrp research well providing depth material descriptions classic algorithms developed vrp number areas research undertaken thesis could continued include introduce mating process generate new sites interesting extension enhanced bees algorithm would incorporate crossover operation generating new sites version enhanced bees algorithm unpromising sites simply culled alternative approach borrowing concept genetic algorithms tabu search case taillard adaptive memory would replace site recombination two successful sites advantage approach would open new area search space exploration area promising combines components two already successful sites crossover process could simple using existing vrp crossover operator two fittest solutions site although much sophisticated crossover operations imaginable extend combinatorial problems possible follow similar approach taken adapting bees algorithm vrp apply combinatorial problems bees algorithm especially strong providing robust solutions search space contains many local optima imagine many combinatorial problems would advantage obvious starting point would apply job shop scheduling problem shares many characteristics vrp include real world constraints although literature filled variations vrp add additional constraints vrptw pdp etc focus tackling computationally hard constraints time windows multiple deploys etc area often addressed literature deal soft constraints day disruptions differing skill sets across fleet factoring dispatcher assignment preferences unfortunately constraints barrier vehicle route optimisation adopted many logistics companies interesting line research would extend optimisation methods developed vrp include feedback envision interactive process dispatcher feedback optimisation process soft constraints modelled within algorithm would also interesting see supervised learning methods artificial intelligence naive bayes classifier could incorporated optimisation process bibliography bee image http bees algorithm website http biodidac bank digital http resources teaching biology altinkemer gavish parallel savings based heuristic delivery problem operations research balinski quandt integer program delivery problem operations research jean berger mohamed barkaoui hybrid genetic algorithm capacitated vehicle routing problem genetic evolutionary gecco volume lecture notes computer science pages blanton wainwright multiple vehicle routing time capacity constraints using genetic algorithms proceedings international conference genetic algorithms pages bullnheimer hartl strauss applying ant system vehicle routing problem advances trends local search paradigms optimization pages bullnheimer hartl strauss improved ant system algorithm vehicle routing problem annals operations research christofides eilon algorithm vehicle dispatching problem orq christofides mingozzi toth exact algorithms vehicle routing problem based spanning tree shortest path relaxations mathematical programming clark wright scheduling vehicles central depot number delivery points operations research bibliography dantzig fulkerson johnson solution traveling salesman problem operations research dantzig ramser truck dispatching problem management science oct martin desrochers jacques desrosiers marius solomon new optimization algorithm vehicle routing problem time windows operations research march eilon christofides distribution management mathematical modelling practical analysis griffin london fisher jaikumar generalized assignment heuristic solving vrp networks foster ryan integer programming approach vehicle scheduling problem operations research gaskell bases vehicle fleet scheduling operational research quarterly gendreau hertz laporte tabu search heuristic vehicle routing problem management science gendreau laporte potvin metaheuristics vehicle routing problem technical report les cahiers gerad revised gillett miller heuristic algorithm vehicle dispatch problem operations research haimovich bounds heuristics capacitated routing problems mathematics operations research hamilton memorandum respecting new system roots unity icosian calculus philosophical magazine holland adaptation natural artificial systems university michigan press karp reducibility among combinatorial problems miller thatcher editors complexity computer computations pages plenum press kirkman representation polyhedra philosophical transactions royal society london series laporte mercure nobert exact algorithm asymmetrical capacitated vehicle routing problem networks laporte nobert surveys combinatorial optimization chapter exact algorithms vehicle routing problem bibliography laporte semet classical heuristics vehicle routing problem technical report les cahiers gerad chee peng lim lakhmi jain satchidananda dehuri editors innovations swarm intelligence volume studies computational intelligence springer nagata edge assembly crossover capacitated vehicle routing problem evocop lncs road transport forum road transport forum facts website http oliver smith holland study permutation crossover operators traveling salesman problem second international conference genetic algorithms applications pages traveling combinatorial problems relation logistics regional blood banking phd thesis evanston northwestern university osman metastrategy simulated annealing tabu search algorithm vehicle routing problem annals operations research paessens savings algorithm vehicle routing problem european journal operational research pham ghanbarzadeh koc otri rahim zaidi bees algorithm technical report potvin review algorithms vehicle routing algorithms vehicle routing problem pages rao zionts allocation transportation units alternative tripsa column generation scheme subproblems operations research reimann stummer doerner ant system vehicle routing problem proceedings genetic evolutionary computation conference pages renaud boctor laporte fast composite heuristic symmetric traveling salesman problem informs journal computing robinson hamiltonian game traveling salesman problem research memorandum robuste daganzo souleyrette implementing vehicle routing models transportation research bibliography stefan ropke heuristic exact algorithms vehicle routing problems phd thesis department computer science university copenhagen diku alexander schrijver history combinatorial optimization till operations research management elsevier shaw using constraint programming local search methods solve vehicle routing problems pages solomon algorithms vehicle routing scheduling problems time window constraints oper taillard parallel iterative search methods vehicle routing problems networks thangiah nygard juell gideon genetic algorithm system vehicle routing time windows proceedings ieee conference artificial intelligence applications pages toth vigo granular tabu search application vehicle routing problem technical report paolo toth daniele vigo editors vehicle routing problem society industrial applied mathematics philadelphia usa willard vehicle routing using tabu search master thesis management school imperial college london wren holliday computer scheduling vehicles one depots number delivery points operations research quarterly pages yellow computational modification savings method vehicle scheduling operational research quarterly
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extreme learning machine approach predicting near chaotic hcci combustion phasing adam vaughana stanislav bohaca dept mechanical engineering university michigan ann arbor usa may abstract fuel efficient homogeneous charge compression ignition hcci engine combustion timing predictions must contend chemistry physics period doubling bifurcation turbulent mixing model parameters drift mixture state information typically resolved basis especially transients previous work abstract mapping function coupled vector regression shown predict experimentally observed combustion timing wide range engine conditions despite aforementioned difficulties main limitation previous approach partially acausual randomly sampled training dataset used train proof concept offline predictions objective paper address limitation proposing new online adaptive extreme learning machine elm extension named weighted extension enables fully causal combustion timing predictions randomly chosen engine set points shown achieve results good better previous offline method broader objective approach enable new class model predictive control strategies high variability hcci ultimately bring hcci low nox reduced emissions production engines keywords time series chaos theory dynamical system adaptive extreme learning machine introduction since gasoline engines largely operated controlling power output throttle restricts airflow using simple spark control burn timing operating close stoichiometry reliable spark ignition catalysts reduce nox emissions throttle hurts fuel efficiency pumping losses especially stoichiometric mixtures used thermodynamically less fuel efficient mixtures diluted air exhaust gases broad availability enabling technologies variable valve timing relatively new type combustion called homogeneous charge compression ignition hcci received increased research interest past decade hcci uses autoignition burn lean excess air mixtures produce nox quantities require expensive catalyst aftertreatment instead spark combustion timing controlled thermodynamic trajectory mixture complex chemical kinetics nox production freedom stoichiometric shackles spark igi fundamental changes updated version original october paper version includes algebraic simplifications minor corrections improved body text corresponding author email addresses vaughana adam vaughan sbohac stanislav bohac preprint submitted elsevier nition hcci achieves greater fuel efficiency thermodynamically ideal lean mixtures unthrottled operation improved fuel economy relevance sustainability national oil independence greenhouse gas initiatives seek curb petroleum usage primary challenge hcci autoignition ensure burn timing synchronized motion piston important efficient extraction mechanical work mixture avoid unsafe noisy combustion unstable near chaotic combustion oscillations synchronization important combustion researchers use normal units time instead use angle crank describes position piston represents time crank angle takes certain amount time fixed crank rotation speed thermodynamic trajectory mixture driven piston varying cylinder volume function crank angle fig angles measured relative piston top cylinder top dead center tdc engine tdc occurs twice per cycle different regions piston may compressing expanding mixture valve open moving mixture intake exhaust manifolds highlighted cylinder volume curve two regions one exhaust valve open intake valve open note two may cylinder pressure cylinder volume high cyclic variability severely constrains available load limits hcci function mean pressures air residual gases motivation goals primary constraint hcci need keep combustion timing ringing combustion stability limits ringing limit excessive pressure rise rates encountered stability limit high combustion timing observed since limits play key role constraining hcci usable operating range desirable explore new methods predict behavior beyond constraints particular ability predict correct high might enable use late phased combustion mitigate excessive pressure rise rates currently constrain hcci highload operation also potentially addressing high experienced towards end goal expanding hcci load envelope paper builds previous work describing new online adaptive machine learning method enables fully causal combustion timing predictions across randomly chosen engine set point transients include stable near chaotic bifurcation behavior start fuel injection timing quantity fuel pnvo pevo pivc milliseconds rpm exhaust partially tdc nvo tdc crank angle air tdc figure schematic key engine cycle variables valve events separated number crank angle degrees termed negative valve overlap nvo unlike conventional engines nvo prevents hot exhaust gases leaving cylinder typically stores residual gases next cycle offering practical way raise mixture temperature ensure hcci autoignition changing amount nvo one affect mixture temperature dilution ultimately control chemical kinetics behind combustion timing temperature dilution work opposite directions typically temperature dominates nvo instantly adjustable common variable valve timing systems reader cautioned many researchers publish results fully variable lift timing electric hydraulic valve actuation systems expensive implement production engines use nvo residual gases introduces strong coupling top already chemistry physics occur throughout complete engine cycle compounding issues residual gases neither airflow cylinder quantity residual gases cylinder accurately resolved burn happens mean value basis commonly available sensors especially transients beyond residual gas influences also complex secondary influences combustion behavior turbulent mixing manifold resonance effects combustion deposits different varieties fuel even ambient temperature variations hcci already significant challenge given complexity combustion mode also exhibits period doubling bifurcation cascade chaos similar seen high residual spark ignition engines nearly chaotic hcci still deterministic becomes oscillatory sensitive parameter variations residual gas fraction fluctuations oscillatory stability limit behavior commonly referred experimental observations authors previous publication abstract mapping function engine combustion created within framework discrete dynamical system simple abstraction intended convey conceptual understanding experimental behavior seen fig return maps return maps show experimentally observed combustion timing given cycle along abscissa next cycle along ordinate random engine actuator set points value time crank angle degrees fuel net heat release achieved thus measures timing end burn relation piston position crank reader note structure behavior despite random actuator set points used generate fig structure shows deterministic transition oscillatory high behavior combustion moves towards later combustion timing viewed least single period doubling bifurcation sensitive dependence engine set point mathematically interesting oscillatory high structure undesirably constrains practical hcci engine operation thorough description data provided shown similar structure percent burn metrics although less pronounced especially see cyl high next function previous cyl cyl predictive sense injected random unpredictable noise could generate qualitative return map shape similar seen experimentally time series predictions shown cloud possible combustion timings ranging stable oscillatory fact highlighted extension work used random residual noise actual time series disturbances experiments said models useful showing period doubling cascade chaos driven residual gas fraction explain observed high behavior context machine learning provides computationally efficient way capture complex combustion patterns simultaneously avoiding explicit knowledge underlying mixture state composition provided appropriate abstract mapping function chosen clearly benefits machine learning approach key issue machine learning data driven relatively large quantities data needed adequately cover large dimensional spaces shown conceptually fig high dimensional data might viewed porcupine engine operating condition might viewed quill porcupine machine learning algorithms know nothing ideal gas law chemical kinetics ability extrapolate quills limited especially provided sparse data previous work used random sampling cycle time series training ensure data driven model data fit quills assessed model ability predict remaining randomly chosen cycles thus training dataset partially acausual model shown adapt new conditions cyl figure return map probability histograms generated cycles random engine set points outliers omitted total data colormap show order magnitude differences modeling approaches skeletal functional form abstract mapping function built using measurable quantities thermodynamics known correlations unlike approaches usually discussed engine literature machine learning technique vector regression combined abstract functional form provide quantitative predictions primary motivation machine learning approach existing chemical kinetics computational fluid dynamics cfd capture engine behavior see gasoline mechanism validation methods offline mapping function fit original computationally intensive predictions subject experimental uncertainties mean value mixture state composition online adaptation move quills also adjust parameter variation updated figure high dimensional data might viewed conceptually porcupine primary goal paper design online adaptive algorithm fit new data point reference simulation time rpm millisecond engine cycle measured single core modern computer computational complexity extreme loworder approximation models hcci control developed since least early based spark ignition engine knock models developed recently efforts made extend type model high regions hcci injecting random residual gas fraction noise capture uncertainties mixture state composition model tuned limited set conditions contribution primary contribution work development new online learning method provide adaptive fully causal predictions near chaotic hcci combustion combustion timing method called weighted ring extreme learning machine enables robust online updates extreme learning machine elm model trained fit quills offline data developed work weighted least squares extension online sequential elm developed independently similar derivation available difference work classification application use ring buffer data structure chunk online updates offline trained regression model data ring buffer weighted heavily data originally used fit offline model allows emphasis placed recent measurements might quills fig offline trained model result engine parameter variation thus approach allows one prescribe partitioned balance offline model fit need adapt recent conditions also explicitly avoids adaptation local conditions could compromise global generality forgetting old ring buffer data eventually exit buffer fig gives schematic representation approach methods mapping function modifications previous work engine combustion abstracted following mapping function cycle iteration index time net heat release occurred time net heat release occured injection pulse width milliseconds start injection top dead center btdc pressure variables measurements mean pressures specific regions combustion cycle see fig details simplified net heat release algorithm available fuel rail pressure constant however reader note pressure drop cylinder pressure nvo injections varies transient step high regions cylinder pressure variables chosen capture residual coupling air flow without difficulties explicitly modeling quantities meet engine controller timing requirements modified moved previous cycle range mean shortened also moved closer tdc take advantage inherent signal amplification provided compression process subscripts ivc evo nvo refer general timing regions intake valve close exhaust valve open negative valve overlap respectively want predict future output input vector cycle timeline forget weighted ring buffer previous data pairs figure schematic overview overview differences derivation lacks bias vector uses gaussian distribution drops unnecessary logistic function exponential negative uses approximant found empirically computation bias addition step could removed loss fitting performance elements drawn gaussian distribution elm theory requires distribution continuous although ability remove bias likely problem specific primary benefits elm approach method used core algorithm basic goal extreme learning machine elm solve output layer weight vector scales transformed input output elm easily adapted online adaptation elm provides good model generalization data noisy elm extremely computationally efficient hidden layer output matrix given input matrix target vector neurons set data pairs cycle timestep variables given since stronger indicator oscillatory behavior seen fig used model input commonly encountered engine literature thus used model output two related quantities terms model fit statistics significant benefit using one said isolated instances observed better job predicting large oscillations neuron activation function chosen commonly used logistic function without unnecessary negative solution split offline online chunk recent data pairs avoid computational storage burden using offline data directly matrices partitioned subscript denoting offline online updated components respectively using random input weight vector composed random variable samples gaussian distribution input variables gives following similar derivation recursive least squares adding weight matrix inversion portion weighted normal equations terms use random individual neurons initialization main difference extreme learning machine versus conventional neural networks iteratively train vectors collected single input weight matrix held fixed across input row vectors output values logistic works well one modification improves computational efficiency processors without dedicated instruction raspberry modification replace exponential following approximant portion normal equations similarly using existing relations following simple logistic relations small number floating point operations used reused intermediate terms known boundedness normalized inputs known weights make approximant work well application significant degradation model performance found used implementations described hereafter normal equations used solve least squares solution substituting full online solution extend weighted least squares solution one incorporate diagonal weight matrix normal equations yields online solution without need offline dataset trade computational burden sized inverse inverse scales smaller sized ring buffer one let online predictions unlike additional simplification possible algorithm propagate time begin append portion distribute give usage procedure scale columns zero unity variable combustion implementation column variable values percentile saturated respective percentile value normalized zero unity percentile based saturation limits done adequately represent distribution tails avoid scaling issues random transformation enables low computational complexity algorithm may result matrices numerical implementations use double precision additionally one consider using singular value decomposition matrix inversions using gaussian distribution elm input weights hold ize fixed training predictions combustion implementation done function mersenne twister pseudo random number generator used based seed initial trials cylinder individually computed model used identical input weight matrix build previously acquired samples cover wide range conditions using input matrix output target vector formats given eqs respectively combustion implementation initial training data minutes random engine set points covering cycles random engine set points single engine speed however appears minutes data may sufficient pruning training data simplified substitutions distributing provide transforming identity gives form identity applied yield substantially simpler form ring buffer sized inverse finally noting one substitute arrive following algorithm summary offline training reader note needed online adaptation size matrices scales none increasing number neurons original offline data needed additionally note simply reverse recent cycle vector updated weighted ring buffer finally mentioned resulting update law structurally similar kalman filter also uses recursive least squares future work look applying kalman filtering algorithm improvements square root filtering use matrix inversion lemma yield online adaptation include cycles cycles transient set point step provided small model fitting performance improvement specify weight matrix offline measurements combustion implementation simple scalar value chosen using design experiments weight works well proof concept future work rigorously determine weight perhaps optimization techniques note allows weighting applied offline small offline weighting equivalent large online weighting solve offline solution using eqs hold values constant future predictions populate ring buffer size recently completed pairs using input ring buffer output ring buffer solved average rate per combustion cycle per cylinder ghz desktop computer running gentoo online predictions eqs recast loop automatically parallelized code across four worker threads provide predictions average rate per combustion cycle per cylinder level performance adequate although algorithm development main focus paper implementation wrelm algorithm built using custom raspr berry data acquisition hardware video demonstrating predictions control available software system comprised execute update algorithm combustion cycle shown fig combustion implementation taken cycles tuning existing datasets desired vary offline data build using input matrix output target vector specify weight matrix combustion implementation identity matrix chosen since weighting already applied offline data step gradually increased weighting recent time steps ring buffer explored however net significant improvement model fitting performance simple scalar value offline data although explored current implementation vary solve updated solution using eqs cycle input vector fully populated transform vector using solve predicted target value using repeat steps new time step caching results hidden layer outputs previous time steps reduce computational requirements user space application runs heat release calculations software leverages eigen matrix library custom assembly code matrix raspberry double precision floating point unit asynchronous fasync notification used specific crank angle events published fiq code synchronize user space software execution crank rotation kernel preempt patched patches completely disable fast interrupt request fiq usage usb driver arm assembly code pressure data acquisition using fiq kilosamples per second total cylinder channels code kernel module contains fiq assembly code memory allocation standard mmap ioctl fasync hooks user space second user space thread uses standard websockets libwebsockets code library stream processed data user interface coded javascript library minimal raspbian distribution adaptation routine run possible perform model predictive control predictions within worst case task context switch calculation latency window level performance needed ensure control authority actuator immediately measured experimental setup table provides summary experimental setup conditions visited pressure acquired implementation collection unoptimized software routines developed using techniques described previous sections offline solution provided eqs code benchmarked faster openblas generic matrix multiply gemm faster eigen basis pegged thermodynamically cycle ivc using polytropic exponent exponent chosen closely match pegging results achieved using single intake runner high speed pressure sensor cylinder purpose computing net indicated mean effective pressure imep cycle defined starting btdc firing ending atdc firing reference cam lift timing duration table airfuel ratio range indicated table measured postturbine represents mixture four cylinders fuel mass per cycle estimated using fuel lower heating value assuming gross heat release greater net heat release combustion efficiency fixed used part fig model training testing presented total cycle counts reported outliers removed outlier criteria detailed intended remove misfires partial burns criteria fairly permissive remove data offline solution trained using minutes test cell time covering cycles random engine set points rpm two random subsequences subsequences comprised random transient set point steps occurring approximately every seconds occasional misfires covering nominal variable ranges given table training data pruned include cycles cycles transient set point step small model fitting performance improvement online solution run separate random subsequence fed unseen cycles similar would experienced implementation online dataset comprised consecutive cycles random engine set points longer online sequences also tested achieved similar results dataset description full collection cycles comprised five minute random test subsequences random subsequence covers nominal ranges listed table however one subsequence holds fixed sequence table experimental setup test conditions results discussion engine fitting performance cycle dataset excluding outliers defined shown table figs minimum coefficient determination given table shows least variance explained model currently defined dataset random transient steps occurring approximately every seconds occasional misfires better achieved vector regression dataset however comparison fully predicting entire cycle dataset whereas training strategy ensured model partially seen operating points trying predict root mean squared error rmse table assessed single set point mean atdc net indicated mean effective pressure imep bar transient sequence started make model lnf ecotec cylinder layout overall displacement bore stroke geometric compression ratioa cam lifta cam durationa cam phaser type hydraulic fuel injector type direct side mounted wall guided fuel designation haltermann epa tier eee description federal emission cert gasoline research octane number motor octane number astm heating value aromatic olefin saturate fractions volume test conditions throttle position turbocharger supercharger residual retention strategy ivo set point rangeb evc set point rangeb set point rangeb set point rangeb net imep values visitedb ratios visitedb estimated fuel per cycleb intake runner cyls fuel injection pressure coolant temperature engine speed wide open wastegate open bypassed negative valve overlap atdc atdc btdc bar bar bar rpm rpm table model error statistics cylinder overall rmse rmse consecutive cycles random transient steps occurring approx every sec occasional misfires modified stock engine first percentile model error deg cyl model error deg frequency cycles cyl frequency cycles frequency cycles frequency cycles cyl predicted positive bias largely midrange values fig shows time series predictions computed early btdc firing missing segments outliers described earlier fig provide qualitative insight model weights online adaptation neurons sorted respective input weight vector transformation specified used cylinder differences due different characteristics cylinder fig shows imep fig shows random engine actuator inputs driving engine set point transient model predictions fig generally show good agreement however occasional tracking errors unclear source tracking errors fundamental model limitation need influence misfiring cylinders going harsh reignite need weight tuning offline training data perhaps something else future work try answer questions overall however authors believe level fit shown table figs good considering dataset includes transients operating points high right complete engine misfire model error deg cyl model error deg figure error histograms model across consecutive cycles random transient steps occurring approx every seconds occasional misfires cyl model predicted model predicted measured summary conclusion work presents new online adaptation algorithm named weighted ring extreme learning machine approach uses weighted ring buffer data structure recent measurements recursively update offline trained extreme learning machine solution shown wrelm used approximate combustion mapping function developed provide reasonably accurate causal predictions near chaotic combustion behavior combustion application pressure crank encoder sensors needed predictions predictions computed early btdc firing algorithm fast implemented raspberry platform twominute video demonstrating available future work explore optimal selection weight try better understand situations lead occasional model tracking errors finally broader objective new modeling approach enable new class model predictive control strategies could potentially bring hcci low nox reduced emissions higher fuel efficiency production gasoline engines measured cyl model predicted model predicted measured cyl cyl measured figure predicted versus measured model across consecutive cycles random transient steps occurring approximately every seconds occasional misfires late combustion timing predicted almost prediction outliers capture correct directionality fig shows distribution model errors clear slight positive bias predictions fig provides insight tails fig shows model errors still generally capture correct directionality fig also shows late combustion timing adding additional inputs necessarily practical computationally experimentally even advisable given occam razor model track near chaotic high cyl model predicted sorted neuron measured misfire region model tracks harsh cyl different transients misfire cyl despite measured model predicted cycle cyl weight cyl injection pulse width cyl weight cyl imep bar indicated mean effective pressure weight cyl soi sorted neuron sorted neuron model often tracks tracking errors cyl occur occasionally measured model predicted sorted neuron model predicted measured weight cyl start injection cyl approx evc ivo evc cyl cyl ivo cyl cycle cycle figure model track transients every seconds operating points near chaotic high particularly harsh region cycle dataset includes misfires colormaps linearly scaled provide qualitative insight level adaptation model differences acknowledgments stefanopoulou jiang cyclic variability dynamical instabilities autoignition engines high residuals ieee transactions control systems technology kantor dynamical instability spark ignited engines science manofsky vavra assanis babajimopoulos bridging gap hcci compression ignition sae paper dec babajimopoulos assanis comparing enhanced natural thermal stratification retarded combustion phasing smoothing hcci rates sae paper mehl pitz dec detailed kinetic modeling heat release prf fuels hcci engine sae paper experimental surrogate modeling study gasoline ignition rapid compression machine combustion flame shaver modeling control hcci engines using variable valve actuation thesis stanford university livengood correlation autoignition phenomena internal combustion engines rapid compression machines symposium international combustion jade larimore stefanopoulou jiang controlled load speed transitions multicylinder recompression hcci engine control systems technology ieee transactions cherkassky mulier learning data concepts theory methods wiley liang huang saratchandran sundararajan fast accurate online sequential learning algorithm feedforward networks neural networks ieee transactions huang zhu siew extreme learning machine theory applications neurocomputing mirza lin toh weighted online sequential extreme learning machine class imbalance learning neural processing letters simon optimal state estimation kalman infinity nonlinear approaches john halleck matrix identities also ring identities url http online accessed version raspberry engine control adaptive extreme learning machine url https online accessed minimal raspbian unattended netinstaller url https online accessed material based upon work supported department energy national energy technology laboratory award number work performed part access project consortium robert bosch llc avl emitec stanford university university michigan direction hakan yilmaz oliver robert bosch llc authors thank vijay janakiraman providing raw data analyzed paper authors also thank jeff sterniak test cell project support vaughan thanks nisar ahmed many helpful discussions advisors bohac claus borgnakke freedom explore interesting topic near chaotic combustion conflict interest university michigan filed provisional patent work described publication vaughan named inventor includes royalty rights bohac declares conflict interest references vaughan bohac extreme learning machine approach predicting near chaotic hcci combustion phasing arxiv babajimopoulos challa lavoie assanis assessment two variable cam timing strategies hcci engines recompression rebreathing proceedings asme internal combustion engine division spring technical conference soto vavra babajimopoulos assessment residual mass estimation methods cylinder pressure heat release analysis hcci engines negative valve overlap journal engineering gas turbines power olesky vavra assanis babajimopoulos effects charge preheating methods combustion phasing limitations hcci engine negative valve overlap journal engineering gas turbines power saxena bedoya fundamental phenomena affecting low temperature combustion hcci engines high load limits strategies extending limits progress energy combustion science vaughan bohac method predict hcci combustion phasing proceedings asme internal combustion engine division fall technical conference lacey effects advanced fuels additives homogeneous charge compression ignition combustion deposit formation thesis university michigan vaughan delagrammatikas high performance continuously variable engine intake manifold sae paper understanding transition conventional combustion hcci gasoline engine proceedings combustion institute disclaimer report prepared account work sponsored agency united states government neither united states government agency thereof employees makes warranty express implied assumes legal liability responsibility accuracy completeness usefulness information apparatus product process disclosed represents use would infringe privately owned rights reference herein specific commercial product process service trade name trademark manufacturer otherwise necessarily constitute imply endorsement recommendation favoring united states government agency thereof views opinions authors expressed herein necessarily state reflect united states government agency thereof
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jan vol regularity properties simulations gaussian random fields sphere cross time jorge clarke cerda departamento universidad federico santa avda chile ceremade umr cnrs psl research university place lattre tassigny paris france clarkemove alfredo newcastle university school mathematics statistics emilio newcastle university school mathematics statistics chair spatial analytics methods center departamento universidad federico santa avda chile url abstract study regularity properties gaussian fields defined spheres cross time particular consider two alternative spectral decompositions gaussian field decomposition establish regularity properties sobolev interpolation spaces propose simulation method study level accuracy sense method turns fast efficient msc subject classifications primary secondary keywords phrases gaussian random fields global data big data covariance expansion spherical harmonics functions schoenberg functions supported proyecto fondecyt beca doctorado nacional supported proyecto fondecyt regular supported clarke simulations grfs spheres cross time introduction variability major importance many fields particular anthropogenic natural processes earthquakes geographic evolution diseases income distributions mortality fields atmospheric pollutant concentrations hydrological basin characterization precipitation fields among others many natural phenomena involving instance climate change atmospheric variables many branches applied sciences increasingly interested analysis data distributed whole sphere representing planet earth evolving time hence need random fields models spatial location continuously indexed sphere time either continuous discrete common consider observations partial realization random field usually considered gaussian see thus dependence structure governed covariance spatiotemporal gaussian field refer reader significant contributions direction specifically let positive integer let kxk unit sphere euclidean space denotes euclidean norm denote gaussian field tour force characterizes isotropic gaussian random fields sphere expansions respect spherical harmonics functions angular power spectrum show smoothness covariance connected decay angular power spectrum discuss relation sample continuity sample differentiability random fields present paper extends part work extension depends two alternative spectral decompositions gaussian field spheres cross time particular propose either hermite classical expansions show regularity properties evolve dynamically time crux arguments rely recent advances characterization covariance functions associated gaussian fields spheres cross time see notably representation terms schoenberg functions inspires proposal alternative spectral decompositions temporal part become crucial establish regularity properties associated gaussian field second part paper devoted simulation methods computationally fast keeping reasonable level accuracy resulting notable step forward efficient simulation methods random fields defined sphere cross time currently almost unexplored cholesky decomposition appealing alternative since exact method however method order computation denoting sample size clarke simulations grfs spheres cross time makes implementation computationally challenging large scale problems called big problem therefore mandatory investigate efficient simulation methods propose simulation method based suitable truncation proposed double spectral decompositions establish accuracy sense illustrate model keeps reasonable level accuracy considerably fast even number spatiotemporal locations high remainder article follows section provides basic material exposition expansions kernel covariances random field presented section section presents regularity results kernel covariance functions terms weighted sobolev spaces weighted spaces section simulation method developed accuracy studied also provide numerical experiments illustrative purposes appendix also provide rather general version theorem manuscript intended random fields except section simulations considered random fields preliminaries section largely expository devoted illustration framework notations major use throughout manuscript tools presented section valid particular references case exposed use section spherical harmonics functions gegenbauer polynomials spherical harmonics restrictions unit sphere real harmonics polynomials also eigenfunctions operator deeper overview spherical harmonics along properties listed subsection found let denote space square integrable functions denotes surface area measure denotes surface area gamma function clarke simulations grfs spheres cross time let hjd denote linear space spherical harmonics degree different degrees spherical harmonics orthogonal respect inner product hjd hyj dim hjd kronecker delta function identically equal one zero otherwise corollary shows dim hjd dim let dim hjd orthonormal basis hjd family dim hjd constitutes orthonormal basis theorem shows hjd besides addition formula spherical harmonics states dim denotes inner product cjr gegenbauer ultraspherical polynomials degree order defined cjr denotes jacobi polynomial parameters degree denotes pochhammer symbol rising factorial defined provided negative integer gegenbauer polynomials constitute basis satisfy orthogonality relation section cjr clarke simulations grfs spheres cross time stirling inequalities fixed exists constants hence assuming relation becomes cjr besides observe see section follows denotes standardized gegenbauer polynomial identically equal one straightforward see dim hjd remark case legendre polynomial degree see deeper overview spherical harmonics detailed description gegenbauer jacobi legendre polynomials isotropic stationary gaussian random fields sphere cross time let complete probability space consider manifolds contained respectively definition geodesic distance great circle spherical distance defined arccos geodesic distance defined clarke simulations grfs spheres cross time throughout unless explicitly presented different way write instead surface area measure equivalent uniformly distributed measures haar measure lebesgue spherical measure analogously write instead lebesgue measure definition mapping called random field random field called gaussian random vector rez rez imz imz multivariate gaussian distributed denotes transpose operator function positive definite finite systems pairwise distinct points constants positive definite function strictly positive definite inequality strict unless call function spatially isotropic temporally stationary exists function cos hence spatially isotropic temporally stationary function depends arguments via great circle distance time lag equivalently via inner product time lag definition call random field isotropic stationary constant covariance spatially isotropic temporally stationary function associated function called covariance kernel simply kernel remark gaussian random field grf isotropic stationary fact isotropic spatial variable stationary time variable see hence invariant distribution rotations spatial variable translations temporal variable throughout manuscript work random fields loss generality kernel covariance functions sphere cross time seminal paper characterized class continuous functions cos positive definite product space clarke simulations grfs spheres cross time defined equation recently extended schoenberg characterization considering product space locally compact group defined class continuous functions cos positive definite particular case offers characterization covariance functions centred isotropic stationary random fields sphere cross time let denote set continuous positive definite functions consider class continuous functions associated spatially isotropic temporally stationary function positive definite following result rephrased allows identify class covariances functions isotropic stationary random fields theorem let let continuous mapping exists sequence cos cos dim hjd called schoenberg functions dim hjd given series uniformly convergent remark comments order unit vector may consider mapping given arguments page show hence schoenberg functions given may understood orthogonal projection onto hjd note comparison representation covariance functions isotropic random fields sphere representation consider schoenberg coefficients schoenberg functions play fundamental role subsequently clarke simulations grfs spheres cross time expansions isotropic stationary grfs kernel covariance functions according theorem kernel isotropic stationary grf admits following representation dim hjd sequence functions series uniformly convergent expression allows consider different expansions kernel introducing representations present expansion random field motivates simulation methodology expansions isotropic stationary grfs sphere cross time following representation isotropic stationary grf proposed sequence mutually independent stochastic processes set forms denumerable infinite dimensional stochastic process completely defines process sphere elements orthonormal basis representations covariance random field allow introduce following family grfs definition let random field defined mean square sense dim stationary gaussian process cov represents schoenberg functions associated mapping equation clarke simulations grfs spheres cross time remark examples random field satisfying definition found appendix proposition let random field definition isotropic stationary grf covariance function given proof proof follows straight using properties process addition formula equation process following expansion see appendix sequence independent complexvalued random variables defined also eigenvalues eigenfunctions respectively integral operator associated defined remark alternative way write dim expressions represent way construct isotropic stationary grfs sphere cross time suggest spectral simulation method however yet proved isotropic stationary grf sphere cross time written way double expansion kernel covariance functions stationarity process theorem clarke simulations grfs spheres cross time cov hence slight abuse notation allows reformulate last expression therefore kernel covariance function also admits expansion dim hjd following call series angular power spectrum theorem implies sequence continuous positive definite functions hermite expansion kernel covariance functions well known satisfies see therefore ensures finite measure particular gaussian measure let standard gaussian measure schoenberg functions associated equation belong class expanded terms normalized hermite polynomials exist constants series converges normalized hermite polynomial degree given duk clarke simulations grfs spheres cross time consequently kernel reformulated dim hjd call series hermite power spectrum regularity properties section devoted study behaviour kernel covariance functions associated isotropic stationary grf shown regularity kernels closely related decay hermite power spectrum angular power spectrum moreover latter characterizes also term truncation grf equation recall introduced two expansions kernel covariance function isotropic stationary grf power spectrum using double expansion according formula valid kernel isotropic stationary grf lebesgue measure hermite power spectrum using hermite polynomials according formula valid kernel covariance function isotropic stationary grf standard gaussian measure recall cov cos remark considering relations gegenbauer legendre polynomials case kernel turns follows present regularity analysis kernel terms behaviour two proposed expansions address first relation hermite power spectrum regularity analysis hermite expansion part manuscript consider measure space standard gaussian measure clarke simulations grfs spheres cross time let spaces standard sobolev spaces extend proposal consider function spaces closure respect weighted norm given note decreasing scale separable hilbert spaces abuse notation writing instead consider canonical partial order relation theorem norm equivalent first last element sum derive another equivalent norm terms summability spectrum first observe normalized hermite polynomials constitute orthonormal basis fixed gengenbauer polynomials cjr basis apparent basis therefore expanded series cjr putting cjr get dim hjd written covariance kernel type clarke simulations grfs spheres cross time allows tackle problem different way instead using spectral techniques regularity kernels might shown isomorphism spaces weighted square summable bisequence spaces denotes sequence weights sake simplicity consider weighted sobolev spaces weighted square summable spaces obtained special case equation order extend isomorphism spaces integer following introduce interpolation spaces defined equipped norm given functional defined inf kvkw kwkw definition interpolation spaces carried analogous way interpolation property see section implies spaces isomorphic isomorphic theorem let expansion given equivalent norm clarke simulations grfs spheres cross time equivalence reduced hermite rephrased proof assume first claim already proved isomorphic weighted space fix let set interpolation theorem see theorem weights given prove isomorphism equivalent prove second formulation theorem dim hjd dim hjd dim iej iej dim dim clarke simulations grfs spheres cross time standard properties normalized hermite polynomials show one hand stirling inequality implies hand hence exists constants therefore expressions allow conclude therefore deduce concludes proof clarke simulations grfs spheres cross time regularity analysis double expansion exception minor details definitions spaces part manuscript follows similarly section hand consider measure space lebesgue measure consider function spaces vtn vtn closure respect weighted norms kvtn given vtn decreasing sequence separable hilbert spaces vtn look weighted square summable spaces section consider interpolation spaces proof next result follows exactly lines theorem hence omitted theorem let given equivalent norm vtn equivalence reduced angular rephrased clarke simulations grfs spheres cross time remark taking account normalizing constants previous results encompasses results section spectral simulation study spectral simulation method random fields denotes time horizon neater exposition along section omit subscripts associated spatial dimension must first introduce notation let associated legendre polynomials defined dxm spherical harmonic basis functions defined cos exp represents spherical coordinates hand let collection stochastic processes thus consider random field order obtain field must impose conditions stochastic processes throughout assume mutually independent identically equal zero clarke simulations grfs spheres cross time note using standard algebra complex numbers coupled condition equation written consider independent processes following fourier expansions cos cos sin sin aqj sequences independent centred gaussian random variables var var var aqj var summable coefficients direct calculation shows covariance function spatially isotropic temporally stationary precisely cov cos finally given two positive integers truncate expression index respectively thus simulate gaussian random clarke simulations grfs spheres cross time field using explicit approximation cos cos cos sin sin sin pej cos cos cos cos pej cos sin sin cos cos pej assess error associated truncated expansion given equation terms positive integers follow scheme used spatial context extend result case next state main result section theorem let suppose exist positive constants positive integers following inequality holds positive constants equation two independent proof decompose terms cos pej cos cos sin clarke simulations grfs spheres cross time defined cos cos sin cos cos cos sin pej cos sin cos sin second term used identity satisfied summable independence implies shown exists positive constant depending hand since cos sin cos kpej cos cos kpej sin cos therefore clarke simulations grfs spheres cross time positive constants depending particular last inequality follows integral bounds corresponding series see proof completed simple examples generated following angular power spectrum illustrate realizations spatial locations coefficients two cases figures show corresponding realizations scenarios respectively case truncate series using note parameter responsible spatial scale smoothness realization realizations illustrated using similar spectrum merely spatial context compare empirical theoretical convergence rates cases described experiment consider study log error terms log taking exact solution note choice bound implies order convergence min following instead calculating take maximum error points grid empirical errors calculated basis independent samples studies reflect theoretical results see figure clarke simulations grfs spheres cross time figure realization spectrum figure realization spectrum clarke simulations grfs spheres cross time figure empirical versus theoretical log simulation method terms log consider spectrum two cases conclusions discussion present work provided deep look regularity properties gaussian fields evolving temporally spheres hope effort put basis facing important challenges related processes fact many open problems related mathematical modeling well statistical inference optimal prediction list open problems included recent survey amongst paper certainly related problem construction processes spheres cross time work could also put basis solve problem related construction multivariate processes might interesting extend study regularity properties vector valued case would imply use pretty different machinery problem closely related approach regularity properties crucial study gaussian fields infill asymptotics hand question arise naturally possible make inference representation like respective spectral decomposition answer priori establishing clear relation parameters random field spectrum obvious task fact today familiar stochastic process known spectrum brownian motion closer generalization fractional brownian motion yet known spectrum however relatively weak hypotheses covariance kernel grf turns mercer kernel opens alternative negative answer previously mentioned considering eigenvalue problem associated integral operator induced kernel clarke simulations grfs spheres cross time appendix theorem recent results functional analysis see allow construct mercer kernels general contexts proper interpretation results allows generalize classic theorem neat way first introduce framework basics notations let nonempty set positive definite kernel function satisfying inequality whenever subset subset set positive definite kernels denoted endowed measure denote class kernels associated integral operator positive following conditions holds finally define mercer kernel according continuous kernel mercer kernel possesses series representation form basis continuous functions decreases series converges uniformly absolutely compact subsets rest manuscript consider topological space endowed strictly positive measure complete borel measure two properties hold every open nonempty subset positive measure every belongs open subset finite measure besides section denotes complete probability space clarke simulations grfs spheres cross time theorem let centred stochastic process continuous covariance function kernel associated covariance function mercer kernel therefore admits expansion orthonormal basis exists sequence real numbers var series expansion converges lim series expansion converges lim convergence series expansion absolute uniform compact subsets sense proof let kernel associated covariance stochastic process let associated integral operator hypothesis direct see mapping belongs since positive definite usual sense matrix clarke simulations grfs spheres cross time positive definite hence determinant nonnegative thus consider classic tensor product functions inner product given apparently thus inequality obtain last condition allows use fubini theorem hence conclusion kernel continuous definite mapping belongs theorem mercer kernel rest proof concerns expansion process follows arguments reproduce convenience reader series representation form clarke simulations grfs spheres cross time decreases basis convergence series absolute uniform compact subsets condition exists set mapping define random coefficients note hence inequality guarantees also fubini theorem allows see var fixed clear orthogonality observe therefore condition clear hence dominated convergence theorem allows conclude clarke simulations grfs spheres cross time fix fubini theorem observe thus proof concluded remark expansion theorem usually require extra hypothesis like compactness associated space kind invariance field line stochastic theorem theorem introduced may understood theorem isotropic fields topological compact group associated clarke simulations grfs spheres cross time haar measure unit mass theorem require condition continuity covariance function acknowledgments indebted editor three referees whose thorough reviews allowed considerably improved version manuscript references adams fournier sobolev spaces second edition pure applied mathematics amsterdam bateman higher transcendental functions vol new york berg porcu schoenberg coefficients schoenberg functions constr bergh interpolation spaces introduction grundlehren der mathematischen wissenschaften berlinnew york bonan clark estimates hermite freud polynomials approx theory christakos random field models earth sciences elsevier christakos hristopulos spatiotemporal environmental health modelling tractatus stochasticus foreword william roper kluwer academic publishers boston christensen measures analogous haar measure math dai approximation theory harmonic analysis spheres balls springer monographs mathematics springer new york dimitrakopoulos geostatistics next century springer netherlands ferreira menegatto eigenvalues integral operators defined smooth positive definite kernels integral equations operator theory ferreira menegatto positive definiteness reproducing kernel hilbert spaces beyond ann funct anal gneiting nonseparable stationary covariance functions spacetime data amer statist gneiting strictly positive definite functions spheres bernoulli jones stochastic processes sphere ann math kozachenko kozachenko modelling gaussian isotropic random fields sphere math clarke simulations grfs spheres cross time lang schwab isotropic gaussian random fields sphere regularity fast simulation stochastic partial differential equations ann appl marinucci peccati random fields sphere representation limit theorems cosmological applications london mathematical society lecture note series cambridge university press cambridge morimoto mitsuo analytic functionals sphere translations mathematical monographs american mathematical society providence porcu furrer modeling temporally evolving spatially globally dependent data porcu bevilacqua genton covariance functions great circle distance sphere amer statist positive definite definitizable functions mathematical topics akademie verlag berlin schoenberg positive definite functions spheres duke math stein covariance functions amer statist triebel theory function spaces monographs mathematics verlag basel triebel interpolation theory function spaces differential operators wiley zastavnyi porcu characterization theorems gneiting class covariances bernoulli
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sep sampling reconstruction using bloom filters neha sengupta amitabha bagchi bagchi srikanta bedathur sbedathur maya ramanath ramanath abstract paper address problem sampling set reconstructing set stored bloom filter best knowledge work first address question introduce novel hierarchical data structure called bloomsampletree helps design efficient algorithms extract almost uniform sample set stored bloom filter also allows reconstruct set efficiently case hash functions used bloom filter implementation partially invertible sense easy calculate set elements map particular hash value propose second method called hashinvert reconstruction study properties two methods analytically well experimentally provide bounds run times methods sample quality bloomsampletree based algorithm show extensive experimental evaluation methods efficient effective introduction bloom filters introduced bloom structures setmembership problem found numerous applications diverse array settings tremendous advantages offer terms space broder mitzenmacher surveyed host applications since usage bloom filters grown diversified typically applications rely query answered correctly good probability able deal drawback probability false positive occur however one fundamental question yet addressed sample element set stored bloom filter related question retrieve set stored bloom filter also addressed believe addressing two problems open possibility using bloom filters applications need store retrieve sample large number sets example storing subsequently sampling large number dynamic online communities form social networks twitter flickr etc could help advertisers determine target products storing retrieving call records associated specific locations investigations note compact structures sketches used compact storage structures samples later obtained however limitation approach sketches proposed created specifically problem sampling tend output sensitive design support reconstruction work hand shows draw samples well reconstruct sets generic synopsis structure bloom filter also useful several applications problem statement formally speaking given set drawn universe name space stored bloom filter referred query bloom filter denote elements false positives query answered returns yes algorithm samples one returns element chosen uniformly random algorithm reconstructs set stored returns set since bloom filters hide information elements stored providing partially correct answers membership queries natural way trying sample set stored bloom filter fire membership queries different elements name space bloom filter method referred dictionary attack scalable since running time linear size name space may huge solution overview paper outline method approaches task much efficiently conceptually design data structure bloomsampletree organizes namespace binary search tree node tree stores subset namespace level tree union subsets yields entire namespace root tree stores elements namespace leaf stores small subset namespace binary search tree constructed key idea locate leaves potentially contain elements present given query bloom filter done intersecting starting root search tree working way towards leaves entire subtrees pruned away yield empty intersections thus eliminating large parts namespace identify relevant leaves efficiently sample reconstruct original set using dictionary attack method explained note search tree needs constructed repeatedly used different query bloom filters drawback approach storing entire namespace bloomsampletree even though small part may actually occupied sparse occupancy namespace regular occurence especially consider keys strings namespace typically order actual occupancy likely order little billion perhaps less therefore construct tree entire namespace large number nodes going empty order address present dynamic version bloomsampletree call takes account occupancy dynamically change size structure occupancy changes algorithms provide sampling reconstruction one important feature require hash functions used bloom filter invertible method needs able use hash functions work given implementation bloom filter used store set also important note distinguish true elements set stored bloom filter false positives created process insertion approach bloom filter without prior knowledge inserted without method distinguishing true elements false positives summary method designed work efficiently scenario namespace potentially large even dynamic interesting subsets large millions billions may continue grow indefinitely iii need either sample reconstruct subset set interesting subsets stored form bloom filters specifically query bloom filters present methods aid engineer chosen use bloom filters particular application optimised parameters achieve given level accuracy ratio true elements elements return true answer membership query way dealing false positives contributions introduce novel called bloomsampletree used sample set stored bloom filter well reconstruct set bloomsampletree takes account occupancy namespace change size occupancy changes provide theoretical bounds runtime quality samples generated algorithm show iii show extensive evaluations algorithms efficient provide good quality samples organization section review literature provide brief background bloom filters outline framework methods operate section section outlines two baseline techniques sampling bloom filters along discussion limitations need bloomsampletree method bloomsampletree sampling reconstruction described detail sections results detailed experimental analysis presented sections related work bloom filters one widely used data structures approximately answering set membership queries compact storage efficient querying simple bit operators made valuable many different settings thorough survey bloom filters applications available despite widespread use aware work systematically addresses problems generating provably uniform samples using bloom filters reconstruct original set given level accuracy efficient way problem identifying least one true positive bloom filter considered adverserial setting study resilient bloom filters attacks given bloom filter adversary mount attack obtain elements original set repeatedly posing queries bloom filter potentially obtaining large number false positives also true positive elements work operate adversarial assume complete knowledge domain values represented bloom filter hash functions used given accuracy level aim efficiently generate provably uniform random samples original set well reconstruct set per accuracy requirements systematically solve problems back solutions detailed analysis time complexity accuracy sketches handling large datasets bloom filters belong general class approximation datastructures called sketches data synopses compactly represent massive volumes data preserving vital properties data needed analysis sketches used frequently databases community include histograms wavelets samples frequency based sketches however synopses datastructures used assumption underlying database always accessible case histograms samples required streaming scenarios reconstructing underlying set data values given level accuracy efficient manner objective begin recently results show sketches used generating samples called generalize earlier work inverse sampling goal maintain synopses structure stream updates addition deletion counts given domain size time possible sample high accuracy elements probability proportional number occurrances unlike techniques approaches focused streaming setting designed specific forms sampling proposed bloomsampletree approach used generate uniform samples bloom filters generic synopsis structure trees bloom filters paper present bloomsampletree comprises complete binary tree bloom filters stored every node purposes sampling reconstruction yoon also propose structure comprises complete tree bloom filters every node address multiset membership problem similar flavor yoon structure bloofi proposed crainiceanu lemire also address multiset membership problem representing set bloom filter stored leaf tree building tree combining bloom filters hierarchically flavour structures similar bloomsampletree concern problem multiset membership testing principle trees built completely different principle build tree contents bloom filters stored node bear relationship store node another work combines bloom filters trees athanassoulis ailamaki authors modify placing bloom filters leaves create approximate tree indexes seek exploit data ordering improve storage performance structure completely different intent design preliminaries section briefly provide necessary background bloom filters subsequently describe framework methods operate bloom filters bloom filter probabilistic data structure used store elements set comprises bit array bits along independent hash functions empty set represented bloom filter whose bits element set array positions indicated set bloom filter supports membership queries bloom filter storing set answer queries form false positive probability depends number bits hashed using hash functions obtain array positions bit positions set result positive since bits could set due insertion elements probability false positive evaluates bloom filter incapable false negatives membership query operation union intersection pair bloom filters also supported implemented using bitwise operations respectively use set hash functions namespace values also use set hash functions probability two fixed disjoint sets represented bloom filters bits hash functions false set overlap predicate true even though false set overlap bloom filter intersection reported probability framework methods operate database subsets elements drawn namespace size instead operating directly assume given compact approximation represented bloom filter given length filter bits set hash functions used construction collections subsets elements commonly seen many application settings including graph databases represent adjacency list vertex information retrieval represent list documents keyword occurs etc first task interested tackling setting generating random sample given specifically given information parameters used building approximation would like obtain provably uniform random sample given original database since operating approximate representation also expected fixed amount inaccuracy measured probability sampling element tolerable specified input system begin noted inaccuracy naturally linked probability false positives bloom filters thus given level inaccuracy accuracy bloom filters used designed second task natural extension reconstruct original entry database high accuracy sampling reconstruction describe two approaches sample element set reconstruct set stored bloom filter first simple dictionary attack method dictionaryattack second uses weakly invertible property certain types hash functions sampling hashinvert methods used sample well reconstruct bloom filter dictionaryattack method suffers high runtime inefficiencies hashinvert method provides guarantees quality sample compare bloomsampletree algorithm two baselines highlight advantages disadvantages approach detail section dictionaryattack sampling membership queries dictionaryattack algorithm relies reservoir sampling guarantee uniform sample equivalent reconstructing input set sampling element proceeds follows membership query fired input set element namespace positive reported element element retained sample diminishing probability proportional size set reconstructed far particular number positives reported positive retained sample probability clearly complexity algorithm size namespace note straightforward use method reconstruct original set hashinvert sampling invertible hash functions method assumes hash functions weakly invertible hash function weakly invertible given value one find set values example weakly invertible hash function constants knowing namespace straightforward find set elements hash given bloom filter exploits weak invertibility hash functions invert randomly sampled set bit candidate sets obtained using different hash function candidate sets subsequently pruned using membership queries bloom filter obtain value sampled uniformly random final sample returned analysis sampling obtained candidate sets done using method reservoir sampling hashinvert method occupies extra space sampling set bit takes time size bloom filter set bit chosen inversion using hash function takes time overall time taken sampling note contrast dictionaryattack provides uniformly random samples bounds given regarding quality samples case hashinvert however algorithm used reconstruct original set exhaustively running hashinvert algorithm set bits bloom filter simple trick gives benefits hashinvert algorithm bloom filter dense number unset bits potentially less number set bits therefore instead inverting set bits invert unset bits results set elements present original set therefore original set recovered set difference operation bloom sample tree section define bloomsampletree data structure help sample reconstruct set stored bloom filter bloomsampletree basically organises entire namespace note bloomsampletree built used repeatedly sample given query bloom filter definition bloomsampletree complete binary tree denoted log levels threshold whose choice discuss later section every node bloomsampletree bloom filter stores subset namespace every level tree contains entire namespace partitioned uniformly amongst nodes level hierarchically speaking organisation laminar sense union subsets namespace stored two sibling nodes gives set stored parent node bloom filters used bloomsampletree parameters number bits set hash functions bloom filters used sets sampling trying reconstruct reason frequently intersecting bloom filter set interest bloom filters stored various nodes bloomsampletree present formal definition definition given namespace size size bloom filter set hash functions used construction form integer parameter bloomsampletree collection bloom filters log bloom filters uses bit vector size hash functions property bloom filter stores elements note collection bloom filters forms tree structure since portion name space stored partitioned equally amongst nodes parent two nodes tree leaves tree store sets size namespace subdivided figure shows example bloomsampletree namespace node tree except root consist bloom filters size storing range elements depicted node set stored bloom filter query set need sample reconstruct note bloom filters constructed bloom filter set bloom sample tree bloom filter figure bloomsampletree levels query bloom filter representing set want sample mentioned introduction even though namespace may large likely small portion occupied therefore building complete bloomsampletree explained previous section potentially wastes huge amount space example data set experimented see section taken twitter contains million user ids distributed namespace size billion fraction namespace occupied order therefore practice build tree portions namespace actually occupied call condensed version tree dynamically change structure based change occupancy namespace namespace assigned tree potentially contains nodes reflect overview algorithm build tree follows let set identifiers currently use namespace initialise queue node log repeat queue empty dequeue node level offset within level check range intersection yes create attach tree insert elements range log enqueue node node create bloom filter corresponding subrange grow next level point nothing algorithm essentially goes tree building subtrees required accommodate elements ignoring subtrees corresponding ranges overlap although algorithm constructs search tree known ahead time easy see evolve grows new twitter accounts made need insert new element already existing nodes tree need create new node potentially subtree time taken build offline proportional size final tree constructed multiplied time range query time taken update tree proportional height tree sampling bloomsampletree given query bloom filter sample algorithm proceeds root following recursive manner relies pruning search space performance gains given node compute intersection bloom filters stored left right children node child nodes intersection empty contain element belonging range associated node therefore subtree rooted node pruned search intersection one child non empty search proceeds along child node child node subtree rooted pruned search intersection child nodes non empty one child nodes selected probability directly proportional estimated number elements corresponding intersections search proceeds along child note possible intersection false positive discovered subtree case search backtracks proceeds along child node estimated number elements intersection two bloom filters given following expression number bits set number bits set size bloom filters number hash functions used number bits set bitwise recall equation gives probability intersection incorrectly estimated two sets stored disjoint discuss issue section leaf node every element range node checked membership sample leaf node value sampled uniformly random set values satisfy membership test none elements within range satisfy membership query indicates search reached leaf node due string false set overlap case sample node algorithm called bstsample formal description algorithm figure shows typical scenario encountered sampling bloomsampletree numbers side node indicate order nodes traversed shown algorithm ultimately generates sample leaf node following one true path several false positive paths may branch multiple places note example node ultimately determined led several false positive paths discovered subsequently false positive path empty intersection potential path true path subtree pruned search subtree visited figure typical scenario sampling bloomsampletree false positive path chosen errors determining empty intersection empty intersection immediately results pruning subtree potential paths left unexplored choice following either subtree true path path actually taken algorithm generate sample subtree contrast whole subtree node immediately pruned search space search reaches leaf brute force search conducted scope false set overlap due bloom filter intersection sampling multiple items algorithm presented sampling outputs single sample sample multiple items could run algorithm multiple times however multiple runs done together one pass bloomsampletree explain given integer less size set stored send independent search paths bloomsampletree according algorithm bstsample paths sent bloomsampletree single pass since paths arriving internal node leaf processed node leaf move node find bloom filters children intersect query bloom filter take paths independently paths choose one children random bstsample send path child continues till paths reaches leaf let take concrete example illustrate process assume given query bloom filter intersect bloom filters stored left right child root bloomsampletree estimate size intersections let say throw three independent coins biased come heads probability suppose two coins come heads one comes tails recursively call two instances multiple sampling method easy see given tree structure bloomsampletree extension algorithm bstsample general perform better times running time case ask single sample output since paths behave like single sampling path bstsample guarantee sample quality maintained finally two paths happen reach leaf sample leaf without replacement depending whether samples generated without replacement summary analyses given bloomsampletree structure algorithm sampling briefly summarise analyses performed effect various parameters quality samples first question answer whether method generates uniformly random sample answer uniformly random sample indeed generated high probability prove property section show empirically well section accuracy given bloom filters approximate data structures possible samples generate actually belong original set recall sample generated membership queries leaf quantify accuracy samples follows acc number elements query set size namespace probability false positives bloom filter implementation accuracy defined simply computes ratio correct outcomes potential outcomes algorithm clearly size bloom filter effect accuracy determine based desired accuracy show performance method various values accuracy section runtime analysis runtime algorithm depends number false paths may follow analytically show expected number nodes visited section given bloomsampletree however also address practical issue regard runtime cost performing intersections node opposed cost performing number membership queries note possible based hash function used cost membership queries may cheaper expensive cost intersections two costs directly related elements stored leaf height bloomsampletree log tradeoff costs follows mcost cost one membership query bloom filter size hash functions icost cost intersection pair bloom filters size current node storing values would like determine whether better perform membership queries perform intersections leaf level log performing membership queries preferred traversing tree truncate tree leaf tree hence determine max icost log mcost empirically show runtime costs throughout section memory requirements memory required store bloomsampletree constructed used repeatedly depends size bloom filter number levels tree log interesting observation framework tradeoff memory accuracy memory runtime tradeoff accuracy runtime explained previous paragraphs therefore set best possible log order optimize runtime memory required may actually reduce creased accuracy cause need use larger bloom filter bloomsampletree increased accuracy would potentially reduce number levels reduce intersection cost described previous paragraph effect end reducing space used increasing accuracy also increasing runtime discuss empirical results detail section sample quality running time first question arises distribution samples bstsample produces aim produce uniform distribution set stored bloom filter present theoretical result shows samples produced near uniform first state result discuss implications proposition given set taken name space size run bstsample bloomsampletree define log log probability sampling algorithm finally samples size stored bloom filter leaf bloomsampletree lies probability least long let random proof probability bit zero insertion elements variable indicating number zero bits theorem set log log estimated size population bloom filter log log log log log also bound probability least log therefore probability least log log log small values substituing log log log log probability least log returning setting bloomtree let number elements intersection query bloom filter root node similarly let number elements intersection query bloom filter left right child root node respectively sampling process probability proceeding along left child directly proportional estimated number elements intersection left child query bloom filter let estimated number elements left right child root node respectively selecting left child probability least log consider given leaf node bloom tree contains subset size probability unbiased process sampling reach leaf estimate probability reaching leaf bloomtreesample method consider path bloomtree given leaf llog sequence subsets stored nodes slog using analysis done argue choice probability moving least probability log repeating argument log levels get log bloomtreesample reaches bloomtreesample reaches log probability least log log log last inequality applies log whenever since set present leaf nodes hence leaf node containing element probability bloomtreesample reaches leaf node least log recall words high probability log bloomtreesample reaches bloomtreesample reaches log leaf contains element whenever log goes grows bounds proving result discussion sample quality note since grows faster condition implies import proposition eventual leaf bstsample selects sampling chosen probability close proportional number elements set belong segment name space stored leaf absence false positives happens limit size bloom filter would lead perfectly uniform sampling move analysing running time algorithm clearly running time depends number factors provide theoretical analysis number nodes bstsample visits moves tree reach leaf generates sample clearly lower bound number height tree log able match lower bound able control number extra nodes visited give result guide choose system parameters ensure good asymptotic running time show following result proposition expected number bloomsampletree nodes visited algorithm bstsample sampling set size using bloomsampletree log proposition path bloomtreesample takes leaf node generates sample length equal height tree log along way many branches caused false set overlaps intersection bloom filter least bits set element name space bits set bloom filter branches need followed since possible distinguish branch takes genuine true false positives set stored bloom filter sampling order estimate number nodes visited log true path need estimate many false set overlap nodes visit proceed showing certain depth bloomtree false set overlap branch leads constant number false set overlap nodes visited hence depth number extra nodes visited constant factor necessary nodes visited along true path level however make claim assume visit every node formally proceed making following claim claim given node depth bloomtree names stored every leaf query set denotes subset namespace represented subtree rooted probability intersection two bloom filters size containing quantity number nodes algorithm bloomtreesample visits subtree conditioned event reaches proof proof claim follows noting since number names stored node child exactly half stored parent bloomtree noting sampling algorithm finds overlap goes subtrees children get taking expectations repeatedly get summed till leaf level since allows say log result follows whenever note argument basically saying sampling algorithms visits nodes level dominated subcritical branching process goes extinct probability yields nodes expectation mean progeny distribution noting since whenever depth condition satisfied whenever log bloomtree levels contains nodes get result discussion running time looking result note ratio name space bloom filter size critical element number nodes visited raising number bits bloom filter benefit running time least point second term running time analysis continues dominate first term number hash functions used size set sampled also correlated running time follows intuition determining empty intersection bloomsampletree data structure algorithm sampling straightforward implement however one practical problem encounter node algorithm visits set intersection needs performed determines whether prune branch unfortunately reliable way determine size set intersection empty since even single set bit results size estimation therefore use thresholding overcome problem estimated size set intersection particular threshold consider intersection null note heuristic potentially affect theoretical guarantee offered proposition effect since probability making wrong decision assuming set empty fact small choose correct threshold wrong decision implies certain elements set never presented samples see section happen practice reconstruction bloomsampletree recursive traversal tree results reconstruction set given bloom filter intersection node empty reconstructed set node empty set subtree rooted node pruned search however intersection empty search continues along left right children final reconstructed set node union reconstructed sets obtained two child nodes intersection leaf node empty conduct brute force search range node however instead sampling value set elements thus obtained return set reconstructed set node note expected number nodes bloomsampletree visited reconstruction algorithm analysed manner similar bloomtreesample expected number come log note extracting single element set treelike structure bloomsampletree would take log worst case assuming different elements set widely distributed name space worst case number nodes visited reconstruction log least algorithm unlike case sampling able meet lower bound exactly asymptotic sense also since second term directly proportional inversely proportional size bloom filters used choose parameters appropriately minimize time taken experimental evaluation static namespace section describe several experiments conducted determine effectiveness techniques sampling well reconstruction namespace static section describe experiments fraction namespace actually used setup experimented synthetic well real datasets made extensive use synthetic datasets generate controlled varied namespace size elements drawn also varied size sets generated either uniformly sampling namespace forming random local clusters details pointed earlier section desired accuracy levels used determine size bloom filter construct bloomsampletree varied accuracy requirements accordingly designed bloom filters simplicity experiments kept number hash functions although experimented different classes hash functions simple table summarizes parameter choices used experiments unless mentioned explicitly default values parameters mentioned table used experimental evaluations generating clustered uniform query sets report results two kinds randomly generated query sets uniform sets constructed generating elements uniformly random without replacement given range idea generating clustered query sets comes observations web graphs neighbour sets vertices typically ids clustered around nodes generate clustered query sets elements iteratively sampled namespace using pdf updated sample drawn initially pdf begins uniform distribution sample drawn identify max pdf min pdf neighbors divide pdf equally pdf pdf set pdf generate aggressively clustered sets one subtract probability element equally divide accumulated probability controls degree clustering experiments used algorithms baseline method dictionary attack referred additionally evaluating performance set reconstruction use hashinvert baselines described section methods compared bloomsampletree bst approach metrics methodology report following number intersections set membership operations main metric interest compute depth bloomsampletree size bloom filters based accuracy relative costs intersection membership operations discussed section uniform clustered query sets report average number intersection membership operations bloom filters samples average time taken setup report average time samples memory analytically computed overall size required bloomsampletree based size bloom filters used tree depth quality uniform samples report samples generated addition also show empirically observed distribution samples sampling experiments figures show number intersections membership operations uniformly random clustered query sets respectively method always uses membership operations intersection operations hand bloomsampletrees try offset large number membership operations intersections bloom filters note sampling accuracy increases size bloom filters increases well resulting expensive intersection operations runtime performance intersection membership operations become expensive intersections membership operations bloom filter size increases bloom filter size turn determined namespace size well accuracy requirements thus overall efficiency bloomsampletree depends careful balance number operations figure shows average time taken bloomsampletree methods namespace size plots show bloomsampletrees achieve improvements efficiency single sampling round another implementation choice significantly affect performance numbers hash functions figure shows time taken generate samples different hash function families dictionary attack suffers cost computing hash function increases instance hash functions performance goes almost order magnitude hand bloomsampletree sampling procedure defers membership queries lower levels tree time tree already pruned search using fast hash functions like simple bloomsampletree automatically leverages efficiency reduce overall time taken memory requirement finally turn attention amount memory footprint needed method memory requirements mbs shown tables number elements query set tables depth log computed discussed section memory analytically computed using number nodes bloomsampletree memory bloomsampletree thus computed affirmed empirical measurement program execution evident table memory requirement might actually reduce increasing accuracy primarily depth bloomsampletree decreases total memory occupied reduces spite increased bloom filter sizes since lower levels much larger memory footprint higher ones since memory reduce increasing accuracy overall accuracy memory one hand running time bloomsampletree allows fast sampling requires small additional storage methods described paper moreover one need store bloomsampletree possible query set one bloomsampletree given size namespace bloom filter size choice hash functions quality sampling use pearson test briefly describe empirically validate sample quality conduct sampling rounds bloom filter storing set let number times element sampled similarly let expected number times element sampled null hypothesis sampling uniform restated goal test would see null hypothesis rejected given observations define random variable follows distribution degrees freedom given observation compute value let value defined clearly smaller higher value indicating greater deviation expectation words smaller indicates observation lesser support falls threshold known significance level rejected otherwise significance level typically set around set use recommended sample size significance level thus obtained reported sets different sizes table entries table therefore null hypothesis rejected case higher values accuracy clear distribution elements close uniform distribution accuracy value determined based accuracy verified accuracy obtained sampling process using expression measured accuracy table cases measured accuracy found close expected value table shows measured accuracy values creation time table measure time taken create bloomsampletree different sizes namespace desired accuracy note case increased accuracy required value increases resulting decreased depth bloomsampletree section consequently lower creation time desired accuracy creation takes application scenarios bloomsampletree works number sets must constructed bloom filters assumed massive changing bloom filter parameters requires creating sets database prohibitively expensive overshadowing time taken create bloomsampletree algorithm sampling bloomtrees algorithm bstsample bst bloomsampletree query bloom filter leaf exhaustively check interval leaf membership bst leaf end end uniformly sampled return else lflag estimatedsize rflag estimatedsize child intersects reached due false positive return lflag rflag return else randomly select one proceed search along probability proportional estimated number elements uniform lag lag sample bstsample case sample found child search along child backtracking sample sample bstsample return sample else sample bstsample sample sample bstsample return sample end end table parameters experiments parameter range default value size namespace size query set sampling accuracy hash families simple intersection membership sampling accuracy membership intersection sampling accuracy membership intersection sampling accuracy figure intersections set membership queries uniformly random query sets xxxbst legend refers cardinality query sets intersection membership sampling accuracy membership intersection sampling accuracy membership intersection sampling accuracy figure intersections set membership queries clustered query sets legend refers cardinality query sets time sampling accuracy uniformly random query set time sampling accuracy clustered query set figure avg time taken sampling accuracy depth nodes table various parameters settings bloom sample tree implementation memory mbs accuracy depth nodes table various parameters settings bloom sample tree implementation memory mbs time sampling accuracy uniformly random query set time sampling accuracy clustered query set figure avg time taken sampling time sampling accuracy figure effect different hash function families performance levels stime utime levels stime utime levels stime utime table system user time taken create bloomsampletree different values desired accuracy accuracy table reconstruction experiments setup reconstruction experiments follow sampling experiments adding hashinvert baseline figures show number intersections set membership queries reconstruct sets uniformly random clustered drawn namespaces size respectively number intersections sampling accuracy see trend similar ones sampling experiments reasons one may note hashinvert procedure performs membership queries bloomsampletree fewer dictionary attack despite overall cost hashinvert seen figures show time taken reconstruction overall cost hashinvert essentially depends number set reset bits bloom filter bloom filter extremely dense reconstructing help reset bits efficiently reconstructs set whereas sparse one reconstruct using set bits however hashinvert inefficient neither cases apply evident line sets bits bloom filter cause fact hashinvert iterates inverted set set reset bit bloom filter since values may already checked save membership queries however given membership query fast simple hash functions directly translate smaller running times accuracy table measured accuracies uniform query sets size intersection membership membership intersection precision uniformly random query set precision clustered query set figure avg operations reconstructing experiments data namespace far presented results settings namespace contiguous fixed turn attention practical settings size namespace need handle small fraction much larger domain potentially spread throughout setup dataset made use twitter crawl consisting million tweets total million user ids tweet set distributed namespace little billion varying namespace fractions note even though million unique ids dataset could distributed across entire namespace billion suppose example built bloomsampletree leaves range billion effectively divided ranges could empty depending distribution unique ids hypothetical bloomsampletree construct namespaces different namespace fractions follows intersection membership precision membership intersection uniformly random query set precision clustered query set figure avg operations reconstructing uniform namespace following example suppose want construct namespace namespace fraction uniformly sample leaves gives set ranges union occupy fraction total namespace clustered namespace namespace fraction need sample leaves clustered way use technique explained section case generating clustered query sets fixed desired accuracy discussed section therefore hypothetical bloomsampletree depth bloom filter size correspondingly depth bloom filter size number nodes therefore space occupancy much smaller query bloom filters identified unique hash tags occurred least times dataset sets users tweeting particular hashtag used construct query bloom filter therefore constructed query bloom filters however experimenting varying namespace fractions simply ignore ids belong namespace currently consideration construct query bloom filters without metrics report following metrics membership intersection precision membership intersection uniformly random query set precision clustered query set figure avg operations reconstructing average time taken namespace fraction run sampling rounds randomly chosen query bloom filters report average time taken generate sample memory occupies much less space full bloomsampletree report space usage namespace fraction accuracy value bloom filter size based desired accuracy bloomsampletree actual accuracy expected better since elements occupy namespace stored report accuracy various namespace fractions sampling experiments average time taken figure shows average time taken generate samples query bloom filters namespace fractions less time taken order magnitude smaller full namespace occupancy also expected sampling time case clustered namespace smaller since leaves share common ancestors far less paths bloomsampletree sampling algorithm follow dictionary attack requires seconds average one sample drawn natural since size namespace extremely large case result included result figure ensure finer variations sampling time taken random clustered namespaces clearly visible time precision uniform random query set time precision clustered query set figure avg time taken reconstruction uniformly random clustered query sets memory figure shows memory usage varying namespace fractions note built full bloomsampletree namespace billion memory required would approximately contrast lower namespace fraction memory usage bloomsampletree uniform case much lower clustered case reason sampling time expect memory requirement bloomsampletree smaller clustered namespace accuracy figure shows sampling accuracy various namespace fractions recall optimized bloomsampletree accuracy uniformly see higher accuracy accuracy depends size namespace mentioned section size effective namespace lower namespace fraction smaller shows bloomsampletree capable producing higher accuracy results overall namespace large actually occupied effective namespace small conclusions paper described efficient method sampling reconstruction sets stored bloom filters particular described bloomsampletree data structure analyzed time precision uniform random query set time precision clustered query set figure avg time taken reconstruction uniformly random clustered query sets properties theoretically experimentally compared technique brute force approach dictionary attack well hashinvert useful using invertible hash functions reconstruct sets extensive evaluation algorithm various settings demonstrated wide applicability significant advantages references bloom hash coding allowable errors commun acm vol broder mitzenmacher network applications bloom filters survey internet vol romero meeder kleinberg differences mechanics information diffusion across topics idioms political hashtags complex contagion twitter www ghosh lerman framework quantitative analysis cascades networks wsdm uniform time clustered namespace fraction memory figure time taken generate uniform sample varying namespace fractions uniform clustered namespace fraction figure memory usage varying namespace fractions uniform clustered accuracy namespace fraction figure sampling accuracy varying namespace fractions cheng adamic dow kleinberg leskovec cascades predicted www macmillan glisson bromby investigating increase mobile phone evidence criminal activities hicss cormode muthukrishnan rozenbaum summarizing mining inverse distributions data streams via dynamic inverse sampling vldb monemizadeh woodruff applications soda jowhari saglam tardos tight bounds samplers finding duplicates streams related problems pods tarkoma rothenberg lagerspetz theory practice bloom filters distributed systems ieee comm surveys tutorials vol bellovin cheswick searches using encrypted bloom filters columbia university tech naor yogev bloom filters adversarial environments corr vol online available http cormode garofalakis haas jermaine synopses massive data samples histograms wavelets sketches found trends databases vol yoon son shin bloom tree search tree based bloom filters multipleset membership testing proc ieee conference computer communications infocom crainiceanu lemire bloofi multidimensional bloom filters inf vol athanassoulis ailamaki approximate tree indexing proc vldb vol october guo chen yuan luo dynamic bloom filters ieee trans knowl data vol jeffrey steffan understanding bloom filter intersection lazy disambiguation spaa new york usa acm vitter random sampling reservoir acm trans math vol online available http papapetrou siberski nejdl cardinality estimation dynamic length adaptation bloom filters dist parallel databases vol online available http mitzenmacher compressed bloom filters podc athreya ney branching processes new york usa acm springer boldi algorithmic gems data miner cave proc fun algorithms springer online available http stamatis six sigma beyond design six sigma crc press vol
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graphs jan january abstract graph satisfying determine positive integers complexity deciding whether given graph also address problem finding function problem class graphs minimum degree less polynomial graphs minimum degree least prove exists value keywords polynomial minimum degree introduction graph partition two disjoint sets let two graph properties graph induces graph property graph property example graph many papers dealing problems graphs examples long list examples problems recognizing bipartite graphs two independent sets split graphs clique independent set well known easy show linear algorithms deciding whether graph bipartite respectively split graph easy exercise show every graph degree vertex half original degree furthermore partition found efficiently greedy algorithm several papers opposite condition studied require vertex least half neighbours inside set belongs partition problem known satisfactory partition problem general graphs partition problem received particular attention finding sufficient conditions graph possess thomassen proved existence function every graph minimum degree least proved max later improved hajnal see thomassen asked whether would hold would best possible complete graph stiebitz proved indeed since result published several groups researchers tried find extra conditions graph would allow smaller minimum degree requirement among others following results obtained theorem integers every graph department mathematics computer science university southern denmark odense denmark email jbj work done first author visiting lirmm montpellier france hospitality gratefully acknowledged research supported danish research council grant number lirmm montpellier france email theorem integers every graph theorem integers except every graph integers every graph two share edge original proof constructive neither theorems bazgan gave polynomial algorithm constructing graph minimum degree least least input main result paper full characterization complexity partition problem theorem let integers polynomial decide whether graph values decide existence partition result implies already graphs results complexity finding partitions lower upper bounds degrees inside partition find anything implies theorem result stiebitz insures minimum degree input graph large enough least always exists conjecture minimum degree large less always trivial solved polynomial time conjecture exists function problem class graphs minimum degree less polynomial graphs minimum degree least next section introduce notions tools used later section give proof theorem section provide partial results concerning conjecture particular prove exists value finally section address partition problems mainly dealing connectivity part partition notice regarding results establish paper first open case conjecture following problem problem complexity problem graphs minimum degree notation definitions preliminary results notation standard follows paper graphs parallel edges loops use shorthand notation set special graphs first define graphs used frequently proofs ensure certain vertices sufficiently high degree let graph obtain subdividing one edge vertex let obtained adding vertex adjacent degree vertex let graph vertices obtain adding new edge linking non adjacent vertices outer cycle new vertex joined vertices outer cycle incident another new vertex let graph defined let let graph obtain adding cycle vertices degree adding new vertex adjacent vertices degree finally let graph obtain deleting one edge adding two new vertices edges graphs depicted figure vertices vertices figure graphs connected instances given instance clauses variables variable occurs least literal least define bipartite graph graph vertex set first set corresponds literals second one clauses edge set containing edge vertex vertices corresponding literals every say connected instance connected lemma instances connected proof suppose connected components fix literal vertex add new variable new clauses let new formula obtained adding variable easy check equivalent connected clauses adding extra variables necessary also obtain equivalent connected instance literal occurs least twice leave easy details interested reader ring graphs first introduce important class graphs play central role proofs directed analogue graphs used ring graph graph one obtains taking two copies complete bipartite graph vertices edges joining circular manner adding path vertex path ith copy etc indices modulo etc proofs reductions variants problems call copies switch vertices start showing associate ring graph given formula let instance consisting clauses set boolean variables clause form belongs negation variable adding extra clauses obtain equivalent formula necessary ensure every literal occurs least twice shall use fact one proofs variable ordering clauses induces ordering occurrences resp clauses let resp denote number times resp occurs clauses let ring graph defined follows vertex set edge set consists following edges edges paths associate clause set consisting three vertices representing occurrences literals occurrence clauses contains vertex occurrence clauses contains vertex two vertices defined similarly proofs often add vertex adjacent vertices example depicted figure figure ring graph corresponding formula grey boxes contain switch vertices white vertices variable vertices added clauses vertices part ring graph following observation forms base many proofs easy prove proof result similar see theorem let formula let corresponding ring graph contains cycle intersects sets cycle yes proof theorem case start trivial observation proposition every graph least vertices except star proposition polynomial algorithm testing whether graph partition proof try every choice adjacent vertices whether solution clearly yes least one attempts succeed hence starting moving vertices one neighbour either end good partition case partition exists choice case following variant satisfiability call known given boolean cnf formula consisting clauses variables clause literals variable occurs clauses literal appears twice decide whether satisfied could find proper reference proof give one presented pages set course lyuu national taiwan university assume instance variable occurs total times formula clauses cir introduce new variables replace first occurrence negated otherwise replace similarly replace occurrence cij finally add new clauses clauses size force variables take value satisfying truth assignment repeating replacement variables original formula obtain equivalent instance need another variant call clauses still size variable allowed occur times times literal following scheme original variable occurring least times adding extra clauses obtain equivalent instance new clauses form cycle bipartite graph easy see connected instance connected instance hence lemma connected theorem decide whether graph partition proof show reduce instance connected polynomial time show extend construction higher values start construction use several disjoint copies graphs achieve construction let connected instance clauses variables https may assume literals occur least follows fact may assume instance normal reduction preserves property construct follows variable introduce three new vertices two edges literal occurs precisely identify vertex private copy occurs precisely twice identify vertex private copy add new vertices corresponds clause join edge vertices correspond literals gets two edges way identify vertex private copy add new vertices edges finally add edges claim satisfied suppose first satisfying truth assignment easy check good take union vertices corresponds false literals note vertex degree via private copy one graphs corresponding literal occurred times conversely assume claim must vertices one degree exactly clearly also however construction literal clause vertices degree induce connected graph use instance connected bipartite graph thus vertices must vertex isolated contradiction hence vertices implies also vertices hence least one vertices also define truth assignment follows put true otherwise put false since must neighbour satisfying truth assignment obtain construction replace copy copy copy copy identify literal clause vertices vertex extra private copy finally identify vertex vertex private copy easy see complete proof case increase degree literal clause vertices identifying vertices private copies repeat proof corollary problem graphs minimum degree proof recall proof vertices corresponding clauses must always belong good partition hence connect vertex edges obtain graph minimum degree good partition satisfiable vertices must belong degree theorem every choice natural numbers decide whether graph proof show reduce given instance clauses variables proceed follows start copy ring graph add following identify vertex vertex private copy add new vertex join edges vertices identify vertex private copy add new vertex join edges vertices identify vertex private copy add new vertex identify vertex private copy add three edges three vertices correspond literals finally add new vertex join vertices via private copies identifying vertex chosen vertex claim final graph satisfiable first make observations every vertex switch vertex degree exactly degree adjacent exactly one identified one vertex private copy switch vertices degree exactly vertices degree exactly vertices copies degree exactly vertices copies degree exactly vertices degree vertex degree may clearly assume least convenience writing define empty graph talk without condition least suppose first satisfiable theorem means cycle intersects neighbourhood another cycle let consist vertices corresponding private copies vertices along private copies finally vertex vertices copies used let contains vertices private copies easy check good partition assume good way connected vertices via copies implies must belong set must otherwise rename sets since degree least one vertices corresponding literal must belong suppose vertex corresponding literal vertices path corresponding literal including two end vertices switch vertices must belong follows fact vertices degree neighbour moreover resp belongs resp degree resp belongs least one vertices resp belongs resp belongs one resp must lie since empty implies restriction cycle consisting paths either path path hence cycle intersecting neighbourhoods vertices hence satisfiable theorem combining results section concludes proof theorem higher degrees study borderline polynomial instances partition problems try see close get bound minimum degree still instance give precise answer combining corollary result proposition polynomial algorithm checking whether graph minimum degree least proof suffices see test given edge whether partition done starting moving vertices vertices least neighbours note process preserves invariant hence process terminates found desired partition otherwise proceed next choice edge start problem also give precise borderline polynomial instances proposition exists polynomial algorithm checking whether given graph minimum degree least proof first test whether two disjoint cycles done polynomial time pair exists assume found pair disjoint cycles put vertices continue move vertices least two neighbours current process stops remaining set induces graph minimum degree least since vertices move one neighbour proceed partitions try raise minimum degree see whether still prove theorem every decide whether graph minimum degree proof give proof explain extend larger let instance variable clauses let obtained adding edge vertices distance one paths replace paths square construct graph starting follows add two vertices join vertices correspond literals add vertices add two edges two edges resulting graph minimum degree claim satisfiable know theorem previous proofs used approach vertex set partitioned two cycles contains neighbour vertices proof easy satisfiable let let easy check contains neighbour assume literal appears least twice insure suppose since adjacent vertices degree share neighbour must belong set set must also contain vertices without loss generality thus subset vertices degree initial terminal vertex path degree using difficult see vertex path vertices path two adjacent switch vertices vertices vertices using observation made vertices would impossible similarly show contain vertices vertices hence switch exactly one vertices exactly one vertices see vertices induces cycle vertices degree outside cycle must contain neighbour hence theorem satisfiable obtain result higher values induction proved base case assume already constructed satisfiable construct two copies joining copies vertex edge easy check theorem deciding whether graph minimum degree pcomplete proof let graph obtain starting ring graph adding following add vertices tree whose internal vertices degree leaves denoted pairs parent pairs vertices correspond clause join vertex vertices correspond literals correspond add edge add new vertices add edges add edges join edges vertices add edge join edges vertices add edge first prove vertices must set note different sets partition one neighbours must sets similar argument vertices must belong set partition vertices must belong set partition easy check implies claim satisfiable theorem find vertex disjoint cycles contains vertex corresponding literal let easy check must contain exactly one vertices exactly one vertices suppose good partition argument must since graph vertices except switch vertices degree thus vertices neighbour also earlier proofs easy check vertex one paths vertices path proof previous theorem conclude switch exactly one vertices exactly one vertices see vertices induce cycle vertices degree outside cycle must contain neighbour hence theorem satisfiable corollary every decide given graph minimum degree least proof follows induction theorem base case way proved last part theorem proposition polynomial algorithm deciding whether given graph minimum degree proof let theorem algorithmic version result may assume denote vertex set done take assume vertex adjacent vertices induce take assume vertex adjacent vertices induce contains cycle conclude starting adding vertices long one least neighbours process stops good partition hence assume acyclic one connected component non trivial spanning tree two leaves share neighbour without loss generality induced cycle formed path disjoint find good partition hence found partition yet must independent set whose vertices joined vertices easy find good partition consisting one vertex remaining vertices finally solution problems thomassen proved every graph connectivity least minimum degree least natural ask complexity deciding whether graph prescribed lower bounds connectivity start simple observation proposition exits polynomial algorithm deciding whether given graph connected proof suppose first two connected components two components yes one hence assume connected easy see good partition tree nontrivial block leaf tree thus assume consider sometimes called start arbitrary cycle let last ear add let end vertices good partition perhaps bit surprisingly require bit connected part problem becomes proofs use reductions given formula describe necessary modifications theorem decide whether undirected graph vertex partition connected proof add vertices edges follows clause add vertex join three edges three literal vertices corresponding several proofs add new vertices edges add new vertices edges add new vertices edges claim resulting graph vertex partition connected satisfiable note construction every good partition every vertex must belong particular path contains vertex vertices since want connected edges imply every one vertices one vertices belong implies exactly one exactly one vertices belong otherwise would empty easy check desired partition exists contains cycle uses precisely one paths avoids least one literal vertex every clause thus theorem satisfiable good partition since decide whether graph two vertex disjoint cycles polynomial time following result whose easy proof leave interested reader implies polynomial decide whether graph two connected graphs proposition graph connected cycle pair disjoint cycles either connected exactly two connected components contain cycle theorem decide whether graph vertex partition proof let formula let graph constructed proof let graph obtained adding following vertices edges add new vertices add edges add edges path complete path cycle adding edges add edge vertices claim two graphs satisfiable first observe good partition must vertices inside follows fact degree needs neighbours set without loss generality cycle deleting vertices exactly graph proof theorem easy see vertices must belong good partition implies desired graph partition connected problem theorem proof complete inspecting proof difficult see following holds theorem decide whether graph graphs may worth try extend results section higher edge references alon splitting digraphs combin prob cohen havet finding good digraphs enumerable properties theor comput gutin digraphs theory algorithms applications london edition havet finding good digraphs hereditary properties theor comput bazgan tuza satisfactory partition problem discrete appl bazgan tuza vanderpooten decompositions graphs bounded treewidth planarity theor comput bazgan tuza vanderpooten efficient algorithms decomposing graphs degree constraints discrete applied mathematics bensmail complexity partitioning graph connected subgraphs combin extremal graph theory academic press london bonamy dabrowski feghali johnson paulusma recognizing graphs close bipartite graphs international symposium mathematical foundations computer science mfcs august aalborg denmark pages bonamy dabrowski feghali johnson paulusma independent feedback vertex sets graphs bounded diameter information proc bondy murty graph theory volume graduate texts mathematics berlin recognizing decomposable graphs graph theory dyer frieze complexity partitioning graphs connected subgraphs discrete appl hammer split graphs congress gerber kobler algorithmic approach satisfactory graph partitioning problem european operation research gerber kobler classes graphs partitioned satisfy vertices australasian grigoriev sitters connected feedback vertex set planar graphs graph theoretical concepts computer science volume lect notes comp pages springer verlag berlin hajnal partition graphs condition connectivity minimum degree combinatorica kaneko decomposition graphs degree constraints graph theory osthus partitions graphs high minimum degree connectivity combin theory ser splitting graph disjoint induced paths cycles discrete appl liu partitions graphs degree constraints disc appl liu bipartition graph degree constraints science china mathematics graphs containing independent circuits hungarian matlapok yang decomposing graphs degree constraints misra philip raman saurabh sikdar fpt algorithms connected feedback vertex set combin stiebitz decomposition graphs digraphs kam series discrete charles university prague stiebitz decomposing graphs degree constraints graph theory suzuki takahashi nishizeki linear algorithm bipartion biconnected graphs inform process thomassen graph decomposition constraints connectivity minimum degree graph theory thomassen paths circuits subdivisions selected topics graph theory vol pages academic press van hof paulusma woeginger partitioning graphs connected parts theor comput xiao nagamochi complexity kernels bipartition induced graphs theor comput
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crossover speeds assembly genetic algorithms nov dirk sudholt department computer science university sheffield united kingdom abstract fundamental question effective crossover genetic algorithms combining building blocks good solutions although discussed controversially decades still lacking rigorous intuitive answer provide answers royal road functions every bit building block latter show using crossover makes every genetic algorithm least twice fast fastest evolutionary algorithm using standard bit mutation terms moderate crossover beneficial effectively turns mutations improvements combining right building blocks later stage compared evolutionary algorithms makes mutations useful introducing crossover changes optimal mutation rate holds uniform crossover crossover experiments statistical tests confirm findings apply broad class functions keywords genetic algorithms crossover recombination mutation rate runtime analysis theory introduction ever since early days genetic algorithms gas researchers wondered crossover effective search operator folklore crossover useful combine building blocks schema high fitness give better solutions watson jansen put considerable difficulty demonstrating rigorously intuitively many attempts understanding crossover made past mitchell forrest holland presented royal road functions example supposedly genetic algorithms outperform search algorithms due use crossover royal roads divide bit string disjoint blocks block makes positive contribution fitness case bits therein set blocks thus represent schemata configurations building blocks optimal solutions however authors later concluded simple randomized hill climbers performed better gas role crossover studied multiple angles including algebra markov chain models infinite population models dynamical systems see chapter overview statistical mechanics see chapter also biology role crossover far settled population genetics exploring advantages recombination sexual reproduction famous open question called queen problems evolutionary biology graham bell others evolutionary processes found harder analyze using asexual reproduction represent quadratic dynamical systems recent work population genetics focussed studying speed adaptation describes efficiency evolution similar vein research evolutionary computation furthermore new theory mixability proposed recently perspective theoretical computer science arguing recombination favours individuals good mixers individuals create good offspring recombined others several researchers recently independently reported empirical observations using crossover improves performance evolutionary algorithms eas simple function unable explain fact even settings simple well understood demonstrates need solid theory serves motivation work runtime analysis become major area research give rigorous evidence proven theorems however studies far eluded fundamental setting functions crossover proven superior mutation constructed artificial examples like jumpk real royal road functions problem coloring problems inspired ising model physics computing unique sequences finite state machines selected problems optimization allpairs shortest path problem ising model trees consist hierarchical building blocks none papers addresses singlelevel building blocks setting simple royal roads watson jansen presented constructed function proved exponential performance gaps eas using mutation however definition internal structure building blocks complicated artificial used tailored get necessary diversity regard question gas combine building blocks approach give intuitive explanation one hoping paper presents intuitive explanation supported rigorous analyses consider royal roads functions composed building blocks monotone polynomials special case every bit building block give rigorous proofs show main proof arguments transfer broader classes functions experiments support latter main results follows show section every uniform crossover standard bit mutation least twice fast every evolutionary algorithm uses standard bit mutations terms precisely dominating term expected number function evaluations decreases holds provided parent population offspring population sizes moderate inertia large population slow exploitation reason speedup store neutral mutation mutation altering parent fitness population along respective parent use crossover combine good building blocks two individuals improving current best fitness bipartite graphs problem equivalent classical graph coloring problem colors words crossover capitalize mutations beneficial disruptive effects building blocks use uniform crossover leads shift optimal mutation rate demonstrate section simple greedy always selects parents among current best individuals eas optimal mutation rate greedy optimal mutation rate ignoring terms introducing crossover makes neutral mutations useful larger mutation rates increase chance neutral mutation optimality proved means matching lower bound expected optimization time greedy applies crossover operators bit value taken either parent using optimal mutation rate expected number function evaluations log log results limited uniform crossover absence linkage section shows results hold gas using crossover arbitrary slightly stronger conditions crossover probability set appropriately small value reasoning carries functions clear building block structure experiments section reveal similar performance differences royal road functions random polynomials unweighted positive coefficients largely confirmed statistical tests evidence findings also transfer weighted functions like linear functions provided population store solutions different fitness values different building blocks crossover able combine case greedy simple significantly faster random linear functions optimal class functions first result analysis uniform crossover remarkably simple intuitive gives direct insight working principles gas simplicity also makes well suited teaching purposes work extends preliminary conference paper parts results results restricted one particular greedy extended version presents general analytical framework applies gas subject mild conditions includes greedy special case end provide tools analyzing parent offspring populations gas believe independent interest moreover results crossover improved leading constant larger upper bound crossover additive term uniform crossover mutation rates left open question whether crossover effective uniform crossover assembling building blocks provide new refined analysis gives affirmative answer mild conditions crossover probability related work sudholt theile considered search behaviour idealized highlight potential benefits crossover ideal circumstances able recombine two individuals equal fitness independent evolutionary lineages fitness gain order idealized would therefore able optimize expected time however idealization reasonably achieved realistic eas common search operators hence result regarded academic study potential benefit crossover related strand research deals analysis simple simple one best known best researched gas field uses generational model parents selected using selection generated offspring form next population neumann oliveto witt showed simple without crossover high probability optimize less exponential time reason population typically contains individuals similar fitness selection similar uniform selection oliveto witt extended result uniform crossover simple uniform crossover population size still needs exponential time even needs exponential time reach solution fitness larger arbitrary constant authors relaxed condition population size work exclude crossover advantageous particularly since right circumstances crossover may lead large increase fitness advantage noticeable simple crossover still fails badly one year published doerr doerr ebel presented groundbreaking result designed proven optimise simple transformation thereof time log spectacular result search algorithms using unbiased unary modifying one individual exhibiting inherent search time log shown lehre witt shows crossover lower expected running time constant factor call algorithm starting one parent first creates offspring mutation random potentially high mutation rate selects best mutant crosses times original parent using parameterized uniform crossover probability taking bit first parent always parameter algorithm leads number log expected function evaluations decreased scheme adapting according current fitness cleverly designed work efficiently similar functions uses design two phases environmental selection differences mutation performed crossover mutation fully independent offspring number flipping bits random variable determined standard bit mutations number flipping bits used offspring focus work different goal understand standard eas operate crossover used speed assembly commonly used eas preliminaries measure performance algorithm respect number function evaluations performed optimum found refer optimization time algorithms equals number generations apart initialization eas offspring populations eas gas optimization time factor larger number generations note number generations needed optimize fitness function often easily decreased using offspring populations parallel evolutionary algorithms significantly increases computational effort within one generation number function evaluations fair widely used measure looking function evaluations often motivated fact operation dominates execution time algorithm number function evaluations reliable measure wall clock time however wall clock time might increase introducing crossover additional search operator also increasing mutation rate numbers might required jansen zarges point case effect leads discrepancy number function evaluations wall clock time concern must taken seriously aiming reducing wall clock time however implementation must checked individually respect therefore keep concern mind still use number function evaluations following uniform crossover makes eas twice fast show mild conditions every least twice fast counterpart without crossover latter evolutionary algorithms using standard bit mutation author recently proved following lower bound running time broad class eas covers possible selection mechanisms parent offspring populations even parallel evolutionary algorithms slightly rephrase result theorem sudholt let every uses standard bit mutation mutation rate create new solutions expected optimization time least min every function unique optimum constant least log fact author proved among evolutionary algorithms start one random solution use standard bit mutations expected number function evaluations minimized simple theorem also mutation rate best possible choice leading lower bound special case doerr fouz witt recently improved bound towards show range eas defined following introducing uniform crossover cut dominant term running time half standard mutation rate requirement parent selection mechanism selection favor inferior solutions fitter ones formally maximizing fitness function prob select prob select particular implies equally fit solutions selected probability condition satisfied common selection mechanisms uniform selection selection tournament selection cut selection mechanisms class eas covered work defined algorithm eas therein create offspring crossover mutation mutation pick best previous search points new offspring algorithm scheme mutation rate uniform crossover crossover probability maximizing initialize population size true let probability select operator respecting let uniform crossover otherwise select operator respecting end flip bit independently probability add end let contain best individuals break ties towards including individuals fewest duplicates end case ties pick solutions fewest duplicates among considered search points strategy already used jansen wegener groundbreaking work real royal roads ensures sufficient degree diversity whenever population contains different search points fitness stating main result section provide two lemmas showing analyse population dynamics lemmas independent interest may prove useful studies eas following lemma estimates expected time individuals fitness least take whole population generalizes lemma turn goes back witt analysis note lemma applies arbitrary fitness functions arbitrary values arbitrary crossover operators merely relies fundamental universal properties cut selection standard bit mutations lemma consider implementing algorithm crossover operator fitness function assume current population contains least one individual fitness expected number function evaluations needed individuals current population fitness least log holds rule used environmental selection proof call individual fit fitness least estimate expected number generations population taken fit individuals call expected takeover time fit individuals always preferred individuals environmental selection expected takeover time equals expected number generations fit individuals created starting one fit individual offspring created chance simply create clone fit individual happens creation offspring decides perform crossover selects fit individual parent mutated mutation flip bit probability event least number fit individuals population since fit individual selected parent probability least divide run phases order get lower bound number fit individuals certain time steps phase starts first offspring creation first generation number fit individuals least ends first generation number increased min let describe random number generations spent phase starting new generation fit individuals parent population consider phase offspring creations disregarding generation bounds let denote random number new fit offspring created phase classical chernoff bounds see chapter prob phase called unsuccessful consider another phase offspring creations expected waiting time successful phase expected number offspring creations since phases start generation bounds may need account offspring creations phases implies expected takeover time log also provide following simple handy lemma relates success probabilities created offspring expected number function evaluations needed complete generation event first happened lemma consider implementing algorithm assume offspring creation probability least specific event occurs expected number function evaluations complete generation event first occurs proof expected number trials event probability occur complete generation function evaluations required able prove main result section theorem expected optimization time every implementing algorithm constant mutation probability log constant log log log bound simplifies statements hold arbitrary initial populations main difference upper bound gas lower bound eas additional factor denominator upper bound factor even larger gain larger mutation rates default value shows introducing crossover makes eas least twice fast fastest using standard bit mutation also implies introducing crossover makes eas least twice fast counterparts without crossover proof theorem bound derived using estimate well note log log conditions hence terms absorbed term order prove general bound consider canonical fitness levels fitness level contains search points fitness estimate time spent level best fitness current population fitness level consider three cases first case applies population contains individuals fitness levels less second case population contains copies single individual level third case occurs population contains one individual level population contains different building blocks recombined effectively crossover cases capture typical behaviour albeit cases even whole fitness levels may skipped obtain upper bound expected optimization time summing expected times may spend cases fitness levels case population contains individual level least one individual lower fitness level sufficient condition leaving case individuals population obtain fitness least since never accepts worsenings case left good time individuals reaching fitness least already estimated lemma applying lemma fitness levels overall time spent cases log log case population contains copies individual level case offspring created standard mutation obvious offspring decides use crossover crossover used pick create crossover hence perform mutation leaves case good either better search point created creates another search point ones latter case create population two different individuals level note due choice rule environmental selection always maintain least two individuals level unless improvement larger fitness found probability creating better search point one mutation least suitable flips probability creating different search point level least sufficient flip one flip one flip bit probability either event happening one offspring creation thus least lemma expected number function evaluations case expected number functions evaluations made cases hence last sum estimated follows separating summand use equation simplify integral get plugging yields expected time cases case population contains individuals level identical case rely crossover recombining two different individuals level different building blocks different bits set good chance crossover generate offspring higher number probability performing crossover two different parents one offspring creation least worst case population contains copies one particular individual assuming two different parents selected crossover let hamming distance let denote number among positions offspring note binomially distributed parameters expectation estimate probability getting surplus leads improvement fitness estimate holds since prob prob prob prob mutation keeps probability least together probability increasing current best fitness one offspring creation least lemma expected number function evaluations case total expected time spent cases hence summing expected times yields total time bound log log remark conditions second statement theorem requires log log log order establish upper bound condition seems necessary larger values inertia large population slows exploitation least absence crossover note eas covered theorem removing crossover optimize time log witt showed uniform parent selection expected optimization time log log lower bound log jansen jong wegener showed needs time log log log log log log log badkobeh lehre sudholt showed every algorithm creating offspring using standard bit mutation unary unbiased operators needs time log log log log indicates threshold condition log log log tight polynomials log log remark conditions theorem assumes constant reflects common choices applications eas theorem extended towards smaller larger values follows upper bound time spent cases increases contains factor cases remain unaffected log log still get upper bound high crossover probabilities cases need revisited time cases derived lemma adapted follows probability increasing number fit individuals least number fit individuals population suffices select two fit individuals generate average number offspring happens probability least time bound lemma becomes log time bound theorem becomes log constant log also establishes upper bound remarkable waiting time successful crossovers cases order small values time spent cases negligible compared overall time bound order log shows effective crossover recombining building blocks also note proof theorem relatively simple uses elementary arguments along lemmas fully analysis therefore lends teaching purposes behavior evolutionary algorithms benefits crossover analysis revealed mutations mutations creating different search point fitness help escape case population identical individuals even though mutations immediately yield improvement terms fitness increase diversity population crossover efficient exploiting gained diversity combining two different search points later stage means crossover capitalize mutations beneficial disruptive effects building blocks interesting consequence affects optimal mutation rate eas using standard bit mutations witt recently proved optimal mutation rate linear functions recall optimal sense theorem eas neutral mutations neither helpful detrimental crossover neutral mutations become helpful increasing mutation rate increases likelihood neutral mutations fact easily derive better upper bounds theorem slightly larger mutation rates thanks additional term denominator upper bound dominant term minimized golden ratio leads following asymptotically best running time bound theorem obtained choice dominant term becomes optimal mutation rate corollary gives mutation rate yields best upper bound running time obtained proof theorem however establish mutation rate indeed optimal exclude another mutation rate leads smaller expected optimization time following show simple algorithm upper bound theorem indeed tight terms establishes optimal mutation rate proving lower bounds expected optimization times often notoriously hard task hence restrict simple captures characteristics gas covered theorem easy analyze latter achieved fixing many parameters possible upper bound theorem grows pick smallest possible values parent selection made simple possible select parents uniformly random current best individuals population words define parent population set individuals positive probability chosen parents parent population contains individuals current best fitness call parent selection greedy greedy strategy choose current best search points parents context proof theorem greedy parent selection implies cases never reached parent population never spans one fitness level time spent cases also allows eliminate one parameter setting lower values beneficial cases setting minimizes estimate time spent cases theorem extends towards see also remark call resulting greedy main characteristics greedy parent selection greedy defined algorithm algorithm greedy mutation rate maximizing initialize population size true select let crossover flip bit independently probability let contain best individuals break ties towards including individuals fewest duplicates end following result applies greedy using kind crossover crossover recombination operator bit value taken either parent possible introduce bit value represented parent common crossovers crossovers uniform crossover including parameterized uniform crossover well crossovers following result even includes biased operators like induces tendency increase number theorem consider greedy mutation rate log using arbitrary crossover operator expected optimization time least min log log maxk giving proof note constant maxk hence maximum attained lower bound theorem log log note greedy defined slightly differently duplicate genotypes always rejected algorithm equivalent greedy following reasons current population contains two different individuals equal fitness duplicate one parents created algorithms reject duplicate genotype population contains two individuals different fitness behave like population contained fitter individual matches upper bound small order terms showing greedy new term denominator bound theorem coincidence lower bound least together establishes optimal mutation rate greedy greedy uniform crossover mutation rate minimizes expected number function evaluations terms proof theorem use following technique based fitness levels author theorem sudholt consider partition search space sets search algorithm say level best individual created far probability traversing level level one step expected hitting time least prob starts proof theorem prove lower bound following instead original greedy whenever creates new offspring fitness different bit string current best individual assume following algorithm automatically performs crossover two also assume crossover leads best possible offspring sense bits parents differ set algorithm performs search points hamming distance resulting offspring due assumptions end generation always single best individual reason model algorithm markov chain representing current best fitness analysis follows lower bound eas theorem consider following partition focuses last fitness values let min log let contain remaining search points know initialized probability least log large enough probability makes transition fitness fitness equals prob flip prob flip according lemma considered fitness levels former probability bounded latter probability bounded prob flip prob flip together need find variables along conditions theorem fulfilled define max observe every order fulfill second condition theorem consider following norpn malized variables proves first condition theorem following proof theorem easy show log get calculations carry replacing establishes third last pcondition equivalent get implies using well max max log max log maxk log invoking theorem recalling first fitness level reached probability least log get lower bound log log maxk log maxk log last step used factors log log logc log positive constants bounding min log absorbing terms log log term statement gives claimed bound also ran experiments see whether outcome matches inspection dominating terms running time bounds realistic problem dimensions chose bits recorded average optimization time runs mutation rate set result shown figure one see every mutation rate greedy lower average optimization time predicted performance difference becomes larger mutation rate increases optimal mutation rates algorithms match minimal average optimization times note also deviation much lower higher mutation rates preliminary runs bits gave similar results experiments statistical tests given section crossover crossover operator picks cutting points uniformly random without replacement cutting points divide parents segments assembled alternating parents parents cutting number evaluations mutation rate figure average optimization times greedy uniform crossover onemax bits mutation rate set thin lines show mean standard deviation points offspring suffix odd even uniform crossover seen populations containing different search points equal fitness beneficial uniform crossover easily combine good building blocks holds regardless hamming distance different individuals position bits individuals differ crossover harder analyse probability crossover creating improvement depends hamming distance parents position differing bits consider parents differ two bits bit positions quite close crossover high probability taking bits parent order recombine building blocks cutting point chosen two bit positions similar effect occurs crossover also two bit positions opposite ends bit string following lemma gives lower bound probability crossover combines right building blocks two parents equally fit differ two bits lemma proof may independent interest lemma consider two search points probability crossover number possible cutting points creating offspring larger number least exactly proof identify cutting points bits cutting point results two strings say cutting point separates note prefix always taken claim follows showing number separating cutting points odd claimed probability let random variables describes number cutting points separating variable follows hypergeometric distribution hyp illustrated following urn model red white balls urn contains balls red draw balls uniformly random without replacement describes number red balls drawn define probability odd prob odd odd note following recurrence holds imagine drawing first cutting point separately probability cutting point separating cutting point need even number separating cutting points among remaining cutting points drawn random variable remaining probability number remaining cutting points must even number drawn random variable hence assume induction true using combining yields upper bound follows similarly induction claim follows setting lemma probability crossover creating improvement depends distance two differing bits fortunately search points result mutation one another distance favourable distribution made precise following lemma lemma let result mutation flipping one one positions bits chosen uniformly among respectively random variable min stochastically dominates uniform distribution proof first show following fixed index integer exactly two positions min fixed values result either note two values hence feasible values every let denote number assume first chosen uniformly random consider uniform random choice corresponding without loss generality assume fixed chosen uniformly random worst case distribution min attained distributed feasible bit positions worst case hence uniform distribution stochastically dominates uniform distribution case symmetrical exchanging roles well roles zeros ones yields uniform distribution set worst case stochastically dominates uniform distribution taken together lemma lemma indicate crossover good chance finding improvements recombining right building blocks however based population containing potential parents equal fitness differ two bits following analysis shows population likely contain favourable pair parents however pair might get lost individuals fitness created duplicates removed population parents differ bits lemma apply hence estimate likely crossover find improvement order avoid problem consider detailed rule individuals fewer duplicates preferred case still ties number evaluations mutation rate figure average optimization times bits runs greedy crossover using different rules individuals tied regard fitness number duplicates breaks ties randomly whereas algorithm prefers older individuals mutation rate set considering number duplicates retain older individuals refined rule shown algorithm shown remainder implies favourable pair parents hamming distance created pair never get lost algorithm refined rule let contain best individuals break ties towards including individuals fewest duplicates still ties break towards including older individuals rule called differs one used experiments figure section broke ties uniformly random case individuals tied respect fitness number duplicates call latter rule experiments greedy comparing rules runs indicate performance differences small see figure note however functions plateaus like royal road functions retaining older individuals prevents performing random walks plateau population spread duplicates individual case expect performance deteriorate breaking ties towards older individuals refined rule performance gas follows even though differences small tests reveal statistically significant differences crossover significantly faster significance level mutation rates least two exceptions still contrarily significantly faster crossover mutation rates range theorem expected optimization time every implementing algorithm rule algorithm log constant crossover bound equals upper bound gas uniform crossover improves upon previous upper bound greedy larger reason rem whose dominant term additive term favourable parents could get lost prevented rule conditions conditions well useful allow estimate probability single good individual takes whole population copies remainder section work towards proving theorem assume chosen asymptotic statements require large enough value hold true nothing prove statement holds trivially bounded estimate time spent fitness level best fitness current population end focus higher fitness levels log probability creating offspring level estimated nicely time reaching higher fitness levels constitutes term compared claimed running time bound following lemma proves claim general setting needed proof theorem particular holds arbitrary rules crossover operators lemma every implementing algorithm constant using initialization crossover operator expected time fitness level log reached first time log proof given appendix remainder section focus higher fitness levels log specify different cases fitness level cases similar ones uniform crossover additional conditions similarity individuals cases also additional error state accounts undesirable unexpected behavior pessimistically assume error state left towards cases level case population contains individual level least one individual lower fitness level case population contains copies individual level case population contains two search points current best fitness resulted mutation hamming distance case error state reached case best fitness none prior cases applies difference analysis uniform crossover case rely population collapsing copies single individual helps estimate probability creating favourable pair case effectively performs mutations case lemma consider defined theorem parameters log constant total expected time spent cases across log log proof already analyzed expected time cases across fitness levels proof theorem use lemma get expected time spent cases log case algorithm behaves like one using uniform crossover described theorem crossover operators working identical individuals case left either better offspring created different offspring ones created latter case either case error state reached proof theorem know expected time spent cases across levels bounded estimate total time spent cases time turns comparably small allow ignore fact cases actually reached case implies population contains pair hamming distance consider mutation created offspring note mutation flips probability denote bit positions differ random variable support law total expectation prob first bound conditional expectation considering probabilities improvements crossover successful crossover performed probability search point bit selected first parent probability least remaining search point selected second parent probability least cutting points chosen lead fitness improvement latter event probability least lemma finally need assume following mutation destroy fitness improvements probability least probability successful crossover least using min max min min min another means escaping case using crossover mutation create improvement probability least constant applying lemma min note upper bound min therefore pessimistic replacing min different random variable stochastically dominated according lemma min dominates uniform distribution assume multiple combining yields last sum estimated follows along log get log sum following log log log log log log log integral completes proof remainder proof devoted estimating expected time spent error state end need consider events take course deviating situations described cases since case based offspring hamming distance parents one potential failure offspring fitness hamming distance greater parent created probability estimated following lemma lemma let denote probability standard bit mutation mutation rate search point creates different offspring probability additionally offspring hamming distance larger parent proof found appendix another potential failure occurs population collapse copies single search point transition case case made first estimate probability mutation unexpectedly creating individual fitness lemma probability standard bit mutation mutation probability creates search point ones parent less ones note special case lemma gives upper bound highest probability jump fitness level attained parent level however larger mutation probabilities longer true general cases probability jumping level maximized parents lower fitness levels hence closer inspection transition probabilities different fitness levels required see proof appendix using lemma estimate probability collapsing copies single search point described case lemma consider defined theorem parameters log constant fix fitness level probability reach population containing different individuals fitness either reaching population containing copies individual level reaching higher fitness level log proof show good probability repeatedly creating clones individuals fitness finding improvement avoiding following bad event bad event happens individual fitness level created one offspring creation means cloning existing individual level probability bad event bounded follows case crossover used happens probability bound probability bad event trivial bound otherwise individual needs created mutation either worst fitness level mutating parent level probability former bounded lemma probability latter necessary flip one using probability bad event level hence bounded constant reach population containing different individuals fitness stated bad event happens population collapsed copies single search point moved higher fitness level consider first generation individual fitness reached first time since might possible create several individuals one generation consider offspring creations executed sequentially consider possibility bad events offspring creations following first offspring level let number function evaluations following generation individuals population fitness least lemma log log considering offspring creations first generation leading level completing generation end function evaluations less trials bad events probability one bad bounded prob log absorbing yields claimed result prepared estimate expected time spent error states log lemma consider defined theorem parameters log constant expected time spent states log log proof spends time error state actually reached first calculate probability state reached either case lemma states probability reaching population different individuals level reaching case better fitness level log pessimistically ignore possibility case might reached happens thus upper bound probability reaching case recall case individuals identical crossover effect performs mutations first consider case note log along implies hence lemma force according lemma probability leaving case creating different individual fitness least probability offspring hamming distance greater parent second statement lemma conditional probability reaching error state leaving case towards another case level case note case reached case single offspring fitness hamming distance parent offspring guaranteed survive assume offspring many duplicates removed first thus case several offspring fitness differing parent created need hamming distance larger order reach case probability decreases increasing hence probability bound also holds finally case implies exists pair hamming distance new generation two offspring least one copy always survive individuals multiple duplicates removed first case among current parents offspring individuals exist duplicates preferred newly created offspring probability reaching error state case case error state reached according probability least finding better individual one offspring creation constant using lemma translates expected function evaluations expected time spent case total expected time across error states theorem follows previous lemmas proof theorem claimed upper bound follows adding upper bounds expected time smaller fitness levels lemma expected times spent considered cases lemma lemma believe technical conditions theorem involving could relaxed possible generalize lemmas towards differing bits individuals equal fitness mutation rate mutation rate royal road mutation rate random polynomials figure average optimization times runs greedy various crossover operators functions bits royal road function block size random polynomials unweighted monomials degree mutation rate figure discussed following section presents experiments statistical tests includes comparison uniform crossover crossover greedy extensions functions royal roads monotone polynomials far theorems proofs focused strong results performance eas hand however insights gained stretch far beyond royal road functions generally consist larger blocks bits bits block need set order contribute fitness otherwise contribution blocks contribute amount fitness fitness sum contributions royal road random polynomial uniform uniform uniform uniform table summary results tests data figure function table shows pairwise comparisons greedy uniform crossover resp output statistics package version constant mutation rate cell describes rule subject minimum value gives number exceptions rule applicable fundamental insight gained neutral mutations also applies royal road functions mutation completes one block destroys another block neutral mutation offspring stored population crossover recombine finished blocks way difference destroyed block may evolve neutral mutations occur alter bits destroyed block population dominated many similar solutions becomes harder crossover find good pair recombination however crossover generally high probability finding improvements last effect probably plays minor role theoretical analysis general royal roads level detail harder impossible far results royal roads monotone polynomials mostly asymptotic recently doerr presented tighter runtime analysis offspring populations royal road functions may lend generalization results future work use experiments see whether performance similar use royal roads bits block size pairwise disjoint blocks bits also consider random monotone polynomials instead using disjoint blocks use monomials degree conjunctions bits monomial made bit positions chosen uniformly random without replacement leads function similar royal roads blocks broken share bits bit positions completely random figure shows average optimization times runs functions greedy uniform crossover chose last two crossovers odd treat ends bit strings differently even odd two bits close opposite ends bitstring high probability taken different parents whereas even high chance taken parent lemma special case consistency simplicity use rule settings ties fitness broken towards minimum numbers duplicates remaining ties broken uniformly random perfectly match conditions theorem require lower crossover bility rule experiments show crossover still effective conditions met crossovers better slightly worse uniform crossover accordance observation analyses improvements crossover might harder find case differing bits close proximity royal roads curves similar difference greedy bit smaller random polynomials visible differences albeit smaller tests confirm wherever noticeable gap curves statistically significant difference significance level outcome tests summarized table small mutation rates tests significant mutation rates less differences greedy gas statistically significant apart exceptions random polynomials difference uniform crossover crossover significant royal roads majority comparisons showed statistical significance number exceptions however random polynomials majority comparisons statistically significant comparisons crossover show statistical significance findings give strong evidence insights drawn analysis transfer broader classes functions building blocks need assembled linear functions another interesting question far theoretical analyses work extend cases building blocks different weights simplest case class linear functions defined positive weights doerr doerr ebel provided empirical evidence faster linear functions weights drawn uniformly random open question whether also holds common gas implementing algorithm experiments greedy found random linear functions advantage visible provide explanation observation reveal well suited weighted building blocks whereas gas might reason behaves like presence weights case current population contains two members different fitness ignores inferior one behaves population contained fitter individual since select fitter individual twice crossover followed mutation essentially mutates fitter individual behavior equals working fitter individual efficient buildingblock functions building blocks equally important easily mutation rate mutation rate random linear functions figure average optimization times runs uniform parent selection various crossover operators functions bits random linear functions weights drawn independently uniformly random anew run erate store individuals equal fitness population recombine different building blocks however presence weights chances creating individuals equal fitness might slim behaves like theorem long population contain two different individuals fitness equivalent functions search points different fitness values equivalent includes linear functions extreme weights like binval generally functions denotes largest weight also includes almost surely random linear functions weights drawn interval proof first two statements established preceding discussion functions search points bit weight higher fitness search points bit provided bits larger weights fixed follows inductively search points different fitness values random linear functions consider function constructed sequentially adding new bits randomly drawn weights assume adding bits bit patterns different fitness values trivially true bits adding new bit fitness value duplicated bits weight equal selection weights first bits since selections finite weight almost surely different statement follows induction sense able benefit crossover settings theorem since greedy parent selection suppresses diversity population order benefit crossover population needs able maintain select individuals different building blocks slightly different fitness values long enough crossover good chance combining building blocks achieves using cleverly designed twostage offspring creation process mutation first creates diversity best among mutants retained recombined parent times however explain crossover beneficial common designs promising common design need sophisticated figure shows already simple uniform parent selection performs significantly better hence greedy benefit crossover smaller main qualitative observations average optimization time smaller crossover mutation rates slightly larger improve performance tests significance level showed uniform crossover significantly faster random linear functions mutation rates less crossovers gave mixed results slower low mutation rates except crossover crossover faster high mutation rates crossover crossover shows uniform crossover speed assembly weighted building blocks albeit gas particular greedy proving rigorously random arbitrary linear functions remains challenging open problem identifying characteristics gas crossover beneficial cases conclusions future work demonstrated rigorously intuitively crossover speed building block assembly evidence holds broad class functions basic insight mutations create new building blocks destroying others still useful mutants stored population lead successful recombination parents later generation effect makes every cut selection moderate population sizes twice fast every words adding crossover halves expected optimization time terms applies uniform crossover crossover arbitrary values furthermore demonstrated analyze parent offspring populations eas gas long moderate exploitation slowed obtained essentially results arbitrary gas simple greedy analyzed work provides novel techniques analysis algorithms including lemmas may prove useful studies eas another intriguing conclusion following naturally analysis optimal mutation rate gas greedy changes using uniform crossover simply neutral mutations hence mutations become useful experiments perfect accordance theoretical results functions like royal roads random polynomials indicate performance differences also hold much general sense empirical evidence might also extend linear functions weighted building blocks general albeit apply greedy discussion section shown population must able store individuals different building blocks long enough crossover combine even though individuals might inferior fitness values subject replacement results give novel intuitive rigorous answers question discussed controversially decades plenty avenues future work would like extend theoretical analysis gas royal road functions monotone polynomials also investigating weighted building blocks like linear functions interesting challenging topic future work gas benefit crossover increased mutation rate cut selection removes offspring inferior fitness cut selection counteracts disruptive effects crossover increase mutation rate situation entirely different generational gas ochoa harvey buxton reported introducing crossover decrease optimal mutation rate future work could deal complementing different settings investigating balance selection pressure replacement selection optimal mutation rate acknowledgments author partially supported epsrc grant member cercia university birmingham research leading results received funding european union seventh framework programme grant agreement sage author would like thank reviewers detailed constructive comments helped improve manuscript references abramowitz stegun handbook mathematical functions formulas graphs mathematical tables dover new york ninth dover printing tenth gpo printing edition arora rabani vazirani simulating quadratic dynamical systems proceedings acm symposium theory computing stoc pages auger doerr editors theory randomized search heuristics foundations recent developments number series theoretical computer science world scientific badkobeh lehre sudholt unbiased complexity parallel search international conference parallel problem solving nature ppsn volume lncs pages springer barton charlesworth sex recombination science bell masterpiece nature evolution genetics sexuality univ california press jong evolutionary computation unified approach mit press dietzfelbinger naudts van hoyweghen wegener analysis recombinative ieee transactions evolutionary computation doerr doerr ebel lessons fast genetic algorithms proceedings genetic evolutionary computation conference gecco pages acm doerr fouz witt sharp bounds functions variable drift proceedings annual genetic evolutionary computation conference gecco pages acm press doerr happ klein crossover provably useful evolutionary computation theoretical computer science doerr johannsen winzen multiplicative drift analysis algorithmica doerr royal road functions evolutionary algorithm almost larger offspring populations ieee congress evolutionary computation cec pages doerr sudholt witt evolutionary algorithms optimize separable functions parallel foundations genetic algorithms foga pages acm eiben smith introduction evolutionary computing springer fischer wegener ising model mutation versus recombination theoretical computer science forrest mitchell relative building block fitness building block hypotheses proc foga pages morgan kaufmann jansen analyzing evolutionary algorithms computer science perspective springer jansen jong wegener choice offspring population size evolutionary algorithms evolutionary computation jansen wegener analysis evolutionary proof crossover really help algorithmica jansen wegener real royal road crossover provably essential discrete applied mathematics jansen zarges analysis evolutionary algorithms computational complexity analysis algorithm engineering proceedings workshop foundations genetic algorithms foga pages acm sudholt theile crossover helps optimization proceedings annual conference genetic evolutionary computation gecco pages acm press personal communication sudholt general upper bounds running time parallel evolutionary algorithms evolutionary computation lehre yao crossover constructive computing unique output sequences soft computing lehre witt search unbiased variation algorithmica livnat papadimitriou dushoff feldman mixability theory role sex evolution proceedings national academy sciences livnat papadimitriou pippenger feldman sex mixability modularity proceedings national academy sciences mitchell forrest holland royal road function genetic algorithms fitness landscapes performance proc european conference artificial life pages mit press mitchell holland forrest genetic algorithm outperform hill climbing advances neural information processing systems pages morgan kaufmann mitzenmacher upfal probability computing cambridge university press neumann oliveto witt theoretical analysis selection landscapes efficiency genetic evolutionary computation conference gecco pages acm press neumann theile crossover speeds evolutionary algorithms problem international conference parallel problem solving nature ppsn pages springer neumann witt bioinspired computation combinatorial optimization algorithms computational complexity springer ochoa harvey buxton error thresholds relation optimal mutation rates floreano nicoud mondada editors advances artificial life volume lecture notes computer science pages springer oliveto witt improved runtime analysis simple genetic algorithm proceedings genetic evolutionary computation conference gecco pages acm oliveto witt runtime analysis simple genetic algorithm theoretical computer science qian zhou analysis recombination evolutionary optimization artificial intelligence rabani rabinovich sinclair computational view population genetics random structures algorithms rowe genetic algorithms kacprzyk pedrycz editors handbook computational intelligence springer appear rowe vose wright group properties crossover mutation evolutionary computation storch wegener real royal road functions constant population size theoretical computer science sudholt crossover provably essential ising model trees proceedings genetic evolutionary computation conference gecco pages acm press sudholt impact parametrization memetic evolutionary algorithms theoretical computer science sudholt crossover speeds assembly proceedings genetic evolutionary computation conference gecco pages acm press sudholt new method lower bounds running time evolutionary algorithms ieee transactions evolutionary computation sudholt thyssen running time analysis ant colony optimization shortest path problems journal discrete algorithms vose simple genetic algorithm foundations theory mit press watson jansen royal road crossover provably essential proceedings genetic evolutionary computation conference gecco pages acm wegener witt optimization monotone polynomials simple randomized search heuristics combinatorics probability computing weissman barton limits rate adaptive substitution sexual populations plos genetics weissman feldman fisher rate crossing sexual populations genetics witt runtime analysis simple functions evolutionary computation witt tight bounds optimization time randomized search heuristic linear functions combinatorics probability computing appendix appendix contains proofs lemmas omitted main part proof lemma current population best individual fitness lemma expected number log function evaluations individuals fitness least one offspring creation results improvement crossover used mutation flips exactly one probability event constant due conditions using lemma expected time fitness level log reached first time therefore log log log log log log proof lemma order create different search point fitness level must integer min flip flip necessary sufficient condition min case yields claimed lower bound upper bound bound term using bound binomial coefficients min applying yields formula hence claimed upper bound second statement follows upper bound fact offspring hamming distance case probability proof lemma search point ones created parent ones value flip flip sought probability therefore max max max max max using bound max term max max max maximum attained case get probability bound claimed trivially bound sought probability
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modeling musical context using queen mary university london london university north florida jacksonville usa jun present semantic vector space model capturing complex polyphonic musical context model based representation negative sampling used model slices music dataset beethoven piano sonatas visualization reduced vector space using stochastic neighbor embedding shows resulting embedded vector space captures tonal relationships even without explicit information musical contents slices secondly excerpt moonlight sonata beethoven altered replacing slices based context similarity resulting music shows selected slice based similar context also relatively short tonal distance original slice keywords music context music neural networks semantic vector space introduction paper explore semantic similarity derived looking solely context musical slice appears past research music often modeled recursive neural networks rnns combined restricted bolzmann machines term rnn models eck schmidhuber sak markov models conklin witten statistical models using representation incorporates musical information pitch pitch class duration intervals research focus modeling context content vector space models rumelhart typically used natural language processing nlp represent embed words continuous vector space turney pantel mcgregor agres liddy within space semantically similar words represented geographically close turney pantel recent efficient approach creating vector spaces natural language processing mikolov herremans chuan modeling musical context using proceedings first international workshop deep learning music joint ijcnn anchorage may although music language possesses many types characteristics besson discuss similarity music language terms among others structural aspects expectancy generated word note therefore use model nlp specifically model negative sampling used create train model captures musical context attempts modeling musical context semantic vector space models example huang use model chord sequences order recommend chords ordinary novice composers paper aim use modeling musical context generic way opposed reduced representation chord sequences represent complex polyphonic music sequence slices without additional processing musical concepts beat time signature chord tones etc next sections first discuss implemented model followed discussion music represented finally resulting model evaluated refers group models developed mikolov used create train semantic vector spaces often consisting several hundred dimensions based corpus text mikolov vector space word corpus represented vector words share context geographically close space architecture based two approaches continuous continuous model cbow former uses context predict current word whereas latter uses current word predict surrounding words mikolov models low computational complexity easily handle corpus size ranging billions words matter hours cbow models faster observed performs better small datasets mikolov therefore opted work latter model negative sampling architecture model represented figure word corpus size position network tries predict surrounding words window figure training objective thus defined log whereby term calculated softmax function calculating gradient term however computationally expensive alternatives circumvent problem include hierarchical softmax morin bengio noise contrastive estimation gutmann model used research implements variant latter namely negative sampling figure model word position herremans chuan modeling musical context using proceedings first international workshop deep learning music joint ijcnn anchorage may idea behind negative sampling well trained model able distinguish data noise goldberg levy original training objective thus approximated new efficient formulation implements binary logistic regression classify data noise samples model able assign high probabilities real words low probabilities noise samples objective optimized mikolov cosine similarity used similarity metric two vectors vector space two vectors dimensional space angle defined tan similarity cos research port discussed model techniques field music replacing words slices polyphonic music manner done discussed next section musical slices words order study extend model musical context polyphonic musical pieces represented little injected musical knowledge possible piece simply segmented slices duration slices calculated piece based distribution time note onsets smallest amount time consecutive onsets occurs cases selected slices capture pitches sound slice onset slice played held slice slicing process depend musical concepts beat time signature instead completely vocabulary words thus consist collection musical slices addition label pitches chords sounding pitches including chord tones tones ornaments recorded slice reduce pitches pitch classes either pitches considered different pitches musical knowledge use global key transpose pieces either major minor segmentation enables functional role pitches tonality stay across compositions turn causes repeated slices dataset allows model better trained less data next section performance resulting model discussed results order evaluate well proposed model captures musical context experiments performed dataset consisting beethoven piano sonatas resulting dataset consists words total unique occurrences discussed models efficient train within minutes model trained cpu macbook pro trained model number times different number dimensions vector space see figure dimensions accurate model becomes herremans chuan modeling musical context using proceedings first international workshop deep learning music joint ijcnn anchorage may however time train model also becomes longer rest experiments decided use dimensions second experiment varied size skip window many words consider left right current word results displayed figure show skip window ideal dataset results varying number results varying size skip sions vector space window figure evolution average loss training step represents training windows visualizing semantic vector space order better understand evaluate proposed model created visualizations selected musical slices dimensionally reduced space use stochastic neighbor embedding technique developed maaten hinton visualizing data previously used music analysis context visualizing clusters musical genres based musical features hamel eck case identified chord slice dataset belongs based simple method expect tonally close chords occur together semantic vector space figure confirms hypothesis examining slices contain chords perfect fifth apart space looks dispersed often see figure occurs chord pair figure hand looking tonally distant chord pair figure see clusters appear reduced vector space happens tonally distant chords figure content versus context order examine captures semantic meaning music via modeling context modify piece replacing original slices similar one captured cosine similarity vector space model really able herremans chuan modeling musical context using proceedings first international workshop deep learning music joint ijcnn anchorage may green blue black green gray green blue green blue figure reduced vector space different slices labeled close chord capture modified piece sound similar original allows evaluate effectiveness using modeling music figure shows first measures beethoven piano sonata moonlight movement measures modified pitch slices dashed box audio version score available modified slices produced replacing original slice highest cosine similarity based embeddings tonal distance original modified slices presented slice pair calculated average number steps pair pitches two slices tonnetz representation cohn extended pitch register observed even thought cosine similarity around tonal distance http herremans chuan modeling musical context using proceedings first international workshop deep learning music joint ijcnn anchorage may selected slice remains relatively low cases example tonal distance third dashed box modified slice major triad pitches original slice single pitch however notice necessarily model musical context voice leading example better voice leading achieved pitch last dashed box replaced pitch figure excerpt beethoven piano sonata movement modified measures replacing slices report highest cosine similarity figure number notes marked different color orange held notes onsets played previously notes remain played current slice notes create unique situation music generation using example orange note pitch first dashed box held note indicates pitch played previous slice however capture relation considers similarity original modified slices conclusions model negative sampling used build semantic vector space model complex polyphonic music representing resulting vector space reduced twodimensional graph show musical features notion tonal proximity captured model music generated replacing slices based context similarity also presents close tonal distance compared original future embedded model combines instance longshort term memory recurrent neural network based musical features would offer complete way completely model music tensorflow code used research herremans chuan modeling musical context using proceedings first international workshop deep learning music joint ijcnn anchorage may available acknowledgements project received funding european union horizon research innovation programme grant agreement references kat agres stephen mcgregor karolina rataj matthew purver geraint wiggins modeling metaphor perception distributional semantics vector space models workshop computational creativity concept invention general intelligence proceedings international workshop essli pages mireille besson daniele comparison language music annals new york academy sciences nicolas yoshua bengio pascal vincent modeling temporal dependencies sequences application polyphonic music generation transcription arxiv preprint richard cohn operations parsimonious trichords tonnetz representations journal music theory darrell conklin ian witten multiple viewpoint systems music prediction journal new music research douglas eck juergen schmidhuber finding temporal structure music blues improvisation lstm recurrent networks neural networks signal processing proceedings ieee workshop pages ieee yoav goldberg omer levy explained deriving mikolov negativesampling method arxiv preprint michael gutmann aapo estimation unnormalized statistical models applications natural image statistics journal machine learning research feb philippe hamel douglas eck learning features music audio deep belief networks ismir volume pages utrecht netherlands anna huang david duvenaud krzysztof gajos chordripple recommending chords help novice composers beyond ordinary proceedings international conference intelligent user interfaces pages acm elizabeth liddy woojin paik edmund ming multilingual document retrieval system method using semantic vector matching december patent http herremans chuan modeling musical context using proceedings first international workshop deep learning music joint ijcnn anchorage may laurens van der maaten geoffrey hinton visualizing data using journal machine learning research nov stephen mcgregor kat agres matthew purver geraint wiggins distributional semantics conceptual spaces novel computational method concept creation journal artificial general intelligence tomas mikolov kai chen greg corrado jeffrey dean efficient estimation word representations vector space arxiv preprint tomas mikolov quoc ilya sutskever exploiting similarities among languages machine translation arxiv preprint tomas mikolov ilya sutskever kai chen greg corrado jeff dean distributed representations words phrases compositionality advances neural information processing systems pages frederic morin yoshua bengio hierarchical probabilistic neural network language model aistats volume pages citeseer david rumelhart geoffrey hinton ronald williams learning representations errors cognitive modeling hasim sak andrew senior beaufays long memory recurrent neural network architectures large scale acoustic modeling interspeech pages tan michael steinbach vipin kumar introduction data mining first edition longman publishing boston usa isbn peter turney patrick pantel frequency meaning vector space models semantics journal artificial intelligence research herremans chuan modeling musical context using proceedings first international workshop deep learning music joint ijcnn anchorage may
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nov regularity monomial ideals takayuki hibi kazunori matsuda bstract let denote polynomial ring variables field deg homogeneous ideal dim hilbert series form known one reg deg reg regularity present paper given arbitrary integers monomial ideal reg deg constructed furthermore give class edge ideals graphs reg ntroduction let denote polynomial ring variables field deg homogeneous ideal dim hilbert series form proposition say let reg denote regularity fact lemma one reg deg converse false following example found herzog let monomial ideal strongly stable dim depth reg stage reasonable discover natural class monomial ideals reg deg furthermore one escape temptation present following conjecture given arbitrary integers exists strongly stable ideal reg deg follows lemma pure resolution deg reg dim depth mathematics subject classification key words phrases regularity graph result desired ideal found class squarefree lexsegment ideals purpose present paper give affirmative answer weak version conjecture say monomial ideal reg deg constructed theorem given arbitrary integers exists monomial ideal reg deg basic process order obtain required ideal theorem find monomial ideal reg deg arbitrary integer proof theorem achieved section hand section give class edge ideals graphs reg deg roof heorem giving proof theorem several lemmata prepared let denote polynomial ring variables field deg lemma follows immediately definition regularity terms graded betti numbers lemma let proper homogeneous ideal reg reg lemma lemma let polynomial rings field let nonzero homogeneous ideal write regard homogeneous ideals reg reg reg reg reg reg reg reg reg lemma lemma let monomial ideal variable appears monomial belonging unique minimal system monomial generators reg max reg reg first step proof theorem given integers construct monomial ideal reg mensioned introduction desired ideal found class squarefree lexsegment ideals let lex denote lexicographic order induced monomial ideal called squarefree lexsegment generated squarefree monomials squarefree monomials squarefree monomials deg deg lex one fix integers consider squarefree lexsegment ideal polynomial ring variables field proposition one reg thus particular deg proof since follows lemma reg hence lemma says reg desired let thus short exact sequence yields required proposition guarantees conjecture true second step proof theorem turn discussion finding desired monomial ideal let polynomial ring variables field introduce monomial ideals defined follows remark one regarded edge ideal ferrers graph associated partition appearing regarded partition see using theorem one reg reg lemma one reg proof since belongs unique minimal system monomial generators follows reg claim reg using lemma together remark one reg reg reg furthermore lemma together remark says reg hence reg follows lemma let consider short exact sequence remark yield furthermore remark yeilds hence desired lemma one reg proof since belongs unique minimal system monomial generators one reg claim reg using induction since assertion trivial let lemma together remark guarantees reg reg moreover virtue lemma remark well induction hypothesis follows reg reg reg hence lemma says reg consider short exact sequence follows remark furthermore remark well induction hypothesis guarantees hence one desired monomial ideal plays important role proof theorem proposition one reg hsn thus particular deg hsn proof virtue lemma sufficient show reg since belongs unique minimal system monomial generators one reg claim reg follows lemma remark together lemma reg reg reg using lemma remark together lemma one reg reg reg hence lemma says reg desired considering short exact sequence remark together lemma yields hsn hsn furthermore remark together lemma yields hsn hsn follows hsn hsn required position finish proof theorem proof proof theorem let positive integers virtue proposition case discussed let follows reg virtue example proposition one reg let lemma together proposition yields reg hence deg desired monomial ideal xamples purpose section give class edge ideals graphs reg deg let finite simple graph vertex set edge set finite graph called simple possesses loop multiple edge edge ideal monomial ideal generated quadratic monomials general quite difficult compute regularity edge ideal however one compute reg easily graph notion cameronwalker graph introduced refer reader classification graphs example fix write star triangle joining triangles one common vertex theorem hence reg deg reg deg example fix write graph drawn vertex set rrr theorem says however reg deg hand graph interest viewpoint sequence coefficients known theorem gorenstein ring symmetric converse false routine computation shows hence symmetric odd necessary unimodal example unimodal general unimodal acknowledgment first author partially supported jsps kakenhi second author partially supported jsps kakenhi eferences aramova herzog hibi squarefree lexsegment ideals math bruns herzog rings revised cambridge stud adv vol cambridge university press cambridge benedetti varbaro dual graph algebras int math res imrn corso nagel monomial toric ideals associated ferrers graphs trans amer math soc cameron walker graphs maximum induced matching maximum matching size discrete math dao huneke schweig bounds regularity projective dimension ideals associated graphs algebraic combin herzog hibi monomial ideals graduate texts mathematics springer london hibi higashitani kimura keefe algebraic study graphs algebra hoa tam invariants mixed product ideals arch math basel stanley hilbert functions graded algebras adv math takayuki ibi epartment ure pplied athematics raduate chool nformation cience echnology saka niversity uita saka japan address hibi azunori atsuda epartment ure pplied athematics raduate chool nformation cience echnology saka niversity uita saka japan address
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correntropy maximization via admm application robust hyperspectral unmixing feb fei zhu abderrahim halimi paul honeine badong chen nanning zheng abstract hyperspectral images spectral bands suffer low ratio due noisy acquisition atmospheric effects thus requiring robust techniques unmixing problem paper presents robust supervised spectral unmixing approach hyperspectral images robustness achieved writing unmixing problem maximization correntropy criterion subject commonly used constraints two unmixing problems derived first problem considers unmixing constraints second one deals abundances corresponding optimization problems solved efficiently using alternating direction method multipliers admm approach experiments synthetic real hyperspectral images validate performance proposed algorithms different scenarios demonstrating unmixing robust outlier bands index terms correntropy maximum correntropy estimation alternating direction method multipliers hyperspectral image unmixing problem zhu institut charles delaunay cnrs technologie troyes france halimi school engineering physical sciences university honeine litis lab rouen france badong chen nanning zheng institute artificial intelligence robotics jiaotong university china chenbd nnzheng ntroduction pectral unmixing essential issue many disciplines including signal image processing wide range applications classification segmentation material identification target detection typically hyperspectral image corresponds scene taken many continuous narrow bands across certain wavelength range namely pixel spectrum assuming spectrum mixture several pure materials unmixing problem consists two tasks identifying pure materials endmembers estimating proportions abundances pixel practice two steps performed either sequentially simultaneously wellknown endmember extraction algorithms include ones vertex component analysis vca well ones minimum simplex analysis minimum volume constrained nonnegative matrix factorization endmember extraction relatively easy geometry abundance estimation remains open problem indeed abundances estimated using methods geometric approaches tackling issues nonlinearity paper consider abundance estimation problem linear mixture model lmm investigated past decades underlying premise linear combination endmembers physically interpretable two constraints often enforced estimation problem abundance constraint anc abundance constraint asc pixel considering constraints method fcls presented recently proposed unmixing algorithm sunsal sparse unmixing variable splitting augmented lagrangian addresses optimization problem taking advantage alternating direction method multipliers admm sunsal also proposed solve constrained sparse regression problem constraint asc relaxed regularizer added unmixing algorithms hugely suffer noisy data outliers within bands indeed real hyperspectral images remote sensing considerable proportion spectral bands noisy low snr due atmospheric effect water absorption bands need removed prior applying existing unmixing method otherwise unmixing quality drastically decreases sensitivity outliers due investigated cost function fcls sunsal algorithms well unmixing algorithms explore solutions worth noting nonlinear unmixing algorithms also suffer drawback including kfcls nonlinear fluctuation methods methods information theoretic learning provides elegant alternative conventional minimization problems considering maximization correntropy due stability robustness noise outliers correntropy maximization based theoretical foundations successfully applied wide class applications including cancer clustering face recognition recently hyperspectral unmixing name works resulting problem optimized technique either supervised manner unsupervised nonnegative matrix factorization paper consider hyperspectral unmixing problem defining appropriate criterion thus taking advantage robustness large outliers opposed conventional criteria including constraints commonly used physical interpretation propose solve resulting constrained optimization problems alternating direction method multipliers admm algorithms indeed admm approach splits hard problem sequence small handful ones relevance solve nonconvex problems studied section show admm provides relevant framework incorporating different constraints raised unmixing problem present socalled cusal unmixing variable splitting augmented lagrangian study particularly two algorithms solve anc asc unmixing problem solve unmixing problem rest paper organized follows first provide succinct survey classical unmixing problems section section iii propose unmixing problems subject aforementioned constraints study robustness resulting optimization problems solved admm algorithms described section experiments synthetic real hyperspectral images presented sections respectively finally section vii provides conclusions future works lassical nmixing roblems linear mixture model lmm assumes spectrum expressed linear combination set pure material spectra termed endmembers consider hyperspectral image let denote matrix spectral bands let column row representing band pixels notation simplicity denote lmm written xrt matrix composed endmembers mlr xrt abundance vector associated pixel additive noise matrix form pixels noise matrix following endmembers assumed known either information using endmember extraction technique spectral unmixing problem consists estimating abundances pixel often solving optimization problem min kyt denotes conventional solution conventional problem given tall endmember matrix optimization problems often written single optimization problem using following matrix formulation min denotes frobenius norm solution finally optimization problem also tackled considering image pixels spectral band yields following optimization problem min denotes row argument problem formulations solution suffer two major drawbacks first one several constraints need imposed order physical meaning results second drawback sensitivity noise outliers due use fitness measure two drawbacks detailed following physically interpretable abundances nonnegative anc satisfy constraint asc considering constraints problem formulated min kyt subject denotes column vector ones applied matrix form min subject since solution dealing constraint several iterative techniques proposed active set scheme lawson hanson algorithm multiplicative iterative strategies fcls technique recently alternating direction method multipliers admm applied success hyperspectral unmixing problem sunsal algorithm recent work hyperspectral unmixing advocated sparsity abundance vectors case spectrum fitted sparse linear mixture endmembers namely abundances respect small number endmembers nonzero end regularization included cost function yielding following constrained sparse regression problem min kyt subject parameter balances fitness solution sparsity level worth noting asc relaxed included problem often considered using following matrix formulation min kxt subject sensitivity outliers aforementioned algorithms rely solving constrained optimization problem thus inheriting drawbacks using fitness measure major drawback sensitivity outliers outliers spectral bands largely deviate rest bands indeed considering image pixels optimization problems take form min subject aforementioned constraints formulation easy see squared gives weight large residuals namely outliers predicted values far actual observations moreover common hyperspectral images present unusable spectral bands due low ratio essentially atmospheric effects water absorption following section overcome difficulty considering correntropy maximization principle information theoretic learning yields optimization problem robust outliers iii based nmixing roblems section examine correntropy write unmixing problems correntropy maximization ones algorithms solving problems derived section correntropy correntropy studied nonlinear local similarity measure two random variables estimation using defined expectation operator kernel satisfying mercer theorem practice joint distribution function unavailable sample estimator correntropy adopted instead employing finite number data estimated normalization factor gaussian kernel kernel correntropy leads following expression correntropy exp denotes bandwidth gaussian kernel maximization correntropy given max termed maximum correntropy criterion noteworthy statistics mean square error mse depends heavily gaussian linear assumptions however presence noise particular large outliers observations greatly deviated data bulk effectiveness algorithms significantly deteriorate contrast maximization correntropy criterion appropriate signal processing robust particular large outliers shown next underlying robustness correntropy criterion section study sensitivity outliers correntropy maximization principle showing robustness underlying mechanism end examine behavior correntropy terms residual error defined thus correntropy becomes exp compared statistics mse correntropy robust respect outliers shown fig illustrating correntropy objective functions terms residual error residual error increases function keeps increasing dramatically contrary correntropy sensitive within region small residual errors region controlled kernel bandwidth large magnitudes residual error correntropy falls zero consequently correntropy criterion robust large outliers function correntropy correntropy correntropy residual error fig illustration objective function solid line correntropy objective function dashed lines different values kernel bandwidth unmixing problems unmixing problem consists estimating unknown abundance matrix minimizing objective function negative correntropy given exp gaussian kernel considered equivalently exp ylt xrt considering anc asc constraints correntropy unmixing problem becomes min subject sake promoting sparse representations objective function augmented penalty abundance matrix leading following problem min kxt subject admm olving based nmixing roblems first briefly review alternating direction method multipliers admm following expressions chap consider optimization problem form min functions closed proper convex admm solves equivalent constrained problem min subject particular constraint instance formulation may seem trivial optimization problem tackled using augmented lagrangian method objective function separable alternating variable separately admm repeats direct update dual variable scaled form admm algorithm summarized algorithm algorithm admm algorithm input functions matrices vector parameter initialize repeat arg minx kax arg minz stopping criterion unmixing following apply admm algorithm solve unmixing problem case presented main steps summarized algorithm rewrite variables optimized vector rrt stacked columns matrix namely rewrite also following vectors rrt zrt urt following formulation admm algorithm set identity matrix rrt zero vector indicator function set defined otherwise case subproblem line algorithm addresses nonconvex problem without form solution overcome difficulty apply inexact admm variant lines algorithm solves subproblem iteratively using gradient descent method instead solving exactly explicitly eliminate equality constraints constraints replacing xrt xrt xrt let reduced vector unknowns estimated stacked means objective function transformed exp ylt mlr mlp mlr xpt gradient respect stacked entries given mlr mlr exp similarly function expressed respect entries gradient xpt zrt urt xpt zpt upt given xrt xpt zrt zrt urt urt solution line algorithm becomes projection onto first orthant shown line algorithm algorithm unmixing initialize repeat repeat convergence reform using max stopping criterion unmixing algorithm order apply admm algorithm express constrained optimization problem follows analogy previous case line algorithm solved iteratively gradient descent method given algorithm lines gradient respect xis stacked exp ylt xrt mlr line algorithm involves solving arg min admm applied solve various problems including lasso difference lasso latter term enforced case lasso soft thresholding operation soft thresholding operator defined following straightforward project result onto nonnegative orthant order include constraint thus yielding max maximum function results lead unmixing algorithm sparsitypromoting summarized algorithm algorithm unmixing initialize repeat repeat convergence max stopping criterion initialisation bandwidth determination apply stopping criterion algorithms according primal dual residuals small enough namely primal residual starts increase kxk iii maximum iteration number attained bandwidth gaussian kernel note small value parameter punishes harder outlier bands thus increasing robustness algorithm outliers note study admm applied address nonconvex objective function thus convergence guaranteed theoretically according considering issues propose fix bandwidth empirically summarized algorithm described next following first initialize bandwidth parameter function reconstruction error given solution case result apart solution parameter augmented condition xkf satisfied algorithm divergence occurs stopping criterion satisfied namely primal residual increases iterations case either parameter large due overestimated initialization small accordingly either decrease increase admm converges algorithm tuning bandwidth parameter initialize using cusal algorithm algorithm stopping criterion iii satisfied condition else increase line satisfied optimal value end else due overestimated decrease line else increase line end end xperiments ynthetic data section performance proposed algorithms evaluated synthetic data comparative study performed considering six methods proposed linear nonlinear unmixing models fcls fcls developed linear model enforcing anc asc constraints technique yields optimal abundance matrix sense sparse unmixing variable splitting augmented lagrangian sunsal method based admm several variants developed including different constraints bayesian algorithm generalized bilinear model baygbm method estimates abundances generalized bilinear model gbm adds interactions endmembers linear model yielding model xit xjt controls interactions endmembers product baygbm considers anc asc bayesian algorithm polynomial mixing model bayppnmm algorithm estimates parameters assuming pixel reflectances nonlinear functions endmembers using nonlinear terms characterized anc asc required kernel kfcls method generalizes fcls replacing inner product kernel function following gaussian kernel applied simulation robust nonnegative matrix factorization rnmf capture nonlinear effect outliers method introduces regularization term linear model accounting constraints problem optimized descent strategy fair comparison paper endmembers fixed real ones regularization parameter set degree sparsity suggested first compare presented methods series experiments performed mainly considering influences four aspects mixture model noise level iii number corrupted bands number endmembers reflectance reflectance bands fig bands left right usgs signatures chosen simulation image pixels generated using either linear mixing model polynomial mixing model ppnmm gaussian noise snr endmembers shown fig drawn usgs digital spectral library endmembers defined continuous bands wavelength ranging abundance vectors uniformly generated using dirichlet distribution ppnmm values generated uniformly set according imitate noisy bands real hyperspectral images several bands generated data corrupted replacing corresponding rows random values within number corrupted bands varies set unmixing performance evaluated using abundance root mean square error rmse defined kxt rmse estimated abundance vector fig illustrates average rmse realizations respectively lmm ppnmm data easy see presence outlier bands proposed algorithm outperforms comparing methods terms rmse different mixture models noise levels numbers endmembers also shown performance proposed algorithm improves increasing snr performance proposed presented compared sparsitypromoting well fcls series data sparse abundance matrices influences number corrupted bands sparsity level abundances studied image pixels generated linear mixture model endmember matrix composed usgs signatures angle two different endmembers larger nonzero entries abundance vector generated dirichlet distribution value indicator sparsity level ranges number rmse rmse fcls baygbm bayppnmm kfcls rnmf number corrupted bands number corrupted bands snr snr rmse rmse fig number corrupted bands number corrupted bands snr snr lmm data root mean square error rmse respect number corrupted bands averaged ten realizations different number endmembers snr corrupted bands varies set gaussian noise snr level commonly present real hyperspectral images according algorithms regularization parameter adjusted using set unmixing performance algorithms evaluated using error measured decibels according defined sre kxt kxt results averaged ten realizations illustrated fig considering abundance matrix estimation sparse different levels conclude following concerning case without outlier bands outperforms fcls number outlier bands increases proposed algorithm generally provides best unmixing quality highest sre value especially rmse rmse fcls baygbm bayppnmm kfcls rnmf number corrupted bands number corrupted bands snr snr rmse rmse fig number corrupted bands number corrupted bands snr snr ppnmm data root mean square error rmse respect number corrupted bands averaged ten realizations different number endmembers snr xperiments eal data section presents performance proposed algorithms real hyperspectral image consider taken cuprite mining image acquired aviris sensor flying las vegas nevada usa image widely investigated literature raw data contains bands covering wavelength range among relatively noisy ones low snr namely bands geographic composition area estimated include minerals neglecting similar signatures consider endmembers often investigated literature vca technique first applied extract endmembers clean image bands starting bands noisy bands randomly chosen bands gradually included form series input data therefore experiments conducted bands fcls sre sre corrupted band corrupted bands sre sre corrupted bands fig corrupted bands lmm data averaged error sre respect sparsity level averaged ten realizations comparison various number corrupted bands snr since abundances unknown performance measured averaged spectral angle distance sad illustrated fig sad defined input spectra reconstructed ones sad arccos kkb estimated abundance maps using bands given fig fig fig respectively absence noisy bands bands considered methods lead satisfactory abundance maps bayppnmm providing smallest sad number noisy bands increases especially unmixing performance methods deteriorates drastically proposed cusal yields stable sad obtained results confirm good behavior proposed cusal algorithms robustness presence corrupted spectral bands sad fcls baygbm bayppnmm kfcls rnmf fig number bands cuprite image averaged spectral angle distance sad using different number bands computed without noisy bands vii onclusion paper presented supervised unmixing algorithm based correntropy maximization principle two correntropybased unmixing problems addressed first constraints second constraint term alternating direction method multipliers admm investigated order solve unmixing problems effectiveness robustness proposed unmixing method validated synthetic real hyperspectral images future works include generalization correntropy criterion account multiple reflection phenomenon well incorporating nonlinear models acknowledgment work supported french anr grant hypanema fig cuprite image estimated abundance maps using clean bands left right sphene alunite buddingtonite kaolinite chalcedony highway top bottom fcls baygbm bayppnmm kfcls rnmf fig cuprite image estimated abundance maps using bands clean bands legend fig fig cuprite image estimated abundance maps using bands clean bands legend fig eferences keshava mustard spectral unmixing ieee signal processing magazine vol jan honeine richard geometric unmixing large hyperspectral images barycentric coordinate approach ieee transactions geoscience remote sensing vol jun nascimento vertex component analysis fast algorithm unmix hyperspectral data ieee transactions geoscience remote sensing vol apr winter algorithm fast autonomous spectral determination hyperspectral data algorithm fast autonomous spectral determination hyperspectral data proc spie imaging spectrometry vol agathos zaharie plaza minimum volume simplex analysis fast algorithm linear hyperspectral unmixing ieee transactions geoscience remote sensing vol sept miao endmember extraction highly mixed data using minimum volume constrained nonnegative matrix factorization ieee transactions geoscience remote sensing vol march chen richard honeine nonlinear unmixing hyperspectral data based model ieee transactions signal processing vol nonlinear estimation material abundances hyperspectral images spatial regularization ieee transactions geoscience remote sensing vol may heinz chang fully constrained least squares linear spectral mixture analysis method material quantification hyperspectral imagery ieee transactions geoscience remote sensing vol mar huck guillaume minimum dispersion constrained nonnegative matrix factorization unmix hyperspectral data ieee transactions geoscience remote sensing vol jun plaza dobigeon parente gader chanussot hyperspectral unmixing overview geometrical statistical sparse approaches ieee journal selected topics applied earth observations remote sensing vol figueiredo alternating direction algorithms constrained sparse regression application hyperspectral unmixing ieee workshop hyperspectral image signal processing evolution remote sensing whispers boyd parikh chu peleato eckstein distributed optimization statistical learning via alternating direction method multipliers foundations trends machine learning vol zelinski goyal denoising hyperspectral imagery recovering junk bands using wavelets sparse approximation ieee international conference geoscience remote sensing symposium igarss july broadwater chellappa banerjee burlina kernel fully constrained least squares abundance estimates ieee international geoscience remote sensing symposium igarss chen richard honeine estimating abundance fractions materials hyperspectral images fitting mixing model proc ieee workshop hyperspectral image signal processing evolution remote sensing jun liu pokharel correntropy properties applications signal processing ieee transactions signal processing vol principe information theoretic learning renyi entropy kernel perspectives springer science business media wang wang gao matrix factorization maximizing correntropy cancer clustering bmc bioinformatics vol zheng maximum correntropy criterion robust face recognition ieee transactions pattern analysis machine intelligence vol wang pan xiang zhu robust hyperspectral unmixing metric ieee transactions image processing vol nov nikolova analysis minimization methods signal image recovery siam journal scientific computing vol lawson hanson solving least squares problems classics applied mathematics society industrial mathematics roche cuevas aime general method devise signal restoration multiplicative algorithms constraints signal processing vol may iordache plaza sparse unmixing hyperspectral data ieee transactions geoscience remote sensing vol june total variation spatial regularization sparse hyperspectral unmixing ieee transactions geoscience remote sensing vol vapnik nature statistical learning theory new york usa chen maximum correntropy estimation smoothed map estimation ieee signal processing letters vol peng chen zhao robust hammerstein adaptive filtering maximum correntropy criterion entropy vol tibshirani regression shrinkage selection via lasso journal royal statistical society series methodological halimi altmann dobigeon tourneret nonlinear unmixing hyperspectral images using generalized bilinear model ieee transactions geoscience remote sensing vol unmixing hyperspectral images using generalized bilinear ieee international conference geoscience remote sensing symposium igarss altmann halimi dobigeon tourneret supervised nonlinear spectral unmixing using postnonlinear mixing model hyperspectral imagery ieee transactions image processing vol fevotte dobigeon nonlinear hyperspectral unmixing robust nonnegative matrix factorization ieee transactions image processing vol dec nascimento hyperspectral subspace identification ieee transactions geoscience remote sensing vol aug halimi dobigeon tourneret unsupervised unmixing hyperspectral images accounting endmember variability ieee transactions image processing vol december yokoya chanussot iwasaki nonlinear unmixing hyperspectral data using matrix factorization ieee transactions geoscience remote sensing vol yuan yan manifold regularized sparse nmf hyperspectral unmixing ieee transactions geoscience remote sensing vol fan miller comparative study new nonlinear model common linear model analysing laboratory simulatedforest hyperspectral data international journal remote sensing vol halimi honeine hyperspectral unmixing presence endmember variability nonlinearity mismodelling effects http
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sep detecting binomiality carsten conradi thomas kahle abstract binomial ideals special polynomial ideals many algorithmically theoretically nice properties discuss problem deciding given polynomial ideal binomial methods general main motivation source examples simplification steady state equations chemical reaction networks homogeneous ideals give efficient algorithm binomiality detection based linear algebra inhomogeneous input algorithm give sufficient condition binomiality remedy construct heuristic toolbox lead simplifications even given ideal binomial contents introduction criteria binomiality homogeneous case heuristics inhomogeneous case references introduction algebra mainstay modern applied mathematics across sciences often comes form polynomial equations much flexible linear equations modeling complex phenomena price paid mathematical algebra algebraic much involved linear algebra fortunately polynomial systems applications often special structures paper focus sparsity polynomials terms sparsest polynomials monomials systems monomial equations big topic algebraic combinatorics view modeling much help date august mathematics subject classification primary secondary key words phrases polynomial systems biology binomial ideal steady states chemical reaction networks authors supported research focus dynamical systems state carsten conradi thomas kahle solution sets unions coordinate hyperplanes next interesting class binomial systems polynomial allowed two terms binomials flexible enough model many interesting phenomena sparse enough allow specialized theory strongest classical results binomial systems require one seek solutions algebraically closed field complex numbers however objects applications think concentrations probabilities assumption prohibitive one often works real numbers leads fields real geometry new theory combinatorial commutative algebra shows binomial equations field assumptions skirted dependence binomial systems coefficients quite weak binomial equations one hope results depend explicit values parameters thus robust presence uncertainty main theme paper detect binomiality decide given polynomial system equivalent binomial system common way decide binomiality compute basis since ideal generated binomials reduced basis binomial corollary polynomial systems arising applications however computing reduced basis often demanding parameter values unknown computations performed field rational functions parameters even though computationally feasible time consuming usually yields output hard digest humans added complexity comes fact bases contain lot information may needed specific task deciding binomiality polynomial system hence methods desirable methods bases started generalization gauss elimination polynomials since come back roots linear algebra advent type algorithms try arrange computations sparse linear algebra exploited method draws linear algebra bases monomials inspired developments computer algebra deciding set polynomials brought binomial form using linear algebra question whether coefficient matrix partitioning kernel basis definition proposition deciding property requires row reductions hence computationally cheap compared bases shown coefficient matrix suitably extended polynomial system admits partitioning kernel basis polynomial system generated binomials first insight show converse need hold example general computer algebra profits homogeneity true bases example hilbert function driven algorithms used convert basis cheap compute one expensive order lex also observe phenomenon approach satisfying answer detecting binomiality binomial detection problem found given system polynomials homogeneous section discuss case eventually leads algorithm inhomogeneous case things complicated basis computations reduced homogeneous case easy trick detection binomiality example address problem collecting heuristic approaches best case establish binomiality without bases recipe approaches also used system entirely binomial binomials example demonstrate polynomial system binomial steady state ideals binomiality detection applied polynomial system motivation comes chemical reaction network theory ordinary differential equations polynomial sides used model dynamic processes systems biology mathematics systems extremely challenging particular since realistic models huge involve uncertain parameters consequence latter studying dynamical systems arising biological applications often amounts studying parameterized families polynomial odes first order business concern large part work area determine steady states thus real zeros families parameterized polynomial equations moreover structure polynomial odes entails existence affine linear subspaces invariant solutions hence questions concerning existence uniqueness steady states existence multiple steady states equivalent questions regarding intersection zero set parameterized family polynomials family affine linear subspaces polynomial equations describing steady states equivalent binomial equations generate binomial ideal mathematical analysis becomes much easier main theme system binomial instance one decide efficiently positive steady states exist monomial parametrization found using linear algebra integers section steady states toric positive real part toric variety sufficient criterion toric steady states appears theorem since zero sets general polynomial systems need parametrizations view task detecting binomiality important step analyzing systems biology areas like algebraic statistics control theory economics etc frequent challenging problem analysis dynamical systems biology decide multistationarity existence parameter values leading one isolated steady state variety results precluding multistationarity appeared recent years see instance methods employing jacobian similarly several sufficient criteria multistationarity emerged example general problem remains hard however case binomial steady state equations question multistationarity often answered carsten conradi thomas kahle effectively example positively theorem negatively theorem results require study systems linear inequalities notation paper work polynomial ring variables coefficient field usually field real rational functions set parameters methods agnostic towards field system polynomial equations variables encoded ideal polynomial homogeneous terms total degree ideal homogeneous generated homogeneous polynomials binomial polynomial two terms particular monomial binomial important distinguish binomial ideals binomial systems binomial system polynomial system binomial contrast ideal binomial ideal exist binomials generate ideal thus general form binomial system even generate binomial ideal sake brevity give introduction commutative algebra refer standard text books like modest amount matroid theory necessary section picked first pages acknowledgment thank david cox discussions role homogeneity computer algebra grateful alicia dickenstein pointing crucial error earlier version paper supported cds center dynamical systems university criteria binomiality basic criterion decide ideal binomial compute basis works buchberger algorithm binomials binomial since reduced basis unique must computable binomial generators consists binomials ideal binomial however bases hard compute criteria using linear algebra also desirable linear algebra enters write polynomial system product coefficient matrix entries vector monomials clearly use row operations matrix bring form row two entries ideal generated binomials monomials criterion naive detect binomial ideals since allows combinations given polynomials show section least homogeneous ideals extended characterization embark details formalize condition matrix definition matrix partitioning kernel basis kernel admits basis vectors disjoint supports exists basis ker supp supp detecting binomiality following proposition allows one check partitioning kernel basis linear algebra underlying reason restricted structure kernel expressed best matroid language proposition following equivalent matrix partitioning kernel basis column matroid direct sum uniform matroids corank one possibly several coloops reduced row echelon form two entries row proof let partitioning kernel basis supports basis satisfy circuit axioms thus equal circuits column matroid indeed circuit elimination satisfied trivially overlap two circuits element ker partitioning kernel basis property supp supp either proportional one support properly contains support circuit circuit columns appear circuit coloops remaining columns form direct sum uniform matroids corank one column matroid direct sum matroids unique reduced row echelon form block structure corresponding direct sum decomposition therefore suffices consider single block kernel full support coloops blocks ignoring zero rows reduced row echelon form matrix rank rows reduced row echelon form exactly one entry correspond positions every element kernel zero thus assume none row exactly two entries let column entries restriction corresponding pivotal columns yields block containing zero rows unique kernel vector corresponding dependencies block orthogonal kernel remaining columns procedure applied column thus constructed basis partitioning kernel basis remark proposition shows complexity deciding matrix partitioning kernel basis essentially elimination one needs field operations larger dimensions matrix remark direct sum arbitrary uniform matroids called partition matroid translate proposition polynomial systems proposition partitioning kernel basis vector monomials appropriate length ideal binomial carsten conradi thomas kahle system transformed binomial system using combinations partitioning kernel basis proof coloops first part theorem coloops give monomials second statement clear since combinations polynomials row operations coefficient matrix change kernel general strategy suitably extend given system redundant polynomials proposition yields binomiality extended system happens following example example let ordering monomials system linearizes coefficient matrix reduced row echelon form partitioning kernel basis proposition algorithm takes problem account working degree degree ideal binomial since binomial fact zero quotient ring first hint extend following theorem due theorem theorem let polynomial system exist monomials system fim coefficient matrix partitioning kernel basis binomial theorem true since additional generators change ideal system generates together explicit description binomial generators case partitioning kernel basis theorem yields result condition theorem also necessary test binomiality could built trying systematically identify monomials however converse theorem true example let homogeneous binomial ideal generated choice monomials coefficient matrix system partitioning kernel basis proof example first need following curious little fact detecting binomiality lemma ideal contain binomial proof assume algebraically closed since contains binomial extension algebraic closure assume product binomial assume divisible variable indeed variable divides divides find lower degree binomial since homogeneous also assume homogeneous potentially renaming variables assume since algebraically closed solution equation generator vanishes vanish contradiction shows contain binomial proof example let highest total degree among monomials system consider restriction involved polynomials degree since highest degree part equals monomial multiples contribute degree part system whole coefficient matrix partitioning kernel basis also submatrix columns degree monomials one proposition case row reductions submatrix would yield binomial degree ideal generated impossible lemma example may seem contrived kind trivial obfuscation binomials happen applications course humans obvious one first isolate linear binomial search implied quadratic binomials reduce trinomial next aim algorithm implements idea least homogeneous case homogeneity assumption skirted unfortunately true ideal binomial homogenization binomial corollary homogenization accessible without basis would superb purposes homogenizing generators binomial ideal would always yield binomial ideal unfortunately case example shows homogeneous case given ideal homogeneous graded vector space structure quotient allows one check binomiality degree degree need basic facts quotients modulo binomials see section details set binomials induces equivalence relation set monomials hbi space quotient ring spanned equivalence classes monomials linearly independent proposition binomials homogeneous situation particularly nice example carsten conradi thomas kahle equivalence classes finite elements quotient degrees notions monomial binomial polynomial extended quotient ring example binomial polynomial uses two equivalence classes monomials unified mathematical framework treat quotients modulo binomials monoid algebras refrain introducing notion consequence discussion polynomial system considered modulo binomials coefficient matrix quotient system arises coefficient matrix original system summing columns monomials equivalence class example let among monomials total degree three well become equal thus degree three part quotient one basis vector per equivalence class consequently trinomial maps binomial coefficient matrix matrix arises matrix summing columns corresponding well reduction modulo lower degree binomials example done general lemma let homogeneous polynomials degree set homogeneous binomials degree quotient ring ideal binomial coefficient matrix images partitioning kernel basis proof graded version nakayama lemma see corollary together exercise implies ideal welldefined number minimal generators degree therefore minimal generating set consists degree polynomials proposition applied finitedimensional vector space degree polynomials correctly decides binomiality lemma basis following binomial detection algorithm algorithm input homogeneous polynomials output yes binomial generating set one exists otherwise let empty let fmin set elements minimal degree redefine fmin compute reduced row echelon form coefficient matrix fmin detecting binomiality row three entries output stop find set binomials whose images generate hfmin redefine redefine redefine image output yes proof correctness termination termination obvious fact maximum number iterations loop equals number distinct total degrees among step relies proposition step binomials generate hfmini read reduced row echelon form via proposition preimages binomials suffice lemma shows loop either exhausts binomial stops case remark homogeneous case natural choice vector spaces work polynomials fixed degree iteration loop algorithm rows span vector space polynomials degree ideal modulo binomials hbi general inhomogeneous situation extra work needed construct suitable vector space particular one needs select infinite list binomials ideal many enough reduce given polynomials binomials whenever possible interesting problem future adapt one selection strategies algorithm bases task remark coefficient matrices polynomial systems typically sparse efficient implementation algorithm take account remark algorithm could also written completely polynomial ring without quotients new degree one would consider coefficient matrix fmin together binomials degree ideal hbi list grows quickly list monomials appearing binomials thus elegant work quotient also efficient implement algorithm completely without bases refinements necessary simply using make compute basis effectively work quotient purposes however necessary proposition algorithm implemented without bases proof critical step algorithm reduces modulo binomials already found following step elements fmin need written terms basis vector space rdeg fmin degree deg fmin monomials modulo binomials equivalence relation introduced beginning carsten conradi thomas kahle section also thought graph monomials thus reductions carried graph enumeration algorithms like breadth first search restricting monomials degree deg fmin connected components vector space basis rdeg fmin thus used gather coefficients step remark feasibility computations cases bases computed demonstrated example contains binomial ideal whose basis computed whose proved using computation yielded negative answer question radicality conditional independence ideals algebraic statistics remark using bases one represents connected component graph defined hbi least monomial respect term order philosophy necessary one work connected components per bother picking finding specific representative component representative works implementation one could choose data structure monomial stores index connected component belongs remark trivial generate classes examples bases methods fail algorithm quick example take set binomials whose basis computed add polynomial ideal algorithm immediately goes work reducing polynomial modulo binomials implementation bases embarks hopeless task remark remark highlights spirit method basis ideal contains much information binomiality one avoid expensive computation decide simple question heuristics inhomogeneous case ideals one encounters chemical reaction network theory often homogeneous results section apply first idea one may inhomogeneous case work partial homogenization bases quite robust relation homogenization example compute basis ideal suffices homogenize generators compute basis homogeneous ideal dehomogenize although intermediate homogeneous ideal generally equal homogenization original ideal dehomogenized basis basis dehomogenized ideal exercise unfortunately notion binomiality lend kind tricks geometrically homogenizing polynomials ideal yields projective closure dehomogenizing restricts one affine piece homogenizing generators creates extra components infinity components need binomial even detecting binomiality binomial intersection need binomial see also problem case following example example ideal hab binomial equals homogenizing generators however yields ideal hab present alternatives give complete answers quick check applied resorting expensive basis computation quickest least likely successful approach try linear algebraic manipulations given polynomials equivalently one applies row operations coefficient matrix instance computing reduced row echelon form partitioning kernel basis ideal binomial generators combinations binomials may seem much ask happen family networks section linear algebra successful one homogenize generators run algorithm resulting homogeneous ideal comes binomial original ideal binomial following simple fact proven instance corollary proposition let ideal homogeneous ideal generated homogenizations generators using variable generated dehomogenization generating set illustrate phenomenon leading failure heuristics example consider network example steady states real zeros following polynomials binomials used eliminate one every pair eliminate checked eliminating instead lead binomials immediately although leads carsten conradi thomas kahle linear trinomials using linear relations remaining system recognized consist two independent binomials analysis shows steady state ideal consideration equals binomial ideal basis computation example also yields result arguably less instructive note also naive homogenization yield binomiality element linear homogenization algorithm would pick element fmin stop since binomial effect example motivates final method term replacements using known binomials expect useful applications system biology following reasons often happens generators linear combinations binomials example steady state ideals networks complexes always binomial generators complexes produced one reaction thus rate change binomial mapk networks describe certain types cellular signaling one often finds binomials form kxa frequently binomials steady state ideals linear equivalently concentrations equal scaling may depend kinetic parameters happens examples detecting binomiality illustrate term replacements larger example comes network erk activation embedded two negative feedback loops see section pointers relevant biology example consider following steady state ideal generated polynomials obvious factorization following elements binomials system seven conservation relations found linear algebra according strategy use binomials simplify system eliminate possible using conservation relations always possible conservation relations stem duplicate equations like eliminate remaining part consists dividing reaction constants binomials form kxj rational expression involving reaction constants using yields carsten conradi thomas kahle particular find five new binomials adding yields trinomial consequently original system equivalent system consisting binomials two trinomials relatively simple shape comparison computed basis rational functions reaction rates coefficients although computation finished minutes result practically unusable basis consists elements huge rational functions coefficients structure observed completely lost lesson learned example term replacements using binomials useful solving polynomial system even end result binomial especially case notion minimal generators absent computations binomials package probably assist automatically useful reductions example natural general choice would replace higher degree monomials lower degree ones would directly reveal binomiality example finally summarize possible strategy deal inhomogeneous ideals example instance solved already item also item recipe try linear algebra proposition homogenize given ideal run algorithm algorithm returns binomials proposition original dehomogenized ideal binomial homogenization carried linear algebra reductions possibly detect homogeneity already earlier stage compare example algorithm returns negative answer dehomogenize use known binomials term replacements example potentially homogenize enlarged generating set compute reduced basis detecting binomiality references banaji craciun criteria injectivity unique equilibria general chemical reaction systems adv appl math bachmann greuel lossen pfister singular introduction commutative algebra springer berlin craciun feinberg multiple equilibria complex chemical reaction networks graph siam appl math craciun feinberg multiple equilibria complex chemical reaction networks semiopen mass action systems siam appl math conradi flockerzi multistationarity mass action networks applications erk activation math biol cox little shea ideals varieties algorithms springer new york eder survey basis computations preprint eisenbud sturmfels binomial ideals duke math eisenbud commutative algebra view toward algebraic geometry springer new york new efficient algorithm computing bases pure appl algebra grayson stillman software system research algebraic geometry available http joshi shiu simplifying jacobian criterion precluding multistationarity chemical reaction networks siam appl math johnston translated chemical reaction networks bull math biol kahle miller decompositions commutative monoid congruences binomial ideals algebra number theory kahle rauh sullivant positive margins primary decomposition commut algebra kahle decompositions binomial ideals softw algebra geom feliu regensburger conradi shiu dickenstein sign conditions injectivity generalized polynomial maps applications chemical reaction networks real algebraic geometry found comput published onlineu dickenstein shiu conradi chemical reaction systems toric steady states bull math biol turjanski mapk networks capacity multistationarity due toric steady states math biosci oxley matroid theory oxford university press oxford wiuf feliu kinetics determinant criteria preclusion multistationarity networks interacting species siam appl dyn syst hochschule technik und wirtschaft berlin germany address magdeburg germany website 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repetitive scenario design giuseppe calafiore feb abstract repetitive scenario design rsd randomized approach robust design based iterating two phases standard scenario design phase uses scenarios design samples followed randomized feasibility phase uses test samples scenario solution give full exact probabilistic characterization number iterations required rsd approach returning solution function desired levels probabilistic robustness solution novel approach broadens applicability scenario technology since user presented clear tradeoff number design samples ensuing expected number repetitions required rsd algorithm plain scenario design becomes one possibilities sitting one extreme tradeoff curve one insists finding solution single repetition comes cost possibly high possibilities along tradeoff curve use lower values possibly require one repetition keywords scenario design probabilistic robustness randomized algorithms random convex programs ntroduction purpose approach described paper obtain probabilistically reliable solution design problem affected uncertainty concept probabilistic design discussed extensively control community last decade well accepted standard tool tacking difficult robust design problems refer reader survey paper book many pointers related literature essential elements probabilistic design approach following ones spec function associates real value pair design parameter uncertainty instance rnq function represents design constraints specifications problem particular shall say design robust design paper make standing assumption convex probability measure prob defined describes probability distribution uncertainty equipped two essential elements given given design vector position define probability violation spec function prob say robust design holds designer also typically seeks minimize cost function considered linear form without loss generality see section guaranteeing finding robust design amounts solving optimization problem computationally hard general perhaps harder finding classical deterministic robust design optimization problems solved exactly restrictive cases linear specific distribution normal see deterministic convex approximations problems discussed special classes problems affine entries independent also sampling average approximation saa method replaces probability constraint one involving empirical probability violation based sampled values see optimization problem resulting saa however remains intractable general giuseppe calafiore dipartimento automatica informatica politecnico torino italy standard scenario theory effective approximation schemes optimization problems remain date hard tackle numerically alternative efficient randomized scheme emerged last decade finding robust designs technique technology see recent surveys area robust control called scenario design introduced scenario design one considers random samples uncertainty builds scenario random convex program rcp min given convex compact domain given objective direction optimal solution problem exists random variable depends multiextraction consequence violation probability relative scenario solution priori random variable scenario design lies somewhere robust design minimized subject design minimized subject indeed optimal objective value resulting scenario design lower optimal objective high probability higher suitable optimal objective see section moreover fundamental feature scenario design optimal solution feasible high probability problem key result recalled next sake clarity shall work following simplifying assumption routinely made literature scenario design see assumption probability one respect problem feasible attains unique optimal solution also need following standard definition see definition definition let denote optimal objective value problem also define min constraint said support constraint key fact regardless problem structure number support constraints problem exceed number decision variables see theorem instance problem happens precisely support constraints problem instance said fully supported see definition definition instances problem fully supported almost surely respect random extraction constraints say problem fully supported one following key result holds see theorem corollary theorem let assumption hold given holds probn moreover bound tight since holds equality class problems form fully supported probability one remarkable feature result holds irrespective probability distribution assumed depends problem structure dimension parameter scenario problems bernoulli trials given let consider following bernoulli variable associated problem otherwise definition event happens one interpretation thus time solve scenario problem priori probability realizing successful design finding solution robust design probability realizing failure finding solution robust classical scenario theory usually prescribed choose make small values low common guarantees event happen practical words regime scenario problem return robust solution practical certainty moreover key feature scenario theory high level confidence reached relatively cheap computational price indeed considering condition given desired probability level using fairly standard techniques bounding binomial tail see corollary details one prove condition satisfied since appears bound logarithm indeed see grows gracefully required certainty level however cases number constraints prescribed reaching desired confidence levels high practical numerical solution convex optimization solvers certainly efficient practical limits number constraints deal limits depend actual type convex problem say linear program semidefinite program sdp one deals critical situation instance problem semidefinite program formally taken maximum eigenvalue function matrices describing linear inequality constraints dealing sdp many thousands lmi constraints pose serious practical issues contribution paper discuss variation scenario approach used obtaining robust solution high confidence using small values precisely interested using scenario optimization regime side close one shall solving repeatedly instances scenario problem checking result via suitable violation novel approach named repeated scenario design rsd discussed section contains relevant results section iii describes two numerical examples robust control design proposed approach applied improving readability technical proofs reported appendix notation preliminaries shall make intensive use beta related probability distributions definitions standard facts recalled next denote beta beta density function parameters beta gamma function integers holds also denote fbeta cumulative distribution function beta density fbeta beta fbeta regularized incomplete beta function standard result establishes integers holds fbeta notice expression exact minimal value easily found numerically searching may conservative least integer number successes independent bernoulli trials success probability random variable binomial distribution denote bin cumulative distribution given bzc prob prob bzc fbeta bzc bzc fbeta fbeta bzc denotes largest integer larger number successes binary trials trial success probability random variable beta distribution random variable distribution cumulative distribution random variable given see bzc prob generalized hypergeometric function epetitive scenario design section develops main idea paper repetitive scenario design rsd mean iterative computational approach iteration scenario problem solved ensuing solution checked violation oracle either deterministic randomized illustrated next oracle returns false another iteration performed instead oracle returns true algorithm terminated current solution returned rsd user selects desired probabilistic feasibility level number scenarios used theorem iteration holds probn denotes multisample elementary terms iteration rsd method thought biased coin toss probability success toss getting least setting probability need close one simple idea behind rsd method repeat coin toss obtain success success detected violation oracle one may easily argue intuitively probability obtaining success point algorithm much higher probability obtaining success single toss similar idea recently proposed authors solve repeatedly scenario problem followed randomized test feasibility approach results however distinctively different ones proposed scenario problems solved using number scenarios grows iteration count value plain corresponds plain scenario design major shortcoming approach analysis number iterations bounded either deterministic probabilistic sense tradeoff curve proposed choice function expected running time algorithm result guarantee algorithm reach final iteration equals plain hence complexity algorithm worse one plain scenario design method actual reduction number design samples theoretically guaranteed shall next analyze precisely probabilistic features rsd algorithm two cases first case assume ideal exact feasibility oracle available checking current solution case may unrealistic general serves providing insightful preliminary analysis rsd approach second case analyze rsd approach practically implementable randomized feasibility oracle used violation oracles deterministic oracle black box given input value design variable returns output flag value true false otherwise oracle may realizable computationally practice since computing probability numerically hard general reason next also introduce randomized oracle defined means randomized scheme described next randomized oracle input data integer level output data logic flag true false generate samples according prob forpi let otherwise return true else return false simply evaluates empirical probability violation samples returns true false otherwise similar type randomized feasibility oracle previously introduced used probabilistic design setting also see also section validation step proposed however propose paper different one used cited references latter exits false flag soon one infeasible sample found whereas allows infeasible samples exit also kind priori analysis develop repetitive scenario design based entirely novel repetitive scenario design ideal oracle consider following rsd algorithm repetition consists plain scenario optimization step followed feasibility check ensuing solution performed exact feasibility oracle algorithm rsd input data integer level output data solution initialization set iteration counter scenario step generate samples according prob solve scenario problem let resulting optimal solution step set flag true else set false exit condition flag true exit return current solution else set goto following theorem holds theorem let assumption hold given define running time algorithm value iteration counter algorithm exits solution returned algorithm robust design expected running time algorithm equality holds scenario problem running time algorithm probability equality holds scenario problem see section appendix proof theorem remark potential limits rsd approach preliminary results theorem show potential rsd approach suppose chosen say means plain scenario approach least chance returning good solution robust design however see point theorem probability algorithm returns robust design within iterations eventual outcome algorithm robust probability one expected number iterations rsd algorithm worst case problem theorem also shows fundamental limit rsd approach decrease hence increase respect plain scenario design approach decrease much otherwise expected number iterations algorithm tends thus fundamental tradeoff reduction reduces effort needed solving scenario problem increase number iterations algorithm tradeoff fully captured plotting expected running time bound versus number scenarios repetitive scenario design randomized oracle section contains main contribution paper consider realistically implementable version rsd approach randomized oracle used instead ideal deterministic one algorithm rsd input data integers level output data solution initialization set iteration counter scenario step generate samples according prob solve scenario problem let resulting optimal solution step call current input set flag true false according output exit condition flag true exit return current solution else set goto generic iteration stage algorithm illustrated figure fig generic stage algorithm next analyze algorithm along two directions first contrary algorithm present algorithm may exit solution robust due randomized nature oracle may detect false positive misclassifying good solution show probability bad exit event made arbitrarily small second fully characterize probabilistic running time iterations exit algorithm start following key preliminary lemma backbone whole paper lemma let assumption hold given iteration define events true returns true goodtrue returns true badtrue returns true let fbeta fbeta iteration algorithm holds probn true prob goodtrue prob badtrue fbeta moreover problem one bounds hold equality probn badtrue see section appendix proof lemma state main result concerning algorithm theorem let assumption hold let given let notation set lemma let denote product probability probn probn define event badexit algorithm exits returning bad solution badexit algorithm returns following statements hold badexit fbeta problem one actually holds badexit equality holds scenario expected running time algorithm problem running time algorithm probability equality holds scenario problem see section appendix proof theorem asymptotic bounds key quantity related expected running time algorithm upper tail distribution quantity related hypergeometric function ratios gamma functions may delicate evaluate numerically large values arguments therefore useful manageable albeit approximate expression following corollary gives asymptotic expression see section appendix proof corollary holds interesting consequence corollary large conclude last equation gives approximate asymptotic expression upper bound expected running time algorithm also tells bound better smaller corresponding bound ideal algorithm practical dimensioning scenario oracle blocks typical probabilistic design problem given dimension decision variable level probabilistic robustness require design intend use randomized approach also set confidence level level probability randomized approach successfull returning robust design plain non repetitive scenario design setting requires dimensioning number scenarios guarantee done instance using bound via simple numerical search however required turns large practice ensuing scenario optimization problem becomes impractical deal numerically switch repetitive scenario design approach case suggest following route designing scenario oracle blocks let first select level used oracle qualitatively decreasing increases expected running time decreases required converse happens increasing suggest set range dimensioning scenario block dimension scenario optimization block choosing achieve good tradeoff complexity scenario program grows expected number iterations required rsd approach decreases choice made instance plotting approximate expression becomes exact upper bound expected running time algorithm versus selecting value running time acceptable dimensioning oracle block selected according approach described consider point point theorem dimension block searching numerically side problem remark observe general bound used design block however expression easier deal one hence advisable use former preliminary dimensioning phase values verified actual bound another advantage using bounding technique analogous one described section invert condition finding manipulation condition satisfied choice pair satisfied guarantee priori randomized algorithm may fail returning robust design desired rigorously holds assumption scenario problem one nice feature highlighted workload necessary achieve desired failure level subdivided samples scenario problem samples oracle lower complexity scenario problem employed long paired randomized oracle suitable notice however making choice pair expected running time algorithm also taken consideration places lower limit small see also discussion section remark observe typical cases dealing large milder problem dealing large due fact merely checking satisfaction inequality generally easier solving related optimization problem many constraints also remark algorithm inherently parallel speedup potentially gained processors available parallel randomized feasibility test actually whole approach formulated fully parallel instead sequential form workers solve parallel instances scenario problems worker parallel used randomized oracle parallel version rsd method easily analyzed using probabilistic tools developed paper iii umerical examples exemplify steps rsd approach algorithm dimensioning numerical results using two examples robust control design first example deals robust input design uncertain linear system second example deals robust performance design positive linear system robust input design consider system form scalar input signal rna interval uncertain matrix form qij vector zeros except one entry given final time target state problem determine input sequence state robustly contained small ball around target state input energy large write matrix system given formally express design goals form minimization level tradeoff parameter letting problem formally stated framework setting assuming uncertain parameter random uniformly distributed hypercube scenario design problem takes following form min dimensioning rsd algorithm set thus size decision variable scenario problem set desired level probabilistic robustness require level failure randomized method require randomized method return good solution practical using plain scenario approach imposing would require scenarios let see reduce figure resorting repetitive scenario design approach let fix thus plot asymptotic bound expected number iterations function shown figure see plot instance fig example section plot choice corresponds value upper bound expected number iterations algorithm let choose value scenario block simplified condition tells let choose samples used oracle choices thus algorithm upper bound average running time notice upper bound tight problems conservative problems necessarily thus general may expect performance practice better one predicted theoretical bound numerical test considered nominal matrix dimension matrix shown top page target state run algorithm times test run recorded number iterations solution returned upon exit figure shows number repetitions test runs see algorithm exited times single repetition maximum repetitions figure predicted upper bound practical performance thus better predicted suggests problem hand fully supported figure shows level empirical violation probability evaluated oracle upon exit finally figure shows optimal level returned algorithm test runs figure shown optimal input signal returned algorithm averaged test runs computational improvements example rsd approach permitted substantial reduction number design samples samples required plain scenario method samples price moderate number repetitions average number repetitions test runs numerical experiments carried intel xeon machine using cvx matlab average test experiments rsd method required return solution comparison purposes also run plain scenario optimization scenarios required attain desired level time required obtaining solution using rsd approach instead plain scenario design thus yielded reduction computing time one order magnitude reason improvement due fact scenario optimization problem rsd approach uses scenarios took solved typical run subsequent randomized oracle test computationally cheap taking number iterations test run test run fig example section repetitions algorithm test runs levels empirical violation probability evaluated oracle upon exit test runs test run fig example section optimal level returned algorithm test runs average test runs optimal input returned algorithm fig example section network model uncertain linear transportation network second example consider variation transportation network model introduced section see figure model described state equations states represent contents four buffers parameters represent rate transfer buffer buffer input flow min min second buffer take output total content buffers see consider situation vector parameters designed uncertainty term assumed truncated normal random vector zero mean covariance matrix system form matrix metzler entries nonnegative theorem states given system stable gain smaller given exist vector ones taking samples robust scenario design problem one one seeks minimize gain subject constraint scenarios see problem robustified version one discussed section problem stated convex due product terms entries however introducing new variables rewrite problem variables see dimensioning rsd algorithm size decision variable scenario problem previous example set desired level probabilistic robustness require level failure randomized method using plain scenario approach imposing would require scenarios next reduce figure resorting repetitive scenario design approach let fix thus plotting asymptotic bound expected number iterations function figure previous example see choice corresponds value upper bound expected number iterations algorithm let choose value scenario block simplified condition tells samples used randomized feasibility oracle choices thus algorithm upper bound average running time notice general may expect performance practice better one predicted theoretical bound since actual problem may fully supported numerical test computational performance first solved problem via plain scenario approach using scenarios computational time resulting following optimal solution next run rsd method algorithm times test run recorded number iterations solution returned upon exit figure shows number repetitions test runs see algorithm exited times single repetition maximum repetitions average repetitions figure shows level empirical violation probability evaluated oracle upon exit finally figure shows optimal level returned algorithm test runs average test trials running time rsd method since plain scenario approach required slower newly proposed rsd approach test example repetition rsd method required solving scenario problem randomized oracle check observe oracle time much lower scenario optimization time number iterations test run test run fig example section repetitions algorithm test runs levels empirical violation probability evaluated oracle upon exit test runs onclusions repetitive scenario design generalizes scenario approach robust design setting iterative procedure whereby scenario design trials followed randomized check feasibility level solution expected number repetitions trials procedure dictated key quantity well approximated large recover extreme situation standard scenario design valid solution found single repetition cost possibly large smaller values trade complexity solution scenario problem additional iterations rsd algorithm extent reduced however limited upper bound impose test run fig example section optimal level returned algorithm test runs expected running time since tells numerical examples showed proposed rsd approach may lead improvements computational time one order magnitude compared plain scenario approach ppendix proof theorem first point theorem obvious sice algorithm terminates true returned deterministic oracle happens condition satisfied point two let bernoulli variables representing outcome step iteration oracle returns true otherwise oracle returns false observe probability since algorithm terminates soon true returned oracle running time algorithm defined random variable iteration true returned first time clearly geometric distribution denotes product probability measure mean geometric distribution whence proves second point note equality holds scenario problem one cumulative geometric distribution function increasing thus implies proves third point proof lemma given iteration algorithm let consider sequence binary random variables appearing inside otherwise definition prob random variable cumulative distribution function given therefore given form bernoulli sequence success probability however random variable cumulative distribution therefore form conditionally bernoulli sequence directing finetti measure simpler terms described compound distribution first success probability extracted random according directing distribution pno conditional generated according bernoulli distribution success probability let random variable binomial distribution bin thus probno bzc fbeta bzc bzc fbeta bzc bzc fbeta considering next let fbeta unknown function fbeta observe identically zero scenario problem one consider event goodtrue true leting probn goodtrue probn probno dfv using fbeta dfv using defined fbeta beta fbeta next analyze two terms first term fbeta beta using beta beta def beta fbeta observe holds fbeta fbeta therefore obtain following bound fbeta fbeta particular fbeta fbeta next consider term fbeta fbeta integrating parts fbeta beta since forall expression shows identically zero problems considering fact permits conclude probn goodtrue probn proves also obtain probn true probn probn proves using probn badtrue probn probn true goodtrue scenario problem hence probn badtrue problem using proves upper bound probability badtrue supported case reason instead follows probn badtrue probno dfv fbeta dfv integrand decreasing fbeta dfv fbeta dfv fbeta probn fbeta fbeta since fbeta proves proof theorem let define event badexitk one algorithm reaches iteration exits bad solution solution probability event probability returns false precisely times guarantees reach iteration event badtrue happens iteration therefore letting denote probability badtrue denote probability true events defined lemma badexitk event badexit algorithm terminates bad solution union events badexitk therefore badexit badexitk prob badtrue probn true use upper bound use conclude badexit fbeta proves fully supported case instead use upper bound hence conclude badexit proves let next denote running time algorithm value iteration count algorithm terminates since algorithm terminates soon true event happens since true events statistically independent among iterations geometric probability probability true therefore expected value inequality follows proves point theorem via reasoning probability hence conclude proves third point theorem proof corollary fbeta beta recall beta density mean peak mode variance observe fbeta fbeta fbeta cumulative distribution beta density peak density tends variance distribution goes zero permits argue large function fbeta inflection point near decreases rapidly value value crosses function fbeta tends step function one zero therefore integral beta fbeta proves eferences aldous exchangeability related topics hennequin editor xiii volume lecture notes pages springer calafiore ghaoui optimization models cambridge university press calafiore random convex programs siam journal optimization calafiore campi scenario approach robust control design ieee transactions automatic control calafiore dabbene probabilistic analytic center cutting plane method feasibility uncertain lmis automatica calafiore dabbene tempo research probabilistic methods control system design automatica campi garatti exact feasibility randomized solutions uncertain convex programs siam journal optimization chamanbaz dabbene tempo venkataramanan wang sequential randomized algorithms convex optimization presence uncertainty garatti campi modulating robustness control design principles algorithms ieee control systems magazine michael grant stephen boyd cvx matlab software disciplined convex programming version http march lee sabavala bayesian estimation prediction model journal business economic statistics luedtke ahmed sample approximation approach optimization probabilistic constraints siam journal optimization nemirovski shapiro convex approximations chance constrained programs siam journal optimization pagnoncelli ahmed shapiro sample average approximation method chance constrained programming theory applications optim theory petersen tempo robust control uncertain systems classical results recent developments automatica probabilistic programming shapiro editors stochastic programming volume handbooks operations research management science elsevier amsterdam rantzer distributed control positive system tempo calafiore dabbene randomized algorithms analysis control uncertain systems applications communications control engineering eric weisstein beta binomial distribution wolfram web resource
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modeling duct flow molecular communication wayan tobias arman vahid adam robert institute digital communications electrical engineering computer science university ottawa nov school known taylor dispersion via effective diffusion coefficient particle distribution derived regime large distances interaction diffusion flow averaged uniform particle distribution gaussian spread along axis molecular communication authors adapted model keep analysis analytically tractable conditions simplifications justified considered detail particular clear channel behaves described outside dispersion regime moreover clear conditions taylor dispersion satisfied impact molecular communication potentially theoretical ntroduction insight phenomena could exploited molecular using molecules conveying digital messages recently communication system design focus paper twofold first introduce recognized key communication strategy nanoscale devices artificial cells cooperatively fighting disease notion dispersion systematic manner contrast entities involved molecular communication previous works second analyze highlight major nanoscale diffusion plays significant effects particle transport molecular communication systems two different regimes namely role transport messages however diffusion limited effective range renders dispersion regime regime latter molecular communication inefficient extended distances considered literature prevailing limitation overcome exploiting fluid flow example blood vessels addition diffusion example blood vessels two key messages paper follows interplay fluid flow diffusion governs regime diffusion supply oxygen lungs tissues consequently constant drift accurately capture channel charthe molecular communication literature considered basic acteristics means effective diffusion coefficient models fundamental mechanisms particular mean velocity regime basic channel characteristics diffusion initial spatial release pattern transmitter unbounded space uniform flow context affect resulting particle distribution molecular communication investigated example flow encountered ducts cause model might applicable boundaries significant intersymbol interference isi especially far nanonetwork previous work considered regime diffusion tends decrease isi environment uniform flow study impact enabling particles move away bounded detail hand boundary diffusion drift studied paper structured follows section clear simplified model applicable molecular introduce system model present preliminaries section communication since flow blood vessels microfluidic iii analyzes duct channel different flow regimes channels ducts especially microscale far numerical results presented section finally uniform hence general reduction reality section draw conclusions model justified particular marginal axial particle distributions inherently coupled ystem odel reliminaries makes mathematical analysis channel response system model difficult blood vessels simulations shown notion dispersion interaction diffusion consider straight impermeable cylindrical duct laminar flow principally investigated infinite axial extent radius described transport sought molecular communication extend coverage improve reliability mitigate interference one active mechanism inherent many liquid environments fluid flow flow models often assuming diffusion constant drift however diffusion flow usually encountered threedimensional bounded environments flow highly blood vessels microfluidic channels qualitative understanding relevant physical effects inherent channels systematic framework provided based number ratio distance duct radius review relevant laws physics highlight simplified models uniform flow transport applicable several molecular communication setups highlight effect different flow scenarios channel impulse response point release uniform release fig system model geometry along axis red shading reflects flow velocity maximum center vanishes boundary corresponding parabolic shape released particles reside diffusing uniform release sketched three different time instances point uniform transmitter release shown black dot blue line respectively cylindrical coordinates points axial position radial distance azimuth angle duct filled fluid viscosity subject steady laminar flow flow velocity function given parabolic function see fig consider transmitter releases ntx molecules either uniformly randomly distributed cross section point moreover assume transparent receiver specified points satisfying receiver mounted duct wall axial distance given radial extent spanning angle see fig model particle release instantaneous released particles transported fluid flow brownian motion simplicity assume particles interact influence flow field small size forces gravity acting particles negligible preliminaries molecular communication information conveyed mass transfer mass transfer fluids mediated flow brownian motion referred advection diffusion respectively thereby mass transfer described spatial probability density function pdf interpreted normalized concentration gives average fraction particles within differential volume time pdf found solution following partial differential equation pde also refer advectiondiffusion equation denotes partial derivative respect nabla operator moreover diffusion coefficient velocity vector point solve need know velocity field general velocity field obtained solving navierstokes equation provides fundamental description flow relating velocity field local pressure applied rigid straight channels boundary conditions velocity boundary zero subject flow velocity profile referred poiseuille flow thereby assuming newtonian fluid fluid described viscosity obtain veff mean velocity channel function applied pressure gradient particular veff obtained veff note maximum velocity vmax occurs center found using thermodynamic reasoning assuming spherical particles diffusion coefficient satisfies einstein relation viscosity temperature fluid respectively particle radius note strictly valid solvent modeled continuous respect solute particles einstein relation gives physical insight understood absolute diffusion coefficient fundamental characteristic diffusion process governs microscopic particle motion important parameter number gives estimate importance diffusion advection dimensionless number defined veff considering duct radius length scale interest intuitively increases decreases veff increase importance particle transport flow diffusion becomes relevant respectively iii nalysis uct hannel equation environment fig simplifies following pde velocity field independent axial position boundary hold initially given uniform point release respectively still difficult solve general nonlinear velocity inherent coupling interestingly solved enough time passed diffusion flow profile averaged refer behavior dispersion regime trivially regime includes special case veff pure diffusion another special case solved regime regimes seek observation probability pob dvrx expense larger required distance dispersion take place fact rearrange veff approximates minimum diffusion coefficient required dispersion occur small large neglect first second parts sum side respectively substituting minimizing respect obtain minimum possible effective diffusion coefficient veff vrx deff min vrx volume also refer pob impulse response impulse response fundamental dmin veff given deff characteristic molecular communication channel deff min two values dmin dmin heavily influences example symbol error rate achieve effective diffusion coefficient values given dispersion regime deff deff deff min dispersion result interaction diffusion advection due flow profile observation probability obtained integrating interaction lead particle distribution uniform receiver volume given spatial pdf written veff regime particle pob distribution depend initial difference point uniform release veff behavior occurs chapter gaussian interest derive time pob attains distance large diffusion coefficient maximum maximizing pob respect large duct radius small naturally includes cumbersome resort maximizing special case pure diffusion flow present yields satisfied written following veff deff tmax equation veff deff deff veff approximation peak height follows pmax pob tmax note diffusion tmax effective diffusion coefficient deff mean velocity veff time particles moving mean eff instantaneous uniform release solution velocity reach given regime veff exp subsection directly determine observation probability pob pob neglecting diffusion distribution without resorting uniform release first assume uniform release following effective diffusion case particles lie surface paraboloid extends along axis time coefficient deff obtained exhibits rotational symmetry thereby marginal distribution deff remains uniform flow geometric given also refer diffusion shape given fig hence time coefficient pure diffusion taylor points lie paraboloid simply effective diffusion coefficient write deff given also obtain following note general deff moreover deff auxiliary relationship decreased small values increases however decreasing comes obtained probability pob pob ntx fraction particles within receiver volume particles independently observed within determined particles lie within segment volume time respectively reason angular extent within inner outer uniform point release initial particle radii respectively following uniform release position understood independently uniformly observation probability pob consequently random distributed within available area whole written pob variables cross section one point depend paraboloid intersects umerical valuation volume fact three scenarios using simulation validate derived shown fig first case paraboloid yet reached green line analytical expressions explore regimes fig first intersection occurs mathematical analysis difficult thereby unless explicitly paraboloid reaches points stated otherwise employ following physical parameter second case values diffusion coefficient choose paraboloid intersects reasonable estimate small proteins two orange line fig time values duct radius characterized paraboloid intersecting considered reasonable small capillaries points microfluidic ducts respectively thereby two txlast case occurs red line fig distances considered values case moreover choose receiver dimensions require order obtain impulse receiver size scales duct radius microscopic simulation time response obtain step set fluid flow mean velocity assumed veff reasonable small capillaries show fig adapted considered parameter values terms number ratio summary impulse response distance duct radius particular regime following uniform release written shaded two regimes obtained analytical results section iii expected applicable separates two regimes shown black line derived analytical results valid parameter values well pob within dispersion regime however analytical results expected accurate close boundary set two duct radii pob maximized also time two distances show fraction pob observed tail resulting four combinations black dots impulse response decays polynomial time fig see two scenarios lie close may give rise significant isi molecular communication boundary regions parameter values chosen simulations reveal deviations systems point release point release either regime hand scenarios coordinates lie well within regime expect within observe deviations developed theory note changing particles certainty duct radius affects whereas change influences considering parameter values chosen impulse response given paper fig conclude dispersion pob rect regime applicable small microscale ducts hand also see large set parameters rect zero otherwise regime appropriate release point within especially medium large ducts fig numerically evaluate effective diffusion impulse response zero times eqs intuitively valid coefficient given function molecular generally solution applicable diffusion coefficient different duct radii thereby duct radius large distance small show deff deff limiting expressions note still give observation large small respectively furthermore distinguish probability even though flow deterministic small shown orange line large shown dispersion fig sketch regions different transport regimes adapted four simulation scenarios shown black dots fig snapshot particle positions uniform release shown different colors starting left right respectively total ntx released deff largest time shown red lines show standard deviation positions veff mean assuming gaussian distribution due dispersion small times particles follow parabolic profile closely slightly larger times deff particles start spread diffusion minimum large times particles become uniformly distributed along dimension within duct due dispersion note veff mean particle position arrived respectively summary small times release regime large times release dispersion regime accurately model actual behavior figs show impulse response respectively case consider fig effective diffusion coefficient deff function well uniform point molecular diffusion coefficient release point release position blue line satisfy satisfy chosen particles arrive receiver respectively finally deff min shown diffusing simulate impulse responses investigate red dot value highlighted developed analytical models provides best fit black vertical line increases deff also increases case time seen transition orange fig comparison pob applicable blue line soft threshold given delineating point uniform release pob applicable validity dispersion regime also increases uniform release shown thereby obtain deff pob peaks also highlighted large dots value otherwise unattainable simulated impulse response following point diffusion coefficient small proteins release significantly larger simulated uniform release case simulated impulse responses neither deff meaningful anymore fig dmin match pob pob however especially considering increase leads increase dispersion behavior simulated data hand dmin increase leads less dispersion points tend better described pob pob flow broaden particle distribution along deviations simulated impulse axis effectively responses point uniform release pob gain basic understanding particle evolution much smaller dispersion regime provides much towards dispersion fig show three snapshots better fit despite fact strictly satisfied particle positions corresponding three different time fig consistent green particle cloud instances distinguished different colors following fig appears uniform comparing pob uniform release particular plot pob see peak pob larger smaller motivated fact marginal distribution pob small large respectively uniform distribution within circular disk uniform larger times pob side effect regime pob coincide expected becomes simple linear function considered independent conclusion parameter shown blue line furthermore values close boundary fig still applicable according fig dispersion occurs vicinity dispersion model note uniform point release peak values impulse responses fig least order magnitude smaller fig moreover uniform point release simulated impulse responses fig decay faster slower peak values fig respectively pob pob sim unif sim point peak onclusion fraction particles time pob pob sim unif sim point peak fraction particles time fig impulse responses simulation results shown uniform point release additionally simulated analytical impulse responses due point release scaled constant factor better visualization simulation results ntx fig pob pob shown former peak times highlighted uniform point release simulation results also shown fig dispersion approximation applicable scenario clarity shown considering point release simulated curve matches rectangular shape pob reasonably well hand simulated impulse response significantly deviates rectangular shape diffusion enough time result spread pulse expected fig observe general good agreement pob simulation results case uniform release regime provides reasonable description channel nevertheless larger times small deviation residual particles close particles already passed comparing impulse responses point release uniform release find tail impulse response strongly depends initial distribution considerable isi especially fraction particles released close duct wall however long reached particles initial release close center duct might reduce isi comparing figs see channels behave completely different duct radius changed dispersion generalizes concept diffusion crucial molecular communication thereby flow accounted effective diffusion coefficient effective diffusion coefficient multiple orders magnitude larger molecular diffusion coefficient however description relies large distance hand many practical scenarios microscale fall within regime dispersion insignificant regime initial release pattern drastically influences isi uniform release dispersion regime might preferable flowdominated regime reduced isi given duct radius either regime applicable depending distance number eferences farsad yilmaz eckford chae guo comprehensive survey recent advancements molecular communication ieee commun surv tutorials vol noel cheung schober diffusive molecular communication disruptive flows proc ieee icc jun wicke ahmadzadeh jamali unterweger alexiou schober molecular communication using magnetic nanoparticles apr srinivas eckford adve molecular communication fluid media additive inverse gaussian noise channel ieee trans inf theory vol jul kim eckford chae symbol interval optimization molecular communication drift ieee trans nanobioscience vol probstein physicochemical hydrodynamics introduction john wiley sons bruus theoretical microfluidics oxford university press felicetti femminella reali simulation molecular signaling blood vessels software design application atherogenesis nano commun networks vol taylor dispersion soluble matter solvent flowing slowly tube proc soc lond vol aris dispersion solute fluid flowing tube proc soc lond vol apr mao liu yang channel modelling molecular communications across blood vessels nerves proc ieee icc may chahibi pierobon akyildiz pharmacokinetic modeling biodistribution estimation molecular communication paradigm ieee trans biomed vol bicen akyildiz analysis leastsquares design microfluidic channels molecular communication ieee trans signal vol sun yang liu channel capacity modelling blood molecular communication blood flow drift proc acm nanocom
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new tools connections approximation nikhil parinya bundit danupon jesper aug eindhoven university technology netherlands aalto university finland weizmann institute science israel kth royal institute technology sweden danupon eindhoven university technology netherlands abstract paper develop new tools connections exponential time approximation setting given problem instance parameter goal design algorithm fastest possible running time show following results maximum independent set exp time chromatic number exp log time minimum vertex cover exp time minimum vertex cover exp time throughout omit polyloglog factors polynomial input size respectively best known time bounds problems bourgeois cygan maximum independent set chromatic number bounds complemented exp lower bounds exponential time hypothesis eth chalermsook laekhanukit thesis results show bounds tight problems key algorithmic results sparsification procedure reduces problem variant allowing use better approximation algorithms bounded degree graphs obtaining first two results introduce new randomized branching rule finally show connection pcp parameters approximation algorithms connection together independent set algorithm refute possibility overly reduce size chan pcp chan also implies significant improvement result refute conjecture dinur manurangsi raghavendra acm subject classification nonnumerical algorithms problems keywords phrases approximations algorithms pcp exponential time algorithms digital object identifier introduction independent set vertex cover coloring problems central problems combinatorial optimization extensively studied classical results concern either approximation algorithms run polynomial time exact algorithms run sub algorithms useful scenarios lack flexibility sometimes wish better approximation ratio worse running time nikhil bansal parinya chalermsook bundit laekhanukit danupon nanongkai jesper nederlof licensed creative commons license leibniz international proceedings informatics schloss dagstuhl informatik dagstuhl publishing germany new tools connections approximation computationally powerful devices faster algorithms less accuracy particular running time approximation ratios needed settings algorithmic results approximation ratio studied already literature several settings notably context polynomialtime approximation schemes ptas instance planar graphs baker celebrated approximation scheme several problems gives independent set time exp time graphs small treewidth czumaj give exp time algorithm given graph along tree decomposition width find independent set general graphs approximation results several problems studied several works see basic building block lies behind many results partition input instance smaller parts optimal sub solution computed quickly least faster fully example obtain independent set one may arbitrarily partition vertex set blocks restrict attention independent sets subsets blocks get exp time algorithm first sight one might think algorithm easily improvable via advanced techniques shown almost pcps imply independent set coloring requires least exp time assuming popular exponential time hypothesis eth setting sophisticated approximation schemes planar graphs marx showed algorithm planar independent set run time exp assuming eth implies algorithm czumaj improved run time exp lower bounds despite interesting far tight means answer question whether known approximation improved significantly fact many settings far understanding full power exponential time approximation example exclude algorithms time function see know algorithms run asymptotically faster fastest exact algorithm runs time time paper aim advance understanding study question designing fast possible approximation algorithms independent set coloring vertex cover general hyper graphs results independent set result following use omit log log factors theorem randomized algorithm given graph integer outputs independent set constant probability size least denotes maximum independent set size algorithm runs time exp prove result introduce new randomized branching rule introduce put context towards previous results follows sparsification technique reduces maximum degree given number technique already studied given graph integer answer yes independent set size least independent set size least bansal setting exponential time approximation algorithms independent set cygan see paragraph search tree techniques bourgeois see section authors obtain running times specifically sparsification technique branch select vertex try include independent set discard recurse possibilities vertices sufficiently high degree key property decide include vertex independent set may discard neighbors generate instances keep branching vertices degree least maximum degree smaller exp log instances created instance maximum independent set easily greedy argument cygan note gives worse running times algorithm works along line incorporates two simple ideas first observation instead solving leaf instance greedy algorithm one use recent approximation algorithm bansal independent set bounded degree graphs choose immediately gives improvement time essentially exp log improve use present additional innovative idea introducing randomization idea relies fact sparsification step unexploited slack aim specifically whenever branch consider include branch probability lower expected number produced leaf instances sparsification step exp log preserves approximation factor good probability via fairly standard methods see show also gives faster algorithm coloring following sense theorem randomized algorithm given graph integer outputs constant probability proper coloring using colors algorithm runs time exp log final indication sparsification powerful tool obtain fast exponential time approximation algorithms show combination result halperin sparsification lemma gives following result vertex cover problem hypergraphs edges size set cover problem frequency theorem every every exp time algorithm vertex cover problem hypergraphs edges size note vertex cover graphs gives exp running time gives exponential improvement denominator exponent upon approximation bonnet runs time recently brought attention williams independently unpublished results hypergraph vertex cover independent set using sparsification techniques similar connections pcp parameters question approximating maximum independent set problem time close connections three important parameters pcps size gap discuss implications algorithmic results terms pcp parameters observation already made bourgeois exploit new way new tools connections approximation roughly speaking gap parameter ratio completeness soundness freeness parameter number distinct proofs would cause verifier accept simply logarithm freeness convenience continue discussions terms freeness instead freebit freebit gap dependency freeness gap played important role hardness approximation notably existence pcps freeness gap parameter equivalent hardness approximating maximum independent set result building block proving hardness approximation many combinatorial problems coloring disjoint paths induced matching cycle packing pricing fair say pcp parameter captures approximability many natural combinatorial problems better parameter implies stronger hardness results existence pcp arbitrarily large gap freeness lowest possible fact equivalent inapproximability vertex cover best known due chan pcp gap freeness log yielding best known approximating maximum independent set sparse graphs approximating maximum independent set graphs size freebit gap approximation algorithm main concern polynomial size pcps thing matter comes exponential time approximability another important parameter size pcps come play size freebit gap tightly captures sub exponential time approximability many combinatorial problems instance log moshkovitz raz constructs pcp size freeness implies independent set requires time approximation result independent set implies following tradeoff results corollary unless eth breaks freebit pcp gap freeness size must satisfy particular implies chan pcp made smaller size log unless eth breaks light equivalence freebit pcps freeness approximation vertex cover result shows pcp must size least remark results known pcps knowledge first result kind related results best known results independent set log regime log hardness also holds coloring vertex cover best known hardness approximation hardness assuming unique games conjecture three problems independent set coloring vertex cover admit exact algorithms run time unless eth fails besides aforementioned works sparsification techniques exponential time approximation studied bonnet paschos mainly hardness results obtained roughly speaking existence pcp freeness gap implies hardness approximating independent set graphs bansal preliminaries first formally define three problems consider paper independent set given graph say independent set edge endpoints goal independent set output independent set maximum cardinality denote cardinality maximum independent set vertex cover given graph say vertex cover every edge incident least one vertex goal vertex cover output vertex cover minimum size generalization vertex cover called vertex cover cover defined follows given hypergraph hyperedge cardinality goal find collection vertices hyperedge incident least one vertex minimizing degree hypergraph maximum frequency element coloring given graph proper function goal coloring compute minimum integer admits proper number referred chromatic number denote graph denotes set neighbors denotes let denote graph subgraph induced use exp denote order avoid superscripts use suppress factors polynomial input size use suppress factors polyloglog respectively upper lower bounds write functions faster approximation via randomized branching sparsification maximum independent set section prove theorem key lemma lemma suppose approximation algorithm dis runs time outputs independent set size maximum degree algorithm running expected time exp log outputs independent set expected size proof consider following algorithm algorithm draw random boolean variable true true return largest else return else return dis figure approximation algorithm independent set using approximation algorithm dis works bounded degree graphs new tools connections approximation convenience let fix start analyzing expected running time algorithm per recursive call algorithm clearly uses time remains bound number recursive calls made vertices bound log induction note chosen exp log use inequality base case induction note condition line hold algorithm use recursive calls statement trivial clearly positive inductive step see true exp exp exp exp exp using exp exp exp using exp exp continue analyzing output algorithm clearly returns valid independent set neighbors discarded included line independent set returned line remains show induction base case recursive call made note line indeed obtain maximum degree inductive case let maximum independent set let vertex picked line distinguish two cases based whether inductive step follows induction hypothesis otherwise least false true required first inequality uses induction hypothesis twice invoke lemma using algorithm dis bansal implied following theorem theorem theorem approximation algorithm dis independent set graphs maximum degree running time exp proof theorem may apply lemma virtue theorem exp obtain exp log log expected time algorithm outputs independent set expected size since size output upper bounded obtain independent set size least probability least may boost probability repetitions bansal markov inequality repetitions together run exp time probability theorem statement follows union bound repetitions run claimed running time simultaneously repetition finds independent set size least probability least deterministic algorithm additionally also show deterministic algorithm runs time exp log algorithm utilizes feige algorithm blackbox deferred appendix graph coloring use approximation algorithm independent set subroutine approximation algorithm coloring prove theorem follows proof theorem algorithm combines approximation algorithm section independent set exact algorithm optcol coloring see follows algorithm chr let log log let optcol optimum coloring remaining graph return figure approximation algorithm chromatic number claim chr returns high probability proper coloring using colors prove theorem invoke chr asymptotic running time first note iteration loop log line algorithm decreased multiplicative factor must independent set size least therefore log last iteration thus number iterations must satisfy log log log exp implies consequently number colors used first phase algorithm line line claimed upper bound follows number colors used second phase line clearly upper bounded upper bound running time note line runs time log exp log exp log log new tools connections approximation implementing optcol using time algorithm line also takes log time running time follows vertex cover hypergraph vertex cover section show application sparsification technique vertex cover obtain theorem sparsification step applied explicitly instead utilize sparsification lemma impagliazzo blackbox subsequently solve instance using algorithm halperin sparsification lemma due impagliazzo shows instance vertex cover problem reduced exponential number lemma sparsification lemma algorithm given hypergraph edges size real number produces set systems edges size time every subset vertex cover vertex cover every degree exp next tool approximation algorithm vertex cover problem input graph low degree due halperin theorem polynomial time algorithm vertex cover problem hypergraphs edges size every element degree large enough complete proof theorem applying lemma parameter number instances produced lemma exp exp note graph degree plugging value halperin algorithm gives approximation factor thus gives running time exp translates running time exp pcp parameters approximation hardness approximation connections questions three parameters pcps size freebit gap formally quantify connection define new terms formally illustrating ideas already around literature original formulation set cover problem popular formulation problem equivalent direct transformation bansal define class languages fgpcp stands freebit pcp let positive real functions language fgpcpc constant constants verifier input access proof satisfies properties verifier runs time proof accepts probability proof accepts probability random string verifier accepting configurations parameters log referred gap size freebit pcps respectively convenience call freeness pcp intuitive way view pcp class pcps parameterized gap interesting question pcps hardness approximation literature find smallest functions theorem sat function least linearly growing constant independent set input graph done time unless eth fails think fixed number therefore seen function single variable prove theorem later section corollary assuming sat randomized algorithm sat must case poly log log proof otherwise imply approximation algorithm poly log log poly log log theorem would contradicting existence independent set let phrase known pcps framework fgpcp chan pcps stated sat poly log applying results means one wants keep freebit parameters given chan pcps size must least log another interesting consequence connection vertex cover freebit pcps polynomial time setting theorem vertex cover hard approximate sat poly intended pcps theorem arbitrary small soundness freeness remains corollary implies pcp must size least proof theorem step creating hard csp need following lemma creates hard csp fgpcp csp used later construct hard instance independent set lemma sat randomized reduction sat csp following properties number variables number clauses freeness satisfiable val otherwise val new tools connections approximation proof let number corresponding verifier input create csp follows proof bit variable set variables perform iterations iteration verifier picks random string create predicate xbq proof bits read verifier random string predicate true assignment verifier accepts local assignment first assume satisfiable proof verifier accepts probability let assignment agrees proof satisfies predicate probability therefore expected number satisfied predicates chernoff bound probability satisfies less predicates next assume satisfiable assignment fraction random strings satisfied corresponding proof pick random string probability accepts choices strings expected number satisfied predicates chernoff bound probability satisfies predicates union bound possible proofs length proofs probability step fglss reduction fglss reduction standard reduction csp independent set introduced feige reduction simply lists possible configurations partial assignment clause vertices adding edges conflict two configuration detail predicate partial assignment true vertex pair vertices variable appearing edge lemma fglss reduction algorithm given input csp clauses variables freeness produces graph val val denotes maximum number predicates satisfied assignment combining everything assume sat let constant verifier sat gives gap invoking lemma csp variables clauses moreover freeness gap respectively applying fglss reduction graph assume algorithm gives approximation time notice ngf therefore algorithm distinguishes time contradiction hardness dinur manurangsi raghavendra made conjecture sat admit approximation scheme runs time observe hardness independent set time constant proof uses standard amplification technique deferred appendix research work leaves ample opportunity exciting research obvious open question derandomize branching whether theorem proved without randomized bansal algorithms probabilistic approximation guarantee easily derandomized using random partition vertex set parts splitters seems harder strengthen expected running time bound running time bound improve running times algorithms mentioned introduction use partition argument possibly using randomized branching strategy specifically independent set planar graphs time log independent set time log mentioned introduction result marx still leaves room lower order improvements another open question category fast set goal find independent size example time algorithm known function distinguishes graphs graphs partition argument gives running time strong lower bounds known problem finally big open question area find exclude vertex cover graphs subexponential time fixed constant acknowledgment supported nwo vidi grant erc consolidator grant supported isf grant grant supported european research council erc european union horizon research innovation programme grant agreement swedish research council reg supported nwo veni grant references brenda baker approximation algorithms problems planar graphs acm nikhil bansal anupam gupta guru guruganesh theta function independent sets sparse graphs symposium theory computing stoc pages mihir bellare oded goldreich madhu sudan free bits pcps nonapproximabilitytowards tight results siam andreas thore husfeldt mikko koivisto set partitioning via inclusionexclusion siam bonnet michael lampis vangelis paschos tradeoffs inapproximable problems symposium theoretical aspects computer science stacs pages bonnet vangelis paschos sparsification subexponential approximation acta informatica pages nicolas bourgeois bruno escoffier vangelis paschos approximation max independent set min vertex cover related problems moderately exponential algorithms discrete applied mathematics chris calabro russell impagliazzo ramamohan paturi duality clause width clause density sat conference computational complexity ccc pages parinya chalermsook bundit laekhanukit danupon nanongkai independent set induced matching pricing connections tight subexponential time approximation hardnesses foundations computer science focs pages siu chan approximation resistance subgroups acm new tools connections approximation marek cygan lukasz kowalik marcin pilipczuk mateusz wykurz approximation hard problems corr marek cygan lukasz kowalik mateusz wykurz approximation weighted set cover inf process marek cygan marcin pilipczuk exact approximate bandwidth theor comput artur czumaj andrzej lingas johan nilsson approximation algorithms optimization problems graphs superlogarithmic treewidth inf process irit dinur pcp theorem gap amplification acm irit dinur mildly exponential reduction gap electronic colloquium computational complexity eccc uriel feige approximating maximum clique removing subgraphs siam discrete uriel feige shafi goldwasser shmuel safra mario szegedy interactive proofs hardness approximating cliques acm uriel feige joe kilian zero knowledge chromatic number comput syst eran halperin improved approximation algorithms vertex cover problem graphs hypergraphs siam johan clique hard approximate within annual symposium foundations computer science focs pages russell impagliazzo ramamohan paturi francis zane problems strongly exponential complexity comput syst subhash khot dor minzer muli safra independent sets games grassmann graphs electronic colloquium computational complexity eccc subhash khot ashok kumar ponnuswami better inapproximability results maxclique chromatic number automata languages programming international colloquium icalp pages subhash khot oded regev vertex cover might hard approximate within comput syst subhash khot igor shinkar hardness approximating parameterized clique problem innovations theoretical computer science itcs pages new york usa acm bundit laekhanukit inapproximability combinatorial problems phd thesis mcgill university pasin manurangsi prasad raghavendra birthday repetition theorem complexity approximating dense csps corr marx optimality planar geometric approximation schemes foundations computer science focs pages dana moshkovitz ran raz pcp subconstant error acm jaroslav svatopluk poljak complexity subgraph problem commentationes mathematicae universitatis carolinae ryan williams huacheng personal communication bansal deterministic algorithm independent set section give deterministic algorithm runs time log algorithm simple consequence feige algorithm restate slightly different form logk theorem let graph independence ratio parameter one find independent set size time poly algorithm proceeds follows enumerate sets size independent time log log log otherwise independence ratio least choose log feige algorithm finds independent set size least logk logk log log log log running time log log log redefine log log algorithm algorithm runs time log log log hardness independent set sketch sketch proof given formula perfect completeness soundness first perform standard amplification sparsification get gap parameter number clauses freeness perform fglss reduction get graph therefore time would lead algorithm satisfies fraction clauses formula time words algorithm independent set turned approximation algorithm approximating time
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oct les groupes mathieu aussi sporadiques labib haddad yves sureau guide haute montagne pays des hypergroupes abstract plea reopening building site classification finite simple groups order include finite simple hypergroups hypergroups first introduced marty congress stockholm confused later quite different notion name given inopportunely well aware probably quite mathematicians must already felt uncomfortable presence sporadic simple groups large tableau classification finite simple groups might wrote though reference mention follows try explain step step hypergroup suggest notion simplicity hypergroups simple natural way match notion case groups hoping fruitful examples constructions included introduction qui suit est plaidoyer pour chantier classification des groupes simples finis afin inclure les hypergroupes simples finis notion hypergroupe introduite par marty lors stockholm pas confondre avec une notion plus tardive tout fait laquelle malencontreusement nom suis bien conscient bon nombre ont sentir mal aise sujet que nomme les groupes simples sporadiques dans grand tableau classification certains ont coucher par bien que puisse mentionner aucune dans qui suit commencer par rappeller pas pas est hypergroupe puis comment introduire simple utile qui avec notion classique dans cas des groupes donnant une pour construction hypergroupes donne des conditions pour ils soient simples fournit nombreux exemples pose enfin question peine clos chantier classification des groupes simples finis rouvrir ses portes afin entreprendre classification des hypergroupes simples finis quelques hypergroupe est ensemble muni une binaire associative multivalente dont les celles des groupes plus une binaire multivalente sur ensemble est une application qui chaque couple fait correspondre une partie groupes mathieu sporadiques pour alors par toutes les parties parcourt parcourt maniement des multivalentes tout petit effort attention plus que usage des classiques univalentes demande mais habitude finit par estomper lorsque risque confusion est minime simplement lieu lieu enfin une des parties est singleton lieu plus simplement retrouve classique des binaires univalentes hypergroupe est ensemble non vide muni une binaire multivalente qui les deux suivantes pour tous dans ces deux impliquent que sorte que produit est jamais vide bien entendu tout groupe est hypergroupe dont est univalente tout hypergroupe dont est univalente est groupe puisqu alors pour chaque couple les des solutions exemple soient groupe quelconque ses soit ensemble des classes droite modulo ensemble xhyh est des classes droite xhyh parcourt posant xhyh une binaire multivalente sur ensemble qui fait hypergroupe comme sans cette est univalente seulement est invariant dans cas hypergroupe est groupe quotient classique par lorsque est pas invariant dans obtient alors exemple hypergroupe qui est pas groupe traditionnellement appelle tout hypergroupe isomorphe hypergroupe classes droite forme morphismes morphisme est alors par une application hypergroupe dans hypergroupe qui homomorphisme appelle isomorphisme entre toute application bijective qui est morphisme ainsi que comme dans cas des groupes morphisme bijectif est isomorphisme car dans cas est aussi morphisme effet pour aura autrement dit bien entendu lorsque sont des groupes retrouve ainsi les notions classiques homomorphismes isomorphismes groupes nombres premiers groupes simples comme pour deux entier est premier lorsqu exactement deux diviseurs cela revient dire est seulement multiple deux entiers distincts groupe est simple lorsqu exactement deux invariants cela revient dire seulement deux images homomorphes distinctes isomorphisme afin aux hypergroupes notion faut trouver substitut convenable notion image homomorphe groupe voici que nous proposons reflets dira que hypergroupe est reflet hypergroupe lorsqu existe une application surjective qui les suivante dira alors que application est tout est morphisme effet surjective observera aussi sans une application est isomorphisme seulement est injectif bien entendu est groupe hypergroupe est reflet groupe seulement est une image homomorphe chaque hypergroupe toujours naturellement les deux reflets suivants hypergroupe trivial seul groupes mathieu sporadiques ces deux reflets sont isomorphes seulement est trivial dira hypergroupe est simple lorsqu isomorphisme exactement deux reflets cette permet les arcanes des peut que cette nouvelle notion soit bon choix est clair groupe est simple seulement est classe des hypergroupes simples finis contient ainsi celle des groupes simples finis elle contient prolonge voir effet invariance aura besoin suivante etant groupe ainsi que deux dira que sousgroupe est invariant modulo lorsque kxk hxk kxh pour tout cela notion classique invariant car bien entendu est invariant sens classique seulement est invariant modulo trivial observera passant aux inverses que chacune des deux conditions suivantes est autre kxk hxk pour tout kxk kxh pour tout chacune elles est ainsi condition outre lorsque est invariant modulo puisque pour condition hxk kxk condition peut donc comme ceci hxk pour tout soient groupe deux quelconques ses est invariant modulo est reflet tout reflet est isomorphe ces est invariant modulo afin pas couper fil nous renvoyons appendice tout reflet est donc est simple seulement est distinct que les seuls invariants modulo sont particulier est simple lors que est maximal dans ainsi lorsque est maximal groupe est pas invariant dans hypergroupe est simple est pas groupe existe donc des hypergroupes simples finis qui sont pas des groupes classes gauche semblablement hypergroupe des classes gauche modulo les hypergroupes isomorphes ces hypergroupes classes gauche les que les par exemple est groupe seulement est invariant dans est ainsi les deux groupes bien entendu deux hypergroupe est reflet est invariant modulo ses transposent particulier est simple seulement est hypergroupe soit hypergroupe avec son comme dans cas des univalentes muni cette nouvelle est nouveau hypergroupe que par que appelle hypergoupe que dire hypergroupe hypergroupe groupes mathieu sporadiques soient groupe ses les deux hypergroupes sont isomorphes autre pour que les deux hypergroupes soient isomorphes faut suffit que soit invariant dans autrement dit lorsque est pas invariant les deux hypergroupes sont pas isomorphes lorsque sousgroupe est invariant sont seul groupe quotient plus application qui chaque classe droite fait correspondre classe gauche est isomorphisme hypergroupe sur hypergroupe comme une simple montre est invariant alors sont tous deux groupe quotient classique par soit isomorphisme entre ces deux hypergroupes dans chacun eux est seule classe qui donc plus quelque soit dans hypergroupe pour chaque existe tel que aura alors ainsi dans produit est singleton cela revient dire que hyh sorte que pour tout donc est invariant ils vont par paires soit groupe hypergroupe est simple sait que hypergroupe est simple lorsque plus sont pas des groupes sait ils sont pas isomorphes ainsi comme les racines imaginaires ces hyergroupes simples vont par paires particulier prenant tous les couples est sousgroupe maximal non invariant indice fini dans groupe obtient une famille paires hypergroupes simples finis deux deux non isomorphes cette famille prolonge ainsi famille des groupes cycliques simples finis les plus simples des exemples prend pour groupe toutes les permutations ensemble non vide fini infini ayant pour cardinal distingue point particulier prend des permutations qui laissent point fixe est maximal pour est pas invariant sorte que est alors simple qui est pas groupe dont cardinal est celui par classe que table multiplication alors ainsi pour tout pour observera encore ceci pour tout ainsi contrairement cas des groupes ordre chacun ces hypergroupes est toujours trois avec son cardinal que pour particulier prenant pour successivement chacun des entiers obtient une suite simples finis ayant qui sont pas des groupes pas plus construction des hypergroupes classes droite gauche forme peut grandement une large que voici donne ensemble une partie ainsi une binaire univalente partiellement pour les seuls couples que appellera les couples composables plus souvent simplement lieu dira que est une trame groupes mathieu sporadiques soit une relation sur ensemble pour chaque soit classe modulo alors une multivalente naturelle sur ensemble des classes suivante dira que est une structure quotient lorsque est hypergroupe dira que relation est peut donner facilement des conditions suffisantes pour que relation soit sous forme ensemble premier ordre est pas besoin rentrer dans ici voir pour cela appendice pour moment suffit savoir que ces conditions explicites existent une heureuse circonstance bien entendu chaque hypergroupe classes forme une naturelle dont trame est est groupe tous les couples sont composables celle groupe relation est par xry respectivement xry plus donc chaque hypergroupe isomorphe quelconque ces hypergroupes classes une cette est cependant pas due une quelconque ces hypergroupes elle est lot tous les hypergroupes quels ils soient voici comment tout hypergroupe une effet soit hypergroupe quelconque chaque triplet pour lequel associe couple composable pose cela une est ensemble tous les couples composables obtient ainsi trame prend pour relation sur dont les classes sont les parties sans que est hypergroupe isomorphe hypergroupe cqfd une nouvelle famille exemples voici une illustration qui fournit des hypergroupes qui sont pas des prend ensemble une famille ensembles non vides deux deux disjoints suppose que distingue point particulier introduit ensemble toutes les applications injectives condition suivante existe moins une permutation telle que ait pour tout dira que couple deux est composable lorsque prend pour application sur partie cette application voit sans appartient bien obtient ainsi une trame application dans est surjective une relation sur dont les classes sont les parties application permet identifier ensemble des classes modulo ensemble multivalente correspondant est par facilement table multiplication sous forme suivante quels que soient pour chaque indice pour tout pour tout pour tout indice dans cas particulier exemple simple plus haut groupes mathieu sporadiques plus toutes les parties ont cardinal est encore effet voit facilement introduisant une nouvelle trame une trame fait que les parties sont chaque prolonge qui est une application bijective autrement dit une permutation ensemble prolongement est cependant pas unique ensemble toutes ces permutations est groupe par restriction par des permutations qui laissent fixe point sans grand que structure est isomorphe structure isomorphe cqfd trouvera davantage sujet ces structures dans appendice pour moment notera cependant encore ceci pour obtient une structure qui est hypergroupe mais pas voir appendice scholie explorer les liens qui peuvent exister entre deux ayant une trame donne deux relations sur ensemble respectivement par classe modulo classe modulo ainsi deux structure quotient structure quotient respectivement qui sont munies des multivalentes correspondantes bien entendu lorsque dans cas est une application surjective plus suffit reporter aux pour constater que aura alors toujours inclusion pour avoir faut suffit que ait pour examiner plus attentivement cette situation par lui donner nom commode convenons dire que est invariante modulo lorsque que condition est satisfaite pour tous dans supposons donc que soit invariante modulo trois importantes pour tous dans aura est sera autrement dit est hypergroupe sera reflet car est alors particulier est groupe sera une image homomorphe fait aussi par nous faut qui donc par implications successives tel que groupes mathieu sporadiques tel que donc pour utilise condition sous forme suivante tel que donc cela point est hypergroupe fait que est surjective effet multivalente est alors associative car celle est elle comme pour cela point dont point aura sans doute similitude avec faite appendice pour cas particulier des vient suivante qui les reflets des soit une hypergroupe quelconque tout reflet une est une invariante modulo soient projection sur ensemble quotient sur par application autrement dit xsy hypergroupe est ainsi hypergroupe par adopte les notations posant montrer que condition donne ainsi puisque est existe tel que cela veut dire existe tel que par suite corollaire soit une hypergroupe non trivial cet hypergroupe est simple seulement les deux seules invariantes modulo sont totale une remarque insistons tout hypergroupe une aide une trame convenable plus haut fait encore plus que voici soit ensemble muni une multivalente quelconque autrement dit certains des produits peuvent vides nul besoin associative reproductive construction plus haut fournit une trame une relation sur telles que soit une autrement dit est isomorphe structure quotient peut concevoir cette comme multivalente sur une univalente sur mais partiellement que gagne passant multivalent univalent perd tout petit peu passant une partout une uniquement pour certains couples mal est pas grand car cela notion une univalente pour laquelle certains produits lieu des singletons seraient simplement vides groupes mathieu sporadiques dans chaque est pour ainsi dire une partie peut aussi concevoir cette comme structure par structure peu comme pour les topologiques cette faire est loin rare elle est par nombreux exemples uniquement deux groupe lie connexe par groupe lie simplement connexe ensemble analytique droite par projection plan les chemins qui distinguent pas bien dans groupe voient bien plus clairement une fois dans complexe souslin qui ensemble analytique est aide plan comme une nappe que sur une corde linge bien entendu vient donner pour les hypergoupes applique toutes les structures multivalentes univalentes applique particulier aux groupes groupe est simple seulement une ses condition suivante les deux seules invariantes modulo sont totale plus est ainsi alors toutes les groupe cette condition question pose alors plus simple groupe est simple directement tenant plus bien simplifier preuve parfois prenant hauteur groupe aide une trame convenable constructions utumi donne hypergroupe dont par signe elle est pas ncessairement commutative donne une partition ensemble pour chaque par partie partition laquelle appartient suppose que ayant les suivantes pour tout notera que alors sur une nouvelle multivalente par cette est toujours reproductive puisque elle est associative seulement pour tous cela voit simplement sans grand dira alors que cet hypergroupe est construit partir hypergroupe partition exemple utumi voici premier exemple cette construction par utumi prend pour groupe cyclique ordre huit partition des trois classes les conditions pour construction satisfaite est hypergroupe utumi que pas bien suivante pour chaque collection est une partition les cardinaux sont tous pour eaton nom cogroupes droite cette classe hypergroupes hypergroupe utumi fait partie aussi lui nom cogroupe utumi pour plus reportera ici nouveau que voici cogroupe utumi est hypergroupe simple effet soit pour chaque partie posons particulier pour chaque les traduisent ici suivante groupes mathieu sporadiques que tire particulier deux cas seulement peuvent premier cas alors pour chaque est isomorphisme second cas contient alors contient aussi produit donc produit ainsi que toutes les sommes une simple montre que donc quel que soit voit que hypergoupe est donc trivial singleton cqfd plus brodant sur canevas obtient donc ceci hypergroupe construit partir groupe une partition est simple que pour chacune des parties singleton une moins des sommes est appendice lorsque traite hypergroupes adopte souvent une convention tacite chaque fois que produit est singleton pourvu que risque confusion soit minime reflets lem soient deux groupe suppose que est invariant modulo est alors reflet pour posons par est associant chaque classe classe obtient donc une application surjective canonique que est sorte que xhyh xkyk ces deux ensembles sont effet soit xkyk alors xhyk puisque kyk hyk autrement dit intersection xhy est pas vide soit cette intersection alors xhyh ainsi sorte que xkyk xkyh xhyk ces quatre ensembles sont application est donc bien deux remarques utiles pour lem suivant aura besoin des deux que voici tout effet soit existe alors deux tels que donc autrement dit conclut remarquant que cqfd groupes mathieu sporadiques donne des hypergoupes ainsi que des applications leur sont des morphismes est clair que est morphisme sont des morphismes pourvu que soit surjective application est aussi morphisme effet pour dans existe dans tels que ainsi donc autrement dit est bien morphisme cqfd lem soient groupe quelconque ses sousgroupes reflet existe alors sousgroupe invariant modulo tel que soit isomorphe soit par goupe puis pour chaque posons soient par commence par montrer que est puis montre est invariant modulo enfin que est isomorphe hypergroupe pour sorte que donc commence par utiliser fait que est morphisme pour xhyh donc sorte que pour aussi tire sorte que pour qui est singleton donc particulier pour vient donc plus alors donc sorte que cela prouve bien que est par successives autrement dit utilise les aux xhuh ainsi que existe ainsi une application qui chacune des classes associe est une application bijective hxk kxk donc pour tout hxk kxk cela prouver que est invariant modulo soit application canonique qui fait correspondre classe classe est autrement dit est particulier est morphisme surjectif morphisme donc est morphisme puisque est bijectif voir est donc isomorphisme comme conditions soient une trame une relation sur pour chaque par classe modulo sorte que toutes les relations suivantes sont synonymes xry yrx afin simplifier les pour conviendra ceci lorsque suppose implicitement que couple est composable autrement dit que voici deux relatifs structure quotient ils sont tous deux premier ordre groupes mathieu sporadiques voit sans que condition est suffisante pour assurer condition est suffisante pour assurer autement dit pour que soit faut suffit elle satisfasse les deux sur nouvelle famille exemples donne ensemble une famille ensembles non vides deux deux disjonts suppose que distingue point particulier sur cet ensemble une multivalente dont voici table quels que soient pour chaque indice pour tout pour tout pour tout indice est clair que structure ainsi sur uniquement isomorphisme famille des cardinaux finis infinis pour dira elle est type cela quelque peu les structures type correspondant cas introduite dans article lorsque pour tout indice pour indice structure est hypergroupe qui est pas effet pour voir que structure est hypergroupe suffit assurer que les conditions sont satisfaites pour condition est sans quant condition reprend trame qui sert construire structure soient alors dans ainsi que dans tels que voit simplement que sorte que soit ainsi est que montre existe alors point tel que qui application est point point est des points prend sinon construit posant avec soumis aux trois conditions suivantes cela est possible car chacune des parties famille moins trois points enfin ces conditions impliquent que bien pourrait aide table multiplication avec peu patience que est associative reproductive enfin pour les deux parties sont pas alors elles seraient dans cqfd peut ainsi ranger les structures dans quatre classes distinctes classe pour tout indice cela comprend bien entendu tous les cas toutes ces structures sont des dhypergroupes comme dit plus haut classe pour tout indice pour indice ces structures sont toutes des hypergroupes qui sont pas des hypergroupes classes classe pour indice des produits est vide particulier groupes mathieu sporadiques classe tous les autres cas ces structures sont pas des hypergroupes car est alors comme peut que les trois cas suivants pour indice pour indice pour indice donc car donc car donc car pour petite histoire les versions texte remontent dernier souvent partie groupe travail des nombres blaise pascal sur invitation lundi juin sous titre encore possible avancer emile mathieu avec suivant promenade travers champs suivie bref plaidoyer faveur liste des guise conclusion provisoire sait faire agir les hypergroupes sur des ensembles comme fait pour les groupes plus sait les faire agir sur ensemble des racines est encore une autre histoire peut lors poser question suivante raisonnable non existence hypergroupes mathieu simples peine clos chantier classification des groupes simples finis rouvrir ses portes afin entreprendre classification des hypergroupes simples finis bibliographie eaton theory cogroups duke math haddad labib sureau yves les cogroupes les algebra haddad labib sureau yves les cogroupes construction utumi pacific krasner sur ramification des corps nongaloisiens nombres acad belgique sci coll pages marty sur une notion groupe scand math stockholm marty sur les groupes hypergroupes une fraction rationnelle ann sci norm ser iii utumi yuzo hypergroups group right cosets osaka math rue charonne paris france address
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optimal task scheduling mobile edge computing systems wireless virtual reality xiao zhiyong kuikui yaping hongming medianet innovation center shanghai jiao tong university shanghai china corporation china email zhiyongchen kuikuili yapingsun aug cooperative edge computing mec expected effective solution deliver virtual reality videos wireless networks contrast previous mec framework reduces consumption mobile device increasing consumption develop mec framework reduce consumption increasing consumption exploiting caching resources mobile device paper specifically according task modularization mec server deliver components stored device device uses received components corresponding cached components construct task resulting low consumption high delay mec server also compute task reduce delay however consumes communicationresource due delivery entire task therefore propose task scheduling strategy decide computation model mec server operates order minimize consumption delay constraint finally discuss tradeoffs communications computing caching proposed system ntroduction virtual reality wireless networks gaining unprecedented attention due ability bringing immersive experience users application leading fact wired cables current wireless systems lte cope latency throughput requirements wireless application video address problem effective solution mobile edge computing mec enables cloud computing capabilities edge wireless network base stations deploying computation resource network edge mec performs computation tasks closer device improving quality computation experience mec user offload computation tasks device mec server due limited computation capability device attracted significant attention recently dynamic computation scheduling algorithm based lyapunov theory proposed minimize execution fig task modularized recovered user delay task failure similarly proposed optimization offloading framework minimize execution latency user energy consumption order minimize mobile energy consumption constraint execution delay optimized offloading policy based channel gains consumption local computing energy general mec exploits communication resource reduce requirement user computation resource therefore quite suitable application low bandwidth consumption online picture processing video data rate online videos already many times greater high definition even though networks maybe able satisfy requirement meanwhile latency system online videos also use mec solution improve network responsiveness reduce latency however try reduce computation resource increasing communications resource overhead contrast try reduce consumption communications resource taking advantage computation caching resources mobile devices term solution mec paper paper present mec framework develop optimal task scheduling policy minimize average transmission data per task system average transmission data per task adopted consumption communications resource metric major contributions summarized follows propose mec denoted assume fig proposed mec system caching enabled mobile device work reduce consumption communications resource caching resource increasing consumption computing resource mobile device address issue consider video task modularized today media transport mmt standard mpeg shown fig besides contents composing whole task popularity result one intended task mec server delivers corresponding components stored device device uses received components corresponding cached components construct task exploiting local computation resource develop optimal task scheduling policy minimize average transmission data per task course mec server also select mec computation model mec server combines corresponding con tents target task deliver task device mec computation mode reliable way latency due fast computation capability reduce mec server model delivers data per task user therefore formulate transmission data consumption minimization problem delay constraint propose task scheduling strategy based lyapunov theory discuss tradeoffs communications computing caching present joint allocate communications computing caching resources proposed mec system achieve target latency ystem odel shown consider mec system mobile device access mec server obtain task mec server abundance computing caching resource mobile device limited computing ability cache capacity task model consider task consists number contents mmt assets contents composing task come set possible contents denoted note one content may used task popularity distribution contents contents equal size mec server contents cache capacity mobile device store contents adopt popular caching strategy stored content set denoted system time slot length let task scheduled time slot consists contents denote hkt content index vector task hkt indicates content fhkt thus size let gtn denote whether requested cached mobile device given hkt otherwise computation model task either executed mec server mobile device let denote required cpu cycles computing one bit cpu frequency mec server mobile device respectively wireless transmission throughput mec computation mode mobile device task request mec server computes contents task deliver task size transmission data dct size computation data dcc therefore dct time slots required complete task similar use model mec state means mec server idle indicates task processing mec server time slots required complete computation local computation mode model mec server delivers contents stored mobile device mobile device computes received contents cached contents task mec server transmit stored contents gtn task size transmission data dlt size computation data also dlc result dlt time slots required complete similarly let denote mobile device state system allocates task mobile device time slot updates task queueing model task arrival process modeled bernoulli process probability task arrivals task first enters task queue infinite capacity number task waiting queue queue state updates according following equation denotes whether task arriving time slot thus denotes task scheduling decision time slot notice two tasks scheduled time slot first task scheduled second task first task scheduled computation mobile device mec server second task scheduled operate local computation mode mec computation mode cpu occupied first task yielding otherwise first task local computation mode mec computation mode could scheduled second task result five possible states task scheduling decision time slot iii task cheduling trategy roblem ormulation mec server reliable way reduce computation latency due consumes communication resource due hence mec server needs make task scheduling decision time slot minimize average transmission data per task average delay constraint task scheduling strategy mec server mobile device idle task scheduled mec computation mode local computation mode queue state denotes task queue empty task scheduled time slot indicates infinite tasks task queue yielding unstable system according describe system state case mobile device mec server idle system process two tasks two tasks least one task processed mobile device mec server task remains wait task queue precessed mec server mobile device task scheduling policy expressed follow one task task processed mobile device mec server remains wait task queue thus case case mec server idle mobile device busy system process one task mec server task scheduling policy case case mobile device operates idle mode mec server occupied one task scheduled mobile device task scheduling policy expressed following case mec server mobile device busy task task queue task scheduled time slot expressed nli nci denote task respectively nli means mobile device occupied task time slot busy follow nli time slots similarly nci problem formulation length task queue infinite total number task close therefore average transmission data per task expressed nxx dlt uic dct lim dlt dct denote dlt dct task respectively system model know task requires transmission time waiting time processing time computation processing time mec server mobile device dominant influence execution delay based little law execution delay including waiting time processing time proportional average queue length task buffer execution delay written lim denote thus consumption minimization problem formulated nxx dlt uic dct min lim lim indicates delay constraint ensure task requires completed finite delay unfortunately stochastic optimization problem system state changes offloading decision made impossible solved convex optimization methods ptimal task cheduling lgorithm based lyapunov heory problem transform simplify consider lyapunov optimization theory first define lyapunov function consider initial state queue unstable volatile thus expectation system stable therefore lyapunov drift function given see maintain stability queue minimize time slot therefore expectation would tend infinite result following lemma lemma let define scheduling rate order ensure cmax state also approach lower length however minimization cause minimization thus define lyapunov function use cmax proof please refer appendix according lyapunov theory make task scheduling decision minimize queue control parameter denotes system sensitive communication cost system sensitive delay increase lyapunov becomes sensitive communication cost notice optimal task scheduling decision minimizing right side also minimize queue length stability constraint therefore solve time slot min algorithm optimal task scheduling algorithm based lyapunov theory obtain queue state mobile device state mec server state beginning time slot find system case discussed section iii obtain system case determine solving min set update according respectively time slot obtain based possible choices solve time slot enumeration method thus propose optimal task scheduling algorithm based lyapunov theory shown algorithm lemma equivalent control parameter sufficiently large solution close let average transmission data obtained solving optimal value respectively proof please refer appendix tradeoffs communications computing caching subsection reveal tradeoff average transmission data per task computing ability mobile device tradeoff average transmission data per task cache capacity also discussed proof due page limitations skip proof dlt cache capacity proof due page limitations skip proof proposition see decreases increase caching capacity mobile device know decreases decrease dlt based proposition hence increase decrease dopt similar proposition large approaches decrease increase umerical esults section evaluate performance proposed optimal scheduling policy simulations number contents time slot length content bits average arrival rate mobile device ghz cpu frequency cache capacity transmission throughput bps mec server ghz cpu frequency unless otherwise specified distributed uniformly assume content popularity distribution identical among elements task follows zipf distribution parameter set simulations consider mec computation policy baselines executes tasks mec server shows average transmission data per task achieved proposed optimal scheduling policy decreases cache capacity means average consumption traded cache capacity keep queue length stable verifies tradeoff presented proposition moreover average transmission data per task mbits proposition define dct mec computation policy optimal schedling optimal schedling optimal schedling fig consumption cache capacity according lemma large approaches based dlt large observe decreases increase average transmission data per task mbits proposition let dct dlt denote average transmission data mec computation model local computation model respectively let dlc denote average computation data local computation model thus inf easible use denote exception time slots required complete task mec computation model local computation model respectively mec computation policy optimal scheduling optimal scheduling optimal dopt cpu frequency mhz fig consumption computing ability mobile device scheduling policy always outperforms mec computation policy even contents cached mobile device optimal scheduling always executes part tasks local computation policy redundant transmission contents needed tasks avoided however mec computation policy delivers contents requested task presents average transmission data per task versus mobile device computing ability increase mobile device computing ability decrease average consumption reason increase decreases computing delay local execution tasks executed local computation policy large system sensitive consumption tasks scheduled mobile device dalg close dopt sufficiently large verify lemma sufficiently large local computation mode becomes optimal schedule strategy tradeoff communications computing caching fig average latency proposed system different resources allocation bps bps one see communication throughput decreases increasing proof lemma assume feasible exists least one satisfying constraints satisfies following condition transmission throughput mbps positive value according little theorem average arriving rate larger average service rate queue length tends infinite increase time slot therefore solved proposed algorithm following condition satisfied cpu frequency mhz caching capacity fig incorporation communications computing caching achieve target delay set puting capability caching capacity increases task scheduled mobile device yielding lower communication cost similar therefore system needs smaller observe computing ability impact consumption caching capacity cache capacity impact performance local computation mode hand system small computing ability caching capability large transmission throughput required result reconfirm importance simultaneous exploitation wireless networks onclusion paper investigated communicationconstrained mobile edge computing systems wireless virtual reality transmission data consumption minimization problem execution delay constraints formulated proposed task scheduling strategy based lapunov theory tradeoffs communications computing caching proposed system also dicussed finally simulation results shown proposed scheduling strategy achieve significant reduction average transmission data consumption ppendix proof lemma according first substituting rewritten task arriving rate independent finally according lemma positive value substituting obtain dalg taking iterated expectation using telescoping sums get dalg divide let eferences bastug bennis medard debbah toward interconnected virtual reality opportunities challenges enablers ieee communications magazine vol liu chen qian three primary colors mobile systems ieee communications magazine vol september wang zhang zhang wang yang wang survey mobile edge networks convergence computing caching communications ieee access vol zhang huang yang sun zhang content classification scheme vod service ieee international symposium broadband multimedia systems broadcasting june huang chen ceylan jin videos single ieee virtual reality march mao zhang chen letaief dynamic computation offloading computing energy harvesting devices ieee journal selected areas communications vol dinh tang quek offloading mobile edge computing task allocation computational frequency scaling ieee transactions communications vol huang chae kim resource allocation computation offloading ieee transactions wireless communications vol march liu mao zhang letaief computation task scheduling computing systems ieee isit july jian qinghai kyungsup ramesh tradeoff wireless powered communication networks ieee transactions vehicular technology july ross introduce probability models academic press
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quantum algorithms graph bipartiteness connectivity agnis oct university latvia riga latvia abstract span program model computation used design quantum algorithms boolean function exists span program leads quantum algorithm optimal quantum query complexity general finding span programs easy task work given query access adjacency matrix simple graph vertices provide two new quantum algorithms algorithm testing graph bipartite uses quantum queries algorithm testing graph connected uses quantum queries introduction concept span program model computation new introduced karchmer wigderson many applications classical complexity theory span programs used evaluate decision problems reichardt spalek introduced new complexity measure span programs witness size reichardt showed later strong connection quantum query complexity quantum algorithm evaluating span programs two complexity measures essentially equivalent difficulty come span program good witness size complexity authors dealt span programs evaluating boolean functions bits posed open question exist interesting quantum algorithms based directly asymptotically large span program belovs used span programs construct learning graphs also used span program approach matrix rank problem ambainis came simple yet powerful span program graph collision problem paper extend family algorithms based span programs present two new quantum algorithms algorithm graph bipartiteness problem algorithm graph connectivity problem algorithms quantum query sense work supported erc advanced grant methods quantum computing optimal witness sizes match quantum query complexity lower bounds problems thus demonstrate span programs useful also problems asymptotically large input possibly algorithms could building blocks bigger span programs future graph connectivity problem studied ready exists quantum query algorithm requires qubits quantum memory advantage algorithm uses log qubits quantum memory span program uses vector space dimensions similarly graph bipartiteness problem solved search method uses qubits quantum memory approach span program requires log qubits quantum memory preliminaries paper present algorithms work simple graphs given adjacency model given graph vertices input size algorithm assume input variable corresponds value entry row column adjacency matrix span programs definition span program tuple hilbert space called target vector finite set vectors denote span program said compute function domain span basically definition says input variable two sets vectors span program authors define vectors advance say vectors set available vectors set available vector included sets say free vector always available function returns iff target vector expressed linear combination available vectors otherwise returns definition positive witness vector positive witness size max min witness negative witness vector negative witness size max min witness witness size program wsize witness size function denoted wsize minimum witness size span program computes theorem wsize coincide constant factor exists constant depend wsize wsize span program testing graph bipartiteness bipartite graph graph whose vertices divided two disjoint sets edge connects vertices set undirected graph bipartite iff odd cycles algorithm exists span program graph vertices detects graph bipartite wsize proof make span program detects graph odd cycle let number vertices given graph span program follows span program testing graph bipartiteness dimensional vector space basis vectors target vector every make available free vector every every edge input make available vectors states span program mostly form vertex index represents vertex started search odd length cycle represents parity current path length first subindex state also considered subspace index subspace span vectors corresponding edges form consisting sum two states belong subspace span program target vector expressed linear combination available vectors least one vectors form expressed without loss generality odd length cycle target vector expressed taking vectors corresponding edges cycle alternatingly plus minus sign therefore span program always return given graph bipartite side odd length cycle none vectors form expressed using available vectors cancel state using vector corresponding edge vertex adjacent vector contains state move state state possibly coefficient parity bit flipped similarly cancel state using vector corresponding edge going vertex stop process need reach state done odd cycle path must closed parity bit restricts odd length odd cycle span program always return conclude indeed computes expected function remains calculate witness size case odd cycle need calculate positive witness size odd cycle length target vector expressed way positive witness consists entries therefore cycle also cycle therefore target vector also expressed way follows vertices cycle target vector therefore expressed atleast different ways combine ways taken coefficient get positive witness size estimate negative witness size must find negative witness derive defining acts basis vectors definition every must therefore lets pick way repeat following steps changes happen every available vector defined define every available vector defined define yet defined define given vector span program value total number vectors exceed therefore negative witness size combining positive negative witness sizes obtain upper bound witness size also corresponds quantum query complexity wsize span program testing graph connectivity graph said connected every pair vertices graph connected undirected graph one vertex connected vertices transitivity graph connected algorithm exists span program graph vertices detects graph connected wsize proof let number vertices given graph span program follows span program testing graph connectivity dimensional vector space basis vectors target vector every make available free vector every every edge input make available vector vertices reachable vertex index given graph connected use belov span program connectivity subroutine subroutine checks given graph path vertex vertex span program target vector edge input use vector span program using subroutine times check vertices connected vertex index create separate subspace span subroutine call avoid interference common technique compose span programs span program returns vertices connected otherwise returns case given graph connected need calculate positive witness size subroutine shortest path length vertex vertex larger threfore vector set requires vectors express subroutines positive witness size estimate negative witness size must find negative witness derive defining acts basis vectors definition need talk negative witness vertex connected vertex index vertex belongs different connected component vertex index lets name connected component let count vertices connected component lets pick way vertex set vertex set must therefore set repeat following step changes happen every available vector defined define yet defined basis vectors set negative witness choice overall negative witness size get increased vectors correspond nonexistent graph edges connects connected components graph border edges edge border edge border edge correspond vectors vectors vector increases negative witness size value border edges therefore overall negative witness size combining positive negative witness sizes obtain upper bound witness size also corresponds quantum query complexity wsize acknowledgements grateful andris ambainis suggestion solve graph problems span programs many useful comments development paper references ambainis balodis iraids ozols smotrovs parameterized quantum query complexity graph collision corr http belovs quantum algorithm rank problem corr http belovs quantum algorithm kdistinctness ieee annual symposium foundations computer science focs ieee http belovs span programs functions proceedings symposium theory computing acm http belovs reichardt span programs quantum algorithms stconnectivity claw detection epstein ferragina eds algorithms esa lecture notes computer science vol springer berlin heidelberg http berzina dubrovsky freivalds lace scegulnaja quantum query complexity graph problems van emde boas eds sofsem theory practice computer science lecture notes computer science vol springer berlin heidelberg http heiligman mhalla quantum query complexity graph problems sannella eds automata languages programming lecture notes computer science vol springer berlin heidelberg http furrow panoply quantum algorithms quantum info comput sep http karchmer wigderson span programs proceedings eighth annual structure complexity theory conference ieee http reichardt span programs quantum query complexity general adversary bound nearly tight every boolean function annual ieee symposium foundations computer science focs ieee http reichardt reflections quantum query algorithms proceedings annual symposium discrete algorithms siam http reichardt spalek quantum algorithm evaluating formulas proceedings fortieth annual acm symposium theory computing stoc acm new york usa http
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decentralized hypothesis testing energy harvesting wireless sensor networks jun alla tarighati james gross senior member ieee joakim senior member ieee consider problem decentralized hypothesis testing network energy harvesting sensors sensors make noisy observations phenomenon send quantized information phenomenon towards fusion center fusion center makes decision present hypothesis using aggregate received data time interval explicitly consider scenario messages sent parallel access channels towards fusion center avoid limited lifetime issues assume sensor capable harvesting energy needs communication environment sensor energy buffer battery save harvested energy use time intervals key contribution formulate problem decentralized detection sensor network energy harvesting devices analysis based model battery propose sensor decision design method considering long term energy management sensors show performance system changes different battery capacities numerically show findings used design sensor networks energy harvesting sensors ntroduction distributed detection problem formulations traditionally addressed detection sensor networks considering network performance measures like error probability receiver operating characteristic setups spatially separated sensors make observations phenomenon send summary observations towards fusion center channels sensor viewed quantizer quantizes observation according network arrangement sends output either another sensor many applications sensors send outputs multiple access channel parallel access channels commonly known parallel topology survey early works decentralized hypothesis testing wireless sensor networks found large number sensors small batteries limited often used wireless sensor networks major limitation sensors finite lifetime words sensors work long battery last implies also network limited lifetime many solutions increase lifetime sensor nodes proposed see references therein methods aim find energy usage strategy maximize lifetime network lifetime remains bounded finite alternative way dealing problem use energy harvesting devices sensor nodes energy harvesting device capable acquiring energy nature sources energy harvesting technologies provide promising future wireless sensor networks self sustainability effectively perpetual network lifetime limited sensor battery lifetime acquiring energy environment makes possible deploy wireless sensor networks situations impossible using conventional sensors poses new challenges related management harvested energy new challenges due fact amount energy available sensor random since source energy might available times may want use sensor nodes address problem detection networks sensors arranged parallel sensor energy harvesting device time sensors send message towards state current hypothesis makes decision hypothesis time sensors communicate using energy asymmetric keying ook low communication rate scheme distributed detection applications positive message sent cost one unit energy negative message conveyed cost energy previously shown ook energy efficient modulation scheme rayleigh fading transmission though primarily concerned explicitly modeling fading channels sensors herein assume sensor equipped internal battery allowed use long term energy conversion policy assume observations sensors conditioned true hypothesis independent goal design sensors transmission rules way error probability time instance minimized end use bhattacharyya distance conditional distributions input frequently used past performance measure design distributed detection systems using bhattacharyya distance essentially reduces joint design decision rules sensors design decision rules single sensors may risk capturing aspects joint design decision rules peripheral sensors however due analytical tractability follows choice design rule bhattacharyya distance frequently considered performance metric literature novelty work formulation fig binary asymmetric channels sensors fusion center harvesting sensor formulate problem section iii study performance energy harvesting sensor different battery capacities section illustrate results design sensor networks presenting numerical simulations finally section conclude paper fig decentralized hypothesis testing scheme parallel network centralized detection problem system costs coupled random behavior energy available sensors concretely find depletion probability sensor batteries evaluate performance network different battery capacities buffer sizes show problem formulation changes compared unconstrained case consider energy features problem designing sensors network distributed inference using energy harvesting agents gained lot interest last years see references therein context distributed estimation nayyar studied structure optimal communication scheduling zhao huang nourian studied optimal power allocation schemes studied performance energy harvesting cooperative network fading channels hong considered case peripheral node sends version observation towards required energy transmission available sensor otherwise sensor remains silent also conveys information regarding magnitude observed signal hua liu considered adaptive quantization distributed estimation using approach energy harvesting relay systems also considered nodes relays harvest energy need transmit amplified received messages towards receiver medepally mehta considered relay selection scheme multiple active relays available one selected transmit context distributed detection however finding optimal decision rules remote energy harvesting sensors largely open studied structure decision rules agents infinite battery size communication channels paper shall extend results general case erroneous communication channels outline paper follows section describe structure parallel network energy reliminaries section first present system model define energy harvesting sensor formulate problem designing energy harvesting sensors network limitations proposed system model influence obtained results discussed concluding remarks section system model consider decentralized hypothesis testing problem sensors arranged parallel according fig time interval defined sensor makes observation phenomenon sends message towards note assumed element abstract space could single scalar measurement vector measurements made time interval sensor consider case different observations sensor conditioned hypothesis independent identically distributed random variable corresponding observations sensor time interval phenomenon modeled random variable drawn binary set apriori probabilities respectively gives rise conditionally independent observations conditional distribution fxn sensors paper assume present hypothesis changes time fashion fixed time interval also assume sensors allowed use long term energy usage policy managed together internal battery words consider long term behavior sensor battery state operates steady state communication channels sensors links sensors communication sensor nodes sensors communicate using energy asymmetric keying positive message labeled conveyed transmission message negative message labeled conveyed model channel sensor general binary discrete memoryless channel called binary asymmetric channel bac error occurs probability error occurs probability shown fig received message corresponding sensor output channel binary symmetric channel bsc channel reduce important stress although consider channel unreliable also consider fixed subject alteration sensors optimization sensor decision rules encompass design air transmission protocol bac also relevant model case channel fading channel receiver uses energy detector detect input otherwise energy received resulting sensor positive threshold ook transmission fading channel energy threshold detector may reasonable assume reflecting belief deep fades causing received energy meet specified threshold likely noise induced fluctuations push decision metric threshold boundary overall problem considered structurally similar classical binary hypothesis testing problem one bit channels words sensor given realization computes message using decision function sends message towards based aggregate received bac channel outputs makes decision corresponding random present hypothesis time interval variable using decision function note directly access sensor outputs access corresponding bac channel outputs overall objective paper design decision functions sensors way performance measure optimized problem considered extensively classical distributed detection literature energy always available sensors send messages channels channels however problem designing sensors decision functions sensor energy harvesting device largely open shall herein consider problem designing decision functions sensors sensor energy harvesting device follows first define performance metric section define energy harvesting device let random variable corresponding output messages sensor due independence observations conditional pmf associated message vector obtained according fxn set observations satisfy let btot total bhattacharyya distance time instance given sensor decision functions btot log random variable corresponding bac output bayesian error probability minimized applies maximum map rule corresponding error probability upper bounded total according context distributed hypothesis testing acknowledged bayesian error probability criteria cases lead tractable design procedure decision functions even small sized networks led authors consider bhattacharyya distance performance measure network addition observations sensors conditioned true hypothesis independent total bhattacharyya distance simplifies greatly decouples btot log pyn pyn words time instance total bhattacharyya distance parallel network summation bhattacharyya distances delivered different sensors therefore network maximizes bhattacharyya distance network individually optimized sensors suffices maximize received sensor separately focus single energy harvesting sensor drop subscript denote time throughout paper interchangeably use terms bhattacharyya distance energy harvesting sensors consider energy harvesting sensor fig let assume time interval energy arrives given fig model energy harvesting node left example time interval right stochastically node stationary ergodic random process see let battery state let corresponding random process general correlated random process time even energy harvesting process note actions sensor affect future battery state sensor knows battery state assume battery size capacity amount available energy battery transmission time min amount energy used send message time corresponding random variable assume arrival energy time interval used time interval see fig assume energy arrives packets time interval sensor capable harvesting one packet energy also assume drawn bernoulli distribution assume sending message costs packet energy energy making observation processing negligible assumptions rather simplistic repeatedly used literature see references therein energy harvesting system adopt model sake analytical tractability insight thus sent time interval otherwise noted received sensor time log ratio given observation let let shown unconstrained channels quantizers optimal decision rules local sensors chen willett generalized result case channels shown thresholds optimal nonideal channels well see conditions result holds furthermore scalar observations observation sensor monotone increasing threshold directly translated sensor observation immediate consequence theorem throughout paper denote optimal threshold energy unconstrained sensor words threshold maximizes arg max log note false alarm detection probability unconstrained sensor respectively say observation model sensor separable perfect exists threshold lim say sensor decision function quantizer threshold otherwise situation unconstrained sensor channel infinity detection problem trivial however applications observation models exist observation models nonseparable threshold conditions satisfied however condition may hold asymptotically high snr limit thus let define ratio snr sensor say observation model sensor asymptotically separable snr goes infinity sensor observations exists threshold lim therefore corresponding unconstrained sensor channel goes infinity many observation models considered literature asymptotically separable following introduce one observation models consider case observation rayleigh distribution unconstrained mean situation energy always available sensor scale parameter rician distribution scale parameter noncentrality parameter conditional distributions sensor therefore exp exp modified bessel function first kind order zero relevant observation model low complexity sensors wireless sensor network used detect presence known signal gaussian noise based received power simplicity assume definition letting snr infinity threshold conditions satisfied therefore observation model asymptotically separable shall herein assume energy constrained sensor also quantizer applies following threshold test otherwise words sensor time compares observation threshold battery empty sends message towards otherwise remains would like note setup static device sense makes decision time based input messages time use previous input messages sensors words take account correlation received messages sensor could possibly introduced memory battery sensor follows find threshold maximizes delivered bhattacharyya distance sensor show problem hand optimum decision rule depends observation model also depends battery charge state arrival energy features capacity battery show resulting thresholds general differ unconstrained case following section different battery capacities study depletion probability energy harvesting sensor furthermore show one design energy harvesting sensor different battery capacities note designing energy harvesting sensors mean selection decision threshold iii esign nergy arvesting ensors formulate problem let first find conditional mass probabilities resulting sensor decision observe energy harvesting sensor time depends observation note condition consequence assumption distribution sensor battery depletion probability assumptions energy harvesting probability bernuoulli time space probability observations sensor time say battery markovian behavior sense conditioned state time depends state time sequence previous states markovian assumption steady state probability battery charge state derived allows consider long term performance energy harvesting sensor equipped battery let state probability vector battery charge time superscript indicates transposition let define transition probability matrix battery steady state state probability vector satisfy steady state probability battery state lim found using fact summation probabilities must equal unity steady state dropping subscript conditional probabilities given plug find resulting bhattacharyya distance log observe resulting energy constrained case steady state depends depletion probability battery function energy features battery capacity therefore following study performance energy harvesting sensor different battery capacities end arbitrary battery capacity lemma find depletion probability find threshold maximizes end consider energy harvesting sensor assume transmission time battery charge state since time interval sensor capable harvesting consuming one packet energy state transmission time either two exceptions rule battery charge state state former case state battery using using using using bcbc using using using using bcrs bcbcrs bhattacharyya distance rsrs bhattacharyya distance rsrs bcbcrs parameter parameter fig bhattacharyya distance energy harvesting sensors different threshold tests noiseless left noisy right communication channels function parameter transmission time either zero one since battery charge negative latter case state battery transmission time either since space save energy packets lemma depletion probability energy harvesting sensor given proof see appendix remark depletion probability energy harvesting sensor always zero however admits following expression pqe otherwise seen noting summation converges otherwise sum diverges makes depletion probability equal zero depletion probability expression infinite capacity battery sensor would like note difference model queue model battery charge state transition either change one remain unchanged line expression found modeling battery state process note condition results zero depletion probability follows intuition sense probability energy arrival higher probability energy consumption probability sensor observation threshold decides send message battery accumulate energy probability one long term empty situation problem unconstrained setup plugging expression depletion probability energy constrained sensor find expression sensor follows note since function observation models simplify follows considering different battery capacities numerically find threshold maximizes corresponding bhattacharyya distance sensor capable saving one packet energy incoming energy saved battery battery empty otherwise sensor discards incoming energy packet sensor optimal threshold found maximizing use grid search find threshold maximizes sensor capable saving two packets energy incoming energy saved battery battery empty one packet energy buffer otherwise sensor discards incoming energy packet maximizing find optimal threshold sensor fig illustrates resulting energy harvesting sensors using adapted unconstrained thresholds figure left considers communication channels figure right considers bac channels parameters observe unconstrained threshold always cases bhattacharyya distance bounded snr goes infinity theorem find expression upper bound arbitrary battery size find conditions bhattacharyya distance upper bounded theorem consider energy harvesting sensor assume probability harvesting energy time interval probability hypothesis channels bac channels fig sensor input exceed log proof see appendix remark upper bound achievable separable observation model selecting also asymptotically achievable sensor asymptotically separable observation model asymptotes also shown fig dotted lines note possibly unexpected insight due result sensors make asymptotically separable observations sensors sure true hypothesis optimal sensor act greedily always transmit whenever internal battery allows transmission situation still receive transmission subset sensors battery depletion events independent across sensors due independence energy arrival process remark upper bound increasing function battery capacity line intuition performance energy harvesting sensor improved increasing battery capacity upper bound also increasing function probability harvesting energy decreasing function probability also follow intuition sense increasing probability energy available battery performance sensor improved affects amount available energy directly affects battery content complicated way according optimally designed sensor aims send message consume packet energy observation increasing probability hypothesis likely sensor aims send consume energy increases depletion probability decreases performance sensor would also like note unlike previous works design sensor decision functions distributed detection network using bhattacharyya distance performance metric problem formulation comprises affect apriori probabilities depletion probability remark finite never grows unboundedly unconstrained case seen grow unboundedly separable asymptotically separable observation models channel increasing snr using one analyze performance sensor asymptote fig optimum energy harvesting sensor function sensor battery capacity shown observation model sensor according observe figure maximum upper bound discussed increasing functions sensor battery capacity remark infinite battery capacity sensor conditions distance grows unboundedly asymptotically separable model snr increases concretely communication channels noiseless bhattacharyya distance optimal threshold test increases unboundedly separable asymptotically separable observation models snr increases fig bhattacharyya distance sensor shown using different thresholds different setups observe figure parameter snr increases case bhattacharyya distance increases unboundedly otherwise upper bounded asymptote shown dotted line found following section compare error probability performance networks energy harvesting sensors designed using conventional unconstrained formulation using proposed formulation considering error probability performance designed networks consider choice using sensor makes optimal decision sense bhattacharyya distance case required energy transmission positive message always available sensor however energy always available sensor sensor act conservatively sense sensor remain silent preserve energy future time slots unless receives observation presence hypothesis high reliability formulation bhattacharyya distance threshold determines received observation high reliability presence hypothesis according simulation results always obtain confirms discussion bhattacharyya distance bhattacharyya distance upperbound using upperbound using bcbc battery capacity fig bhattacharyya distance energy harvesting sensor function battery capacity noisy noiseless communication channels corresponding upper bounds according theorem rror robability erformance etworks section illustrate benefit results numerical examples consider sensor network energy harvesting sensors suppose sensors make observations phenomenon time interval send ook message let assume sending positive message consumes packet energy negative message conveyed energy cost let observation model sensor conditioned true hypothesis observations independent sensors use threshold test note though section assume observation model sensors results conclusions drawn work generalized observation models sensors respectively sensor case using results previous section design sensor decision rules different battery sizes compare error probability performance unconstrained case expected error probability uses map criterion time found using max numerically computed without need simulations fig shows error probability performance designed sensor networks sensors sensors cases adapted threshold leads better performance unconstrained threshold line results based bhattacharyya distance adapted threshold leads higher also observe error probability converge zero lower bounded shown bhattacharyya distance rsut bcrs using using using using using using bcbc bcrs bcbc rsrs parameter fig bhattacharyya distance energy harvesting sensor different threshold tests different function noncentrality parameter upper bounded single sensors also observe figure using optimal threshold threshold increasing battery capacity one two improve error probability performance network sensors fig error probability performance network sensors shown different sets channels erroneous error probability network converges fixed value channels error probabilities adapted threshold energy unconstrained threshold rapidly zero observations also line results terms bhattacharyya distance channels increases unboundedly otherwise converges asymptote note observations choices oncluding emarks paper studied problem decentralized hypothesis testing network energy harvesting sensors sensors network make observations phenomenon harvest energy need environment consider case sensors different battery capacities save harvested energy considering bhattacharyya distance performance metric formulated problem designing sensors network considering constraints imposed energy harvesting proposed method design sensors decision rules studied performance sensors different battery capacities presented conditions bhattacharyya distance upper bounded therefore error probability lower bounded converge zero paper considered case observations energy charging processes sensors using using using using bcrs utrsbc utrs bcbc bcrs rsrs bcrs using using using using using using fig error probability performance networks energy harvesting sensors different battery capacities noisy communication channels independent possible extension work consider case observations sensors energy charging processes correlated sake analytical tractability insight studied case sensor decision rule simple singlethreshold test known sensors result better performance concretely battery state dependent thresholds improve performance energy harvesting sensor extension study study performance threshold tests although presented herein numerically found optimal threshold tests optimal threshold using grid search case observation model results confirm multiple thresholds improve performance sensor gains small compared introducing single threshold first place would also like note members distance like also used performance metric design decision rules sensors lot similar derivations paper chose use performance metric since according one analytically tractable members distances reported efficient metric among studied work represents first attempt introduce energy harvesting considerations context distributed detection using established arguably simple design metrics bhattacharyya distance approach furthermore utilizes simplified models energy arrival model taken simplifying assumptions make results tractable interpretable also naturally come strong limitations apart extension batterydependent decision threshold one could also consider case sensors optimize ook transmission energy parameter parameter fig error probability performance networks energy harvesting sensors different order influence error probability transmission probability fig based given physical channel model would however arguably relevant conjunction refined energy arrival storage model presently handle directly given simplicity assume energy arrives quanta matched energy needed positive transmission furthermore although decoupling sensor design caused considering bhattacharyya distance often used simplify sensor design problem distributed detection literature approach scrutinized adding energy harvesting aspect due fact jointly designed sensor decision rules potentially lead tangible benefits advanced joint energy conservation rules across sensors implicitly provided differences random battery state across sensors acknowledgment authors would like thank associate editor anonymous reviewers valuable comments suggestions led improvement paper authors would especially like thank reviewers prompted consider communication channels ppendix roof emma using state probabilities found using transition probabilities obtain replacing respectively qpe equations replacing transition probabilities get following equations system equations described homogeneous difference equation characteristic polynomial follows find another upper bound maximizing equivalent minimizing minimizing equivalent maximizing since decreasing function replacing depletion probability obtain whose roots lead general solution difference equation max applying auxiliary conditions obtain thus solution difference equation found given considering optimization problem objective function decreasing function note appears argument summation maxima attained therefore problem reduces following problem max describes state probability terms depletion probability using fact summation probabilities must equal unity obtain desired depletion probability expression pke ppendix roof heorem prove theorem first consider following function taking derivative respect observe increasing function minima attained words min follows show objective function increasing function therefore maxima attained proves theorem aim show evaluating summation introducing equivalently obtain using conclude given upperbounded log according chain rule using first principles straightforwardly show complete proof need show end note thus proving equivalent proving prove first show optima setting first equal zero derivative show optima maxima finding second derivative completes proof theorem therefore proof eferences varshney distributed detection data fusion new york dai distributed detection wireless sensor networks using multiple access channel ieee trans signal vol ciuonzo romano rossi decision fusion distributed mimo wireless sensor networks ieee trans wireless vol ciuonzo rossi dey massive mimo decision fusion ieee trans signal vol feb longo lookabaugh gray quantization decentralized hypothesis testing communication constraints ieee trans inf theory vol mar tarighati bayesian design decentralized hypothesis testing communication constraints proc ieee int conf acoustics speech signal process icassp may veeravalli varshney distributed inference wireless sensor networks phil trans math phys eng vol chamberland veeravalli wireless sensors distributed detection applications ieee signal process vol chen tong varshney channel aware distributed detection wireless sensor networks ieee signal process akyildiz sankarasubramaniam cayirci survey sensor networks ieee commun vol baek veciana minimizing energy consumption sensor networks distributed data compression hierarchical aggregation ieee sel areas vol nuggehalli srinivasan rao energy efficient transmission scheduling delay constrained wireless networks ieee trans wireless vol sharma mukherji joseph gupta optimal energy management policies energy harvesting sensor nodes ieee trans wireless vol april ulukus yener erkip simeone zorzi grover huang energy harvesting wireless communications review recent advances ieee sel areas gunduz stamatiou michelusi zorzi designing intelligent energy harvesting communication systems ieee commun magazine vol sudevalayam kulkarni energy harvesting sensor nodes survey implications ieee communications surveys tutorials vol rago willett censoring sensors scheme distributed detection ieee tran aerosp electron vol april berger guerriero zhou willett pac mac decentralized detection using noncoherent modulation ieee trans signal vol ciuonzo romano rossi optimality received energy decision fusion rayleigh fading diversity mac sensors ieee trans signal vol evans dey decision fusion noncoherent fading multiaccess channels ieee trans signal vol poor thomas applications distance measures design generalized quantizers binary decision systems ieee trans vol tarighati optimality rate balancing wireless sensor networks ieee trans signal vol july nayyar teneketzis veeravalli optimal strategies communication remote estimation energy harvesting sensor ieee trans autom control vol sept zhao chen zhang optimal power allocation energy harvesting estimation system proc ieee int conf acoustics speech signal may huang zhou jiang zhang cui power allocation joint estimation energy harvesting constraints proc ieee int conf acoustics speech signal may nourian dey distortion minimization multisensor estimation energy harvesting ieee sel areas vol march fan letaief outage probability energy harvesting cooperative networks rayleigh fading channel ieee trans vehicular vol feb hong distributed estimation analog forwarding wireless sensor networks proc ieee int conf commun syst iccs liu liu liu chen adaptive quantization distributed estimation wireless sensor networks approach int distributed sensor networks vol chalise zhang amin energy harvesting ostbc based mimo relay system proc ieee int conf acoustics speech signal process icassp medepally mehta voluntary energy harvesting relays selection cooperative wireless networks ieee trans wireless vol tarighati gross decentralized detection energy harvesting wireless sensor networks proc european signal process conf eusipco berger note error detection codes asymmetric channels inf control vol tsitsiklis decentralized detection large number sensors math signals vol kailath divergence bhattacharyya distance measures signal selection ieee trans vol valentini levorato santucci aging aware random channel access wireless networks ieee wireless commun letters vol michelusi zorzi optimal adaptive random multiaccess energy harvesting wireless sensor networks ieee trans vol michelusi stamatiou zorzi transmission policies energy harvesting sensors energy supply ieee trans vol tutuncuoglu ozel yener ulukus improved capacity bounds binary energy harvesting channel proc ieee int symp inf theory isit tsitsiklis extremal properties quantizers ieee trans vol chen willett optimality test local sensor decision rules presence nonideal channels ieee trans inf theory vol feb liu chen quantizers decentralized detection sensor networks ieee trans inf theory vol july lehmann romano testing statistical hypotheses springer science business media gross shortie thompson harris fundamentals queueing theory john wiley sons diniz silva netto digital signal processing system analysis design cambridge university press
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stability phenomena homology tree braid groups sep eric ramos abstract tree study changing behaviors homology groups varies uconf prove ranks homologies described single polynomial construct polynomiallexplicitly terms invariants tree accomplish prove group endowed structure finitely generated graded module integral polynomial ring prove naturally decomposes direct sum graded shifts squarefree monomial ideals following spend time considering methods might generalized braid groups arbitrary graphs make various conjectures direction introduction introductory remarks statements main theorems recent years push towards understanding mechanisms connecting various asymptotic stability results topology algebra instance let connected oriented manifold dimension interior manifold boundary write conf configuration space conf natural action conf symmetric group may therefore define unordered configuration space uconf conf classical theorem mcduff mcd theorem implies group uconf independent contrast work mcduff seen analogous statement true ordered configuration spaces conf true however perhaps next best thing follows work church ellenberg farb cef theorem polynomial betti number dimq conf agrees results type fall heading one might call asymptotic algebra modern philosophy asymptotic algebra roughly stated follows family algebraic objects display asymptotic stability phenomena often times encoded single object finitely generated appropriate abelian category case mcduff group uconf realized piece finitely generated graded module result church ellenberg farb involves showing conf constituents finitely generated representation category finite sets injections see cef notion philosophy also heavily featured author supported grant eric ramos sam snowden recent resolution stembridge conjecture goal paper apply similar techniques homologies unordered configuration spaces trees note problem considered master thesis results work disjoint current work paper graph always refer connected compact dimension tree graph contractible topological space essential vertex vertex degree valency least essential edge connected component space obtained removing essential vertices note essential edges vertices graph unaffected subdivision edges thought topologically essential pieces graph main result relates asymptotic behavior homologies braid group tree state result first need define kind connectivity invariant trees let tree set max dimq vji vji essential words maximum number connected components broken removing exactly essential vertices therefore maximum degree vertex essential vertices number essential edges convention theorem let tree write denote braid group uconf polynomial degree dimq remark follows theorem ghrist theorem given graph homeomorphic strictly greater number essential vertices realized case tree theorem fact cases polynomial theorem explicitly computed throughout course work see theorem computation implies something somewhat surprising homology groups corollary let tree let homology groups depend degree sequence interesting note rank agrees polynomial opposed agreeing sufficiently large case configuration spaces manifolds result kind precedent already discussed result church ellenberg farb states oriented manifold stability phenomena homology tree braid groups interior manifold boundary dimension conf agrees polynomial cef theorem perhaps interesting observation trees thought graphs boundary unclear whether connection work church ellenberg farb goes deeper however outline proof prove theorem rely classical techniques commutative algebra well modern techniques combinatorial topology first key ingredient discrete morse theory forman given regular complex see definition write set discrete morse function map cells satisfying following two hypotheses cells call critical respect sets conditions empty main consequence discrete morse theory critical cells determine homotopy type formally let write subcomplex comprised closures cells critical cell deformation retract otherwise unique critical cell obtained attaching cell moreover complex free critical whose homology homology space call differential morse differential see definition using work abrams farley sabalka able impose discrete morse structure spaces uconf graph spend good amount time recounting construction farley sabalka section accomplished strategy develop strong understanding critical cells specifically work towards understanding changing behaviors critical cells varies let graph essential edges write integral polynomial ring prove following section exists finitely generated graded free basis vectors indexed critical uconf specializing case tree work farley implies morse differential always trivial using complex obtain following theorem let tree essential edges let denote integral polynomial ring variables action critical cells uconf described section imposes structure finitely generated graded abelian group uconf particular exists finitely eric ramos generated graded uconf result ghrist abrams see theorem implies spaces uconf aspherical follows immediately uconf mind first part theorem simply follows existence hilbert polynomial course theorem tell anything degree hilbert polynomial bound obstruction accomplish must first prove structure theorem modules state theorem first recall definition squarefree monomial ideal say ideal squarefree monomial ideal contains generating set monomials none divisible square ideals subject theory many desirable properties instance much known hilbert polynomial see reference subject theorem let tree let denote theorem isomorphic direct sum graded twists squarefree monomial ideals dimension find theorem implies conclusions theorem fact able compute polynomial associated explicitly terms invariants tree see theorem overview paper next section spend time developing necessary background includes short summaries discrete morse theory section configuration spaces graphs sections following use machinery developed preliminary sections prove statement section finally specialize case trees use enumerative combinatorial methods prove theorem via explicit computation hilbert polynomial sections finish paper briefly consider case general graph note explicit results paper limited case tree statement hold graph result like theorem therefore hold general graphs long know morse differential commutes action belief author action perhaps slight alteration indeed commute differential unfortunately known differential become tremendously complicated increases complexity see case methods work therefore provide least strategy proving stability results general graphs final sections discuss implications stability phenomena homology tree braid groups acknowledgments author would like send thanks jordan ellenberg many insights well help editing paper author would also like send special thanks steven sam vital advice approaching primary problem work great amount gratitude must also sent daniel farley whose expertise field invaluable author learning material finally author would like acknowledge generous support national science foundation grant preliminaries discrete morse theory take time briefly summarize key points forman discrete morse theory largely following exposition forman farley sabalka park introduction spent time discussing notion discrete morse function one thing stand definition literal values function immaterial namely classification critical cells unchanged composition strictly monotone function many cases often easier construct relationships cells rather discrete morse function hints towards construction known discrete vector fields use approach exposition future sections begin first must place certain light restrictions spaces working definition let complex cell always refer open cell given cell dimension often write indicate dimension write denote set cells denote set cell said regular face cell given characteristic map homeomorphism closed ball map say complex regular given pair cells regular face equivalently regular attaching map cells homeomorphism assume throughout exposition regular complex original paper discrete morse theory forman proves results without requirement regular spaces uconf graph actually cubical complexes certainly regular complexes therefore condition regular restrictive need definition let regular complex discrete vector field collection partially defined functions satisfying following three conditions injective image disjoint domain domain face eric ramos given regular complex equipped discrete vector field cellular path two cells finite sequence face say path closed say trivial discrete vector field said discrete gradient vector field admits closed cellular paths discrete gradient vector field regular complex call cell redundant domain collapsible image critical otherwise proposition proposition theorem let regular complex equipped discrete gradient vector field consider filtration redundant cells removed redundant critical cells removed obtained attaching along boundaries number critical discrete gradient vector field deformation retracts onto proposition leads one notable corollary record corollary proposition corollary let regular complex equipped discrete gradient vector field homotopy equivalent complex precisely number critical case traditional morse theory decomposition space given corollary used compute homology groups simplicity state construction cubical complexes although general case similar definition let cubical complex write free abelian group define allows define boundary morphism given stability phenomena homology tree braid groups turning chain complex well known fact homology chain complex usual homology space assume equipped discrete gradient vector field map defined collapsible critical otherwise sign definition chosen negative coefficient property closed paths implies set stable value let denote free abelian group basis indexed critical morse complex associated defined boundary map given map known morse differential theorem theorem theorem isomorphisms configuration spaces graphs section review necessary facts configuration spaces graphs next section explain techniques discrete morse theory apply spaces definition graph compact connected complex dimension one tree topologically contractible graph given graph vertex write denote degree vertex note loop contributes degree count say essential say boundary vertex unique edge connected called boundary edge configuration space points topological space conf note natural action conf symmetric group unordered configuration space points quotient space uconf conf eric ramos majority paper work spaces uconf note many structural theorems discussed section apply spaces order apply discrete morse theory questions configuration spaces first need place complex structure accomplish use theorem abrams definition discretized configuration space points subcomplex spanned cells form cell edge vertex whenever write udn denote quotient action symmetric group theorem theorem kkp theorem let graph assume satisfies following two properties path connecting distinct vertices degree length least homotopically essential path connecting vertex length least inclusions conf udn uconf homotopy equivalences remark note first cited source abrams states theorem assuming path connecting distinct vertices degree length least noted proof version theorem stated true brief argument given proven second source kim park give formal argument improvement third source prue scrimshaw provide discrete morse theory argument independent first two sources purposes exact number vertices needed unimportant always subdivide edges needed note subdividing edges graph impact configuration spaces conf uconf much follows often assume without explicit mention subdivided enough homotopy equivalence theorem holds theorem implies conf uconf homotopy equivalent cubical complexes dimension fact able better theorem theorem let graph homeomorphic conf uconf homotopy equivalent complexes dimension number essential vertices remark homeomorphic uconf easily seen homotopy equivalent circle throughout work literature general recurring theme certain theorems apply graphs neither interval interesting observe two graphs precisely stability phenomena homology tree braid groups homeomorphic compact manifolds note ghrist originally proved theorem using classical topological means later see naturally falls discrete morse structure farley sabalka placed udn first noted farley sabalka one remarkable thing note ghrist theorem dimension configuration spaces graphs independent behavior stark contrast behavior configuration spaces smooth manifolds dimension paper ghrist also proves configuration spaces graphs fact aspherical result later reproven abrams theorem shows udn universally covered cat complexes theorem theorem corollary let graph conf hence uconf aspherical say conf note theorem analogous says configuration spaces plane aspherical case fundamental groups ordered unordered spaces artin pure braid groups artin braid groups respectively borrow terminology context well definition let graph braid group strands defined uconf similarly define pure braid group strands conf study braid groups graphs still active area research see kkp groups immediate corollary theorem obtain following corollary let graph isomorphisms conf uconf goal paper establish methodology understanding stability phenomena groups spirit modern trends asymptotic algebra note spend little time considering pure braid groups fact seem much literature homology groups vastly complicated braid groups eric ramos following theorem farley exactly computes homology groups tree theorem let tree group free note theorem slightly general discuss general version later sections expanding upon work farley following theorem park suggests tree case indicative general phenomenon theorem theorem let graph planar torsion free therefore moreover case planar torsion kim park proved planar torsion free kkp theorem also conjectured work theorem true noted theorem far stronger written fact result explicitly computes groups terms combinatorial invariants graph follows computation amount group unvarying moreover biconnected requires removal least vertices disconnect lemma final section paper conjecture extension fact higher homologies see conjecture finish section record result gal euler characteristic spaces note gal proves general theorem computing euler characteristic configuration space simplicial complex theorem theorem let graph set conf number edges remark conf cover uconf follows formula easily used compute euler characteristic uconf well definition let graph least one essential vertex let denote set essential vertices essential edge connected component stability phenomena homology tree braid groups corollary let graph least one essential vertex let denote number essential edges polynomial degree function uconf equal proof first note smoothing degree vertices impact spaces uconf may therefore assume without loss generality vertices case euler characteristic uconf coefficient power series expansion essential straight forward enumerative combinatorics argument implies coefficient power series expansion expression polynomial degree exactly sufficiently large remains show agreement begins assume essential vertices standard fact enumerative combinatorics coefficients power series expansion rational function form agree polynomial long deg case specific instance long theorem largely inspiration work suggests asymptotically betti numbers polynomial work prove suggestion case tree also provide setup case general graph hopefully able illustrate behavior case well discrete morse theory uconf section outline discrete morse structure uconf developed work farley sabalka begin fixing assuming graph sufficiently subdivided theorem hold uconf fix spanning tree well embedding plane label vertices applying search concretely begin choosing boundary vertex root label number continue boundary edge adjacent labeling vertices encountered along way increasing labels point essential vertex encountered travel leftmost relative current direction travel edge whose vertices labeled boundary vertex encountered one returns recently passed essential vertex example correctly labeled tree given figure remainder section previous paragraphs write denote set cells uconf denote set uconf definition given edge write denote largest respect given labeling vertex denote smallest vertex vertex write denote unique edge note eric ramos figure tree properly labeled note tree also sufficiently subdivided apply theorem first edge unique path within let eri udn eri eri say blocked note convention always blocked define partial function inductively following way given eri assume without loss generality unblocked vertex smallest respect labeling assuming image set eri eri unblocked vertices image undefined theorem sections collection partial functions form discrete gradient vector field udn spaces uconf high dimensional still visualize cells indeed often useful think cells uconf subsets edges vertices graph figure see examples critical collapsible redundant taken tree figure examples cell collapsible smallest unblocked vertex cell redundant blocked image indeed cell could possibly map would however case smallest unblocked vertex finally cell critical every vertex blocked collapsible examples one imagine existing way classify cells three types write classification first need bit nomenclature definition let eri udn say edge order respecting label larger label stability phenomena homology tree braid groups figure critical collapsible redundant respectively remark note edges call deleted edges definition never order respecting theorem classification theorem theorem let cell udn let choice spanning tree equipped planar embedding critical contains order respecting edges vertices blocked redundant contains order respecting edges least one vertices unblocked contains order respecting edge thus minimal order respecting edge unblocked vertex label less collapsible contains order respecting edge thus minimal order respecting edge vertex label less blocked hard check previous examples satisfy conditions classification theorem one useful fact discrete gradient vector field critical cells sense restrictive three possible types indeed may expand upon classification critical cells following way lemma let eri critical either erk deleted edge erk essential vertex let critical edges eri number vertices component eri agrees number vertices components proof first statement let edge deleted clear order respecting note boundary vertex case labeling defined therefore remains show degree indeed case vertex eric ramos particular must order preserving proves first claim second claim one uses fact vertices must blocked note observations made lemma originally pointed farley sabalka defined vector field section collected observations lemma could easily referred back future sections find lemma critical defining polynomial ring structure next section find deleted edges chosen remain unchanged increases repeatedly subdivide graph lemma also suggests edges critical cells contained tree unchanging literally case edges subdivided increases lemma tells important information encoded edge essential vertex tail well direction leaving essential vertex reason often bit loose language claim two critical cells different choices edges definition let critical cell discrete gradient vector field let edge classification theorem implies must exist vertex label smaller refer vertex witness witness say necessary proof theorem setup use section fix notation used throughout proofs main theorems note constructions presented previous sections already well established literature prior work knowledge author main construction following sections namely action critical cells uconf appear elsewhere literature case previous sections let denote graph neither circle line segment reserve denote number essential edges denote number essential vertices next must construct spanning tree section first subdivide every edge connects two essential vertices note includes loops done choose spanning tree satisfies following every edge adjacent essential vertex note entirely obvious exists arbitrary graph fact proven farley sabalka proof theorem remark noted farley sabalka one chooses satisfy lemma corollary imply theorem stability phenomena homology tree braid groups chosen spanning tree satisfy observe sufficiently subdivide theorem subdividing follows fact began subdividing edges connected essential vertices often differentiate spanning trees chosen reason modules recall section morse complex free basis indexed critical cells case write denote free basis indexed critical cells discrete gradient vector field udn let denote essential edges graph set xee goal remainder section argue structure finitely generated graded definition fix let denote essential edge given critical eri udn define unique critical udn obtained adding vertex connected component eri containing precisely obtained adding smallest vertex connected component containing well defined lemma well choice tree contain unique representative essential edge construction turns collection graded denote one visualize action described definition following way labeling induces natural flow namely edges flow towards root let eri critical cell imagine vertices particles drifting direction flow edges along endpoints stationary blockades essential edge action involves placing new particle somewhere allowing flow blocked provide illustration action figure lemma module finitely generated proof claim every critical cell obtained critical cell indeed critical cell least one vertex necessary witness edge remove vertex among necessary witness vertices occupies maximal position respect labeling removing vertex leaves critical cell removal create unblocked vertices order respecting edges follows image action appropriate essential edge eric ramos figure illustration multiplication essential edge containing boundary vertex tree figure remark follows lemma number critical grows function like polynomial degree fact claim polynomial must degree strictly less first note action critical cell affect edges follows expressed direct sum graded summand corresponds choice edges critical fixed critical one observes two variables act identically whenever connected component edges removed follows summand containing cell hilbert polynomial degree strictly less whenever hilbert polynomial whole sum polynomials therefore also degree strictly less used final section paper lemma theorem imply asymptotic data homology groups theorem let graph essential edges exists polynomial dimq case trees section begin explore specific case tree work farley allow conclude quite bit able general case begin state refined version theorem theorem let tree morse complex trivial differential particular module theorem state main stability theorem begin following definition stability phenomena homology tree braid groups definition let tree define quantity max dimq vji vji essential example maximum degree number essential edges convention theorem let tree fix betti number dimq equal polynomial degree proof theorem work previous sections imply eventually polynomial remains show polynomial claimed degree begin partitioning basis vectors according collection edges appear action impact edges partitions correspond summand moreover considering one summands observe variables induce distinct operators follows degree hilbert polynomial finish proof show summand whose hilbert polynomial degree fix essential vertices whose absence realizes maximum definition essential vertex fixed list let edge adjacent second leftmost direction relative direction root note critical cell containing edges unique essential edge house witness vertex therefore critical constructed choosing edges along unique witness vertex denote cell submodule generated summand claim submodule desired summand shares edges basis vectors graded piece entirely determined distributing vertices one components thus rankz desired piece theorem remains proven showing agrees polynomial accomplish must first prove theorem first simplification note answer unaffected changing basis may therefore assume working rational polynomial ring grants access many classical approaches solving problem also make heavy use direct sum decomposition described proof theorem definition let tree notation vji eric ramos always refer essential vertices appearing order induced labeling write denote number connected components vji given let denote positive integers vjk also set given pair associate summand generated degree critical cells satisfying following two properties edge vjk vjk precisely essential edges house witness vertices note two pieces data uniquely determine edges allowed appear cells form bases graded pieces summand denote summand observe two variables act identically contained connected component vji let denote quotient particular isomorphic rational polynomial ring variables finitely generated graded module remark empty tuples convention next goal show hilbert function dimq agrees polynomial note already know hilbert function polynomial sufficiently large must show agreement case imply hilbert function accomplish computations next section although spend time set ground work direction following explicitly compute hilbert polynomial begin following proposition module isomorphic squarefree monomial ideal proof recall monomial ideal called squarefree generated monomials none divisible square definition spanned critical cells containing edges determined pair see definition along single witness vertex stability phenomena homology tree braid groups define map suffices specify cells mapped set cell product essential edges containing witness vertex note impossible single essential edge house witness vertex multiple edges monomial indeed squarefree extending map action defines desired isomorphism squarefree monomial ideals well understood objects commutative algebra also subject theory see comprehensive reference subject computing hilbert polynomial note polynomial describing betti number explicitly computed terms invariants tree indeed follows structure theorem park theorem also work farley sabalka hilbert polynomial also computed case tree maximal degree farley aside cases explicit description polynomial known said work paper implies computing polynomial difficult computing hilbert polynomials squarefree monomial ideals words tree work paper reduces task computing finite computation proceed computation section remark proposition reveals monomial ideals appear graded shifts summands quite simple find computation hilbert polynomial falls realm enumerative combinatorics rather theory likely studying cases general graphs require robust commutative algebra therefore proceed case tree using language hope case general graphs completed aid creating uniform means approaching asymptotic behaviors configuration spaces graphs section use decomposition proposition compute hilbert polynomial proposition implies isomorphic squarefree monomial ideal ease computations reorder variables satisfy following begin labeling essential edges adjacent house witness vertices chosen order induced ordering vertices next label essential edges housing witness vertex adjacent variables using rule continue fashion essential edges labeled use remaining variables label final essential edges chosen order first observation following eric ramos lemma isomorphism considered module module proof follows immediately isomorphism definition hilbert series associated module formal power series dimq next goal express rational function standard enumerative combinatorics allow compute hilbert polynomial proposition hilbert series expressed rational function proof prove using fact hilbert series multiplicative tensor products easy see hilbert series augmentation ideal rational polynomial ring variables equal rational function lemma therefore implies factor arises graded shift allow prove result obstruction hilbert polynomial corollary let tree fix dimension agrees polynomial stability phenomena homology tree braid groups proof suffices prove claim summands proposition tells hilbert series takes form implies suffices show total degree numerator proof corollary strictly smaller namely must argue follows fact essential vertex say vjk adjacent least one essential edge never house witness vertex namely essential edge maximal among essential edges adjacent vjk respect ordering vertices moreover essential edge containing root never house witness vertex essential vertex put another way vjk concludes proof proven enough explicitly compute hilbert polynomial consequently following theorem follows results section well combinatorics generating functions theorem let tree fix pair write also let piv polynomial piv dimension equal polynomial piv remark one observes formulation polynomial piv well constants depend degree sequence graph follows homologies braid group tree depends degree sequence recorded corollary specializing theorem case wherein yields much simpler form eric ramos corollary let tree dimension agrees polynomial essential proof case note essential vertex chosen integer therefore polynomial written essential using principle seen desired result kind appears work farley sabalka compute number generators group formula case also found results park another simple case every essential vertex degree case theorem used recover following result farley corollary let tree whose every essential vertex degree dimension agrees polynomial proof major thing note case associated vector therefore otherwise formula becomes completes proof stability phenomena homology tree braid groups concluding remarks section consider work implies homology graph braid groups general graph rather tree let graph essential vertices essential edges also assume homeomorphic neither line segment work section implies carries structure finitely generated tree however longer case morse differential trivial therefore pose following question question action commute morse differential words morse differential morphism graded note altering choice spanning tree influence morse differential therefore unclear whether morse differential commute action choices therefore refine question follows question exist choice spanning tree action commutes morse differential answer questions affirmative immediately conclude many things asymptotic behavior homology groups indeed case hilbert basis theorem would imply finitely generated corollary assume questions affirmative answer write zbi prime exists polynomial degree exists positive integer independent exponent prime positive integer polynomial particular finite set primes polynomials order torsion subgroup precisely mpi proof follows standard facts study graded modules integral polynomial rings work park theorem torsion subgroup annihilated multiplicity torsion eventually constant work also implies rank constant whenever sufficiently connected contrast affirmative answers questions imply top homology always grows fast possible eric ramos theorem assume questions affirmative answer hilbert function dimq hng equal polynomial degree proof theorem implies euler characteristic uconf polynomial degree hand remark implies lower homology groups hilbert functions grow strictly slower follows rank hng must eventually grow like polynomial degree note theorem implies hng therefore seen torsion top homology hng torsion interesting question ask types torsion appear intermediate homologies affirmative answer questions would step right direction work park would suggest action described section perhaps already discussed result theorem implies braid groups biconnected graphs eventually constant first homologies however action defined section detects connectivity chosen spanning tree particular possible one improve action accounting connectivity granted deleted edges question let graph neither circle line segment exist action interacts way deleted edges specifically one define action critical relation whenever two essential edges connected component edges removed existence action would actually suggest something quite strong kind invariants encoded homology braid groups graph recall define maximum number connected components broken removing exactly conjecture let graph neither line segment circle betti number dimq agrees polynomial degree note theorem specific case conjecture also note biconnected implies eventually constant fact proven work park theorem conjecture therefore proposes generalization one aspect work finally think ramifications work spaces uconf action defined purely formal way however study classical stability phenomena homology tree braid groups configuration spaces stability phenomena often arise maps underlying spaces uconf uconf see instance mcd also case question tree action induced map topological spaces uconf uconf general graph questions affirmative answers action homologies induced similar way course question highly relevant questions one understands action topologically might make existence action homologies general graph braid groups apparent references abrams configuration spaces braid groups graphs thesis barnett farber topology configuration space two particles graph algebr geom topol cef church ellenberg farb stability representations symmetric groups duke math ellenberg algebraic structures cohomology configuration spaces manifolds flows farley homology tree braid groups topological asymptotic aspects group theory contemp amer math providence http forman morse theory cell complexes adv math forman user guide discrete morse theory lotharingien combinatoire http farley sabalka discrete morse theory graph braid groups algebr geom topol electronic http farley sabalka presentations graph braid groups forum math http ghrist configuration spaces braid groups graphs robotics knots braids mapping class groups papers dedicated joan birman new york stud adv amer math providence https gal euler characteristic configuration space complex colloq math http kkp kim park graph braid groups artin groups trans amer math soc park characteristics graph braid groups discrete comput geom configuration spaaces graphs masters thesis http mcd mcduff configuration spaces positive negative particles topology volume issue march pages miller sturmfels combinatorial commutative algebra graduate texts mathematics new york eric ramos prue scrimshaw abramss stable equivalence graph braid groups sam snowden proof stembridge conjecture stability kronecker coefficients algebraic combin department mathematics university wisconsin madison address eramos
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analysis design drone flight controller zhuoqun richard craig boston university usa czq richwest einstein feb abstract timing guarantees crucial applications must bound delay sensing processing actuation example flight controller multirotor drone data gyro inertial sensor must gathered processed determine attitude aircraft sensor data fusion followed control decisions adjust flight drone altering motor speeds processing pipeline sensor input actuation bounded drone lose control possibly fail maintain flight motivated implementation multithreaded drone flight controller quest rtos develop composable pipe model based system task scheduling communication abstractions pipe model used analyze two semantics time reaction time freshness time also argue timing properties factored early stage application design thus provide mathematical framework derive feasible task periods satisfy given set timing constraints schedulability requirement demonstrate applicability design approach using port cleanflight flight controller firmware quest intel aero board experiments show cleanflight ported quest able achieve latencies within predicted time bounds derived analysis acm subject classification embedded systems keywords phrases systems timing analysis flight controller introduction past years commercial hobbyist multirotor drones rapidly growing popularity fast development drone technology enables ever widening set applications including aerial photography package delivery search rescue one commonly used control boards drones use today family socs based arm cortex processors include integrated inertial sensors gyroscope accelerometer magnetometer many existing flight control boards perfectly adequate drones operated via radio control fall short processing capabilities needed fully autonomous operation reason developing new approach building autonomous drones using emerging multicore platforms intel aero board qualcomm snapdragon flight development board nvidia jetson boards offer multiple processing cores integrated graphics processing capabilities making capable mission tasks would impossible simpler hardware first step building autonomous drones far involved reimplementation popular racing drone flight control firmware called cleanflight intel aero board also ported quest rtos aero board efficiently predictably manage multiple cores complexity reimplementation cleanflight refactors original code running directly firmware socs application running quest decoupling software components separate threads improves modularity cleanflight provides capability parallel task execution platforms multiple cores flight management tasks able leverage availability increased compute resources potentially improving controllability drone similarly cleaner interfaces software components eases future plans extend cleanflight advanced features camera data processing object detection avoidance simultaneous localization mapping slam necessary autonomous flight management original cleanflight code series tasks executed loops predefined frequencies frequencies based combination capabilities hardware experiences drone developers however multithreaded cleanflight subject extra overheads uncertainty due scheduling communication critical ensure timing correctness cleanflight example gyroscope reading fails correctly influence change motor hence rotor speed within specific time bound drone might able stabilize cleanflight typical many applications process sensor inputs require timebounded changes actuators recent development data fusion algorithms availability open source hardware applications leading revolution areas printing drones robotics driverless cars intelligent home automation systems applications essential guarantee two types timing requirements maximum time takes input sensor reading flow whole system eventually affect actuator output maximum time within input sensor reading remains influential output actuator commands community developed valuable approaches scheduling response time analysis tasks timing analysis received limited attention prior work originated network communication research based communication fifobased buffers drone flight control program however single buffers periodic sampling common paper therefore presents timing analysis drone flight controller based combination task model communication model quest rtos scheduling model also show derived worst case communication time turn used guide design applications contributions paper include proposal composable pipe model capture timing characteristics communication quest rtos demonstration derive task periods given timing constraints application design stage evaluation cleanflight flight controller intel aero board rest paper organized follows section provides background cleanflight corresponding task scheduling communication models adopted throughout paper section describes timing analysis proposed composable pipe model section shows time leveraged application design stage section details evaluation cleanflight aero board related work discussed section followed conclusions future work section execution model section first provide overview application motivated work secondly describe application design model three perspectives task model scheduling model communication model application system overview paper motivated objective implement autonomous flight management system multirotor drones autonomous drone one able reason adapt changes surroundings accomplishing mission objectives without remote assistance human objectives established part effort undertaken port cleanflight firmware traditional soc intel aero compute board aero board quadcore atom processor ram integrated gpu inertial measurement sensors camera connectivity makes capable flight management tasks package delivery aerial photography search rescue would impossible less powerful arm cortex found socs cleanflight targeted racing drones operated humans using radio control core software components cleanflight consist sensor actuator drivers pid controller mahony attitude heading reference ahrs algorithm various communication stacks logging system runtime entities components called tasks tasks total half optional essential ones listed table tasks scheduled highest lowest dynamic priority calculated function task static priority time since last executed task name period system battery alert battery voltage battery current gyro pid accelerometer imu attitude receiver serial magnetometer table list cleanflight tasks static priority medium medium medium realtime realtime medium medium high low low description check system utilization alarm battery runout update battery voltage reading update battery current reading update gyro readings perform motor control update accelerometer readings calculate attitude process commands serial communication update magnetometer readings port cleanflight aero board runs quest rtos developed quest drivers spi gpio uart inertial sensors similar boards quest smp system providing user kernel level threads attitude orientation drone relative reference frame earth well threaded interrupt handlers scheduled virtual cpu vcpu scheduler detailed section task model model flight controller program set periodic tasks task characterized worst case execution time period determined design stage fixed runtime usually profiled worst case execution condition deciding value challenging process major topic paper mainly depends latency constraints schedulability test periodic tasks implemented using quest user level threads paper use term thread task interchangeably apart user level threads kernel threads dedicated interrupts originate primarily spi bus cleanflight quest executes interrupts deferrable thread context corresponding time budget way handling interrupt steal cpu cycles currently running potentially task scheduling model threads quest scheduled scheduling hierarchy threads mapped virtual cpus vcpus mapped physical cpus vcpu specified processor capacity reserve consisting budget capacity period value determined task mapped vcpu vcpu required receive least units execution time every time units runnable long schedulability test passed creating new vcpus way quest scheduling subsystem guarantees temporal isolation threads runtime environment conventional periodic tasks assigned main vcpus implemented sporadic servers scheduled using scheduling rms vcpu smallest period highest priority instead using sporadic server model main tasks bottom half threads special vcpus created threaded interrupt handlers vcpu operates priority inheritance bandwidth preserving server pibs pibs uses single replenishment avoid fragmentation replenishment list budgets caused interrupt service routines isrs using pibs interrupt threads scheduling overheads context switching timer reprogramming reduced communication model control flow within flight controller influenced path data originates sensory inputs ends actuation inputs include inertial sensors optional cameras gps devices actuators include motors affect rotor speeds attitude drone data flow involves pipeline communicating tasks leading communication model characterized interarrival times tasks pipeline buffering tasks access pattern communication buffers periodic aperiodic tasks aperiodic tasks irregular interarrival times influenced arrival data periodic tasks fixed interarrival times operate whatever data available time execution periodic task implements asynchronous communication blocking await arrival new data communication shared buffer used scenarios data history important factor however flight controller data freshness outweighs preservation full history sampled data example motor commands always computed latest sensor data stale data discarded moreover communication results loosely synchronous communication producer suspended fifo buffer full consumer suspended buffer empty communication achieves fully asynchronous communication two communicating parties using simpson algorithm implicit explicit communication explicit communication allows access shared data time point task execution might lead data inconsistency presence task preemption task reads shared data beginning end execution might see two different values preempted two reads another task changes value shared data conversely implicit communication model essentially follows paradigm avoid data inconsistency mandates task make local copy shared data beginning execution work copy throughout execution paper assumes periodic task model simplifies timing analysis applications cleanflight implement periodic tasks sample data perform control operations system also adopts implicit communication data freshness consistency communication timing analysis section first distinguish two different timing semantics communication used basis separate timing analyses secondly develop composable pipe model communication derived separate latencies influence delay lastly use pipe model derive worst case communication time various situations semantics time understanding meaning time consider following two constraints flight controller constraint change motor speed must within gyro sensor reading caused change constraint update gyro sensor value must within corresponding update motor speed values change differ whereas may stay update semantics lead two different constraints appreciate difference imagine two cases table case task reads gyro runs every one controls motors runs every case guaranteed meet constraint motor task runs within matter whether gyro reading changes however fails constraint frequently gyro task likely run even interval conversely case guaranteed meet constraint fails constraint frequently example demonstrates difference two semantics time leads following formal definitions case case gyro period motor period table example periods figure task chain reaction time time takes input data flow system affected period consumer pipeline reaction timing constraint bounds time interval sensor input first corresponding actuator output freshness time time within instance input data influence system affected period producer pipeline freshness timing constraint bounds time interval sensor input last corresponding actuator output constraint constraint reaction time constraint freshness time perform analysis two semantics time section respectively latency contributors communication delay influenced several factors identify part analysis begin first consider communication pipeline illustrated task chain figure task reads input data din channel cin processes produces data task reads produces eventually writes output dout channel cout reading processing task handles data three stages read process write time sum latency stage task chain due asynchrony communication however also need consider one less obvious latency waiting time takes intermediate output read input succeeding task chain summary latency contributors classified follows processing latency represents time takes task translate raw input processed output actual processing latency depends absolute processing time task without interruption also service constraints cpu budget period vcpu associated task communication latency represents time transfer data channel transfer data size bandwidth propagation delay communication channel software overheads communication protocol contribute overall latency since communication model asynchronous described section queuing latency concern work scheduling latency represents time interval arrival data channel sending task receiving task begins reading data scheduling latency depends order execution tasks system therefore significant influence communication delay composable pipe model section identified factors influence communication delay among absolute processing time transfer data size determined nature task question capture rest timing characteristics develop composable pipe model leveraging scheduling approach described section task pipe relationship illustrated figure figure illustration pipe terminology pipe composed three elements pipe terminal pipe one terminal encapsulation data processing power reserved pipe pipe terminal represented vcpu timing characteristics captured vcpu budget period thereby guaranteeing least units execution time every time units pipe terminals associated conventional tasks bound main vcpus kernel control paths including interrupt handlers device drivers bound vcpus described section pipe end pipe two ends one input one output pipe end interface communication channel either bus shared memory theoretically physical timing characteristics pipe end consist transmission delay propagation delay work focuses embedded systems communicating parties typically located within close proximity neglect propagation delay transmission delay modeled bandwidth parameter communication channel also use parameter denote software overheads communication protocol though aware depends data transfer size time difference negligible compared time actual data transfer processing therefore sake simplicity constant model note definition single pipe one terminal two either end pipe differs idea posix pipe comprises task sending receiving ends case pipe represented single terminal takes input produces output example two communicating pipes shown figure representative communication path gyro task attitude calculation cleanflight aero board gyro task mapped pipe whose input end spi bus connected gyro sensor output end region memory shared pipe pipe terminal vcpu gyro task responsible handling interrupts generated spi bus contrary terminal pipe main vcpu ahrs task gyro task reads raw gyro readings pipe input end processes writes filtered gyro readings pipe output end similarly ahrs task reads filtered gyro readings pipe input end produces attitude data output end shared memory figure illustration two communicating pipes notation timing characteristics pipe denoted denote bandwidth software overheads input output ends respectively denotes budget period pipe terminal task also represented denotes size raw data read order perform job denotes size processed data produced denotes uninterrupted processing time takes turn raw data processed data addition denotes mapping task pipe task said mapped pipe data size read input end parameters data size written output end parameters pipe terminal parameters used scheduling accounting read write operations well processing takes time composition chain pipes operator connects pipe output end succeeding pipe input end example figure represented scheduling latency two pipes denoted lastly given task set identity mapped pipe set pipes connected ascending order subscript denotes reaction time pipe chain denotes freshness time reachability mathematically analyzing time introduce concept reachability inspired reachability conditions proposed feiertag necessity introducing reachability due subtle difference asynchronous communication model traditional synchronous communication latter data guaranteed transferred without loss repetition way time derived time interval arrival data input departure corresponding data output unfortunately might result infinitely large time case asynchronous communication every input leads output instead unprocessed input data might discarded overwritten newer input data available explained section infinitely large time mathematically correct lacks practical use therefore following timing analysis ignores input data fails reach exit pipe chain enters instead data inputs result data outputs pipe chain considered define latter class inputs reachable timing analysis alluded execution task divided three stages involving reading processing writing data simplify timing analysis assume tasks able finish read write stages within one period pipe terminal task mapped unrealistic applications flight controller data transferred usually small three stages typically able finish within one period however maintain generality impose restriction length processing stage worst case time single pipe first consider case single pipe two key observations case absence scheduling latency due lack succeeding pipe equivalence two time semantics reaction freshness time due lack preceding pipe therefore use unify notation given task mapped pipe worst case time essentially execution time three stages due timing property pipe terminal guaranteed units execution time within window time units hence latency lwc bounded following mod lwc worst case reaction time pipe chain section extend timing analysis single pipe pipe chain sake simplicity start chain length two show section mathematical framework applicable arbitrarily long pipe chains distinguish tasks mapped two pipes name preceding task producer succeeding consumer producer denoted dpi pipe denoted wip wop similarly consumer task pipe denoted dco wic woc following definition reaction time section investigate time interval specific instance input data denoted read first corresponding output denoted written vital importance recognize time pipe chain simply sum time single pipe chain also need account scheduling latency resulting appended pipe described section scheduling latency depends order execution tasks therefore perform timing analysis two complementary cases case shorter period thus higher priority case shorter period thus higher priority according ordering calculating reaction time case key making use lwc timing analysis find worst case scheduling latency illustrated figure worst case scheduling latency occurs preempts step immediately produces intermediate output dint corresponding preemption uses budget gives cpu back upon resumed immediately produces dint step become runnable read dint step wait budget replenishment waiting time exactly worst case scheduling latency wop figure reaction time case replenishment reads dint processes eventually writes defined time interval arrival departure worst case follows lwc wop note runs budget writing may overwrite dint pipe new data step however implicit communication property guarantees works local copy shared data dint initiates another read lwc figure reaction time case case situation complicated higher priority worst case scenario case hold case primarily might overwrite dint budget replenishment impossible case larger period guaranteed budget replenished able initiate another write words figure step guaranteed happen step problem case reason introducing reachability section find worst case reaction time case find scenario leads worst case scheduling latency also originates reachable input figure illustrates scenario meets requirements figure preempts immediately finishes reading intermediate output step dint corresponding follows longest possible waiting time dint becoming available step reading data step period minus budget execution time read stage waiting time exactly worst case scheduling latency wic reading dint writing might experience one preemption repeatedly overwrites shared data however affect processing dint either spatially temporally thanks vcpu model implicit communication semantic therefore similar case worst case reaction time sum equation pipe equation lwc lwc wic since output end input end share communication channel reasonable assume wop wic proceed unify worst case reaction time using one conditional equation lwc lwc otherwise wop wic special cases systems often profiled offline obtain worst case execution times tasks case would enable cpu resources pipe terminals provisioned task completes one iteration three stages read process write one budget allocation hence period implies equation arbitrarily small positive number account surplus budget completing task stages possible simplify worst case reaction time derived section first equation simplified follows mod lwc using equation equation reduces lwc simplification applied equation case reduces equation otherwise assume communicate data small size shared memory possible discard communication overheads equation simplifies otherwise finally notice appending worst case reaction time increased following otherwise worst case freshness time pipe chain techniques similar section used analyze freshness time avoid repetition abbreviate freshness timing analysis focusing special cases described section recall freshness time defined interval arrival input departure last corresponding output therefore investigate interval specific instance input data read last corresponding output written figure freshness time case case illustrated figure read first instance time intermediate output dint written shared data step produces three outputs corresponding dint steps indirectly last output one preceding write new data dnew step thus worst occurs two consecutive writes case freshness time steps longest possible time interval write happens late possible latest time write immediately second write preempted higher priority figure worst case freshness time communicating shared memory equation simplified case smaller period impossible read intermediate output twice figure step guaranteed happen thus worst case freshness time essentially worst case reaction time shown equation summary worst case freshness latency two communicating pipes represented following conditional equation otherwise composability timing analysis two pipes section extends pipe chains arbitrary length every time extra task mapped appended tail end chain worst case reaction time increases worst case time newly appended pipe plus scheduling latency new pipe tail pipe actual value increase depending relative priority new pipe tail pipe shown equation similarly added freshness time derived equation composability crucial property pipe model since significantly eases time calculation given pipeline provides basis design framework derives task periods given timing constraints detailed following section design significant challenges porting flight control firmware cleanflight run multithreaded application operating system one major issues determine period thread application able meet timing constraints naive approach would start choosing tentative set periods use timing analysis method section validate timing correctness upon failure periods heuristically adjusted validation step repeated timing guarantees met approach however potentially number tasks constraints increase inspired gerber derive task periods timing constraints combining timing analysis pipe model linear optimization techniques section generalize method use broader spectrum control applications problem definition precisely define problem first consider set tasks set pipes dji djo wij woj additionally require following information mapping ease notation assume tasks map pipe subscript hence topology example shown figure value dji djo value wij woj timing constraints namely value aim find feasible set pairs meets specified timing constraints passes task schedulability test ideally necessarily minimizes cpu utilization task run faster needs resources made available additional system objectives solving constraints solution carried process make easier understand use concrete example actual numbers elaborate process consider pipe topology reaction freshness dji djo wij woj figure application task graph schedulability execution times table application timing characteristics graph shown figure six tasks mapped six pipes tasks read inputs sensors tasks write outputs actuators task intermediary responsible complicated processing pid control sensor data fusion timing characteristics tasks pipes shown table note execution times assumed identical tasks practice would necessarily case affect generality approach step use given compute budget pipe terminal budget set value ensures three stages read process write finish one period compute example aggregate times read process write data thus budgets computed similar way input pipe terminal comes multiple sources value aggregated input channels example receives maximum amount data every transfer data pipe terminal necessarily duplicated pipe terminals consumers example generates maximum amount data every transfer placing single copy output shared memory region accessible communication channels involve shared memory data would duplicated step derive list inequations involving period variables given timing scheduling constraints table simplicity scheduling constraint shown utilization bound six pipe tasks however sensor inputs actuator outputs system would map tasks vcpus different utilization bound described earlier work derivation based equations composability property pipe model according conditional equations however every two pipes undetermined priority lead two possible inequations exponentially increases search space feasible periods order prune search space strategy always start case based observation tasks tend inputs sake better overall responsiveness thus reaction constraint example translated inequation derived combining equations possible translate timing constraints inequations periods variables addition periods implicitly constrained given inequations step attempts find maximum value period total cpu utilization minimized left linear programming problem unfortunately polynomial time solution integer linear programming problem known though linear programming solutions still available certain mathematical conditions beyond scope paper instead practice problem simplified usually small number pipe ends task meaning period variable usually involved small number inequations sensor task period usually hardware sampling rate limit example assume known feasible set periods example table easily found ignore integer requirement possible find feasible solution polynomial time using rational numbers rounded integers though rounding may lead constraint violations possible increase time resolution ensure system overheads exceed rounding errors worst case designer always able perform exhaustive search possible constraint solutions evaluation section describes simulations experiments intel aero board atom ghz processor ram simulation experiments developed simulations linux quest predict time using equations section simulation consists three tasks mapped pipes respectively three tasks search prime numbers within certain range communicate one another exchange results communicates communicates communication channel shared memory caches disabled data size set achieve nonnegligible millisecond communication overhead task assigned different search range profiled execution time shown table milliseconds budget pipe set slightly larger execution time corresponding task compensate system overheads settings pipe terminal also shown table milliseconds apart three main tasks system loaded low priority background tasks consume remaining cpu resources serve potential interferences case case table simulation settings case measure reaction time freshness time separately compare corresponding theoretical bounds figure shows results outputs produced reaction freshness times respectively case table case seen observed values always within prediction bounds reference also perform two cases yocto linux shipped aero board kernel version patched patch running simulation system also uncompresses linux source code background places load system background tasks quest figure shows worst case reaction freshness times within first outputs compared linux less variance shown times quest additionally freshness reaction times generally lower quest linux figure summarizes worst case reaction time wcr maximum variance wcr maxrv worst case freshness time wcf maximum variance wcf maxfv quest linux quest reaction linux reaction quest freshness linux freshness observed predicted time time figure observed predicted output figure quest linux cleanflight experiment next experiments apply design approach determining periods task cleanflight decouple software components original cleanflight firmware flight controller multithreaded application running intel aero board hardware software architecture shown figure hardware currently use core run cleanflight quest remaining three cores reserved separation kernel run general purpose linux apart main processor aero board also coprocessor provides interfaces including conversion adc uart serial modulation pwm system currently uses hub send pwm signals electronic speed controllers escs alter motor hence rotor speeds drone modified fpga logic improve timing resolution pwm signals well control duty cycle periods make additional use onboard bosch inertial measurement unit imu hub imu connected main processor via spi bus software minimize engineering efforts currently disable auxiliary features cleanflight telemetry blackbox data logging flight control configuration essential components shown circular tasks figure figure cleanflight data flow ahrs sensor fusion task takes input readings accelerometer gyroscope imu calculates current attitude drone pid task compares calculated target attitudes feeds difference pid control logic original cleanflight code target attitude determined signals human flying drone autonomous setting target attitude would calculated according computations based mission objectives flight conditions output nonetheless mixed desired throttle read pwm task translates motor commands motor commands sent spi bus ultimately delivered pwm signals esc associated separate pair multirotor drone decouple tasks separate threads safety reasons sensor tasks pwm task given individual address spaces simplify experiment setup instrument cleanflight use synthetic radio input value throttle degree pitch roll yaw angle instead reading real radio driver tasks profiled execution times shown table exec times gyro ahrs pid pwm accl radio table task execution times seen figure currently three data paths originating gyro accelerometer radio receiver respectively unfortunately little information available timing constraints imposed path guarantee working drone timing parameters original cleanflight determined trial error hand instead determining optimum timing constraints focus paper guaranteeing given constraints therefore first port cleanflight yocto linux aero board reference implementation linux version remains used estimate desired time example gyro path worst case reaction freshness times measured respectively round use timing constraints quest flight controller implementation using approach determine reaction freshness times accelerometer path set respectively finally radio path set reaction freshness times respectively using execution times table timing constraints apply design approach derive periods results task shown table gyro ahrs pid pwm accl radio table task periods evaluation measure actual time focus longest pipe chain highlighted figure instrument cleanflight code append every gyro reading incrementing also record timestamp gyro input read timestamp stored array indexed every task instrumented maintain translating input data output way preserved along pipe chain input gyro reading output motor command pwm task sends motor commands looks timestamp using compares current time able log reaction freshness time every input gyro reading compare observed time given timing constraints well predicted worst case value results shown figure seen observed values always within predicted bounds always meet timing constraints quest linux time time observed predicted constraint wcr maxrv wcf maxfv figure quest linux times reaction freshness figure cleanflight times related work feiertag distinguish four semantics time provides generic framework determine valid data paths semantic authors perform timing analysis scheduling model assumed hamann also discuss reaction age time work focuses integrating three different communication models including implicit communication model existing timing analysis tools composable pipe model also based implicit communication perform timing analysis using quest rtos task scheduling model large portion reaction time analysis based synchronous graph sdfg communication driven arrival input data recent work singh enhance standard sdfg allow specification latency constraint gerber propose synthesis approach determines tasks periods offsets deadlines timing constraints work relies task precedence constraints scheduling model used analysis work uses scheduling model based quest perform timing analysis derive task periods budgets ensure specific reaction freshness schedulability constraints also efforts develop programming languages prelude giotto able derive tasks periods based timing constraints kirsch use giotto reimplement helicopter control system others developed rather drone flight control using reactive programming languages general operating systems scout exposes paths similar pipe chains model offer quality service guarantees applications paths scout schedulable entities ordered according edf policy lastly programming environments ros orocos lcm already widely adopted design robotics autonomous cars changing development applications paradigms used ros example influenced thinking design systems modularity robustness amongst software components aim augment cleanflight functionality services part ongoing efforts build autonomous drones conclusions future work paper identify two semantics time namely reaction freshness time analyze context quest rtos port cleanflight flight control firmware implemented collection tasks paper describes composable pipe model built task scheduling communication abstractions using pipe model derive worst case time data flow chain tasks various conditions argue timing properties factored early stage application design thus provide mathematical framework derive feasible task periods budgets using quest scheduling framework satisfy given set timing schedulability constraints demonstrate applicability design approach using port cleanflight flight controller quest intel aero board future work integrate design approach qduino development environment provide basis design implementation autonomous multicore flight management system separation kernel enables tasks run legacy linux system parallel tasks running quest rtos references nvidia jetson board march http qualcomm snapdragon flight kit march https amazon prime air http bradley hax magnanti applied mathematical programming addisonwesley publishing company endri bregu nicola casamassima daniel cantoni luca mottola kamin whitehouse reactive control autonomous drones proceedings annual international conference mobile systems applications services mobisys pages new york usa acm bruyninckx open robot control software orocos project proceedings icra ieee international conference robotics automation cat volume pages zhuoqun cheng richard west qduino multithreaded arduino system embedded computing proceedings ieee systems symposium rtss rtss pages washington usa ieee computer society cleanflight http matthew danish richard west scheduling quest operating system proceedings embedded technology applications symposium robert davis sebastian altmeyer leandro indrusiak claire maiza vincent nelis jan reineke extensible framework multicore response time analysis systems nico feiertag kai richter johan nordlander jan jonsson compositional framework path delay calculation automotive systems different path semantics proceedings ieee system symposium workshop compositional theory technology embedded systems barcelona spain november richard gerber seongsoo hong manas saksena guaranteeing requirements calibration periodic processes ieee trans softw july arne hamann dakshina dasari simon kramer michael pressler falk wurst communication centric design complex automotive embedded systems euromicro conference systems ecrts leibniz international proceedings informatics lipics dagstuhl germany rafik henia arne hamann marek jersak razvan racu kai richter rolf ernst system level performance analysis approach 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aug perspective misrepresents frequentist inference nuts bolts learning data aris spanos department economics virginia tech usa august abstract primary objective paper revisit make case merits fisher objections framing frequentist inference argued framing congruent bayesian incongruent frequentist inference provides bayesian approach theory optimal inference misrepresents theory optimal frequentist inference framing inferences solely terms universal quantifier values denoted framing odds primary objective modelbased frequentist inference learn data true value one gave rise particular data frequentist approach relies factual estimation prediction well hypothetical testing reasoning revolve around existential quantifier paper calls question appropriateness admissibility reassesses stein paradox relates capacity frequentist estimators pinpoint paper also compares contrasts lossbased errors traditional frequentist errors coverage type former attached latter inference procedure key words decision theoretic framing bayesian frequentist inference stein paradox estimator loss functions admissibility error probabilities risk functions introduction wald framework widely viewed providing broad enough perspective accommodate compare frequentist bayesian approaches inference despite differences perceived offering neutral framing inference brings focus common features tones differences see berger robert hagan historically wald proposed original variant framework view unify neyman rendering frequentist interval estimation testing problem formulation general contains problems testing hypotheses statistical estimation treated among frequentist pioneers jerzy neyman accepted enthusiastically broader perspective primarily concepts decision rules action spaces seemed provide better framing behavioristic interpretation testing based rules see neyman neyman attitude towards wald framing also adopted wholeheartedly influential berkeley including lehmann lecam foreword collection neyman early papers described wald framing neyman vii natural far reaching extension formulation scope found abraham wald theory statistical decision end argument fisher rejected wald framing grounds seriously distorts rendering frequentist statistics attempt reinterpret common tests significance used scientific research though constituted kind acceptance procedure led decisions wald sense originated several misapprehensions led apparently several exceptions cox tukey birnbaum fisher viewpoint inadequately discussed evaluated subsequent statistics literature primary aim paper revisit fisher minority view taking closer look framework view claim provides neutral framework comparing frequentist bayesian approaches argued fisher view decision theoretic framing germane acceptance sampling misrepresents frequentist inference without merit key argument discussion follows notions loss function admissibility congruent bayesian approach incongruent primary objective underlying reasoning frequentist approach section introduces basic elements decision theoretic view bring links bayesian frequentist approaches calling question conventional wisdom concerning neutrality section takes closer look bayesian approach argues apparatus exist bayesians would forced invent order establish theory optimal bayesian inference section discusses critically notions loss functions admissibility focusing primarily role giving rise stein paradox incompatibility frequentist approach argued frequentist dimension notions loss function admissibility apparent real section makes case framework misrepresents primary objective underlying reasoning frequentist approach section revisits notion loss function dependence information data argued errors different incompatible traditional frequentist errors attached unknown parameters instead inference procedures decision theoretic basic elements framing current three basic elements prespecified parametric statistical model generically specified denotes joint distribution sample rnx denotes sample space parameter space model represents stochastic mechanism assumed given rise data decision space containing mappings rnx denotes set actions available statistician loss function representing numerical loss statistician takes action state nature see ferguson berger wasserman basic idea selects action know true state nature represented however contingent action knows losses gains utilities resulting different choices observes data provides information maps certain action guided solely original wald framing important bring fact original wald framing much narrower rendering due aim formalize approach see ferguson key differences decision action space defined exclusively terms subsets parameter space estimation set singleton points testing null alternative regions respectively original loss weight function zero loss true value discussion follows important note practice unknown general framing introduced wald broadened cam extended scope original generalizing notions loss functions decision spaces follows argued extensions created serious incompatibilities objective underlying reasoning frequentist inference addition historical methodological interest note wald introduced notion prior distribution original machinery reluctantly justified useful tool proving certain theorems situation regarding introduction priori probability distribution entirely different first objection made neyman pointed merely unknown constant variate hence makes sense speak probability distribution second even may assume variate general possibility determining distribution assumptions regarding distribution hypothetical character reason introduce hypothetical probability distribution simply proves useful deducing certain theorems calculation best system regions also first highlight extreme relativism decisiontheoretic notion optimality respect particular loss function best system regions acceptance depend weight function emphasis added shared neutral framework frequentist bayesian approaches share notion statistical model viewing data realization sample key differences three approaches frequentist approach relies exclusively bayesian approach adds prior distribution framing revolves around loss gain utility function loss function often assumed even differentiable convex function take numerous functional forms see wasserman robert bansal inter alia claim perspective provides neutral ground often justified account loss function function sample parameter spaces via two universal quantifiers associated distribution sample frequentist associated posterior distribution bayesian idea allowing values rnx goes beyond bayesian perspective relies exclusively single point obvious whether sufficient justice frequentist approach closer scrutiny suggests frequentist inference misrepresented way quantifiers used framing inference first quantifier plays minor role since relevance defining expectation transforming loss function say risk function place underlying statistical model enters framing hence perspective relevant part behavior affects risk different values undermines pivotal role quantifier determining theory optimal frequentist inference distribution sample takes center stage since sampling distribution statistic estimator test predictor derived via relevant error probabilities calibrate optimality frequentist procedures defined terms second notion optimality revolves around universal quantifier rendering congruent bayesian incongruent frequentist approach specific since different risk functions often intersect optimal rule usually selected risk function reduced scalar two choices relevant risk maximum risk rmax bayes risk hence obvious way choose among different rules find one minimizes relevant risk respect possible estimates case gives rise two corresponding decision rules minimax rule bayes rule inf rmax inf supr inf inf sense decision bayes rule considered optimal minimizes relevant risk matter true state nature happens constitutes key caveat often ignored discussions approaches viewed game nature decision maker selects action irrespective value nature chosen plays role selecting optimal rules since latter nothing true value avoid misreading line reasoning important emphasize true value shorthand saying data constitute typical realization sample distribution see spanos mayo contrasted notion optimality frequentist inference gives center stage sense evaluates capacity inference procedure inform modeler value relevant according reid statistical model family probability distributions central problem statistical inference identify member family generated data bayesian approach shed light affinity framework bayesian approach let take closer look latter bayesian inference primary objective key argument favor bayesian approach often simplicity sense forms inference revolve around single function posterior distribution however half story half posterior distribution utilized yield optimal inferences issue optimality however intrinsically related primary objective bayesian inference outsider looking bayesian approach would surmise primary objective yield probabilistic ranking ordering values modeling begins priori probabilistic ranking based revised observing derive hence key role quantifier indeed hagan argues revised probabilistic ranking inference obtained posterior density final step bayesian method derive suitable inference statements usual inference question seeing data know parameter answer question present entire posterior light question naturally arises revised probabilistic ranking based convey underlying data generating mechanism assumed given rise data ranking learning true value mode obvious highest ranked value feature posterior distribution pinpoints better value mentioned plays role selecting optimal rule since latter revolves exclusively around relevant risk contrast learning data unique value makes perfectly good sense frequentist inference since viewed unknown constant gave rise issue highlights key tension frequentist bayesian approaches primary objective bayesian inference revised probabilistic ranking answer perspective came refocus append bayesian inference hagan echoing earlier views lindley tiao box contrasting frequentist classical inferences bayesian inferences argues classical inference theory concerned constructing good inference rules primary concern bayesian inference entirely different objective extract information concerning posterior distribution present helpfully via effective summaries two criteria process first identify interesting features posterior distribution second criterion good communication summaries chosen convey clearly succinctly features interest bayesian terms therefore good inference one contributes effectively appropriating information conveyed posterior clearly hagan attempt define good bayesian inference begs question constitute effective appropriation information mean apart probabilistic ranking putting aside attempt defend stance entire posterior distribution inference hagan argues criteria optimal bayesian inferences parasitical bayesian approach enter picture via decision theoretic perspective study decision theory two potential benefits first provides link classical inference thereby shows extent classical estimators confidence intervals hypotheses tests given bayesian interpretation motivation second helps identify suitable summaries give bayesian answers stylized inference questions classical theory mentioned potential benefits bayesian approach misleading two reasons first link classical frequentist inference fraught misleading definitions unclarities respect reasoning objectives latter instance quantifier used define optimal estimators respect particular loss functions odds frequentist inference second claim concerning bayesian answers frequentist questions interest misplaced former provides real answers frequentist primary question interest pertains learning bayes rule offers little anything relevant learning value gave rise let unpack answer detail substituting risk function bayes risk one show see bansal second third equalities presume one reverse order integration technical issue treat joint distribution latter raises number questionable moves muddies distinction generic value rnx particular value see spanos light bayesian estimate optimal relative particular loss function minimizes makes clear constitutes optimal bayesian estimate primarily determined schervish bayes estimate mean bayes estimate median iii bayes mate mode practice widely used loss function square whose risk function mean square error mse surprising however definition mse denoted different frequentist mse defined key difference defined point opposed unfortunately statistics textbooks adopt one two definitions mse either ignore seem unaware first sight difference might appear pedantic turns serious implications relevant theory optimality frequentist bayesian inference procedures indeed reliance undermines completely relevance admissibility frequentist inference admissibility estimator inadmissible exists another estimator strict inequality holds least one value otherwise said admissible respect loss function duality loss functions priors affinity bayesian inference cemented duality result loss functions prior distributions see robert duality stems fact minimizing bayes risk equivalent minimizing integral result brings two important features bayesian inference first confirms minor role played quantifier bayesian optimality theory inference second indicates perfect substitutes respect weight function derivation bayes rules modifying loss function prior yields result problem estimating modified weighted loss function identical problem simple loss modified hyperparameters prior distribution form prior distribution srivastava implies practice bayesian could derive particular bayes estimate attaching weight loss function prior distribution depending derivation easier see bansal srivastava bayes rule admissibility argued sequel come surprise learn bayes rules dominate rules admissibility given center stage bayes rule respect prior distribution admissible certain regularity conditions including unique equivalence relative risk function minimax iii admissible estimate either bayes limit sequence bayes rules see wasserman srivastava results used evidence superiority bayesian perspective led intimation effective way generate optimal frequentist procedures find bayes solution using reasonable prior examine frequentist properties see whether satisfactory latter viewpoint see rubin gelman even one agree bayes rules admissible estimators largely coincide importance result hinges appropriateness admissibility frequentist estimators loss functions admissibility revisited claim discussed section notions loss function admissibility incompatible optimal theory frequentist estimation largely framed fisher see savage admissibility minimal property following example used call question notion loss function associated property admissibility optimal frequentist estimators example context simple normal model niid let use notion compare two estimators maximum likelihood estimator mle crystalball estimator compared admissibility grounds estimators admissible thus equally acceptable common sense however suggests particular criterion optimality distinguish strongly consistent unbiased fully efficient sufficient estimator arbitrarily chosen real number ignores data altogether much minimal property moment reflection suggests inappropriateness admissibility stems reliance quantifier admissibility arises fact certain values close say better grounds given primary objective frequentist estimator result seems totally irrelevant gauge capacity achieve example indicates admissibility totally ineffective minimal property filter worst possible estimator instead excludes potentially good estimators like sample median see cox hinkley highlights extreme relativism admissibility particular loss function case absolute loss function however sample median would optimal estimator determines loss function appropriate particular cases despite wholehearted embrace framing lehmann warned statisticians perils arbitrary loss functions argued choice loss function less crucial model exerts important influence nature solution statistical decision problem arbitrary choice squared error may baldly misleading relative desirability competing strong case made key minimal property necessary sufficient frequentist estimation consistency extension law large numbers estimators instance consistency would eliminated consideration inconsistent makes intuitive sense estimator pinpoint infinite data information considered impertinent contrast nothing notion admissibility advances learning data relative particular loss functions efficiency dubious property frequentist estimators pertinent measure finite sample precision frequentist estimators full efficiency defined relative assumed statistical model arbitrary loss function based information data stein paradox admissibility quintessential example bolstered appeal bayesian claims concerning admissibility estimator efron morris gave rise extensive literature shrinkage estimators see saleh let independent sample normal distribution known using notation denoted find optimal estimator respect square overall loss function stein astounded statistical world showing leastbls admissible bls squares estimator inadmissible indeed james stein able come nonlinear estimator bjs kxk bls became known estimator dominates terms demonstrating bjs bls turns also inadmissible dominated modified estimator admissible kxk max see wasserman traditional interpretation result normal independent model estimator reduces overall result seems imply one better overall terms using combined nonlinear shrinkage estimator instead estimating means separately surprising result statistical reason due independence connect inferences pertaining different individual means yet obvious estimator inadmissible argued next result calls question appropriateness notion admissibility respect particular loss function judiciousness frequentist estimation frequentist inference learning data objectives underlying reasoning frequentist inference inadequately discussed statistics literature result key differences bayesian inference remain beclouded frequentist approach primary objective reasoning forms parametric frequentist inference begin prespecified statistical model model chosen set possible models could given rise data selecting probabilistic structure underlying stochastic process way render observed data typical realization thereof light fact value represents different element family models represented primary objective frequentist inference learn data true model denotes true value typicality testable data using misspecification testing see spanos frequentist approach relies two modes reasoning inference purposes factual estimation prediction hypothetical hypothesis testing denotes true value denote hypothesized values associated hypotheses constitute partition frequentist estimator aims optimality evaluated effectively achieves similarly test statistic usually compares good estimator prespecified value behind value assumed generated data hence hypothetical reasoning used testing learn nothing possible values contradicts misleading claims bayesian textbooks robert frequentist paradigm relies criterion risk function compare estimators possible select best estimator reasoning estimators evaluated performance possible values parameter contrary claim relevant value evaluating optimality misleading claims stem apparent confusion existential universal quantifiers framing certain inferential assertions existence formally defined using existential quantifier exists introduces potential conflict existential universal quantifier neither decision theoretic bayesian approach explicitly invoke bayesian rules considered optimal minimize expected loss matter happens different two quantifiers demonstrated using elementary logic logical connective negation used define following equivalence relationships two quantifiers given involve double negations two quantifiers could apart first sight quantifier seems rather innocuous natural context statistical inference seems intuitively obvious one care behavior estimator possible values misleading intuition however since behavior although relevant determines effective frequentist estimator pinpointing matters behavior around assessing effectiveness calls evaluating deductively sampling distribution hypothetical values possible values let unpack details claim frequentist estimation underlying reasoning frequentist estimation factual sense optimality estimator appraised terms generic capacity pinpoint true value whatever sample realization optimal properties like consistency unbiasedness full efficiency sufficiency calibrate generic capacity using sampling distribution evaluated terms instance strong consistency asserts almost surely lim similarly unbiasedness asserts sampling distribution mean equal sense optimal properties defined point achieved using factual reasoning evaluating sampling distribution true state nature without know contrast using loss functions defined terms rendered without knowing example case simple normal model point estimator consistent unbiased fully efficient sufficient sampling distribution usually explicitly stated evaluation distribution factual formally denoted standardized yields pivotal function whose distribution holds true value provides basis constructing confidence interval asserts random interval cover overlay true mean whatever happens probability equivalently error coverage hence frequentist estimation coverage error probability depends sampling distribution attached random interval values without requiring one know evaluation calls question definition unbiasedness frequentist estimation since assertion makes sense values make sense defined similarly appropriate frequentist definition mse estimator initially proposed fisher defined point indeed decomposition meaningful defined point true mean since definition bias thus variance bias involve two values estimator unbiased implies apparent affinity defined variance estimator apparent real latter makes frequentist sense single point estimator frequentist perspective proper frequentist evaluation result important bring conflict overall mse factual reasoning underlying frequentist estimation latter perspective estimator raises several issues concern bls bjs estimafirst tors inconsistent estimators since underlying model suffers incidental parameter problem essentially one observation unknown parameter number unknown parameters increases rate bring futility comparing two estimators clearly consider following simpler example example let sample simple normal model comparing two estimators inferring relatively efficient relative square loss function mse mse totally uninteresting estimators inconsistent second able discuss role admissibility stein result need consider consistent estimator extending original data panel longitudinal data sample xmt case consistent estimators bls xkt enables evaluate notion relatively better objectively admissibility relative overall loss function introduces tradeoff accuracy estimators individual parameters overall expected loss question sense overall mse among group mean estimates provides better measure error learning true values short answer indeed overall mse irrelevant primary objective estimation learn data particular loss function penalizes estimator capacity trading increase bias decrease overall mse latter misleadingly evaluated estimator flouts primary objective favor reducing overall mse summary discussion suggests nothing paradoxical stein original result problematic estimator choice better terms admissibility relative overall mse evaluating accuracy estimators frequentist hypothesis testing another frequentist inference procedure one employ learn data hypothesis testing question posed whether close enough prespecified value contrast estimation reasoning underlying frequentist testing hypothetical nature legitimate frequentist error probabilities testing hypotheses prespecified value one utilizes sampling distribution transforms pivot test statistic replacing specified value yielding however instead evaluating factual evaluated various hypothetical scenarios associated yield two types hypothetical sampling distributions cases underlying reasoning hypothetical sense factual replaced hypothesized values test statistic provides standardized distance hypothesized values true assumed underlie generation data yielding using sampling distribution one define following legitimate error probabilities significance level using sampling distribution one define type error prob power shown test defined test statistic rejection region constitutes uniformly powerful ump test significance level see lehmann type error probability associated test erroneously rejecting accepting type error probabilities evaluate generic capacity whatever sample realization test reach correct inferences contrary bayesian claims error probabilities nothing temporal physical dimension metaphor associated repeated samples relevant feature metaphor repeatability principle dgm represented feature easily operationalized using computer simulation see spanos key difference significance level former latter error probability indeed viewed smallest significance level would rejected data legitimacy error probabilities underlying hypothetical reasoning used beyond rules provide evidential interpretation pertaining discrepancy null warranted data see mayo spanos despite fact frequentist testing uses hypothetical reasoning main objective also learn data true model test statistic like constitutes nothing scaled distance value behind generation hypothesized value replaced best estimator revisiting loss risk functions discussion raises serious questions role loss functions admissibility evaluating learning data particular extraneous information concerning costs associated different parameter values learning sense inconsistent relatively particular loss function efficient optimal estimator learning iii overall mse important learning data true values loss functions come closer scrutiny set reveals loss function needs invoke information sources data usually readily available indeed information available restrictive situations acceptance sampling quality control light proper understanding intended scope statistical inference calls distinguishing special cases loss function part parcel available substantive information information either relevant available tiao box reiterated fisher distinction undoubtedly true one hand situations exist loss function least approximately known example certain problems business sampling inspection sort hand vast number inferential problems occur particularly analysis scientific data way knowing advance use results research subsequently cox went questioned framing even cases inference might involve decision reasons detailed techniques seem fairly limited applicability even fairly clear cut decision element involved may except fields control theory acceptance sampling major contribution statistical technique presenting evidence incisive form discussion rather providing mechanical presentation final decision especially case single major decision involved central difficulty may formulating elements required quantitative analysis rather combining elements via decision another important aspect using loss functions inference practice seem inference since bring problem information data particular statistical inference problem give rise different depending one loss function illustrate consider example chatterjee consider case new drug whose effects studied research scientist attached laboratory pharmaceutical company conclusion study may different bearings action taken scientist whose line investigation would depend company whose business decisions would determined government whose policies health care drug control etc would take shape practice one different agents likely different loss function inferences common denominator scientific evidence relating true stems solely observed data finally extreme relativism loss function optimality renders decisiontheoretic bayes rules highly vulnerable abuse practice one justify estimator optimal however lame terms criteria selecting appropriate loss function example consider manufacturer high precision bolts nuts information buyer checks first last box quality control accepting order suggests minimize losses stemming return products defective appropriate loss function might optimal estimator relative terrible estimator pinpointing inconsistent loss functions inherent distance functions notion loss function stemming information data raises another source potential conflict stems fact within statistical model exists inherent statistical distance function often relating score function hence stemming information contained data see casella berger distribution underlying normal inherent distance function comparing estimators mean square hand distribution laplace relevant statistical distance function absolute distance see shao similarly distribution underlying uniform inherent distance function sup key feature distance functions defined point traditional loss functions question naturally arises might make sense ignore inherent distance functions compare estimators using externally given loss function key difference two assumptions comprise likelihood function testable data underlying loss function moreover likelihood function gives rise global notion optimality known full efficiency defined terms fisher information optimal estimator depends information contained statistical model contrasts admissibility property defined terms local optimality relative loss function based outside information evaluated decisions inferences discussion brings crucial distinction decision inference stemming data even wald introduced perspective fisher perceptively argued field pure research assessment cost wrong conclusions delay arriving correct conclusions conceivably pretence case assessment would inadmissible irrelevant judging state scientific tukey echoed fisher view contrasting decisions inferences like human endeavor science involves many decisions progresses building fairly well established body knowledge body grows reaching conclusions acts whose essential characteristics differ widely making decisions conclusions established careful regard evidence without regard consequences specific actions specific tukey also recognized decision theory distorts frequentist testing replacing error probabilities losses costs wald decision theory given fixed probability errors first kind focused gains losses hacking brought key difference inference pertaining evidence hypothesis decision something result inference conclude hypothesis best supported apparently decide hypothesis question best supported hence decision like inference fallacious deciding something case differs deciding something hence deciding something falls squarely province decision theory deciding something case issue elaborated upon birnbaum two contrasting interpretations decision concept formulated behavioral applicable decisions concrete literal sense acceptance sampling evidential applicable decisions reject research context pattern strength statistical evidence concerning statistical hypotheses central acceptance sampling learning data let bring key features situation decisiontheoretic set makes perfectly good sense situation fisher called acceptance sampling industrial production process objective quality control make decision pertaining shipping substandard products nuts bolts buyer using expected ultimate criterion acceptance sampling context mse risk function relevant evaluate genuine losses associated decision related choice estimate say cost observed percentage defective products nothing type error probabilities acceptance sampling differs scientific enquiry two crucial respects primary aim use statistical rules guide actions astutely use order minimize expected loss associated decision sagacity actions determined respective losses stemming relevant information data cox hinkley key difference acceptance sampling scientific inquiry primary objective latter minimize expected loss costs utility associated different values use data learn true model two situations drastically different mainly key notion true calls question acceptance sampling set indeed loss function defined penalize since reason believe lowest ranked would coincide unless accident consider case acceptance sampling resembles hypothesis testing far final products randomly selected inspection production process situation main objective viewed operationalizing probabilities false view minimize expected losses conventional wisdom situation similar enough testing render latter appropriate framing decision ship particular batch however closer look examples used illustrate situation silvey reveals decisions driven exclusively risk function quest learn data true instance way addressing two types error probabilities fixing small value seek test minimizes type error probability seems utterly irrelevant context one easily think loss function optimal calls much larger type type error probability light discussion different acceptance sampling two types error probabilities determined risk function attempt learn data indeed learning deliberately undermined certain loss function overall mse favor biased estimators type given crucial differences one make strong case objectives underlying reasoning acceptance sampling drastically different pertaining learning data scientific context expected loss legitimate frequentist error key question whether expected loss legitimate frequentist error like bias mse type error legitimate frequentist errors common first stem directly statistical model since underlying sampling distributions estimators test statistics predictors derived exclusively distribution sample via sense relevant error probabilities directly related statistical information pertaining data summarized statistical model second attached particular frequentist inference procedure relate relevant inferential claim error probabilities calibrate effectiveness inference procedures learning data true statistical model light features question sense risk function could potentially represent relevant frequentist errors argument risk function represents legitimate frequentist errors derived taking expectations respect robert misguided two reasons relevant errors estimation including bias mse evaluated respect invoking factual reasoning assumed state nature wald original loss function represents interesting case defined terms renders evaluated since unknown practice contrast errors associated bias mse rendered operational factual reasoning fashioned forgo knowing expected losses stemming risk function attached particular values assignment direct conflict legitimate error probabilities attached inference procedure never particular values expected loss assigned value nothing learning data indeed risk function penalize procedure since latter unknown practice direct conflict main objective frequentist estimation sync acceptance sampling objective inference everything expected losses summary conclusions paper makes case fisher assertions concerning appropriateness framing acceptance sampling inappropriateness frequentist inference closer look framing reveals congruent bayesian approach provides theory optimal inference bayesian rules considered optimal minimize expected loss possible values irrespective true value happens contrast theory optimal frequentist inference revolves around true value since depends entirely capacity procedure pinpoint frequentist approach relies factual estimation prediction well hypothetical testing reasoning revolve around existential quantifier inappropriateness quantifier calls question relevance admissibility minimal property frequentist estimators strong case made relevant minimal property frequentist estimators consistency addition full efficiency provides relevant measure estimator finite sample efficiency accuracy pinpointing properties stem underlying statistical model contrast admissibility relies loss functions based information data argued stein result stems fact admissibility introduces accuracy estimator pinpointing overall expected loss estimator achieves higher overall mse blunting capacity frequentist estimator pinpoint would frequentist care overall mse defined expected losses legitimate errors similar bias mse properly defined well coverage type errors latter attached frequentist procedures calibrate capacity achieve learning data contrast expected losses 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based classical theory probability philosophical transactions royal society london series neyman lectures conferences mathematical statistics probability department agriculture washington neyman selection early statistical papers neyman university california press neyman foundations behavioristic statistics godambe sprott foundations statistical inference holt rinehart winston canada toronto hagan bayesian inference edward arnold london reid statistical sufficiency international encyclopedia social behavioral sciences edited wright edition vol elsevier oxford robert bayesian choice foundations computational implementation springer rubin bayesianly justifiable relevant frequency calculation applied statistician annals statistics saleh theory preliminary test estimation applications savage rereading fisher annals statistics schervish theory statistics shao mathematical statistics springer silvey statistical inference chapman hall london spanos statistical models come revisiting problem 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statistics wald statistical decision functions wiley wasserman statistics springer
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facial expression recognition based complexity perception classification algorithm tianyuan chang guihua wen yang jiajiong school computer science engineering south china university technology guangzhou china crghwen abstract facial expression recognition fer always challenging issue computer vision different expressions emotion uncontrolled environmental factors lead inconsistencies complexity fer variability expression categories often overlooked facial expression recognition systems order solve problem effectively presented simple efficient cnn model extract facial features proposed complexity perception classification cpc algorithm fer cpc algorithm divided dataset easy classification sample subspace complex classification sample subspace evaluating complexity facial features suitable classification experimental results proposed algorithm datasets demonstrated algorithm effectiveness superiority approaches introduction facial expression recognition fer wide range research prospects human computer interaction affective computing including polygraph detection intelligent security entertainment internet education intelligent medical treatment etc know facial expression major way expressing human emotions hence main task determining emotion automatically reliably efficiently recognize information conveyed facial expressions fer research ekman friensen first proposed facial action coding system facs friensen ekman six basic categories expressions surprise sadness disgust anger happiness fear defined facs commonly used basic expression labels fer methods roughly divided three types sun yan methods methods hybrid methods methods describe features mainly geometric relationships key points face face positions shapes methods extract entire facial image whole lastly hybrid methods bine first two methods extract local global facial features face images respectively compared methods utility methods requires improvement difficult accurately label key points effective positions face practical applications fer works focus feature extraction classifier construction divided static dynamic classification methods corneanu static classification methods applicable single static image include svm classifiers bayesian network classifiers random forests softmax classifiers dynamic classification applied facial video sequences taking account features independently extracted frame time basis main dynamic classification models hmm walecki recent years different methods traditional machine learning used fer extract appearance features images including gabor filters local binary patterns lbp savran local gabor binary patterns lgbp zhang histograms oriented gradients hog dahmane feature transform sift berretti traditional methods tend effective specific small sample sets difficult adjust identify new face images brings challenges fer uncontrolled environments extracted features often belong features hard extract find meaningful information distinguishes different categories data however new recognition framework utilizing convolutional neural network cnn ranzato liu kim based deep learning network achieved remarkable results fer used feature extraction recognition multiple convolution pooling layers cnn may extract higher features entire face local area good classification performance facial expression image features present still deficiencies fer many related work focus improvement classification models feature extraction methods easily ignoring relevance several basic expression categories inconsistencies complexity extracting features described lopes difficult tively partition expression feature space expressions like happiness surprise belong highly recognizable categories easily distinguished facial features meanwhile expressions like fear sadness similar situations making difficult distinguish effectively addition facial images easily influenced ethnicity age gender hair uncontrolled factors resulting different facial feature distributions facial feature complexity classification supposing inconsistencies training samples test samples matter suitable feature extraction training samples expression recognition prediction result test samples accurate analogy ask excellent student grasped primary mathematics knowledge tested knowledge higher mathematics university obvious complexity unequal knowledge needs distinguished learning order effectively solve problems paper proposed complexity perception classification cpc algorithm fer referenced simple effective principle algorithm firstly divided dataset two parts difficult classification sub dataset easy classification sub dataset done evaluating complexity facial features expression recognition sample next step separately learn two different feature distributions trained two specific classifiers two sub datasets obtain complexity different sample features instead using uniform classifier predict facial expression algorithm divided test sample corresponding classifiers complete fer facial feature complexity discriminator main contributions summarized follows according scale facial expression dataset proposed simple efficient cnn model based resnets facial feature extraction effectively alleviated problem gradient disappearance enhanced information flow deep network new heuristic classification algorithm named complexity perception classification cpc proposed improved recognition accuracy higher recognizable facial expression categories also alleviated problem misclassified expression categories achieved performance static facial expression recognition benchmarks including dataset dataset work blocks fully connected layers order extract high level facial features selected output second dimensional fully connected layer feature representation learning simple validity occam razor principle proposed complexity perception classification cpc algorithm solve complexity inconsistency problem fer section firstly illustrated evaluate complexity training datasets employed discriminant learining different disjoint sub datasets sample complexity discriminator clearly determine complexity test sample recognition figure shows overview pipeline cpc system feature extraction classical improvement cnn frameworks using resnet densenet huang alleviated problem deep networks prone gradient disappearance backpropagation also remarkably improved performance image classification end motivated network architectures proposed cnn framework facial feature extraction shown figure cnn framework contained multiple convolutional layers modified residual net figure pipeline cnn framework restnets output feature mapping residual block consists transformed composite function identity function combined equation combination may hinder flow information deep networks order improve flow information layers improved combination mode residual block motivated densenet longer summated two inputs concatenated two feature mappings output function feature mapping shown equation figure shows structure traditional residual block modified residual block figure left traditional residual block right modified residual block complexity perception classification algorithm fer figure pipeline complexity perception classification cpc system evaluating complexity features recognition discriminant learning adopted feature extraction using cnn input fer system feature value feature extracted original image feature extraction training samples randomly divided folds sample feature vectors feature dimension sample order improve generalization ability base classifier distinguish easily classifiable samples difficultly classifiable samples first chose fold samples training set remaining folds samples test set resulted base classifiers different training datasets repeated process times obtain trained base classifiers training sample feature vector predicted base classifiers counting correct number prediction expression categories computing ease degree classification sample denotes number base classifiers correct classification number base classifiers training sample evaluated complexity sample classification features ease degree classification addition set parameter named ease threshold boundary distinguish easily classifiable samples difficultly classifiable samples shown equation ease degree classification training sample divided easy classification sample subspace contrast divided difficult classification sample subspace ease degree classification training sample divided training datasets two disjoint sub datasets evaluating feature complexity recognition order achieve discriminant learning different sub datasets trained easy sample classifier easy classification sample subspace trained difficult sample classifier difficult classification sample subspace way could learn different feature distributions two different sample subspaces provide accurate recognition performance test sample easy sample classifier difficult sample classifier follows specified classification method softmax linear svm random forest etc denotes easy classification dataset denotes expression labels similarly denotes difficult classification dataset denotes expression labels sample complexity discriminator order able clearly determine whether test sample belongs easy classification sample subspace difficult classification sample subspace dynamically generated different sample complexity discriminators different test samples sample complexity discriminator model described follows label easy classification dataset label difficult classification dataset sample complexity discriminator model denotes dynamic classification method adopted knn algorithm find neighbors test sample training set dynamically trained sample complexity discriminator nearest neighbor training set dynamic classification method could reflect advantages softmax also improve crimination accuracy feature complexity test sample test sample predicted sample complexity discriminator used easy sample classifier recognize facial expression contrast label used difficult sample classifier recognize facial expression algorithm elaborates implementation method details cpc algorithm algorithm complexity perception classification cpc input feature vector sample classifier complexity discriminator ease threshold output predictive expression label testing sample training partition data training validation test sets partition training data folds apply train training data gets base classifier end end apply base classifiers compute ease degree classification sample training sample divide sample else divide sample end testing training sample end experiment proposed algorithm mainly applied static facial expression recognition evaluated performance proposed complexity perception classification cpc algorithm applying goodfellow lucey datasets datasets dataset facial expression recognition challenge dataset icml launched kaggle dataset contains training images validation images test images image size facial expression classified seven different types dataset extension dataset contains labeled facial videos classified figure confusion matrices facial expressions two datasets baseline result feature extraction cnn predicted softmax cpc result feature extraction cnn predicted softmax adding cpc algorithm one seven categories mentioned categories used dataset extracted four frames sequence dataset contains total facial expressions experimental settings feature extraction phase cnn preprocessed dataset zca whitening global relative normalization effectively remove redundant information input image reduce correlation adjacent pixels image phase feature extraction cnn designed figure used training dataset train networks initial network learning rate set decay learning rate per epoch mini batch size momentum set dropout set activation function used convolutional layers rectified linear unit relu activation function stochastic gradient descent sgd used optimization algorithm addition last fully connected layer used softmax activation function baseline classification cpc firstly observed impact cpc algorithm recognition accuracy different expression categories figure illustrates resulting confusion matrices testing accuracies two datasets employed cross validation ensure stability results effectiveness proposed algorithm verified ing effect complexity perception classification algorithm recognition results feature extraction cnn figure clearly see recognition rates happiness significantly higher expressions belonging easily distinguishable category meanwhile recognition rates fear difficult distinguish adding complexity perception classification algorithm figure found following recognition rate happiness increased case high baseline recognition rate recognition rate fear similarly increased error rate related mistaking fear sadness decreased exciting recognition results demonstrated proposed algorithm improved recognition accuracy easily distinguishable categories also alleviated easy misclassification difficultly distinguishable categories recognition rates anger sadness neutral also increased proposed algorithm improved recognition rates categories none recognition rates expression classes decreased due addition proposed algorithm notice proposed algorithm raise recognition accuracy certain classes cost sacrificing accuracy classes meaningful practical application facial expression recognition figure recognition accuracy natural class increased categories experienced change recognition accuracy upon adding proposed algorithm results confusion matrix dataset also validate conclusion performance cpc algorithm table show recognition accuracy different classifiers two datasets extracted facial features cnn compared recognition effect cpc algorithm baseline recognition experiment respectively trained two normal classifiers baseline recognition employing feature vectors training set two datasets experimental results showed cpc algorithm significant performance facial expression recognition compared baseline recognition rate test set average recognition rate softmax linear svm random forest classifiers enhanced respectively adding cpc algorithm average baseline recognition rate test set three different classifiers reached introduced cpc algorithm average recognition increased comparison methods compared proposed cnn feature extraction based cpc algorithm existing related methods fer table table respectively show recognition accuracies different fer methods proposed table comparison recognition accuracies baseline cpc different classifiers datasets method baseline cpc baseline cpc table recognition rates dataset methods method recognition rate unsupervised goodfellow maxim milakov goodfellow mollahosseini tang liu dnnrl guo kim proposed approach table recognition rates dataset methods method recognition rate mollahosseini liu happy sun yan barman lopes sariyanidi proposed approach algorithm datasets seen table proposed algorithm achieved competitive results dataset table shows proposed algorithm outperformed compared fer methods dataset parameter analysis investigated importance ease threshold parameter complexity perception classification algorithm order determine parameter value produced fer best performance datasets employed cross validation validation set dataset figure figure show recognition accuracy algorithm versus different values ease threshold two datasets compares recognition accuracy baseline seen two figures recognition acknowledgments work supported china national science foundation science technology planning project guangdong province guangdong science technology planning project references figure recognition accuracy proposed algorithm versus different values validation set figure recognition accuracy proposed algorithm versus different values dataset accuracy algorithm exceeded baseline values illustrates robustness algorithm also found random forest classifier stable enough due randomness addition relationship recognition accuracy different values similar linear svm softmax classifiers conclusions paper designed new efficient cnn model facial feature extraction proposed complexity perception classification cpc algorithm facial expression recognition algorithm distinguished easily classifiable subspace difficultly classifiable subspace achieve discriminant learning facial feature distribution experimental results datasets demonstrated algorithm outperformed methods facial expression recognition terms mean recognition accuracy friensen friensen ekman emotional facial action coding system technical report university california san francisico sun wenyun sun haitao zhao zhong jin efficient unconstrained facial expression recognition algorithm based stack binarized binarized neural networks neurocomputing yan haibin yan collaborative discriminative learning facial expression recognition video pattern recognition corneanu corneanu cohn survey rgb thermal multimodal approaches facial expression recognition history trends applications ieee transactions pattern analysis machine intelligence vuong tang huang expression recognition dynamic faces using robust shape features ieee international conference automatic face gesture recognition workshops ieee pages walecki walecki rudovic pavlovic pantic variablestate latent conditional random fields facial expression recognition action unit detection proc ieee int conf autom face gesture savran savran cao nenkova verma temporal bayesian fusion affect sensing combining video audio lexical ieee trans pages dahmane dahmane meunier emotion recognition using dynamic hog features ieee international conference automatic face gesture recognition workshops ieee berretti berretti amor daoudi facial expression recognition using sift descriptors automatically detected keypoints visual computer international journal computer graphics zhang zhang chen local gabor binary pattern histogram sequence lgbphs novel model face representation recognition tenth ieee international conference computer vision ieee pages ranzato ranzato susskind mnih hinton deep generative models applications recognition computer vision pattern recognition ieee pages liu mengyi liu wang chen learning expressionlets manifold dynamic facial expression recognition ieee conference computer vision pattern recognition ieee computer society pages kim kim lee roh hierarchical committee deep cnns decision fusion static facial expression recognition acm international conference multimodal interaction acm pages lopes lopes aguiar afd souza facial expression recognition convolutional neural networks coping data training sample order pattern recognition pages zhang ren sun deep residual learning image recognition computer vision pattern recognition ieee pages huang huang liu lvd maaten weinberger densely connected convolutional networks goodfellow goodfellow erhan carrier challenges representation learning report three machine learning contests international conference neural information processing springer berlin heidelberg pages lucey lucey cohn kanade saragihet extended dataset complete dataset action unit expression computer vision pattern recognition workshops ieee pages mollahosseini ali mollahosseini chan mahoor going deeper facial expression recognition using deep neural networks applications computer vision ieee pages tang tang deep learning using linear support vector machines computer science liu kuang liu zhang pan facial expression recognition cnn ensemble international conference cyberworlds ieee pages guo yanan guo xiong deep neural networks relativity learning facial expression recognition ieee international conference multimedia expo workshops ieee computer society pages kim bokyeong kim roh dong hierarchical committee deep convolutional neural networks robust facial expression recognition journal multimodal user interfaces pages chongliang wang hidden markov model facial expression recognition ieee international conference workshops automatic face gesture recognition ieee pages happy happy routray automatic facial expression recognition using features salient facial patches ieee transactions affective computing pages sun yaxin sun wen cognitive facial expression recognition constrained dimensionality reduction neurocomputing barman asit barman dutta facial expression recognition using distance shape signature features pattern recognition letters sariyanidi evangelos sariyanidi gunes cavallaro learning bases activity facial expression recognition ieee transactions image processing pages
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mobile unmanned aerial vehicles uavs internet things communications mohammad walid mehdi sep wireless electrical computer engineering department virginia tech usa emails mmozaff walids cwc centre wireless communications oulu finland email bennis mathematical algorithmic sciences lab huawei france paris france centralesupelec france email abstract paper efficient deployment mobility multiple unmanned aerial vehicles uavs used aerial base stations collect data ground internet things iot devices investigated particular enable reliable uplink communications iot devices minimum total transmit power novel framework proposed jointly optimizing placement mobility uavs association uplink power control first given locations active iot devices time instant optimal uavs locations associations determined next dynamically serve iot devices network optimal mobility patterns uavs analyzed end based activation process iot devices time instances uavs must update locations derived moreover optimal trajectory uav obtained way total energy used mobility uavs minimized serving iot devices simulation results show using proposed approach total transmit power iot devices reduced compared case stationary aerial base stations deployed addition proposed approach yield maximum enhanced system reliability compared stationary case results also reveal inherent tradeoff number update times mobility uavs transmit power iot devices essence higher number updates lead lower transmit powers iot devices cost increased mobility uavs ntroduction use unmanned aerial vehicles uavs flying wireless communication platforms received significant attention recently one hand uavs used wireless relays improving connectivity coverage ground wireless devices hand uavs act mobile aerial base stations provide reliable downlink uplink communications ground users boost capacity wireless networks compared terrestrial base stations advantage using aerial base stations ability provide communications furthermore high altitude uavs enables effectively establish los communication links thus mitigating signal blockage shadowing due adjustable altitude mobility uavs move towards potential ground users establish reliable connections low transmit power hence provide solution collect data ground mobile users spread geographical area limited terrestrial infrastructure indeed uavs play key role internet things iot composed small devices sensors health monitors devices typically unable transmit long distance due energy constraints iot scenarios uavs dynamically move towards iot devices collect iot data transmit devices communication ranges transmitters case uavs play role moving aggregators base stations iot networks however effectively use uavs iot several challenges must addressed optimal deployment mobility use uavs outlined authors investigated optimal trajectory uavs equipped multiple antennas maximizing uplink communications work maximizes throughput uav system jointly optimizing uav trajectory well transmit power however works considered single uav models investigated optimal deployment movement single uav supporting downlink wireless communications work proposed algorithm optimal deployment multiple uavs provide coverage ground users work provided comprehensive downlink coverage analysis network finite number uavs serve ground users authors used uavs efficiently collect data recharge clusters head wireless sensor network partitioned multiple clusters however work limited static sensor network investigate optimal deployment uavs energy efficiency uplink data transmission communication network investigated presence uavs considered fact none prior studies addressed problem jointly optimizing deployment mobility uavs device association uplink power control enabling reliable communications iot devices best knowledge paper one first comprehensive studies joint optimal deployment aerial base stations device association uplink power control iot ecosystem main contribution paper introduce novel framework optimized deployment mobility multiple uavs purpose uplink data collection ground iot devices particular consider iot network iot devices active different time instances minimize total transmit power iot devices given sinr constraints propose efficient approach jointly dynamically find uavs locations association devices uavs optimal uplink transmit power proposed framework composed two key steps first given locations iot devices propose solution optimizing deployment association uavs case solve formulated problem decomposing two subproblems solved iteratively first subproblem given fixed uavs locations find jointly optimal association devices transmit power second subproblem given fixed device association determine joint uavs locations subproblem transform continuous location optimization problem convex form provide tractable solutions next following proposed algorithm results solving second subproblem used inputs first subproblem next iteration show proposed approach leads efficient solution reasonable accuracy compared global optimal solution requires significant overhead clearly uavs locations device association obtain first step depend locations active iot devices second step analyze iot network time period set active devices changes case present framework optimizing uavs mobility allowing dynamically update locations depending devices activation process first derive expressions time instances update times uavs must move according activation process devices next using update time results derive optimal uavs trajectory total movement uavs updating locations minimized simulation results show using proposed approach total transmit power iot devices significantly reduced compared case stationary aerial base stations deployed results also verify analytical derivations update times reveal inherent tradeoff number updates mobility uavs transmit power iot devices particular shown higher number updates leads lower transmit powers iot devices cost higher uavs energy consumptions rest paper organized follows section present system model problem formulation section iii presents optimal deployment uavs device association section address mobility update time uavs section provide simulation analytical results section draws conclusions ystem odel roblem ormulation consider iot system consisting set iot devices examples devices include various types sensors used environmental monitoring smart traffic control smart parking devices system set rotary wing uavs must deployed collect data ground iot devices uavs dynamically move needed effectively serve iot devices using uplink communication links term served uav implies uplink sinr threshold thus uav successfully collect data ground iot device model assume devices transmit data uavs uplink using frequency division multiple access fdma orthogonal channels let emax maximum energy uav spend movement locations device uav respectively given uav xuav shown fig model consider centralized network locations devices uavs known control center located central cloud server cloud server determine uavs locations device association transmit power iot device analyze iot network within time interval iot devices active different time instances must served uavs time slots beginning slot positions uavs well association updated based locations currently active devices assumed known cloud hereinafter time instance uavs locations associations jointly updated referred update time update times denoted number updates update time based location active devices optimal uavs locations corresponding association must determined effectively serving ground devices iot devices become active served uavs time period note uavs locations device association change next update time clearly since different update times different subset devices might active locations uavs must dynamically change update time therefore uav trajectory consist stop locations uav serves ground devices note model uavs locations necessarily updated set active devices changes instead consider specific time instances update times uavs locations device associations devices transmit power optimized particular considering fact set active devices may continuously change continuously updating uavs locations devices transmit powers associations may feasible lead low reliability high uavs energy consumption need solve complex optimization processes model update times design parameters depend activity devices energy uavs given model objective find optimal joint uavs locations device association update time minimize total transmit power active devices meeting device sinr requirement moreover need develop framework determining update times well uavs mobility handle dynamic changes activity devices end first present channel model activation models iot devices path loss model model optimizing locations uavs information available includes ground devices locations type environment rural suburban urban highrise urban note practical scenarios one additional information exact locations heights number obstacles therefore one must consider consider static iot devices applications fixed locations activation patterns known cloud center uav move uavj uav move devicei control center iot device fig system model randomness associated los nlos links designing communication system therefore communications device typically los view towards specific uav given probability los probability depends environment location device uav well elevation angle one suitable expression los probability given plos exp constant values depend carrier frequency type environment rural urban dense urban elevation angle clearly dij uav dij xuav distance device uav see increasing elevation angle increasing uav altitude los probability increases path loss model los nlos links device uav given dij lij dij los link nlos link carrier frequency path loss exponent excessive path loss coefficients los nlos cases speed light note nlos probability pnlos plos typically given locations uavs devices possible exactly determine path loss type experienced link case path loss average considering los nlos links used communications using average path loss device uav expressed dij dij plos pnlos plos pnlos dij clearly average channel gain uav device note using average channel gain need account los nlos links separately hence sinr expressions become tractable therefore use average channel gain model interference desired links communications computing sinrs iot device activation model indeed activation iot devices depends services supporting instance applications weather monitoring smart grids home automation iot devices need report data periodically however iot devices random activations health monitoring smart traffic control applications therefore uavs must properly deployed collect iot devices data dynamically adapting activity patterns devices naturally optimal locations uavs update times depend activation process iot devices consider two activation models first model iot devices randomly activated smart traffic control applications case concurrent transmissions massive number devices within short time duration lead bursty traffic pointed fact massive iot devices attempt transmit within short time period arrival patterns become bursty thus suggests beta distribution capture traffic characteristic iot devices case iot device active time following beta distribution parameters time interval within iot devices active beta function parameters addition iot devices smart meters typically report data periodically rather randomly devices activation process deterministic assumed known advance case assume device becomes active seconds time duration clearly number activations device channel assignment strategy given devices locations practical channel assignment approach assign different channels devices located proximity approach significantly mitigates possibility strong interference two closely located devices channel assignment problem adopted constrained clustering strategy efficient clustering approach set given points grouped clusters based proximity case given number active devices number orthogonal channels group devices based proximity assign different channels devices group present optimization problem find uavs locations device association transmit power iot devices update time min pmax total number active devices update time set devices index transmit power vector element transmit power device also location uav device association vector element index uav assigned device pmax maximum transmit power iot device noise power furthermore average channel gain device uav function uav location also average channel gain interfering device uav set devices use channel device create interference sinr target must achieved devices represents sinr requirement shows maximum transmit power constraint hereinafter call original problem note transmit power iot devices locations uavs associations unknowns clearly locations uavs impact channel gain devices uavs hence affect transmit power device furthermore given due mutual interference devices transmit power device depends also transmit power interfering devices well associations addition associations depend uavs locations also unknowns therefore mutual dependency fig block diagram proposed solution optimization variables moreover considering constraint see optimization problem highly indeed solving significantly challenging due mutual dependency optimization variables nonconvexity problem next propose framework solving optimization problem essence proposed framework solving proceeds follows update time given fixed uavs locations find optimal association transmit power devices next given fixed uav association previous step determine locations uavs update transmit power devices accordingly procedure done iteratively uavs locations device association transmit power devices found clearly step total transmit power devices decreases hence proposed algorithm converges fig shows block diagram summarizes main steps solving next discuss detail block proposed solution fig iii uav eployment evice ssociation ower control given locations active iot devices minimize total transmit power devices solving clearly uavs locations device association mutually dependent particular find device association locations uavs must known moreover uavs locations optimized without knowing device association therefore decompose two subproblems solved iteratively first subproblem given locations uavs find optimal device association transmit power devices uplink sinr requirements active devices satisfied minimum total transmit power second subproblem given device association resulting first subproblem determine locations uavs transmit power devices minimized note iterative process results subproblem used subproblem next iteration computations performed control center uavs locations device association transmit power devices obtained note given limited number available orthogonal channels interference devices depend number active devices update time clearly interference number active devices time less number orthogonal channels equivalently given scenario one provide tractable analysis deployment association steps therefore investigate interference scenarios separately device association power control given initial locations uavs aim find optimal device association well transmit power iot device total transmit power used successful uplink communications minimized interference scenario presence uplink interference power minimization problem update time given min pmax solve need jointly find optimal device association transmit power active devices sinr constraints given uavs locations clearly given fixed uavs locations optimization variables device association transmit power devices note satisfying sinr requirement device significantly depends distance altitude serving uav therefore feasibility optimization problem depends locations uavs next derive upper bound lower bound altitude serving uav function distance device proposition lower upper bounds altitude uav needed serve device meeting sinr requirement given pmax dij sin dij distance uav device pmax proof let cumulative interference interfering devices device pmax plos pnlos plos plos pmax plos sinri considering using equation dij sin pmax stems also pmax pmax plos plos consider plos equivalent dij finally pmax clearly prove proposition proposition provides necessary conditions uav altitude needed order able serve iot device minimum altitude must increase distance increases words uav altitude needs adjusted based distance elevation angle device uav exceeds furthermore expected maximum altitude uavs significantly depends maximum transmit power devices given given fixed uavs locations problem corresponds problem joint user association uplink power control terrestrial base station scenario algorithm presented leads global optimal solution joint user association uplink power control sinr maximum transmit power constraints result optimal transmit power users base station association total uplink transmit power globally minimized determined problem iot devices correspond users fixed positioned uavs correspond terrestrial base stations case algorithm given algorithm proceed follows start initial value transmit power active devices step step compute iteration case represents minimum required transmit power device reach sinr connecting uav given fixed transmit power devices step find minimum transmit power device connects best uav index best uav assigned device given step step update transmit power device order achieve sinr steps must repeated devices obtain optimal transmit power device association vectors algorithm iterative algorithm joint power control association inputs locations uavs iot devices outputs device association vector transmit power devices set initialize define compute min find arg min update min pmax repeat steps devices converges shown several iterations algorithm quickly converges global optimal solution sinr device equal hence solving able find optimal transmit power devices device association given fixed locations uavs device association transmit power devices used inputs solving second subproblem uavs locations need optimized subsection scenario update time number active devices lower number orthogonal channels equivalently interference devices unlike interference scenario transmit power device computed based channel gain device serving uav therefore considering without interference minimum transmit power device order connect uav case given locations uavs fixed known devices problem simplified hence optimal association problem minimum power scenario min aij aij aij aij pmax aij average path loss device uav known give locations uav device aij equal device assigned uav otherwise aij equal clearly optimization problem integer linear programming ilp general problem solved using standard ilp solution methods cutting plane however solutions might efficient size problem grows particular due potentially high number iot devices efficient technique solving needed transform problem standard assignment problem solved polynomial time assignment problem objective find optimal assignment two sets nodes minimum cost problem devices uavs considered two sets nodes need assigned assignment cost lij nodes however compared classical assignment problem additional constraint results transmit maximum power constraint constraint indicates device assigned uav pmax therefore assignment problem consider lij avoid assigning device uav pmax implies constraint violated subsequently using updated assignment costs lij problem transformed classical assignment problem solved using hungarian method time complexity note absence interference problems solution next present second subproblem original optimization problem order optimize uavs locations optimal locations uavs section given optimal device association goal find locations uavs total transmit power devices minimized words considering mobile nature uavs intelligently update location uav based location associated iot devices interference scenario scenario given associations optimization problem find locations uavs transmit power devices min pmax uav xuav indicates location uav clearly channel gains used depend locations uavs note according function consequently constraint also furthermore transmit power devices uavs locations mutually dependent one hand location uav must determined associated devices connect minimum transmit power hand uav location impact amount interference received interfering devices indeed solving optimization problem challenging problem highly nonconvex particular complexity problem stems mutual dependence transmit power devices locations uavs proposed approach solve based optimizing location uav separately note using results uav associated devices transmit power known proposed solution proceeds follows cloud starts considering single uav optimizing location given fixed transmit power devices cloud updates transmit power associated devices according new location serving uav hence step location uav transmit power associated devices updated iteration finding set pmax next iteration ensures transmit power devices increase iterative process entire process repeated cloud uavs transmit power devices reduced changing uavs locations note step one must determine optimal location uav total transmit power associated devices minimized let set devices index associated uav given optimal location uav determined solving following problem min plos pnlos dij plos pnlos dkj note plos plos dkj dij depend locations uavs also guarantees transmit power device reduced updating location serving uav clearly considering fact objective function constraints twice differentiable convert quadratic form solved using efficient techniques particular adopt sequential quadratic programming sqp method one powerful algorithms solving large scale constrained differentiable optimization problems clearly considering high well large number constraints sqp suitable method solving optimization problem sqp method objective function approximated quadratic function constraints linearized subsequently optimization problem solved solving multiple quadratic subproblems optimization problem find optimal location uav start initial point starting use first order necessary optimality kkt conditions find lagrangian variables particular use lagrangian function vector lagrangian variables vector functions element given determine lagrange variables next step update solution following quadratic programming problem arg min indicate gradient hessian operations clearly inequality constrained quadratic programming moreover shown hessian matrix positive semidefinite hence general case two possible solution approaches active set interior point methods typically active set method preferred hessian matrix dense interior point however suitable approach hessian matrix large sparse problem due potential possible high number active devices number constraints high therefore hessian matrix large sparse hence interior point method used finally based location uav given fixed device association determined next address uavs location optimization scenario scenario absence interference able provide tractable analysis uavs locations optimization considering los propagation optimal locations uavs given error uav altitude fig error objective function approximation min plos pnlos dij pmax plos pnlos dij uav optimization problem xuav however given altitude provide tractable solution problem first given consider following function used dij plos pnlos dij clearly considering fact plos pnlos plos dij see dij bounded two quadratic functions linearly proportional using least square estimation method find coefficients given dij approximated following convex quadratic function dij altitude dependent coefficients note using quadratic approximation solution becomes tractable fig shows error objective function due quadratic approximation see fig obtained based parameters table error less different uavs altitudes constraint consider plos pnlos dij clearly increasing function dij since fixed altitude los probability decreasing function uav distance therefore using dij xuav given write optimization problem uav min xuav uav uav uav xuav plos pnlos pmax next derive solution problem corresponds finding uavs locations uav theorem solution given xuav vector maximizes following concave function max given proof proof see optimization problem quadratically constrained quadratic program qcqp whose general form given min qto qti given also plos pnlos pmax note positive semidefinite matrices hence qcqp problem convex write lagrange dual function inf inf clearly taking gradient function inside infimum respect find result using finally dual problem max proves theorem using theorem fixed altitude find optimal coordinates uav uav xuav optimal uav altitude argument minimizes following function arg min altitude dependent coefficients given note given altitude uav obtained via one dimensional search feasible range altitudes consequently determine optimal location uav note device association presented section iii uavs locations optimization section applied iteratively change location update step clearly iteration total transmit power devices reduced objective function monotonically decreasing hence solution converges several iterations note proposed approach provides suboptimal solution original problem nevertheless solution reasonable accuracy significantly fast compared global optimal solution achieved search corroborated simulations thus far considered iot network one snapshot time duration next analyze iot network considering entire time duration set active devices changes case maintain reliable uplink communications devices uavs must update locations different update times pdate imes obility uav find optimal update time trajectory uavs guarantee reliable uplink transmissions iot devices clearly trajectory uavs well update time depend activation process iot devices furthermore move along optimal trajectories uavs must spend minimum total energy mobility remain operational longer time considered ground iot network set active iot devices changes time consequently uavs must frequently update locations accordingly note uavs continuously move must stop serve devices update locations furthermore mobility uavs also limited due energy constraints hence uavs update locations specific times case time interval need find update times updates framework optimizing mobility uavs different update times tractability assume devices synchronized case synchronization process needs done entire activation period note optimization problems jointly finding optimal uavs locations device association devices transmit power update time depend synchronization assumption update time analysis first propose framework find update times uavs discussed section uav trajectory consists multiple stop locations determined update times uav serves associated ground devices clearly update times depend activation iot devices given time period indeed number update times impacts optimal location trajectory uavs well power consumption iot devices higher number updates leads shorter time interval consecutive updates hence lower number devices active shorter time interval case active devices experience lower interference transmitting data uavs therefore iot devices use lower transmit power meet sinr constraint however higher number updates requires mobility higher energy consumption uavs next provide insightful analysis update time based probabilistic periodic activation models iot devices periodic iot activation applications weather monitoring smart grids smart meters home automation iot devices report data periodically therefore devices activated periodically let activation period device without loss generality assume due periodic nature devices activation find exact number active devices update time proposition exact number active iot devices update time given arg max indicator function equal proof user becomes active exists thus number activations device must greater one considering fact number activations must hence total number active devices need served equal finally considering write arg max proposition gives exact number devices must served uavs update time case update times adjusted according number devices served uavs indeed knowing exact number active devices enables determine update times deterministic efficient way based system requirements probabilistic iot activation certain iot devices probabilistic activations applications health monitoring smart traffic control case iot device becomes active time following beta distribution given scenario next derive specific update times function average number active devices theorem update times average total devices must served uavs given regularized incomplete beta function inverse function total number iot devices time interval devices active proof first find probability device becomes active order send data uav update time discussed system model device needs transmit time becomes active time thus probability device needs served incomplete beta function parameters regularized incomplete beta function average number active devices given lpn lpn used note corresponds mean binomial distribution leads finally considering find clearly update times need determined based iot devices activation patterns fact depends number iot devices activation distribution furthermore according depends also previous update time due fact number active devices need served depends update time difference using theorem update times uavs set based average number active devices typically update time number devices need served uavs high order avoid high interference however considering number available resources orthogonal channels uavs preferable serve maximum number active devices update time hence case number active devices update time must relatively considering system requirements different parameters mutual interference devices acceptable delay serving devices number available channels appropriate must adopted instance using theorem update times set average number active devices lower number channels avoid interference devices next investigate uavs mobility update times uavs mobility thus far determined update times well stop locations update time investigate uavs move stop locations different update times case considering energy limitation uavs emax find optimal trajectory uav guarantee reliable uplink transmissions active iot devices uavs update locations according activity iot devices therefore uavs move initial locations new optimal locations mobility done way uavs spend minimum total energy mobility remain operational longer time fact given optimal sets uavs locations obtained section iii determine move uavs initial new sets locations order minimize total mobility uavs let two sets comprising uavs locations two consecutive update times goal find optimal mapping two sets way energy used transportations two sets minimized model total energy uav use mobility limited emax clearly multiple updates mobilities maximum energy consumption uav update equal remaining energy uav let remaining energy uav location index time write following uavs mobility optimization problem min ekl zkl zkl zkl elk zkl initial new sets uavs locations times assignment matrix element zkl uav assigned location otherwise ekl energy used moving uav initial location index new location index also remaining energy uavs time note guarantees uavs remain operational end period total energy consumption rotary wing uav moving two stop locations computed done distance two stop locations flight duration power consumption vertical movement power consumption horizontal movement clearly altitude difference two stop locations effective vertical according horizontal velocities sin cos composed parasitic power induced power needed overcoming parasitic drag drag parasitic power based equations given effective horizontal velocity cdo drag coefficient air density blade chord number blades reference area frontal area uav note second term represents blade power profile using equations induced power assuming zero tilt angle computed rotor disk radius weight uav angular velocity also given find solving following equation power consumption due vertical climbing descending assuming rapid descent given equations climbing descending windmill state effective vertical velocity finally total mobility energy consumption computed using clearly optimization problem integer linear programming ilp following similar approach used solving transform problem standard assignment problem solved using hungarian method polynomial time complexity table simulation parameters parameter description value pmax maximum transmit power device path loss exponent los links noise power dbm sinr threshold total number iot devices additional path loss free space los additional path loss free space nlos end need remove constraint considering elk constraint satisfied determine satisfied use compute elk compare remaining energy uavs objective function replace elk corresponding unsatisfied constraint elk consequently transformed standard assignment problem result solving assignment matrix optimally assigns uavs destinations therefore locations uavs updated according new destinations destinations uav different update times find optimal trajectory uavs imulation esults nalysis simulations iot devices deployed within geographical area size case consider total number iot devices uniformly distributed area furthermore consider communications urban environment ghz carrier frequency table lists simulation parameters analyze transmit power iot devices energy consumption uavs mobility update times update time analysis unless otherwise stated consider probabilistic activation model iot devices beta distribution parameters applicable compare results stationary aerial base stations uavs scenario adopting optimal device association power control technique subsection stationary case locations uavs assumed fixed target area updated according devices locations statistical results averaged large number independent runs note given iot network serving active devices may possible due limitations number uavs maximum transmit power devices thus stationary aerial base stations proposed approach reliability pmax fig reliability comparison proposed approach stationary aerial base stations using uavs fig show achieved system reliability defined probability active devices served uavs clearly reliability depends locations transmit powers devices well number uavs fig shows reliability maximum transmit power devices pmax varies case uavs deployed serve active iot devices clearly pmax increases reliability also increases fact higher pmax values devices higher chance successfully connect uavs fig see proposed approach leads significantly improved reliability compared case stationary aerial base stations used particular difference reliability stationary case proposed approach significant lower pmax indeed higher reliability achieved dynamically optimizing uavs locations based locations iot devices shown fig increasing pmax reliability increases stationary case increases proposed approach furthermore proposed approach yields maximum improvement system reliability fig shows snapshot uavs locations associated iot devices indicated color resulting proposed approach figure uavs efficiently deployed serve active iot devices uniformly distributed area case devices able send data associated uavs using minimum total transmit power locations uavs well device association determined based locations ground iot devices transmit power fig show total transmit power needed iot devices reliable uplink communications versus number uavs interference scenario clearly total transmit power iot devices reduced deploying uavs instance uav uav uav uav uav devices altitude fig uavs locations associations one illustrative snapshot proposed approach stationary aerial base stations stationary aerial base stations proposed approach total transmit power total transmit power number uavs number uavs fig total transmit power devices number fig total transmit power devices number uavs presence interference uavs scenario considering active devices available channels using proposed approach total transmit power decreases increasing number uavs furthermore using proposed approach total transmit power devices decreases average compared stationary case clearly lower number uavs proposed approach leads higher power reduction compare stationary case words intelligently optimizing locations uavs provides power reduction gains number uavs low fact dense networks high number uavs updating uavs locations obviously longer necessary compared case low number uavs instance see fig power reduction gain achieved deploying uavs around times larger case uavs fig shows total transmit power iot devices function number uavs scenario compared interference scenario devices obviously use lower transmit power sending data uavs instance efficiently deploying uavs devices establish reliable uplink communications total transmit power furthermore fig shows proposed approach total transmit power stationary aerial base stations proposed approach number orthogonal channels fig total transmit power devices number orthogonal channels leads average power reduction compared stationary case fig shows total transmit power devices used meeting sinr requirement number available channels varies result fig corresponds case active devices served uavs clearly total transmit power decreases number channels increases due fact orthogonal resources available interference devices decrease result device reduce transmit power connecting serving uav fig see increasing number channels total transmit power devices reduced proposed approach fact average number interfering devices decreases increase number channels consequently less interference generated devices transmitting uavs fig show average number active devices must served uavs different update times normalized clearly number active devices update time depends activation process devices number update times indicates frequently uavs serve devices fig due beta distributionbased activation pattern iot devices number active devices decreases exceeds fig see higher number update times equivalently shorter time period consecutive updates average number devices need transmit data decreases instance considering average number active devices decreases number updates increases also note lower number active devices leads lower interference devices requires updates mobility uavs fig also verifies analytical results theorem match simulations furthermore fig show average number active devices theory theory simulations simulations update time normalized fig average number active devices update times probabilistic activation exact number active devices updates updates update time normalized fig exact number active devices different update times periodic activation exact number active devices periodic activation case obtained proposition case device becomes active certain activation period expected higher number updates lower number active devices need served uavs instance increasing number updates average number active devices decreases moreover fig shows maximum number active devices updates two times larger case updates therefore order avoid interference devices number orthogonal channels must increased factor number updates decreases fig presents direct result theorem computes update times based average number active devices fig shows set update times order ensure number devices needs served update time exceed specified number see fig achieve lower value updates must occur frequently reduce time interval consecutive updates example seen figure meet update must occur moreover fig shows number updates increases decreases example case reduce number updates needs doubled fig shows impact number updates amount energy uavs use update time normalized update number fig update times different average number active devices move simulations considered intuitively higher number updates requires mobility uavs therefore increasing number updates total energy consumption uavs also increase see fig increasing number updates energy consumption uavs increases factor target area size note mobility uavs also depends size geographical area devices distributed clearly average uavs need move covering larger area interestingly inherent tradeoff number updates mobility uavs transmit power iot devices fact considering fig higher number updates leads higher energy consumption uavs due higher mobility addition shown fig number updates increases lower number iot devices active update time hence lower interference devices result transmit power devices needed satisfying sinr requirement reduced note showed fig devices transmit power decreases interference decreases increasing number orthogonal channels hence higher number updates leads lower devices transmit power requires uavs mobility fig shows overall convergence proposed power minimization algorithm used solving original problem considering uavs see figure case total transmit power iot devices converges iterations fig iteration corresponds joint solution device association uavs locations optimization problems clearly several iterations updating device association total energy consumption uavs area area number updates number iterations relative computational time optimality gap total transmit power fig total uav energy consumption number updates number active devices fig overall convergence algorithm fig proposed approach optimal solution uavs locations longer improve solution fig show example compare accuracy time complexity proposed approach optimal solution obtained exhaustive search perform exhaustive search continuous space discretized space resolution case two uavs deployed serve devices clearly average gap proposed solution optimal solution around however example proposed solution around times average faster optimal solution onclusion paper proposed novel framework efficiently deploying moving uavs collect data uplink ground iot devices particular determined jointly optimal uavs locations device association uplink power control iot devices total transmit power devices sinr constraints minimized addition investigated effective movement uavs collect iot data iot network case based devices activation process derived update time instances uavs must update locations furthermore obtained optimal trajectories used uavs dynamically serve iot devices minimum energy consumption results shown intelligently moving deploying uavs total transmit power devices significantly decreases compared case stationary aerial base stations moreover fundamental tradeoff number updates uavs mobility devices transmit power eferences mozaffari saad bennis debbah unmanned aerial vehicle underlaid communications performance tradeoffs ieee transactions wireless communications vol june zeng zhang lim wireless communications unmanned aerial vehicles opportunities challenges ieee communications magazine vol may jiang swindlehurst optimization uav heading uplink ieee journal selected areas communications vol june zeng zhang lim throughput maximization mobile relaying systems ieee transactions communications vol mozaffari saad bennis debbah mobile internet things uavs provide mobile architecture proc ieee global communications conference globecom washington usa yaliniz yanikomeroglu efficient placement aerial base station next generation cellular networks proc ieee international conference communications icc kuala lumpur malaysia may mozaffari saad bennis debbah optimal transport theory cell association cellular networks ieee communications letters vol kandeepan lardner optimal lap altitude maximum coverage ieee wireless communication letters vol mozaffari saad bennis debbah efficient deployment multiple unmanned aerial vehicles optimal wireless coverage ieee communications letters vol chetlur dhillon downlink coverage analysis finite wireless network unmanned aerial vehicles ieee transactions communications appear chen mozaffari saad yin debbah hong caching sky proactive deployment cacheenabled unmanned aerial vehicles optimized ieee journal selected areas communications vol may mozaffari saad bennis debbah wireless communication using unmanned aerial vehicles uavs optimal transport theory hover time optimization available online lien chen lin toward ubiquitous massive accesses communications ieee communications magazine vol apr lyu zeng zhang lim placement optimization mobile base stations ieee communications letters vol mar pang zhang pan han efficient data collection wireless rechargeable sensor clusters harsh terrains using uavs proc ieee global communications conference globecom austin usa huang algorithms evaluations massive access management cellular based machine machine communications proc vehicular technology conference vtc san francisco usa chen lien communications technologies challenges hoc networks vol july tavana wong congestion control bursty traffic lte networks proc ieee international conference communications icc london june jian zeng jia zhang model machine type communication ieee communications letters vol mar study ran improvements machine type communication gupta nadarajah handbook beta distribution applications crc press selim ismail algorithms generalized convergence theorem characterization local optimality ieee transactions pattern analysis machine intelligence yates framework uplink power control cellular radio systems ieee journal selected areas communications vol sun hong luo joint downlink base station association power control fairness computation complexity ieee journal selected areas communications vol june burkard dell amico martello assignment problems revised reprint siam kuhn hungarian method assignment problem naval research logistics quarterly vol boggs tolle sequential quadratic programming acta numerica vol scheinberg efficient implementation active set method svms journal machine learning research vol oct boyd vandenberghe convex optimization cambridge university press franco buttazzo coverage path planning uavs proc ieee international conference autonomous robot systems competitions icarsc vila real portugal apr zeng zhang uav communication trajectory optimization ieee transactions wireless communications vol june filippone flight performance fixed rotary wing aircraft elsevier
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weak order ideal associated linear codes mijail miguel may edgar abstract work study weak order ideal associated coset leaders linear code set allows incrementally computation coset leaders definitions set leader codewords set codewords nice properties related monotonicity weight compatible order generalized support vector fnq allows describe test set trial set set zero neighbours linear code terms leader codewords keywords linear codes order ideals test set trial set zero neighbours correctable errors introduction pointed common folklore theory binary linear codes ordering coset leaders chosen lexicographically smallest minimum weight vectors provides monotone structure expressed follows coset leader also coset leader nice property proved great value see example used analyzing capability binary linear codes last paper authors introduce concept trial set codewords provide decoding algorithm based set finding weight distribution cosets leaders code classic problem coding theory problem still unsolved many family department department mathematics faculty ciencias naturales exactas universidad oriente santiago cuba cuba emails mijail mborges borgesquintana partially supported scholarship university valladolid erasmus mundus program mundus lindo project institute mathematics imuva university valladolid valladolid castilla spain email edgar partially supported partially supported spanish mineco grants linear codes even codes see set coset leaders also related minimum distance decoding bounded distance decoding problems well set minimal support codewords despite interest generalization ideas known authors communication case paper provide non straightforward generalization outline paper follows section introduces idea generalized support vector section defined weak order ideal associated coset leaders shown computed incrementally theorem establishes coset leaders code belong subsection devoted study set leader codewords code zero neighbour set properties finally section analyze correctable uncorrectable errors defining trial set linear code set leader codewords limitations practical point view results properties studied paper clear size complexity computing set coset leaders anyway main interest study characterizations objects related codes like zero neighbours trial set set correctable uncorrectable errors preliminaries shall denote finite field elements prime linear code length dimension kdimensional subspace fnq call vectors fnq words codewords every word fnq support defined supp hamming weight denoted cardinality supp hamming distance two words fnq minimum distance linear code defined minimum weight among nonzero codewords words minimal hamming weight cosets fnq set coset leaders code fnq denote denote subset coset leaders corresponding coset given coset define weight coset smallest hamming weight among vectors coset equivalently weight one leaders well known given denotes greatest integer function every coset weight unique coset leader let irreducible polynomial degree root element represented word fnq component define generalized support vector support given concatenations expansion component suppgen supp suppgen supp say suppgen corresponding zero set eij denoted represents canonical basis fnq additive monoid fnq respect operation irreducible polynomial used define state following connection fnq nnm fnq nnm mod hand nnm fnq given fnq xij eij yij eij say xij yij using possible relate orders fnq orders nnm vice versa admissible order nnm total order nnm satisfying following two conditions nnm nnm particular admissible order nnm like lexicographical degree lexicographical degree reverse lexicographical orders induces order fnq say representation word standard form denote standard form note therefore standard form also say fnq remark use fnq instead fnq respectively since clear different elections provide equivalent generalized supports definition subset order ideal fashion say subset fnq order ideal order ideal nnm easy check equivalent definition order ideal would suppgen fnq eij instead suppgen condition satisfied least one suppgen say weak order ideal definition subset fnq weak order ideal exists suppgen fnq eij definition voronoi region codeword set fnq set voronoi regions given linear code covers space fnq however words fnq may contained several regions subset fnq define fnq min set words hamming distance boundary defined fnq definition nonzero codeword called zero neighbour voronoi region shares common boundary set coset leaders set zero neighbours denoted definition given linear code set codewords every word either lies exists set zero neighbours test set also set zero neighbours obtained minimal test set according cardinality set weak order ideal coset leaders first idea allows compute incrementally set coset leaders linear code introduced paper used additive structure fnq set canonical generators unfortunately chosen coset representatives may coset leaders weight coset greater theorem theorem assume coset leader fnq supp also coset leader order incrementally generate coset leaders starting adding elements must consider words weight one previous chosen coset leader next result byproduct theorem may characterize vectors need generate weight one coset leader order ensure coset leaders generated theorem let fnq element let supp fnq supp proof theorem proof theorem analogous proof theorem note suppose would imply coset leader contradiction let fnq element suppgen let fnq eij consequence previous theorem situation say coset leader ancestor word descendant binary case definitions behave ones case subtle difference coset leader could ancestor another coset leader ancestor word hamming distance coset leader last case possible binary case set given admissible order nnm define weight compatible order fnq associated ordering given words ordered according weights order break ties class orders subset class monotone fact need little monotonicity purpose work also need every pair fnq note satisfied weight compatible order addition weight compatible order every strictly decreasing sequence terminates due finiteness set fnq binary case behavior coset leaders translated fact set coset leader order ideal whereas non binary linear codes longer true even try use characterization order ideals given order ideals need associated admissible orders definition define weak order ideal coset leaders set elements fnq verifying one following items criterion eij eij fnq criterion wnh eij supp eij remark clear criteria definition weak order ideal theorem let fnq exists proof proceed induction fnq respect order statement true fnq inductive step assume desired property true word fnq exists also smaller arbitrary fixed respect show previous conditions imply also let eij suppgen true induction hypothesis theorem therefore criteria definition guaranteed theorem let fnq proof let suppgen since theorem theorem previous theorem shown contains set coset leaders linear code zero neighbours leader codewords definition set leader codewords linear code defined eij eij fnq note definition bit complex one binary codes due fact general case coset leaders need ancestors coset leaders name leader codewords comes fact one could compute coset leaders corresponding word knowing set adapting algorithm remark algorithm computing based construction theorem guarantees provided associated set leader codewords may computed theorem properties let linear code test set let element covering radius code proof let supp let yik eik let eijt eijz implies eijz addition eijz eijz eijz eijz eijz note supp eijz supp eijz consequently eijz eijz thus test set let exists fnq eij applying definition covering radius thus let eij elements eij fnq eij eij eij implies eij eij define eij clear number set since let let elij elij elij exists two conditions imply hand either coset leader satisfying condition means coset leader coset leader using idea previous paragraph let first condition implies eij fnq suppgen hand eij implies eij therefore eij remark note item theorem implies leader codeword zero neighbour however one differences binary case always true leader codeword although item leader codeword provided condition satisfied furthermore item guarantees set leader codewords contains minimal test set according cardinality see consequence properties theorem could say set leader codewords good enough subset set zero neighbours correctable uncorrectable errors define relation additive monoid describe exactly relation vector space fnq given fnq supp supp note definition translates fnq binary case situation case given fnq words fnq consider elements vector space fnq course relation fnq vector space also true additive monoid true way round set correctable errors linear code set minimal elements respect coset elements set fnq called uncorrectable errors trial set code set following property since monotone fnq set correctable uncorrectable errors form monotone structure namely implies implies general case difference respect binary case may words fnq suppgen suppgen could either correctable error uncorrectable error monotone structure sustained additive monoid fnq let set minimal uncorrectable errors set similar way set maximal correctable errors set elements larger half defined minimal word ordering weight word see details set larger halves codeword denoted set larger halves elements denoted note fnq let theorem characterization set terms larger halves set minimal codewords binary case easy proof theorem corollary also true linear code proposition corollary let linear code following statements equivalent trial set formulate result relates trial sets given weight compatible order set leader codewords theorem let linear code set leaders codewords following statements satisfied trial set given algorithm adapted compute set leader codewords trial set given satisfies following property exists proof proof prove statement proposition let let suppgen since thus coset leader hand let clear leader codeword therefore proof algorithm first step necessary add function insertnext criteria construction whose elements stored listing hand steps construction leader codewords steps enough state condition taking equal coset leader corresponding correctable error add codeword set references barg complexity issues coding theory handbook coding theory vol amsterdam bases structure associated linear codes discret math sci cryptogr computing coset leaders leader codewords binary codes journal algebra applications braun pokutta polyhedral characterization border bases siam discrete math helleseth vladimir capability binary linear codes ieee transactions information theory kurshan sloane coset analysis reed muller codes via translates finite vector spaces information control huffman pless fundamentals codes cambridge university press cambridge macwilliams sloane theory codes holland algebraic structure minimal support codewords set linear codes adv math commun massey minimal codewords secret sharing proceedings joint international workshop information theory mora solving polynomial equation systems macaulay paradigm technology cambridge university press cohen threshold probability code ieee trans inform theory
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deep learning handwritten indic script identification soumya swarnendu kaushik nibaran dept computer science engineering jadavpur university kolkata india dept computer science engineering aliah university kolkata india dept computer science university south dakota vermillion usa dept computer science engineering west bengal state university india jan corresponding authors santosh das nibaran abstract propose novel method uses convolutional neural networks cnns feature extraction limited conventional spatial domain representation use multilevel discrete haar wavelet transform image representations scaled variety different sizes used train different cnns select features precise use different cnns select set features different handwritten scripts identified words per script used test achieved maximum script identification rate using perceptron mlp results outperform techniques keywords convolutional neural network deep learning perceptron discrete wavelet transform indic script identification introduction optical character recognition ocr always challenging field pattern recognition ocr techniques used convert handwritten machine printed scanned document images texts ocr techniques script dependent therefore script identification considered precursor ocr particular case multilingual country like india script identification must since single document postal documents business forms contains several different scripts see fig indic handwritten script identification rich literature often previous works focusing script identification stopping recent work authors introduced script identification performance see whether expedite processing time general works features based structural visual appearances used question relying see use apply features accordingly let machine select features required optimal identification rate inspires use deep learning cnns used extracting selecting features identification task needless say cnns stood well immense contribution field ocr onset ben marked performance cnns mnist dataset recently use cnn indic script bangla character recognition fig two postal document images bangla roman devanagari scripts used reported confused primary goal paper use deep learning concept identify different handwritten indic scripts bangla devnagari gujarati gurumukhi kannada malayalam oriya roman tamil telugu urdu inspired deep concept use cnns select features handwritten document images scanned use multilevel discrete haar wavelet transform addition conventional spatial domain representation image representations scaled variety different sizes representation several different cnns used select features short primary idea behind avoid using features identification using perceptron mlp different handwritten scripts mentioned earlier identified satisfactory performance remainder paper summarized follows section provides quick overview contribution includes cnn architecture feature extraction process section experimental results provided also includes quick comparison study section concludes paper contribution outline mentioned earlier stead using features document image representation goal let deep learning select distinguishing features optimal script identification recent works cnns used successful classification refer observe cnns work especially sufficient data train means data redundancies helpful general cnn takes raw pixel data image training proceeds model learns distinguishing features successfully contribute training process produces feature vector summarize important aspects studied image precisely approach twofold first use cnns three different scales input image secondly use exactly cnns two different scales transformed image wavelet transform merge features make ready script identification follows explain cnn architecture including definitions parameters wavelet transform way produce features fig schematic block diagram handwritten indic script identification showing different modules feature classification cnn architecture general cnn layered architecture consisting three basic types layers namely convolutional layer pooling layer fully connected layer fcl cls consist set kernels produce parameters help convolution operation every kernel generates activation map output pls parameters major role avoid possible data redundancies still preserving significance approach cnns operation corresponding pls addition two different types layers fcl used mlp place fig provide complete schematic block diagram handwritten indic script identification showing different modules feature classification study different cnns used select features variety representations studied image label cnnd every cnnd respectively refer domain representation dimension studied image refers number convolutional pooling layers particular cnn example domain representation expressed refers spatial domain frequency case either taken account note use haar wavelet transform hwt see section certain frequencies removed case dimension simplicity used dimensions signify resolution input images scaled case one two taken cnn means two pairs convolutional pooling layers cnns model two broad cnn architectures cnnd cnnd used summarized table table architecture cnnd architecture parameter layer cnnd channel filter size pad size cnnd channel filter size pad size softmax index convolutional layer pooling layer fcl fully connected layer cnnd six different layers two cls two pls two fcls first two cls followed pls fcls first layer takes image representation generated cls pls reshapes form vector second fcl produces set features cnnd every cnn eight different layers three cls three pls two fcls general architecture much similar previously mentioned cnnd difference lies additional pair follows second pair like cnnd cnns produce set features studied image architectural details aforementioned cnns summarized table follows schematic block diagram system see fig understanding fig provide activation maps means uses spatial domain image representation dimensionality three pairs convolutional pooling layers cnns data representation general since fourier transform may work successfully provide information frequencies present time wavelet transform short time fourier transform typically used help identify frequency components present signal given time hand provide dynamic resolution consider image time signal resized use multilevel discrete image generate frequency domain representation precise use haar wavelet seven different level decomposition generate approximated detailed coefficients since approximated coefficients equivalent zero use detailed coefficients addition modified approximated coefficients reconstruct image modified approximated coefficients consider high frequency components method using variety different wts daubechies several decomposition levels best results observed haar wavelet decomposition level reconstructed image resized fig illustrating activation maps spatial domain image representation dimensionality convolutional pooling layers fed multiple cnns mentioned section like frequency domain representation spatial domain representations fed multiple cnns experiments dataset evaluation metrics protocol evaluate proposed approach handwritten script identification considered dataset named phd indic composed scanned word images grayscale different indic script per script samples shown fig information dataset refer recently reported work primary reason behind considering phd indic dataset test big size data reported literature research purpose till date using exact cnn representation represents count instance label classified accuracy acc particular cnn computed accd precision prec computed precd precid precid precid refers precision label similar fashion recall rec computed recd recid recid recid refers recall label precision recall computed precid recid precid recid following conventional train test evaluation protocol separated images training remaining images testing ran experiments using machine gtx cuda cores gpu ram besides intel pentium ram experimental set mentioned earlier required train cnns first testing words important see training testing performed fig illustrating samples dataset named phdindic used experiment cnns study represented cnnd trained independently using training dataset mentioned earlier clarify cnns either two three pairs consecutive convolutional pooling layers besides three fully connected layers first three layers function input layer number neurons depends size input image specified second layer cnns neurons training apply dropout probability final layer neurons whose outputs used input classifier provides probabilities possible classes training set word images split multiple batches word images cnns trained accordingly optimization adam optimizer used learning rate default parameters helps apply gradients loss weight parameters back propagation computed accuracy cnn training proceeds taking ratio images successfully classified batch total number images studied batch training cnns word images independently test set composed word images specifically input size specification two cnns domain input size cnnd cnnd altogether different cnns since three different input sizes raw image two different input sizes wavelet transformed better understanding refer readers fig note trained cnns extract features one concatenated form single vector ten different cnns employed like conventional machine learning classification features used training testing purpose using mlp classifier fig results terms accuracy precision recall networks cnns possible combinations results comparative study section using dataset evaluation metrics see section experimental setup see section summarize results comparative study follows provide results produced different architectures cnns select highest script identification rate take highest script identification comparative study previous relevant works considered results fig shows comparison individual cnns along effect combining individual cnnd produced maximum script identification rate ensemble networks corresponding domain input size observe positive correlation input size accuracy provides maximum script identification rate category primary reason behind increase accuracy ensemble threelayered networks suggests networks complement study effect spatial frequency domain representation cnnd ensemble networks across input sizes depth network spatial representation cnns produced script identification rate frequency domain representation cnnf escalated however frequency domain representations learning complimented clearly seen combined combination achieved highest script identification rate since fig misclassified samples script names bracket actual scripts system identified incorrectly example first case word image identified gurumukhi actually bangla received script identification rate wise provide samples system failed identify correctly see fig like mentioned section also provide precision recall architectures fig follows highest script identification rate taken comparison comparative study fair comparison widely used deep learning methods lenet alexnet taken addition recently reported work handwritten indic script dataset including baseline results considered table summarize results comparative study focused accuracy precision recall since methods reported accuracy identification rate course fig limited accuracy table method outperforms methods precisely outperforms obaidullah lenet alexnet conclusion paper proposed novel framework uses convolutional neural networks cnns feature extraction method addition conventional spatial domain representation used multilevel discrete haar wavelet transform image representations scaled variety different sizes several different cnns used select features different handwritten scripts bangla devnagari gujarati gurumukhi kannada malayalam oriya roman tamil telugu urdu identified words per script used test achieved maximum script identification rate using perceptron mlp best knowledge biggest data indic script identification work considering complexity size dataset method outperforms previously reported techniques table comparative study method obaidullah features lenet cnn alexnet cnn method multiscale cnn accuracy references ghosh dube shivaprasad script recognitiona review ieee transactions pattern analysis machine intelligence pal jayadevan sharma handwriting recognition indian regional scripts survey offline techniques acm transactions asian language information processing talip singh sarkar nasipuri doermann script identification handwritten indic scripts document analysis recognition icdar international conference ieee hangarge santosh pardeshi directional discrete cosine transform handwritten script identification document analysis recognition icdar international conference ieee pati ramakrishnan word level identification pattern recognition letters obaidullah halder santosh das roy phdindic handwritten document image dataset official indic scripts script identification multimedia tools applications lecun bottou bengio haffner learning applied document recognition proceedings ieee roy das kundu nasipuri handwritten isolated bangla compound character recognition new benchmark using novel deep learning approach pattern recognition letters krizhevsky sutskever hinton imagenet classification deep convolutional neural networks advances neural information processing systems sarkhel das das kundu nasipuri deep quad tree based feature extraction method recognition isolated handwritten characters popular indic scripts pattern recognition smith scientist engineer guide digital signal processing daubechies wavelet transform localization signal analysis ieee transactions information theory portnoff representation digital signals systems based fourier analysis ieee transactions acoustics speech signal processing sundararajan fundamentals discrete haar wavelet transform vonesch blu unser generalized daubechies wavelet families ieee transactions signal processing kingma adam method stochastic optimization arxiv preprint
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nov power flow algebraic system jakub marecek timothy mccoy martin mevissen january abstract steady states circuits studied considerable detail baillieul byrnes derived upper bound number steady states circuit ieee tcas conjectured bound holds circuits general prove indeed case among results studying certain structure algebraisation introduction years steady states circuits studied considerable detail key problem sometimes known power flow load flow problem considers complex voltages buses variables except one reference bus power supplied one denotes complex admittance matrix complex current complex power bus equations based asterisk denotes complex conjugate captures complex nonconvex nature problem model order obtain algebraic system one needs reformulate complex conjugate order one may replace independent variables filter real solutions complex solutions obtained thereby obtain particular structure allows prove variety results particular main contributions paper reformulation equations algebraic system marecek mevissen ibm research ireland ibm technology campus damastown dublin ireland mccoy google analytical results number structure feasible solutions considering losses resolving conjecture baillieul byrnes open three decades empirical results instances including numbers roots conditions optima analytical results rely work morgan sommese multihomogeneous structures empirical results rely bertini leading implementation methods explain section first algebraisation system long history use homotopycontinuation methods problem order make paper restate equations consider circuit represented undirected graph vertices called buses edges called branches admittance matrix real part element called conductance glm imaginary part susceptance blm bus associated complex voltage complex current power demanded generated let correspond reference bus phase magnitude fixed powers buses fixed variety extensions buses denoted generators voltage magnitude phase power fixed branch associated complex power slm plm qlm key constraint linking buses kirchhoff current law stipulates sum currents injected withdrawn bus considering relationship steady state equations hence pkg pkd qgk qdk plm blm glm qlm blm glm additionally one optimise variety objectives steady states one commonly used objective function one approximates costs real power generated reference bus quadratic function cost variety extensions buses power fixed would quadratic function bus quadratic function power would summed across buses loss objective one computes norm vector obtained summing apparent powers usual denoted loss consider objectives section vii results sections apply independently use objective function whatsoever definitions algebraic geometry order state results need definitions algebraic geometry refer reader basics present concepts introduced past three decades yet widely covered textbooks comprehensive treatment please see let integer system polynomial equations support coefficients complex numbers well known polynomials define projective hypersurfaces projective space cpn theorem states either hypersurfaces intersect infinite set component positive dimension number intersection points counted multiplicity equal product degree polynomial call product usual number usual number improved considering definition structure partition index set sets defines structure known group variables set associated degree dij polynomial respect group def dij max say din whenever dij attained call system homogeneous group variables projective space associated group variables structure dimension homogeneous def otherwise definition number assuming number defined coefficient term associated dimension within polynomial dij variables coefficients dij associated degrees dnk consider example wampler usual number considering partition monomial looked polynomial corresponding multihomogeneous number hence minimum across possible structures general number upper bound number isolated roots cpak thereby upper bounds number isolated finite complex roots variety additional methods computing number particular case denote def dij equal multinomial coefficient def coefficient summary number provides sharper bound numberqof isolated solutions system equations usual number famous example eigenvalue problem known number whereas exists structure number hence study structure within steady state equations circuits structure notice order obtain algebraic system equations one needs reformulate complex conjugate order one may replace independent variables later filter solutions complex solutions obtained denote solution real let set slack generators specified assume corresponds reference node phase notice use variables produces structure variable groups example network obtain using algebraic system one formulate number structural results concerning power flows analysis particular structure partition variables several groups bound number isolated solutions theorem exceptions parameter set measure zero alternatingcurrent power flow finite number complex solutions bounded proof equation system linear variables also variables giving rise natural structure since slack bus voltage fixed reference value system equations variables multihomogeneous form theorem see theorem total number solutions space precisely stated bound counting multiplicity subset lie affine patch giving result notice applies also instances alternatingcurrent optimal power flows acopf example instances lesieutre bukhsh single slack bus whose active reactive powers fixed hence result applies notice exception set necessary example illustrate next section bound tight cases deciding whether bound number roots obtained using particular structure tight particular instance nevertheless hard seen theorem theorem malajovich meer exist polynomial time algorithm approximate minimal number polynomial system fixed factor unless however show exists certain structure among solutions corollary exists feasible solution power flow solution even multiplicity greater equal another solution exists proof finite number solutions power flows problem theorem even observe solution system solution implies real solutions solutions necessarily come pairs follows real feasible solutions also even number counting multiplicity result follows note solution multiplicity greater special case highly unlikely real system moreover easily detected since jacobian solution nonsingular solution multiplicity optimal power flow one may also make following observations optimal power flows problem optimising objective steady state remark power flows powers fixed reference bus whenever exists real feasible solution except parameter set measure zero one enumerate feasible solutions finite time indeed theorem know exist finite number isolated solutions system method sommese enumerate roots probability allows pick global optimum trivially notice bertini implementation method sommese makes possible check roots obtained notice addition inequalities accommodated filtering real roots nevertheless method practical may many isolated solutions enumerate generically indeed case whenever two generators variable output buses whose active reactive power fixed corollary optimal power flow problem powers variable outside reference bus additional inequalities complex solution set empty except parameter set measure zero complex solution set smooth real feasible solution exists infinitely many real feasible solutions proof slack bus first system two variables one equation slack buses rank jacobian hence dimension complex solution set least lemma furthermore real feasible solution exists solution set smooth point local dimension complex real solution sets equal therefore since complex solution set real set infinitely many real feasible solutions exist although variety methods studying systems including enumeration point within connected component studying critical points restriction variety distance function points suggest method moments may suitable studying feasible set optimal power flows shown recently allows small errors systems dimension five thousand often case engineering applications one may also interested distance point set feasible solutions considering algebraisation one bound probability distance small using theorem lotz multivariate polynomials could seen converse results method buses upper bound upper bound theorem generic lower bound table maximum number steady states circuit fixed number buses treewidth solutions cost min wrt cost cost cost loss min wrt loss loss loss bukhsh klos wojcicka lavaei low bukhsh mccoy mccoy bukhsh grainger stevenson lesieutre mccoy mccoy mccoy bukhsh mccoy mccoy mccoy bukhsh chow mccoy mccoy mccoy mccoy mccoy mccoy branches source buses instance table properties instances tested figure bertini encoding acpf instance bukhsh impedance single branch computational illustrations order illustrate theorem first present maximum number steady states circuit fixed number buses table compare values upper bound upper bound bkkbased upper bound notice maximum number steady states circuit fixed number buses achieved clique notice generic lower bound obtained number solutions found tracing paths matches upper bound theorem throughout table illustrate proposition enumerated steady states using bertini versatile package methods sommese see figure example bertini input corresponding example constants representing representing results summarised table illustrate theorem present values upper bound collection instances widely known power systems community instances mostly available test case archive optimal power flow opf problems local optima appeared papers available recent distributions matpower benchmark particular present numbers distinct roots instances cases theorem applies number solutions found tracing paths matches upper bound theorem certifying completeness cases one could rely bertini certificates completeness search empirically observe exists unique global optimum instances tested respect objective generation cost objective however number instances global optima unique case particularly good illustration symmetry two generators two demand nodes complete graph results multiple global optima order provide material study structural properties present instances table column notice small instances instance http accessed november figure instance lesieutre instances grow however need case kloks shows treewidth bounded even sparse random graphs high probability complete graphs grows linearly number vertices related work long history study number structure solutions power flows considered bound considered bound based work bernstein kushnirenko derived expression theorem using intersection theory lossless model highlight number solutions alternatingcurrent model losses important open problem note theorem subsumes theorem special case finally note present lower bound without proof recent papers bound certain distinguished solutions solutions also long history applications methods power systems although often homotopy restricted methods heuristic could enumerate solutions power flow recently attracted much interest following work trias see overview conclusions hope structural results provided aid development faster solvers related problems arguably one could using theorem construction start systems homotopycontinuation methods allow larger systems studied extend corollary finding least one point connected component extend methods consider inequalities within tracing rather filtering phase could improve computational performance considerably develop methods optimal power flow problem whose complexity would superpolynomial number buses latter two may important challenges within analysis circuits systems acknowledgements parts work done tim visiting ibm research jakub would like thank isaac newton institute mathematical sciences university cambridge generous support visits dhagash mehta kindly provided variety suggestions related work references aubry rouillier din real solving positive dimensional systems journal symbolic computation baillieul byrnes geometric critical point analysis lossless power system models ieee transactions circuits systems nov baillieul byrnes remarks number solutions load flow equations power system electrical losses decision control ieee conference pages dec baillieul byrnes washburn topological analysis solution equations power system decision control including symposium adaptive processes ieee conference pages dec basu pollack roy new algorithm find point every cell defined family polynomials quantifier elimination cylindrical algebraic decomposition pages springer vienna bates hauenstein sommese wampler numerically solving polynomial systems bertini software environments tools society industrial applied mathematics bernstein number roots system equations funkcional anal bukhsh grothey mckinnon trodden local solutions optimal power flow problem ieee trans power chandra mehta chakrabortty equilibria analysis power systems using numerical homotopy method power energy society general meeting ieee pages july chen mehta network topology dependent solution count algebraic load flow equations arxiv chiang liu varaiya lauby chaos simple power system ieee transactions power systems nov fulton intersection theory springer new york ghaddar marecek mevissen optimal power flow polynomial optimization problem ieee transactions power systems jan guo salam real method computing solutions electric power systems circuits systems iscas ieee international symposium volume pages may hiskens analysis tools power nonlinearities proceedings ieee nov hiskens davy exploring power flow solution space boundary ieee transactions power systems aug kloks graphs bounded treewidth kloks editor treewidth volume lecture notes computer science pages springer berlin heidelberg klos wojcicka physical aspects nonuniqueness load flow solutions international journal electrical power energy systems kushnirenko newton polytopes theorem funct anal lasserre convergent polynomial optimization sparsity siam journal optimization lavaei low zero duality gap optimal power flow problem power systems ieee transactions lesieutre molzahn borden demarco examining limits application semidefinite programming power flow problems communication control computing allerton annual allerton conference pages liu chang jiang yeh toward algorithm compute solutions electric power systems ieee transactions circuits systems regular papers march liu thorp novel method compute closest unstable equilibrium point transient stability region estimate power systems ieee transactions circuits systems fundamental theory applications jul lotz volume tubular neighborhoods real algebraic varieties proc amer math thorp efficient algorithm locate load flow solutions ieee transactions power systems aug malajovich meer computing minimal numbers hard theory computing systems mehta molzahn turitsyn recent advances computational methods power flow equations american control conference acc pages july mehta nguyen turitsyn numerical polynomial homotopy continuation method locate power flow solutions iet generation transmission distribution molzahn lesieutre chen counterexample algorithm finding power flow solutions ieee transactions power systems feb molzahn mehta niemerg toward topologically based upper bounds number power flow solutions american control conference acc pages july morgan sommese homotopy solving general polynomial systems respects structures appl math rouillier roy din finding least one point connected component real algebraic set defined single equation journal complexity salam guo sun parallel processing load flow power systems approach applications decision control proceedings ieee conference pages dec shafarevich basic algebraic geometry springer study edition edition translated russian hirsch revised printing grundlehren der mathematischen wissenschaften vol sommese wampler numerical solution systems polynomials arising engineering science world scientific tavora smith equilibrium analysis power systems ieee transactions power apparatus systems may thorp naqavi load flow fractals decision control proceedings ieee conference pages dec trias holomorphic embedding load flow method power energy society general meeting ieee pages july trias system method monitoring managing electrical power transmission distribution networks patent number type patent version location trias holomorphic embedding loadflow method power systems nonlinear circuits ieee transactions circuits systems regular papers feb wampler bezout number calculations polynomial systems applied mathematics computation wampler efficient start system polynomial continuation numerische mathematik wang chiang number system separations electric power systems ieee transactions circuits systems regular papers may zimmerman thomas matpower steadystate operations planning analysis tools power systems research education power systems ieee transactions
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festung matlab gnu octave toolbox discontinuous galerkin method part advection operator slope limiting balthasar reutera vadym aizingera manuel wielanda florian frankb peter knabnera feb rice university department mathematics erlangen germany university department computational applied mathematics main street houston usa abstract second series papers implementing discontinuous galerkin method open source matlab gnu octave toolbox intention ongoing project offer rapid prototyping package application development using methods implementation relies fully vectorized matrix vector operations comprehensively documented particular attention paid maintaining direct mapping discretization terms code routines well supporting full code functionality gnu octave present work focuses linear advection equation space coefficients provides general order implementation several slope limiting schemes method keywords matlab gnu octave discontinuous galerkin method slope limiting vectorization open source advection operator introduction development milestones matlab gnu octave toolbox festung inite lement imulation oolbox structured rids available run somewhat counter history development discontinuous galerkin methods thus first paper series introduced local discontinuous galerkin discretization diffusion equation using numerical methods introduced current work however enhances package functionality purely hyperbolic original purpose method proposed reed hill analyzed johnson reason behind time inversion numerical software development technology necessary produce fully functional solver hyperbolic equations include upwind fluxes slope tasks complicated solve computationally efficient manner needed pure diffusion equation continued development toolbox still adheres design principles declared design software package using method range standard applications provide toolbox research learning tool open source format supply intuitive ease adoption wider community application engineering professionals relying vectorization capabilities matlab gnu octave optimize computational performance toolbox components demonstrate software development strategies maintain throughout full compatibility gnu octave support users open source software corresponding author email addresses reuter balthasar reuter aizinger vadym aizinger manuel wieland florian frank knabner peter knabner preprint submitted elsevier september refer literature review methods open source packages offering capability present work expands functionality numerical solver published first paper series adding linear advection terms slope limiters general order latter development particularly interesting since best knowledge closed form description slope limiters general order discretizations found literature even less implementations limiters addition hierarchical limiters kuzmin publication accompanying code includes extension standard linear slope limiter general order discretizations new scheme based hierarchical vertexbased limiter using stricter limiting strategy additions work include selection tvd total variation diminishing methods orders one two three employed time discretization instead simple implicit euler method used first paper rest paper organized follows introduce model problem remainder section describe discretization using method sec section introduces slope limiting algorithms first linear discretizations followed general order case implementation specific details reformulation assembly matrix blocks well numerical results given sec routines mentioned work listed documented sec section concludes work gives future perspectives model problem let tend finite time interval polygonally bounded domain boundary consider advection equation conservative form time coefficients prototype application advective transport fluids movement solute due bulk movement fluid case primary unknown denotes solute concentration velocity fluid accounts generation degradation chemical reactions equation complemented following boundary initial conditions inflow boundary denoting outward unit normal outflow boundary defined given initial dirichlet boundary data respectively discretization notation describing scheme introduce notation overview found section index notation let regular family partitions closed triangles characteristic size let denote unit normal exterior let denote set interior edges set boundary edges set edges subscript suppressed interior edge shared triangles define values scalar quantity respectively boundary edge definition left meaningful variational formulation local nature method formulate variational system equations basis multiply smooth test function integrate parts element gives formulation denote space complete polynomials degree let denote broken polynomial space triangulation formulation assume coefficient functions fixed approximated specific way compute approximations given first paper series use standard therefore accuracy approximation improves increasing polynomial order choosing polynomial space functions simplifies implementation done preparation later applications might part solution coupled system incorporating boundary condition semidiscrete formulation reads seek following holds boundary integral calculated using value outflow inflow inflow note use approximate representation velocity boundary integral due fact elements may poor approximation quality edges generally produces different values sides edge ultimately leading different values inconsistent flux approximations instead evaluate normal velocity quadrature point analytically use result numerical integration determination upwind direction demonstrated sec thus far used algebraic indexing style remainder switch mixture algebraic numerical style instance ekn means possible combinations element indices local edge indices ekn lies implicitly fixes numerical indices accordingly used index matrices arrays use bracket notation followed subscript index matrices multidimensional arrays thus array symbol stands component index dimension matlab gnu octave colon used abbreviate indices within single dimension example array matrix local basis representation contrast globally continuous basis functions mostly used continuous finite element method basis functions continuity constraints across triangle boundaries thus standard basis function supported triangle defined arbitrarily ensuring span number local degrees freedom note may general vary triangle triangle simplicity assume uniform polynomial degree every triangle abbreviate clearly number global degrees freedom equals expressions orthonormal basis functions reference triangle sec employed implementation order two found first paper basis functions order four provided routine phi gradients gradphi bases even higher order constructed algorithm using recursion latter unfortunately trivial derive case triangles note modal basis functions posses interpolation properties nodes unlike lagrangian nodal basis functions often used continuous finite element nodal methods local concentration local velocity represented terms local basis ukmj denotes unit vector condense coefficients associated unknowns twodimensional arrays etc symbol called local representation matrix respect basis similar way express coefficient functions linear combinations basis functions use local representation matrices system equations testing yields system equations whose contribution identified boundary integrals reads uklm iii fkl abbreviated written matrix form system given representation vector block matrices side vectors described sections note blocks except mass matrix suppressed time arguments figure two triangles adjacent edge holds contributions area terms matrices remainder section sparse block structure giving definitions blocks tacitly assume zero mass matrix term defined since basis functions supported structure consists local mass matrices henceforth write diag block matrices term given uklm similarly matrices diag local matrices uklm vector resulting obtained multiplication representation vector global mass matrix contributions edge term iii interior edges section consider fixed triangle interior edge shared adjacent triangle associated fixed local edge indices fig fixed index contribution block matrix means depending direction velocity field obtain entries diagonal offdiagonal blocks entries diagonal blocks given ekn ekn entries blocks possibly pairs triangles read boundary edges similarly interior edges contributions boundary edge ekn ekn consist entries block diagonal matrix ekn ekn side vector rkn ekn ekn combine block matrices block matrix since definition entries diagonal blocks eqns matrices differing set edges included sum disregard fact whether interior boundary edges simply assemble entries ekn time discretization system equivalent vector rkn defined discretize system time using tvd total variation diminishing methods orders one two three representatives class ssp strong stability preserving methods advantage using time stepping algorithm type lies guaranteed preservation monotonicity solution discretization also post processed slope limiting method let tend necessarily equidistant decomposition time interval let denote time step size update scheme method given abbreviated coefficients possible choose order spatial approximation order order avoid temporal discretization error dominating spatial one unfortunately optimal ssp kutta methods higher order three known restrict orders one three time discretization slope limiting slope limiters technique prevent onset spurious oscillations violate monotonicity preserving property piecewise constant part solution means restricting degrees freedom generally linear superlinear certain bounds thus eliminating undershoots limiting procedures utilize fact lowest order piecewise constant part solution explicit tvd time stepping schemes guaranteed preserve monotonicity solution produce spurious extrema using physically consistent numerically accurate solution part slope limiters attempt modify full higher order solution suitable one hand prevent oscillations hand preserve much accuracy possible key differences slope limiters affect limiting stencil used edge neighbors node neighbors neighbors neighbors etc presence hoc parameters amount introduced numerical diffusion strict less strict preservation monotonicity degree solution degradation smooth extrema whereas large literature slope limiting piecewise linear discretizations exists limiting solutions much less explored area traditional approach dealing superlinear solutions based ignoring higher order degrees freedom elements linear limiting active methods require much larger stencil provide enough information reconstruction higher order derivatives hierarchical limiters kuzmin represent computationally efficient scheme easily extendable discretization order supporting fully unstructured meshes limiters guarantee strict monotonicity solution violations small may reduced simple modifications described sec taylor basis representation many limiting procedures rely fundamental properties certain choice basis case taylor basis introduce way similar kuzmin consider taylor series expansion local solution xkc xkc centroid xkc use standard notation let form denote integral mean express equivalent xkc xkc note varying terms except affect mean able identify term expansions consecutive index introduce linear index mapping corresponding dim defined implicitly define linear indices polynomial degrees corresponding order four listed table leads following table bottom linear indices middle corresponding polynomial degrees taylor basis top definition local taylor basis xkc xkc opposed basis sec basis defined reference element scaling max min max xki min xki minimum maximum values corresponding spatial coordinates introduced obtain better conditioned operator taylor degrees freedom proportional cell mean values derivatives centroid xkc xkc note taylor basis triangular meshes cell means still decoupled degrees freedom since transform function modal basis representation representation matrix described sec taylor basis representation representation matrix ctaylor employ defined locally taylor choosing obtain taylor taylor ctaylor local mass matrix defined local basis transformation matrix taylor taylor taylor representation vectors ctaylor rkn obtain linear using mdg taylor diag system equations mdg taylor ctaylor employed transform bases figure neighborhood vertex xki red circle considered consists patch elements max containing xki red area bounds cmin cki determined centroid values green squares within neighborhood linear limiter kuzmin aizinger described limiter based limiter improved taking bounds elements containing vertex instead taking edge neighbors cell goal determine maximum admissible slope linear reconstruction form ckc xkc xkc abbreviated function value ckc xkc centroid xkc correction factor chosen reconstruction bounded vertices xki minimum maximum centroid values elements containing xki max cmin xki cki cmin min xki clc cmax max xki clc fig illustration enforce correction factor defined max ckc cki ckc cki cmax min max cki cki cki min cmin cmin cki ckc xkc xki xkc unconstrained linear reconstruction xki limited counterpart solution becomes linear degrees freedom scaled degrees freedom associated higher polynomial degrees set zero particular case linear approximation limiting performed using hierarchical basis opposed solutions require taylor basis representation see sec case linear superlinear degrees freedom remain unchanged hierarchical limiter improvements kuzmin combine standard limiter higher order limiting scheme yang wang limit numerical solution multiplying derivatives order common correction factor instead applying correction linear terms dropping higher degrees freedom kuzmin described scheme detail quadratic representations offer closed form expression limiting procedure discretizations arbitrary orders let set order determine correction factor order computing correction factors using linear limiter linear reconstructions derivatives order taylor taylor taylor xki xki xki indices corresponding degrees freedom given first identified identified formally correction factor defined min max min cmin cmax min cmax cmin max cmin defined avoid loss accuracy smooth extrema lower order derivatives limited factor exceeding higher order derivatives since lower orders typically smoother beginning degrees freedom compute correction factors max correction factor becomes equal one limiting element necessary limited solution becomes xkc stricter form limiter numerical experiments showed implicitly assuming higher order derivatives always smoother lower order derivatives results limiting procedures guarantee strict fulfillment condition especially discontinuities solution modified two key components limiter presented previous section obtained limiter exhibited slightly stronger peak clipping turned always effective instead employing linear reconstruction given replace computation correction factor full reconstruction taylor xki polynomial degree solution begin derivative drop hierarchical limiting condition instead apply correction coefficient immediately coefficients corresponding polynomial degree higher limited coefficients used compute next correction coefficient result stricter limiter slope limiting problems problems slope limiting procedure applied intermediate solution update scheme however due fact taylor basis discussed sec implicit coupling spatial derivatives present leads spatial variations time derivatives reason kuzmin applied slope limiter solution stage also time derivative used addition filtering procedure interpreted selective mass lumping describe technique first discretization taylor basis extend arbitrary basis representations let denote slope limiting operator applies slope limiting procedures global representation vector ctaylor solution taylor basis representation system written taylor basis ctaylor staylor ctaylor replaced ctaylor staylor ctaylor ctaylor diag mii denote full lumped mass matrices taylor basis improve readability drop superscript taylor note case formulations identical consequently update scheme modified replacing taylor ctaylor ctaylor ctaylor selectively lumped limited update ctaylor ctaylor ctaylor taylor taylor staylor although mass matrix modal basis diagonal implicit coupling spatial derivatives still present taylor basis vectors coinciding coordinate directions hence lumping technique directly applied representations bases get rid implicit coupling spatial derivatives time derivative modal basis reformulate lumped time derivative taylor basis taylor taylor taylor taylor taylor taylor taylor taylor taylor taylor taylor using time derivative modal basis transformation obtain taylor mdg taylor mdg taylor mdg taylor thus fully modified version update scheme reads cnh given slope limiting operator formally defined mdg taylor mdg taylor figure affine mapping transforms reference triangle vertices physical triangle vertices xki boundary conditions max problems occur computing bounds cmin cki control points xki dirichlet boundary account boundary data limiting procedure include control points boundary value xki applying slope limiting operator implementation extensive documentation data structures grid etc found first paper series explanations reproduced greater detail provided routines first introduced present work backtransformation reference triangle using back transformation reference triangle defined affine mapping triangle see fig holds det function implies transformation gradient obtained chain rule results transformation formulas integrals element abbreviated edge ekn function ekn numerical integration alternative symbolic integration functions provided matlab implemented quadrature integration functionality triangle edge integrals since transform integrals reference triangle sec sufficient define quadrature rules course rewritten apply every physical triangle quadrature points quadrature weights order quadrature rule largest integer exact polynomials note exclusively rely quadrature rules positive weights quadrature points located strictly interior rules used implementation found routine overview quadrature rules triangles found encyclopaedia cubature formulas edge integration rely standard gauss quadrature rules required order integrals contain integrands polynomials maximum order triangles maximum order edges using quadrature integration one could choose rules integrate terms exactly however sufficient accuracy achieved quadrature rules exact polynomials order triangles edges assembly aim section transform terms required build block matrices reference triangle evaluate either via numerical quadrature analytically assembly block matrices local contributions performed vectorized operations implementation need explicit form components mappings inverses defined recalling det sec obtain obtain rule gradient similarly first paper series make extensive use kronecker product two matrices rma bkl rmb defined rma following present necessary transformation blocks system name corresponding matlab gnu octave routines found sec assembly using transformation rule following holds local mass matrix defined representation local mass matrix reference triangle see global mass matrix expressed kronecker product matrix containing areas local matrix corresponding assembly routine assemblematelemphiphi sparse matrix generated using command spdiags list assembly application product rule gives multidimensional array representing transformed integral reference triangle express local matrix ukl analogously diag vectorize triangles using kronecker product done routine assemblematelemdphiphifuncdiscvec would like point identical assembly first paper except vectorial coefficient function however respective section typos corrected version additionally assembly routine wrong component normal vector used assembly corrected versions code found assembly ease assembly split global matrix given part remainder rdiag roffdiag holds first consider entries given eqns transform integral terms edge reference triangle ekn used transformation rule quadrature rule explicit forms mappings easily derived thus assemble global matrix using kronecker product rdiag diag denotes hadamard product next consider blocks stored roffdiag interior edge obtain analogously note maps since compute line integral integration domain restricted edge edge result integration boiled nine possible maps sides reference triangle expressed arbitrary index pair described expressions nine cases given first paper apply quadrature rule define thus arrive roffdiag sparsity structure blocks depends numbering mesh entities given combination list due upwind flux edge integrals possible include quadrature rule directly opposed assembly element integrals assembly routines edges first paper implementing assembly formulation possible expensive since global matrix would built every quadrature point instead make use fact sparsity structure every quadrature point solely rely element numbering account upwind direction determining structure roffdiag define tensor block vectors rma let rma rmb rmb operator interpreted kronecker product takes different side every row side implemented routine kronvec allows write offdiag offdiag omits expensive assembly global matrix every quadrature point routine assemblematedgephiphivalupwind assembles matrices rdiag roffdiag directly code similar formulation avoid repeated computation normal velocity evaluate quadrature points edge using globally continuous function store dedicated variable vnormalonquadedge employed decision upwind direction using well normal velocity assembly assembly entries transformed using transformation rule mapping ekn ekn ekn ekn implicitly assumed application integral approximated using quadrature rule reference interval ekn allowing vectorize computation triangles resulting routine assemblevecedgephiintfunccontval slope limiters implementation slope limiters described sec three parts must considered assembly transformation matrix mdg taylor slope limiters selective mass lumping limiting explained sec assembly mdg taylor entries mdg taylor transformed using transformation rule integral approximated using quadrature rule reference triangle mdg taylor recall define taylor basis reference triangle since basis functions depend directly physical coordinates element assembly vectorized triangles resulting routine assemblematelemphidiscphitaylor matrices mdg taylor assembled sec used routines transform representation matrix modal taylor explained sec solving system slope limiting operators slope limiters implemented generic manner according sec beginning derivatives evaluate linear full reconstruction derivatives control points xki max values along centroid values used determine minimum maximum values cmin cki see control point correction factors according calculated implemented routine computevertexbasedlimiter depending limiter type value applied certain subset degrees freedom computation repeated next lowerorder derivatives full slope limiting operators provided functions applyslopelimiterdisc applyslopelimitertaylor respectively min overcome numerical problems cases cki cmax cki cki modify eqs max min condition cases modified cki cki cki cki small additionally increase absolute value denominators cases add subtract respectively makes limiter slightly strict one given condition found suitable value double precision computations moreover enforce overcome cases division number close zero might lead values reduce execution time perform necessary evaluation taylor basis functions vertices store result global variable used slope limiting routines implemented limiting obtain selectively lumped required update scheme compute discrete time derivative given sec transform taylor basis representation using implemented routine compute stationary matrix mcorr beginning computationally cheap since diagonal matrix calculate selectively lumped equation backtransformation modal basis cheap since requires inverse diagonal matrix implemented produce selectively lumped intermediate solution update scheme obtained applying slope limiting operator implemented applyslopelimiterdisc numerical results analytical convergence test code verified showing numerically estimated orders convergences match analytically predicted ones prescribed smooth solutions verify spatial discretization restrict stationary version investigate impact different slope limiting schemes convergence order effectiveness slope limiters verified next section choose exact solution cos cos velocity field exp exp domain data derived analytically inserting compute solution sequence increasingly finer meshes element sizes coarsest grid covering irregular grid finer grid obtained regular refinement predecessor discretization error kch time computed difference discrete solution analytical solution described detail first paper table contains results demonstrating experimental order convergence estimated using kch kch interesting points compare table errors different limiting schemes whereas linear approximation get limited finer limiters able distinguish smooth solutions given sufficient mesh resolution case higher order approximation warrants closer look first linear limiter takes heavy toll higher order degrees freedom strict hier vert based linear none limiter kch kch kch kch kch table discretization errors measured estimated orders convergences different polynomial degrees limiter types triangles jth refinement level quadratic cubic quartic solutions produce virtually error thus negating effect accurate solution strict limiter seems flatten cubic approximation thus gaining one order convergence linear limiter hierarchical limiter kuzmin however appears perform well tested meshes approximation orders small effect errors analytical test case solid body rotation benchmark problem use solid body rotation test proposed leveque often used investigate limiter performance consists slotted cylinder sharp cone smooth hump see figure placed square domain transported velocity field counterclockwise rotation choose initial data satisfying slotted cylinder sharp cone cos smooth hump otherwise zero boundary side projected limited initial data linear limiter lumped hier lumped linear limiter lumped time derivative hier limiter lumped time derivative strict limiter lumped time derivative figure visualization solutions end time tend first would like emphasize huge improvement solution applying kuzmin selectively lumped scheme sec reduces numerical diffusion peak clipping visible intersection lines fig visualization figs results smaller error see tab without technique implicit coupling derivatives renders higher order limiter inferior linear limiter results different slope limiters see fig applying lumped scheme similar limiters producing errors range see tab linear vertex based limiter exhibits significantly stronger peak clipping higher order limiters intersection lines fig looking slotted cylinder strict limiter outperforms others producing less slot better preserving shape cylinder also total error lowest strict limiter initial linear linear lumped hier vert based hier vert based lump figure intersection lines solution end time tend without selectively lumped described section large deviation solutions initial data lower right plot due slot cylinder polynomial degrees solution might still violate bounds along edges using control points vertices element slope limiting procedure evaluating numerical fluxes values transported neighboring elements lead cell averages lying outside initial bounds next time step table shows limiter types suffer however small violations bounds usually smooth time due effects numerical diffusion introduce numerical problems linear limiter preserves upper bound comes price stronger effectively restrict solution bounds additional control points edges necessary leads significant increase numerical diffusion hence suitable technique register routines list routines implementation added since first paper alphabetic order reason compactness waive check correct function arguments means routines assert however strongly recommended catch exceptions code extended argument always initial linear hier vert based strict figure intersection lines solution end time tend applying lumped timestepping scheme sec compared solutions different limiters described section mink xkc mink xki mink maxk xkc maxk xki maxk kch kch tend linear hier linear lumped hier strict table minimum maximum values centroids xkc nodes xki edge midpoints entire simulation time two last rows give projection initial data end solution compared analytical initial data struct representing triangulation argument always number local basis functions script demonstrates application presented routines given additionally recommend using laurent sorbers slightly faster implementation kronecker product sparse matrices matlab file speed computation note implementation suitable logical double matrices http datadisc applyslopelimiterdisc datadisc globm globmdisctaylor type implements slope limiting operator given chosen limiter type given string type parameter datadisc representation matrix modal basis logical matrix marking dirichlet boundary nodes boundary data given function datadisc datadisc globm globmdisctaylor type datadisc globm datataylor type datadisc datataylor globm end datataylor applyslopelimitertaylor datataylor type implements slope limiting operator described sec parameter datataylor representation matrix taylor basis parameters applyslopelimiterdisc function datataylor type switch type case linear datataylor case datataylor case strict datataylor otherwise error unknown limiter type end end datataylorlim applyslopelimitertaylorhierarchicalvertex datataylor applies hierarchical limiter described sec input parameters applyslopelimitertaylor function datataylor global size sqrt zeros size alpha zeros ord alphaord ones inddof ord ord ord ord ord mult bsxfun plus ord ind mult ind ord alphatmp ind else alphatmp ind end alphaord min alphaord alphatmp end alpha max alpha alphaord inddof bsxfun times alpha inddof end end datataylorlim applyslopelimitertaylorlinear datataylor applies linear limiter described sec input parameters applyslopelimitertaylor function datataylor global alphae bsxfun times alphae bsxfun times alphae end end datataylorlim applyslopelimitertaylorstrict datataylor applies stricter form hierarchical limiter described sec input parameters applyslopelimitertaylor function datataylor global size sqrt ord alphaord ones inddof ord ord ord ord mult bsxfun plus ord ind mult ind size ind ord alphatmp ind else alphatmp ind end alphaord min alphaord alphatmp end inddof bsxfun times alphaord inddof end end ret assemblematedgephiphivalupwind refedgephiintphiint refedgephiintphiext valonquad bles matrix according sec containing edge integrals products two basis functions multiplied quadrature point specified values stored valonquad value valonquad chosen input arguments refedgephiintphiint refedgephiintphiext provide local matrices multidimensional arrays respectively function ret numt size ret sparse sqrt qord qord rkn length ret ret kron spdiags rkn end sparse length kron rkn sparse end ret ret kronvec end end end assembles two matrices containing integrals products basis function spatial derivative basis function component discontinuous coefficient function whose coefficients specified respectively matrices returned cell variable corresponds matrices according sec input argument refelemdphiphiphi stores local matrices multidimensional array defined computed integraterefelemdphiphiphi coefficients projection algebraic diffusion coefficient broken polynomial space stored input arguments computed ret assemblematelemdphiphifuncdiscvec refelemdphiphiphi function ret iph size ret cell ret sparse ret sparse ret ret kron spdiags kron spdiags ret ret kron spdiags kron spdiags end end ret assemblematelemphidiscphitaylor assembles matrix mdg taylor according sec responds basis transformation matrix one basis function modal taylor basis required transformation modal taylor basis routines function ret global sqrt qord max qord numt ret sparse repmat qord ret ret sparse areat intphiiphij end end end ret assemblematelemphitaylorphitaylor assembles mass matrix taylor basis mtaylor function ret sqrt qord max qord numt ret sparse ret ret sparse areat intphiiphij end end end ret assemblevecedgephiintfunccontval funccont valonquad assembles vector taining integrals products basis function continuous function given value provided quadrature point edge triangles corresponds contributions dirichlet boundaries side according sec marks boundary edges vector assembled funccont function handle valonquad value provided quadrature point computed computefunccontnuonquadedge function ret funccont valonquad global numt sqrt qord qord ones size ones size ret zeros gammamap funccont kkn integral squeeze ret ret kkn integral end end ret reshape ret end ret computefunccontnuonquadedge qord assembles array normal velocity evaluated quadrature points edges triangle function ret qord numt qord ret zeros length gammamap ret bsxfun times bsxfun times end end funccont assembles matrix containing function funccont evaluated node triangle function funccont zeros numt funccont end end ret computefuncdiscatpoints funcdisc phiatpoints assembles matrix containing values discrete function representation matrix stored funcdisc evaluated points evaluated basis functions given phiatpoints function ret funcdisc npoints size phiatpoints size funcdisc ret zeros npoints npoints ret sum funcdisc squeeze end end valcentroid determines matrix min bounds cmax cki vertex triangle required computevertexbasedlimiter function valcentroid cell zeros numt zeros numt abs min abs max vald nan numt vald exist builtin speye size size end min min bsxfun times valcentroidneg shiftcentroidneg vald max max bsxfun times valcentroidpos shiftcentroidpos vald end end evaluates taylor basis functions vertices triangles xki stores global multidimensional array function global zeros numt end end end alphae computevertexbasedlimiter valcentroid computes vector correction factors elements centroid values ckc given valcentroid values unconstrained reconstruction cki specified logical matrix marking dirichlet boundary nodes boundary data given function alphae valcentroid valcentroid repmat valcentroid repmat valcentroid repmat valcentroid tol markneg tol markpos tol ones numt markneg max min markneg markneg tol markpos max min markpos markpos tol alphae min end ret integraterefedgephiintphiextperquad computes multidimensional array functionals ture points edges reference triangle consist permutations two basis functions one belongs neighboring element transformed using corresponds local matrix given function ret global sqrt qord qord ret zeros length edges ret qord qord end end end end end ret integraterefedgephiintphiintperquad computes multidimensional array functionals ture points edges reference triangle consist permutations two basis functions corresponds local matrix given function ret global sqrt qord max qord ret zeros length edges ret qord qord end end end end kronvec computes result rmb given function kronvec size size issparse issparse full full reshape reshape reshape bsxfun times else find kron ones repmat bsxfun plus bsxfun times sparse end end main script solve used template modifications modifiable parameters found lines problem data initial condition velocity side boundary data specified lines function tic start time hmax edge local ordrk min order runge kutta time tend end time numsteps time steps isvisgrid false grid isvissol true true slope type slope linear every files outputtypes cellstr vtk tec check diary log assert order must zero four assert ordrk ordrk order runge kutta must zero three assert hmax maximum edge length must positive assert numsteps number time steps must positive assert slope limiting hmax hmax isvisgrid end numt nchoosek local dofs tau tend numsteps time step size mark local edges mark local edges ismember mark bdr data leveque solid body sqrt cos fcont zeros size cdcont zeros size table basis isslopelim end hatm hatg hatrdiagonquad time globm hatm globmtaylor globmcorr spdiags diag end data cdisc hatm cdcont cdisc cdisc globm globmdisctaylor end fprintf error initial cdisc isvissol cdisc clagrange outputbasename end time fprintf starting time using time step size steps tend tau nstep numsteps omega ordrk tau nstep tau cdiscrk cell length omega cdiscrk reshape cdisc runge kutta steps rkstep length omega fdisc fcont rkstep hatm rkstep hatm rkstep hatm edges rkstep rkstep veloc dot edges time globg hatg globr hatrdiagonquad onq globkd cdcont rkstep ded globl globm reshape fdisc sysa globg globg globr sysv globl globkd time cdiscdot globm sysv sysa cdiscrk rkstep apply slope time reshape cdiscdot globm cdiscdottaylor nan reshape ylo reshape cdiscdot reshape reshape cdiscdottaylor globm end next step cdiscrk rkstep omega rkstep cdiscrk omega rkstep cdiscrk rkstep tau cdcont rkstep cdiscrk rkstep reshape reshape cdiscrk rkstep globm globmdisctaylor end end cdisc reshape cdiscrk end visualization isvissol mod nstep cdisc clagrange outputbasename nstep end end isvissol cdisc clagrange outputbasename nstep end fprintf error initial cdisc fprintf total time seconds toc diary end ind computes linear index corresponding defined function ind sum ind end mult multiindex computes involved representation polynomial solution degree returns array function mult mult zeros mult ord offset ord ord ord mult offset mult ord end end end ret phitaylorphy evaluates ith basis function triangle sec points specified list coordinates coordinates function ret qord ceil sqrt qord ones size ones size length numt nump size repmat max min nump repmat max min nump switch case ret ones nump case ret repmat baryt nump case ret repmat baryt nump case ret repmat baryt nump repmat repmat baryt nump case ret repmat baryt nump repmat baryt nump repmat repmat baryt repmat baryt nump case ret repmat baryt nump repmat repmat baryt nump case ret repmat baryt nump repmat repmat baryt nump case ret repmat baryt nump repmat baryt nump repmat repmat baryt repmat baryt nump case ret repmat baryt nump repmat baryt nump repmat repmat baryt repmat baryt nump case ret repmat baryt nump repmat repmat baryt nump case ret repmat baryt nump repmat repmat baryt nump case ret repmat baryt nump repmat baryt nump repmat repmat baryt repmat baryt nump case ret repmat baryt nump repmat baryt nump repmat repmat baryt repmat baryt nump case ret repmat baryt nump repmat baryt nump repmat repmat baryt repmat baryt nump case ret repmat baryt nump repmat repmat baryt nump end end ret phitaylorref evaluates ith basis function triangle sec points specified list coordinates coordinates function ret ones size ones size ret end datataylor datadisc globmdisc globmdisctaylor converts tion matrix modal basis respective representation matrix taylor basis size solves ctaylor function datadisc globmdisc size datadisc reshape reshape datadisc end datadisc datataylor globmdisc globmdisctaylor converts tion matrix taylor basis respective representation matrix modal basis size solves cdg function datadisc datataylor globmdisc size datadisc reshape reshape datataylor end omega rungekuttassp ord tau nstep provides list time levels weights time step nstep time step size tau according sec order method given parameter ord function omega ord tau switch ord case omega case omega tau case omega tau end end conclusion outlook second installment present paper series implementing matlab gnu octave toolbox introduced performance optimized techniques dealing linear advection operators higher order time discretizations range slope limiters designed support general order discretizations future work plans include nonlinear advection operators coupled systems pdes well applications corresponding coupling mechanisms acknowledgments work reuter supported german research foundation dfg grant index notation symbol definition integral mean matrix blocks diag ctaylor condition ekn tend xkc xki cardinality set euclidean scalar product spatial gradient physical domain composition functions hadamard product kronecker product concentration unknown concentration prescribed initial time concentration prescribed inflow boundary rkn representation vector respect rkn representation vector respect condition true otherwise kronecker delta mth unit vector nth edge physical triangle nth edge reference triangle sets vertices edges triangles set interior edges set boundary edges source sink coefficient function affine mapping mesh fineness diam diameter triangle tend open time interval number triangles unit normal pointing outward unit normal pointing outward number local degrees freedom quadrature weight associated spatial domain two dimensions boundary outflow boundaries polynomial degree ith hierarchical basis function ith hierarchical basis function ith taylor basis function space polynomials degree slope limiting operator respect slope limiting operator respect rth quadrature point number quadrature points set strictly positive real numbers set real numbers time variable nth time level end time mapping time step size kth physical triangle boundary reference triangle velocity coefficient function space variable physical domain space variable reference triangle centroid element ith vertex physical triangle references frank reuter aizinger finite element simulation toolbox unstructured grids url http frank reuter festung finite element simulation toolbox unstructured grids url https frank reuter aizinger knabner festung matlab gnu octave toolbox discontinuous galerkin method part diffusion operator computers mathematics applications cockburn shu local discontinuous galerkin method systems siam journal numerical analysis reed hill triangular mesh methods neutron transport equation tech los alamos scientific laboratory johnson pitkranta analysis discontinuous galerkin method sacalar hyperbolic equation mathematics computation kuzmin slope limiting discontinuous galerkin approximations possibly taylor basis international journal numerical methods fluids gottlieb shu strong time discretization methods math comp gottlieb shu tadmor strong time discretization methods siam review cockburn shu tvb local projection discontinuous galerkin finite element method conservation laws general framework mathematics computation krivodonova xin remacle chevaugeon flaherty shock detection limiting discontinuous galerkin methods hyperbolic conservation laws applied numerical mathematics aliabadi slope limiting procedure discontinuous galerkin finite element method gasdynamics applications international journal numerical analysis modeling michoski mirabito dawson wirasaet kubatko westerink adaptive hierarchic transformations dynamically discontinuous galerkin systems generalized equations journal computational physics yang wang generalized moment limiter methods unstructured grids adv appl math mech zhang xia shu high order discontinuous galerkin schemes conservation laws triangular meshes sci comput kuzmin hierarchical slope limiter adaptive discontinuous galerkin methods journal computational applied mathematics finite element methods engineering science femtec yang wang generalized moment limiter methods unstructured grids aiaa aerospace sciences meeting including new horizons forum aerospace exposition michalak limiters unstructured accurate solutions euler equations aiaa aerospace sciences meeting exhibit luo baum lhner discontinuous galerkin method based taylor basis compressible flows arbitrary grids journal computational physics aizinger geometry independent slope limiter discontinuous galerkin method krause shokin resch krner shokina eds computational science high performance computing vol notes numerical fluid mechanics multidisciplinary design springer berlin heidelberg barth jespersen design application upwind schemes unstructuredmeshes proc aiaa aerospace sciences meeting reno cools encyclopaedia cubature formulas journal complexity cockburn shu discontinuous galerkin method conservation laws multidimensional systems comput phys leveque conservative algorithms advection incompressible flow siam journal numerical analysis sorber kronecker product matlab central file exchange retrieved october url http
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additive approximation phylogeny construction jun pranjal awasthi avrim blum jamie morgenstern sheffet carnegie mellon university pittsburgh forbes pittsburgh usa pawasthi avrim jamiemmt osheffet abstract study problem constructing phylogenetic trees given set species problem formulated finding minimum steiner tree points boolean hypercube dimension known optimal tree found linear time given dataset perfect phylogeny cost optimal phylogeny exactly moreover data phylogeny cost optimal steiner tree known exact solution found running time polynomial number species yet exponential work give algorithm finds phylogenetic tree cost provides best guarantees case log broadening range settings solutions efficiently found also discuss motivation reasoning studying additive approximations introduction phylogenetics subfield computational biology aims construct simple accurate descriptions evolutionary history descriptions represented evolutionary trees given set species represented set features typical choice features single nucleotide polymorphisms snps binary indicator variables common mutations found dna see example challenging problem attracted much attention recent years progress studying various computational formulations problem problem often posed constructing parsimonious tree induced set species formally phylogeny phylogenetic tree set species represented string called taxa length finite alphabet unrooted tree given distance metric work supported part national science foundation grant nsf graduate fellowship center computational thinking define cost tree maximum parsimony dataset tree minimizes cost respect hamming metric optimum steiner tree set metric steiner tree problem known general remains even case binary alphabet metric induced hamming distance extensive recent work experimental theoretical focused binary character set hamming metric version phylogeny problem also focus paper phylogeny called perfect coordinate flips exactly tree representing single mutation amongst set species dataset admits perfect phylogeny optimal tree constructed polynomial time even linear time case alphabet binary work investigate near perfect phylogenies instances whose optimal phylogenetic tree cost near perfect phylogenies studied theoretical experimental settings work given series randomized algorithms find optimal phylogeny running time polynomial exponential clearly log algorithms tractable alternative approach finding phylogenetic tree low cost use generic steiner tree approximation algorithm best current algorithm yields tree cost comment exponential size explicit hypercube respect small representation size requires one implement algorithm using techniques devised especially hypercube alon however notice moderate polylog excess difference cost extremely large compared excess optimal tree cases one would much prefer algorithm whose excess could written function work present randomized poly algorithm finds phylogenetic tree cost theorem given set terminals optimal phylogeny cost exists randomized poly algorithm finds phylogenetic tree cost note theorem provides substantial improvement prior work case log range exact algorithms longer tractable multiplicative approximations yield significantly worse bounds alternatively viewed multiplicative guarantee range tree within factor optimal best knowledge first work give additive approximation either phylogeny problem setting steiner tree problem one immediate without loss generality may assume coordinate flips least since coordinates species agree may discarded front question remains open whether results improved perhaps even rest paper organized follows surveying related work section detail notation preliminaries section presentation algorithm partitioned two parts section present algorithm case pair coordinates identical terminals formal definition section alter algorithm simple case nontrivial way modified algorithm finds phylogeny dataset conclude section discussion motivating problem phylogeny tree different perspective present open problems future research related work mentioned introduction problem constructing optimal phylogeny even restricted binary alphabets schwartz give algorithm based integer linear programming ilp formulation solve problem optimally show experimentally algorithm efficient small instances perfect phylogenies datasets admit tree coordinate changes exactly optimal parsimony trees constructed linear time binary case polynomial time fixed alphabet unfortunately finding perfect phylogeny arbitrary alphabets recent work gives algorithm construct optimal phylogenetic trees binary phylogenies small number coordinates mutate optimal tree however running time algorithm presented work exponential number additional mutations also lot work computing multiplicative approximations steiner tree problem minimum spanning tree mst set terminals achieves approximation ratio long line work led current best bound recent papers use result due borchers showing optimal steiner tree approximated arbitrary precision using steiner trees approximations steiner tree problem immediately extendable problem constructing phylogenetic trees size vertex set phylogeny problem exponential vertices hypercube algorithm works explicit representation graph defined hypercube solve phylogeny problem polynomial time however line work started robins zelikovsky used notion steiner trees efficiently implemented hypercube particular alon showed finding optimal component given set terminals sufficient consider topologies given terminals leaves using able extend work achieve approximation ratio maximum parsimony problem approximation maximum likelihood byrka considered new relaxation steiner tree problem achieved approximation ratio combined topological argument alon achieve ratio phylogenies notation preliminaries dataset consists terminals binary coordinates steiner tree phylogeny consists tree hypercube spans plus possibly additional steiner nodes label edge index coordinate flipped edge cost steiner tree number edges tree given collection datasets define steiner forest problem problem finding minimal steiner tree every separately refer collection partition even though may contain subset original terminal set work consider instances whose minimum steiner tree cost think otherwise constant approximation algorithm steiner problem gives solution cost fix optimal steiner tree optimality leaves must terminals whereas internal nodes may either terminals nonterminals called steiner nodes define coordinate good exactly one edge labeled bad two edges labeled may assume coordinates appear tree otherwise coordinates fixed dimensionality problem less therefore coordinates bad bad coordinate flips least twice thus adds cost least tree given coordinate set terminals define partition call two coordinates interchangeable define cut present following basic facts easy verify see proofs fact let set interchangeable coordinates coordinates appear together optimal tree adjacent one another paths path edges labeled coordinates edge path internal nodes paths terminals degree paths reordering edges yields equivalent optimal tree two good coordinates one side contained within one side equivalently exist values terminals one side jth coordinate set fix good coordinate let good coordinate terminals one side coordinate set endpoints edge labeled jth coordinate set good coordinate bad coordinate define cut immediately follows fact given good coordinate one efficiently reconstruct endpoints edge mutates except coordinates leads following definition given denote set coordinates fixed constant value least one side different coordinates may fixed different sides denote vector corresponding values coordinates pair called pattern coordinate set terminals match pattern set pbi bij simple case coordinate determines distinct cut show main ideas behind algorithm first discuss special case two coordinates define cut terminal set algorithms constructing phylogenetic trees often make assumption preprocess contracting pair interchangeable coordinates however case contractions problematic discuss next section section deal general case deal interchangeable coordinates fashion basic building blocks turn description algorithm high level motivated notion maintaining proper partition terminals definition call partition proper forest produced restricting optimal tree components composed edge disjoint trees equivalently path two nodes component pass node different component clearly initial partition proper goal maintain proper partition current terminals decreasing dimensionality problem step implemented two subroutines detail pluck leaf paste leaf first subroutine works building optimal phylogeny finding good coordinate adjacent leaf terminal tree replacing parent flipped set terminals observe good coordinate removes occurrence leaving terminals new dataset fixed coordinate thus reducing dimensionality problem matching subroutine succeeds returns found steiner forest terminals merely connects edge labeled returns resulting forest omit formal description input partition current terminals exists coordinate terminal set identical except flipping return else fail lemma proper partition succeeds proper partition proof sketch let subtree resides claim leaf attached edge labeled rest terminals indeed case removing means removing edge clearly leaves components forest wlog lies side cut leaf least two disjoint paths connect two terminals since proper terminals means crosses twice replace even less costly tree flipped projecting path two occurrences onto side observe lemma holds underlying alphabet problem binary particular alphabet split merge longer find leaves pluck switch second subroutine one works splitting set terminals two disjoint sets based value coordinate would like split set terminals according recurse side separately order properly reconnect two subproblems need introduce two endpoints edge respective sides split subroutine deals one particular case endpoints easily identified split input partition current terminals coordinate constant every component find component constant denote find pbi set terminals match pattern exists unique terminal matches pattern one side cut pbi pbi flip coordinate let resulting node add side add side cut return else fail matching subroutine split merge assume split succeeds returns assume found steiner forest terminals merge merely connects edge labeled returns resulting forest formal description omitted lemma assume proper partition assume split called good coordinate edge labeled least one endpoint terminal returned partition proper proof sketch since proper induced tree tree forest contains edge lemma follows showing two endpoints edge follows observation endpoints edge must match pattern let endpoint wlog belongs cut coordinates fixed obviously right values coordinates fixed flip traversing set value fixed algorithm introduce algorithm input partition current terminals initially singleton set succeeds returns recurse back return resulting forest number coordinates least pick coordinate invoke split split succeeds recurse merge return resulting forest otherwise fail else every find mst return forest fig algorithm simple case theorem probability algorithm figure returns tree whose cost order prove theorem fix optimal phylogeny initial set terminals partition algorithm creates denote forest induced partition proof theorem relies following lemma lemma proper partition probability split called good coordinate succeeds furthermore split executed times proof theorem proof follows lemmas since start proper partition probability least keep recursing proper partitions reaching base recursion time algorithm reaches base recursion dimensionality problem reduced cost optimal steiner forest msts give optimal steiner tree problem forest cost algorithm reconnects forest adding coordinates edges algorithm removed first two steps algorithm since algorithm removed edges tree outputs overall cost proof lemma let partition first iteration algorithm split invoked assume proper thus forest contains disjoint components call vertex forest degree internal split suppose replace internal split deg many new vertices adjacent one edge breaks forest collections paths call path decomposition tree addition remove path decomposition edges labeled bad coordinate obtain good path decomposition denote number paths good path decomposition first claim call split partition succeeding coordinate lies path length abovementioned decomposition fail assume split called denote adjacent coordinate path choose one arbitrarily two adjacent coordinates path observe decomposition leaves good coordinates good therefore fixed one side fixed one side follows exist binary values every fact node entire tree node connecting recall assume special case define cut follows node terminal use lemma deduce split succeeds split either fail return partition invoked either bad coordinate good coordinate lies path length path decomposition bad coordinates paths length call split fails furthermore calling split good edge lying path length least results endpoints new leaves respective sides result completely unravels path lies therefore successful run algorithm split called times remains bound number paths path decomposition partition failed execute observe forest even single leaf connected rest tree good coordinate would continue leaf definition terminal good coordinate takes certain value follows number leaves bounded number bad edges removing internal splits leaves paths removing bad coordinates edges adds new paths every bad coordinate adjacent leaf removing create new path therefore call split success probability split called times general case interchangeable coordinates may exist describing general case let briefly discuss conventional way initially contracting interchangeable coordinates applying algorithm section might result tree cost analysis first two steps algorithm still holds problem lies base recursion algorithm runs indeed mst algorithm invoked contracted coordinates correspond original coordinates possible using constant approximation entire forest may end tree cost revised algorithm contract edges initially instead let define simple coordinate one split succeeds first alteration make algorithm call split long set simple coordinates sufficiently big however alterations lie base recursion detail algorithm analyze correctness algorithm description coordinate denote set coordinates interchangeable number theorem probability algorithm figure returns tree cost proof theorem follows outline proof theorem observe lemmas still therefore probability algorithm enters base recursion proper partition thus following lemma algorithm outputs tree cost lemma assume base recursion step called proper partition terminals coordinates algorithm returns forest cost full proof lemma deferred appendix however let sketch main outline proof recall good path decomposition used proof lemma partition paths following way clearly split abort might case algorithm picks bad coordinate happen probability input partition current terminals initially singleton set succeeds returns recurse return resulting forest number simple coordinates least pick simple coordinate invoke split split succeeds recurse merge return resulting forest otherwise fail else contract every unique component resides apply pattern matching let pattern simple split else define node node every coordinate set bij every coordinate set define coordinate flipped every find mst retrieve forest every simple add edge labeled else add edge labeled expand contracted coordinates original set coordinates replacing path length return resulting forest fig algorithm general case paths least one terminal paths interchangeable coordinates may appear order coordinates paths simple enter base recursion edges paths paths terminal length paths composed interchangeable coordinates since deduce good therefore endpoints paths fixed bad coordinates therefore contract edges split introduce side cut arbitrary endpoint replacing coordinates zeros side cut cost subtree increases since paths overall cost introducing artificial endpoints paths terminal length paths composed interchangeable coordinates contract since edges paths run mst approximation incur cost edges paths runtime analysis implemented time linear size dataset counting number simple coordinates takes time split takes time naive implementation base case recursion takes time contracting coordinates rest implemented time hence time process node recursion tree since nodes recursion tree total runtime discussion open problems paper presents randomized approximation algorithm constructing phylogenies order achieve obtain steiner tree low additive error however biological perspective goal find good evolutionary tree one give correct answers questions like common ancestor following species two happened earlier questions hope answered finding phylogenetic tree given taxa hence also desirable tree output also captures lot structure optimal tree would like point algorithm fact valuable property notice base case recursion split subroutines construct optimal tree correctly identify endpoints edges remove even algorithm reaches base case recursion declare every edge weight good know endpoints coordinates total algorithm gives structure optimal tree edges edges marked unsure several open problems remain work one whether one devise algorithm outputting phylogenetic tree cost alternatively one may try design exact algorithms efficient even log suspect even case log poses quite challenge finally extending results alphabets intriguing note however even case perfect phylogenies tractable moderately sized alphabets furthermore approach completely breaks alphabets see comment past lemma devising algorithm phylogeny problem alphabets requires different approach altogether references ding filkov gusfield algorithm perfect phylogeny haplotyping pph problem recomb springer blelloch dhamdhere halperin ravi sridhar fixed parameter tractability binary phylogenetic tree reconstruction icalp springer gusfield algorithms strings trees sequences computer science computational biology cambridge university press semple steel phylogenetics oxford lecture series mathematics applications oxford university press hinds stuve nilsen halperin eskin ballinger frazer cox patterns common dna variation three human populations science international hapmap project nature alon chor pardi rapoport approximate maximum parsimony ancestral maximum likelihood trans comput biol bioinformatics jan robins zelikovsky improved steiner tree approximation graphs soda society industrial applied mathematics robins zelikovsky improved steiner tree approximation graphs robins zelikovsky tighter bounds graph steiner tree approximation siam journal discrete mathematics misra blelloch ravi schwartz generalized buneman pruning inferring parsimonious phylogeny berger recomb volume lecture notes computer springer apr lagergren algorithm phylogeny siam comput may karp reducibility among combinatorial problems miller thatcher eds complexity computer computations plenum new york foulds graham steiner problem phylogeny adv appl math sridhar dhamdhere blelloch halperin ravi schwartz algorithms efficient phylogenetic tree reconstruction theory practice trans comput biol bioinformatics oct damaschke parameterized enumeration transversals imperfect phylogeny reconstruction theor comput sci feb agarwala algorithm perfect phylogeny problem number character states fixed sfcs nov byrka grandoni improved approximation steiner tree stoc acm bodlaender fellows warnow two strikes perfect phylogeny icalp takahashi matsuyama approximate solution steiner problem graphs mathematica japonica berman ramaiyer improved approximations steiner tree problem soda pramel steger algorithms steiner problem reischuk morvan eds stacs volume lecture notes computer science springer berlin heidelberg karpinski zelikovsky new approximation algorithms steiner tree problems journal combinatorial optimization zelikovsky better approximation bounds network euclidean steiner tree problems technical report hougardy promel approximation algorithm steiner problem graphs soda borchers ratio graphs stoc acm appendix give short proofs basic properties presented fact many results also found work let set coordinates define cut terminals renaming whenever edge labeled edge lies path edge labeled unique coordinate node path terminal proof assume define cut given edge flips consider shortest path terminal another terminal every node path degree terminal flips suppose node path degree either flips outgoing path terminals define cut since outgoing path terminal flips outgoing paths terminal occurs paths relabel steiner nodes path constant along outgoing paths add one steiner node edge endpoint flips tree cost strictly less tree flipped outgoing path since least two paths flipped two good coordinates one side contained within one side alternatively exist values terminals one side jth coordinate set proof suppose good consider since good may flip tree flips constant flips constant fix good coordinate let good coordinate terminals one side coordinate set endpoints edge labeled coordinate set proof suppose endpoint edge flips constant set since edge allows one coordinate flip across endpoints labelled side constant say pay constant set flip twice contradiction since good constant set labels edge labelled constant setting assuming flips somewhere flips twice contradicting fact good good coordinate bad coordinate define cut proof follows directly fact since occur number times optimal tree good coordinate occurs exactly bad coordinate occurs least twice proof lemma give full proof lemma convenience reiterate lemma lemma assume step entered proper partition terminals coordinates step algorithm returns forest cost proof recall coordinate define weight total number coordinates interchangeable let post denote instance results splitting coordinates weight greater proof lemma reduces showing exists forest post cost forest exists component returns forest cost reconstructing tree plucked split edges weight forest get cost set terminals components post composed terminals belong original new terminals added coordinates weight simple construct forest low cost post taking optimal steiner forest every simple coordinate split upon remove path corresponding coordinates iii every coordinate split upon remove path corresponding coordinates introduce two vertices connecting vertex path resides side cut denote resulting forest post show post contains edges first observe since bad coordinates splitting coordinates weight guaranteed split good coordinates know fact good bad coordinates define cut therefore since proper coordinate resides single component second observe base recursion executed fails therefore proof lemma forest leaves path decomposition contains paths next observation following proposition proposition fix path path decomposition forest exists terminal path coordinates path simple proof let coordinate associated edge path let terminal path closest recall observation two coordinates yield terminal cut must appear optimal tree adjacent one another furthermore order matter see fact therefore may assume adjacent furthermore endpoint exists good coordinate adjacent path follows determine two different cuts set terminals proof lemma simple unique terminal one side cut matching pattern proposition means coordinate corresponds entire path path decomposition edges path terminals determine cut deduce number contracted coordinates weight furthermore number edges path length terminal finally since base recursion executed split fails execute number edges paths terminal follows path decomposition contains edges addition edges paths length terminal path causes algorithm split exactly upper bound number edges post forest post contains bad edges edges paths vertices introduce adding connecting let contracted coordinate split upon let corresponding path removed assume resides side cut clearly identify coordinates path also identify good coordinates good coordinate appears terminals least one side cut value jth coordinate follows path length applies path therefore number edges paths yet upper bound sum thus post contains edges completes proof
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tracking gaze visual focus attention people involved social interaction nov benoit radu horaud visual focus attention vfoa recognized prominent conversational cue interested estimating tracking vfoas associated multiparty social interactions note type situations participants either look object interest therefore eyes always visible consequently gaze vfoa estimation based eye detection tracking propose method exploits correlation eye gaze head movements vfoa gaze modeled latent variables bayesian switching statespace model proposed formulation leads tractable learning procedure efficient algorithm simultaneously tracks gaze visual focus method tested benchmarked using two publicly available datasets contain typical interactions index focus attention eye gaze head pose dynamic bayesian model switching kalman filter dialog interaction ntroduction paper interested computational analysis social interactions addition speech people communicate via large variety cues prosody hand gestures body movements head nodding eye gaze facial expressions example conversation common behavior consists looking either person speaker object current interest computer screen painting wall object lying table particularly interested estimating visual focus attention vfoa looking recognized one prominent social cues used dialog establish communication respect social etiquette attract someone attention signify taking thus complementing speech communication vfoa characterizes pair may defined either line perceiver face perceived target perceiver direction sight gaze direction often referred eye gaze simply gaze indeed one may state vfoa person target perceiver gaze aligned line physiological point view eye gaze depends eyeball orientation head orientation eye head rigid bodies three six degrees freedom respectively head position three coordinates head orientation three angles jointly referred horaud inria grenoble montbonnot france dailymotion paris france work supported erc advanced grant vhia head pose proper choices eyecentered coordinate frames one assume gaze combination head pose eyeball vfoa depends head pose eyeball orientation target location paper interested estimating tracking jointly vfoas group people communicate robot hri interaction may well viewed generalization hri methodological point view former complex latter indeed hri person robot face hence camera mounted onto robot head provides frontal images user face head pose eye orientation easily robustly estimated case hri eyes barely detected since participants often turn faces away camera consequently vfoa estimation methods based eye detection eye tracking ineffective one estimate vfoa indirectly without explicit eye detection propose bayesian switching dynamic model estimation tracking gaze directions vfoas several persons involved social interaction assumed head poses location orientation target locations directly detected data unknown gaze directions vfoas treated latent random variables proposed temporal graphical model incorporates gaze dynamics vfoa transitions yields tractable learning algorithm efficient tracking proposed method may well viewed computational model method evaluated using two publicly available datasets vernissage laeo datasets consist several hours video containing situated dialog two persons robot vernissage interactions laeo particularly interested finding participants either gaze gaze robot gaze object vernissage recorded motion capture system network infrared cameras camera placed onto robot head laeo collected shows note orientation generally refers pan tilt roll angles pose direction refers polar azimuth angles equivalently unit vector since contribution roll angle gaze generally marginal paper make distinction orientation direction supplementary materials include software package examples results available https remainder paper organized follows section provides overview related work gaze vfoa estimation section iii introduces paper mathematical notations definitions states problem formulation describes proposed model section presents detail model inference section derives learning algorithm section provides implementation details section vii describes experiments reports results elated ork already mentioned vfoa correlated gaze several methods proceed two steps gaze direction estimated first used estimate vfoa scenarios rely precise estimation gaze camera like one used detect iris high accuracy eye trackers provide extremely accurate gaze measurements circumstances data used estimate objects interest videos nevertheless invasive instruments hence appropriate analyzing social interactions gaze estimation relevant number scenarios car driving interaction smartphones situations either field view limited hence range gaze directions constrained car driving active human participation ensures device yields frontal views user face thus providing accurate eye measurements scenarios user even asked limit head movements proceed calibration phase even specific constraints imposed scenarios inherently facilitate task eye measurement best knowledge gaze estimation method deal unconstrained scenarios participants facing cameras partially totally occluded eyes etc general eye analysis inaccurate participants faraway camera alternative approximate gaze direction head pose unlike methods head pose estimated images distant cameras methods estimate gaze approximatively since eyeball orientation differ head orientation gaze estimation head orientation benefit observation gaze shifts often achieved synchronously moving head eyes correlation head pose gaze also exploited recently combined head eye features estimate gaze direction using camera method still requires eyes visible several methods proposed infer vfoas either gaze directions head poses example proposed build gaze cone around head orientation targets lying inside cone used estimate vfoa method successfully applied movies limitation resides vagueness vfoa information limited whether two people looking interesting application vfoa estimation analysis social behavior participants engaged meetings meetings characterized interactions seated people interact based speech head movements methods estimate likely vfoa associated head orientation drawback approaches must purposively trained particular meeting layout correlation vfoa head pose also investigated hmm proposed infer vfoas head body orientations work extended deal complex scenarios participants interacting robot hmm proposed enable model following contextual information participants tend look speaker robot object referred speaker robot results show improves performance vfoa estimation nevertheless method requires additional information speaker identification speech recognition problem joint estimation gaze vfoa addressed cooperation task scenario user necessarily face camera cameras hence estimation gaze direct analysis eye regions feasible proposes learn regression space head poses space gaze directions predict unknown gaze observed head pose head pose estimated fitting elliptical cylinder detected face associated gaze direction corresponds line joining head center target center implies learning stage user instructed gaze targets lying table order provide training data regression parameters thus estimated correspond discrete set pairs one target erroneous gaze may predicted latter range gaze directions used training summary proposed bayesian dynamic model experiments vernissage motion capture dataset presented article provide detailed comprehensive description analysis proposed model model inference learning methodology associated algorithms show results motion capture rgb data vernissage additionally show results laeo dataset iii roposed odel proposed mathematical model model inspired psychophysics unconstrained scenarios person switches gaze one target another target possibly using head eye movements quick eye movements towards desired object interest called saccades eye movements also caused vestibuleocular reflex compensates head movements one maintain gaze direction target interest therefore general case gazing object achieved combination eye head movements figure figure illustrates principle method displays observed latent variables associated person indexed two images grabbed camera mounted onto head robot correspond frames left image right image respectively following variables displayed head orientation red arrow hit observed variables well latent variables estimated proposed method namely gaze direction green arrow git vfoa vti head reference orientation black arrow rit coincides orientation example gazes towards robot turns head eventually gaze towards hence vfoa switches robot vti case small gaze shifts reading watching eye movements predominant case large gaze shifts often needed social scenarios head movements necessary since eyeball movements limited range namely therefore proposed model considers gaze shifts produced head movements occur simultaneously eye movements problem formulation consider scenario composed active targets passive targets active target likely move leading role interaction active targets persons passive targets objects wall paintings set targets indexed index designates target let active target person robot passive target object vfoa discrete random variable defined follows vti means person robot looks target time vfoa person robot looks none known targets vti case vti excluded set vfoas time denoted vtn two continuous variables defined head orientation gaze direction head orientation person denoted hit pan tilt angles head respect fixed coordinate frame gaze direction person denoted git also parameterized pan tilt respect coordinate frame namely git although eyeball orientation neither needed used worth noticing difference git hit variables illustrated fig note case robot gaze direction head orientation identical latter easily estimated head motors finally establish link vfoas gaze directions target locations must defined well let xit xit yti zti location target case person location corresponds head center case passive target corresponds target center locations defined coordinate frame also notice direction active target target defined unit vector xij also parameterized two angles already mentioned target locations head orientations observed random variables vfoas gaze directions latent random variables problem solved formulated maximum posteriori map problem argmax since deterministic relationship head orientation gaze direction propose model probabilistically purpose introduce additional latent random variable namely head reference orientation rit choose coincide orientation use following generative model initially introduced linking gaze direction head orientation head reference orientation hit rit hit rit covariance matrix identity matrix diag diagonal matrix mixing coefficients also assumed covariance matrix persons time therefore head orientation observed random variable normally distributed around convex combination associated different people independent given vfoas combining equations independency assumption obtain hit git rit figure graphical representation showing dependencies variables proposed bayesian dynamic model discrete latent variables visual focus attention shown squares continuous variables shown circles observed variables head pose target locations shown shaded circles latent variables gaze reference shown white circles two latent variables gaze direction head reference orientation gaze dynamics dependencies variables embedded variable means covariance matrices estimated via training gaze directions vary lot assume head reference orientations almost constant time enforced imposing total variance gaze much larger total variance head reference orientation namely vfoa dynamics using markov approximation vfoa transition probabilities written following model proposed git vti git xij dgit gaze velocity covariance matrices diag diagonal matrix mixing coefficients therefore person looks one targets gaze dynamics depends direction xij rate equal person look one targets gaze dynamics follows random walk head reference orientation dynamics defined similar way rit rit drit head reference orientation velocity covariance matrices dependencies model variables shown graphical representation figure assumed gaze directions associated different people independent given vfoas crossdependency different people taken account vfoa dynamics detailed section similarly head orientations head reference orientations notice matrix size indeed persons active targets targets one target active targets passive targets case person looks excluded example matrix entries estimation matrix would require principle large amount training data particular presence many symmetries show practice different transitions possible seen following grounds start assuming conditional independence vfoas vti vti let consider vfoa person given vfoas one distinguish two cases either passive target none targets case vti depends person case vti depends summarize write vti vti otherwise based possible count total number possible vfoa transitions notations following possibilities target two possible transitions passive target three possible transitions active target looks target three possible transitions active target looks person three possible transitions active target looks active target different four possible transitions therefore different possibilities vti appendix moreover assuming vfoa transitions depend conclude transition matrix may different entries moreover number possible transitions even smaller passive target number active targets small considerably simplifies task estimating matrix makes task learning tractable nference start simplifying notation namely denotes vertical concatenation emission probabilities become hit clit matrix obtained definition unnecessary dependencies removed combining obtain recursive formulation however model still intractable without assumptions main approximation used work consists assuming local independence posteriors lit vti switching kalman filter approximation several strategies possible depending upon structure commonly used strategies evaluate distribution include variational bayes montecarlo alternatively propose cast problem framework switching kalman filters skf assume filtering distribution gaussian obtain following factorization vti lit thus split components one active target lit vti hit lit aij transition probabilities obtained combining vti lit normalization evidence introduce using sum rule aij matrix vector indices dropped since transitions depend xij map problem derived bayesian framework vfoa variables dlt propose study filtering distribution joint latent variables namely indeed bayes rule yields several algebraic manipulations ijk ijk lit vti lit expression obtained performing constrained kalman filtering transition dynamics defined emission dynamics defined ijk observation hit weights defined constraint comes fact hit achieved projecting mean refer details rephrased follows filtering distribution time possible dynamics lit normal distribution time becomes mixture normal distributions time shown however expect single gaussian lit vti lit done moment matching ijk ijk ijk ijk let introduce finally necessary evaluate following notations cijk cij vti learning vfoa transition probabilities vfoa transitions drawn generalized bernoulli distribution therefore transition probabilities estimated vti vti kronecker delta function section showed different possibilities vfoa transition probability enables derive explicit formula case see appendix consider example one cases namely vti conditional probability person looks target given person looked person person looked target probability estimated following formula follows cij cijk ijk cijk cij applying bayes formula cijk yields vtq cijk ijk cik obtained calculated previous step last factor either equal time active target otherwise cases straightforward compute finally first factor observation component factorized hit introducing latent variable obtain hnt hnt lnt vtn dlnt factors normal distributions hence integrates summary devised procedure estimate online approximation joint filtering distribution vfoas gaze head reference directions earning proposed model two sets parameters must estimated transition probabilities associated discrete vfoa variables parameters associated gaussian distributions learning carried using recordings annotated vfoas recording composed frames contains active targets robot active target persons indexed passive targets addition target locations head poses worth noticing learning algorithm requires vfoa annotations gaze directions still treated latent variables learning gaussian parameters section described derivation proposed model based skf model requires parameters means covariances gaussian distributions defined notice however mean parameterized similarly mean parameterized consequently model parameters remind diagonal matrices covariance covariance assumed matrices common active targets hence total number parameters equal general case skf models discrete variables unobserved learning inference propose learning algorithm takes advantage fact discrete variables vfoas observed learning process namely vfoas annotated propose algorithm adapted case standard kalman filter iteration alternates pass compute expected latent variables maximization expected start describing loglikelihood taking expectation posterior distribution obtain old maximized parameters yields expressions covariance matrices cpt backward pass estimates leads expectations estimated performing forwardbackward pass persons recordings training data yields following formulas mplementation etails clq estimation carried following way old old yield set two linear equations two unknowns proposed method evaluated vernissage dataset looking laeo dataset describe detail datasets annotations provide implementation details analyse complexity proposed algorithm vernissage dataset vernissage scenario briefly described follows fig three wall paintings namely passive targets denoted two persons denoted left person right person interact robot hence robot plays role similarly art guide describing paintings asking questions two persons front recording split two roughly equal parts first part dedicated painting explanation interaction second part consists quiz thus illustrating dialog two participants robot time concerning paintings expectation taken posterior distribution formulas expanded means substituted expressions following terms remain estimated provides estimates expectations via algorithm sake clarity drop superscripts active target index recording index equation introducing notation equations figure vernissage setup left global view exhibition showing wall paintings two participants nao robot right top view room showing vernissage layout scene recorded camera embedded robot head vicon motion capture system consisting network infrared cameras placed onto walls optical markers placed onto robot people heads recorded frames per second fps total ten recordings lasting ten minutes vicon system provided accurate estimates head positions head orientations head positions head orientations also estimated using rgb images gathered camera embedded robot head rgb images processed follows use opencv version detect faces bounding boxes tracked time using next extract hog descriptors bounding box apply estimator yields head positions estimated using line sight face center size provides rough estimate depth along line sight remaining paper referred vicon data rgb data whole setup carefully calibrated vicon rgb data represented coordinate frame experiments assumed passive targets static positions provided advance position robot also known advance one easily estimate orientation robot head motor readings finally vfoas participants manually annotated frames recordings laeo dataset laeo dataset extension tvhid human interaction dataset consists videos extracted shows least two actors appear video engaged four interactions handshake highfive hug kiss videos interaction videos interaction videos grabbed fps video lasts five seconds seconds laeo annotated namely videos split shots separated cuts shots total manually annotated whenever two persons look passive target dataset number active targets corresponds number persons shot practice varies one eight persons faces dataset annotated bounding box coarse label backward vernissage use center size estimate coordinates heads also assigned yaw value one five coarse head orientations also computed finer head orientations using algorithmic details inference procedure summarized algorithm basically iterative filtering procedure update step consists applying recursive relationship derived ijk ijk section cijk using intermediate variables vfoa chosen using map given observations current frame gaze direction mean filtered distribution first two components indeed mean pan tilt gaze angles algorithm inference procedure aze ndvfoa bservations time nitialization argmaxj cij bservations time pdate vti argmaxj cij git return let describe initialization procedure used algorithm probabilistic framework parameter intialization generally addressed defining initial distribution explicitly define distribution initialization based fact repeated similar observation inputs algorithm reaches initialization algorithm uses repeated update method initial observation provide estimate gaze reference directions consequently initial filtering distribution implicitly defined expected stationary state algorithm initialization procedure nitialization cin uniform convergence cin pdate cin return cin algorithm complexity computational complexity algorithm number frames test video number iterations needed algorithm initialization converge computational complexity bservation computational complexity pdate complexity one iteration algorithm depends face detection head pose estimation algorithms hence concentrate section one sees following values need computed hit cijk ijk ijk active target combination targets different possible values possible values factor whose complexity estimated follows part kalman filter algorithm used estimate calculations dominated several matrix inversions multiplications neglecting scalar multiplications matrix additions denoting ckf complexity kalman filter obtain ckf hence ckf vii xperimental results vernissage dataset applied experimental protocol vicon rgb data used strategy training test performed left video used frame recognition rate frr metrics quantitatively evaluate methods frr computes percentage frames vfoa correctly estimated one note however vfoas obtained manually annotating frame data subject errors since annotator associate target person vfoa transition probabilities model parameters estimated using learning method described section appendix provides formulas used estimating vfoa transition probabilities given annotated data notice fifteen transitions probabilities thus estimated identical data vicon rgb gaussian parameters estimated using algorithm section learning procedure requires estimates well targets locations estimated explained since estimates different two kinds data vicon rgb carried learning process twice vicon data rgb data algorithm needs initialization initial parameter values diag matrices defined initialized isotropic covariances particular initialization consistent practice noticed covariances estimated training remain consistent results vicon data frr estimated vfoas vicon data summarized table examples shown figure frr score varies proposed method notice high scores obtained methods recording similarly low scores obtained recording since methods assume head motions gaze shifts occur synchronously explanation could hypothesis valid participants confusion matrices vfoa classification using vicon data given figure similarities results obtained two methods particular wall painting stands behind nao methods always discriminate two targets addition head one persons often aligned painting table frr scores estimated vfoa icon data left right persons recording mean odobez proposed figure confusion matrices vicon data left right proposed algorithm vfoas estimated vfoas diagonal terms represent recall viewpoint person similar remark holds painting consequence methods often confuse vfoa cases seen third image figure indeed difficult estimate whether left person looks finally methods problems recognizing vfoa nothing gaze aversion vti propose following explanation targets widespread scene hence likely acceptable target lies gaze directions moreover nao centrally positioned therefore head orientation used look nao similar resting head orientation used gaze aversion however reference head orientation fixed poorly suited dynamic mapping hence high error rate painting method favors selection target either active passive target nothing case results rgb data rgb images processed described section order obtain head orientations head positions table shows accuracy measurements degrees centimeters compared ground truth provided vicon motion capture system seen head orientation estimates quite accurate error estimating head positions large participants lying front robot recordings particular error increases participant farther away robot cases bounding figure results obtained proposed method vicon data gaze directions shown green arrows head reference directions arrows observed head directions red arrows vfoa shown black circle top row displays image camera top views room show results obtained middle row bottom row last example gazes nothing box larger hence head position average one meter closer true position relatively large errors head position affect overall behavior algorithm frr scores obtained rgb data shown table iii expected loss accuracy correlated head position error results obtained recordings close ones obtained vicon data whereas significant loss accuracy recordings loss notable case right person recordings confusion matrices obtained rgb data shown table ean error head pose estimations rgb data left person right person errors head position centimeters orientation degrees computed respect values provided motion capture system video mean position error pan error tilt error fig case rgb data comparison method method biased use different head orientation head position estimators indeed rgb data results reported obtained unpublished methods estimating head orientations head positions head tracking moreover uses information namely speaker identity based audio track one participants robot well identity object interest also note reports mean frr values obtained test recordings instead frr value recording table summarizes comparison average frr obtained method method yields similar frr score using vicon data first row case head pose inputs table iii frr scores estimated vfoa obtained proposed method rgb data last two columns show head position errors table video mean odobez proposed head pos error comparable dataset quite remarkable knowing estimated model parameters vernissage training data table average shot recognition rate srr obtained proposed method coarse head orientation fine head orientation figure confusion matrices rgb data left right proposed algorithm vfoas estimated vfoas diagonal terms represent recall table ean frr scores obtained proposed method ecording excluded frr means reported oreover uses additional contextual information vicon data rgb data odobez sheikhi proposed used laeo dataset already mentioned section laeo annotations incomplete estimate vfoa frame indeed annotation whether two people looking shot sufficient know frames actually looking moreover two people appear shot annotations specify people look reasons decided estimate parameters using vicon data whole vernissage dataset used pipeline vernissage rgb data estimate head positions face bounding boxes concerning head orientation two cases coarse head orientations manually annotated fine head orientations estimated coarse head orientations obtained following way pan tilt values associated head orientation label namely pan angles assigned labels backwards respectively tilt anble assigned five labels fine head orientations estimated using procedure case vernissage rgb data namely face detection face tracking head orientation estimation using algorithm used compute vfoa frame person thus allowing determine looks fig used two metrics since laeo annotations shot shot recognition rate srr table average precision table note provides scores interesting note proposed method yields results odobez proposed table average precision obtained dobez proposed method coarse head orientation fine head orientation proposed viii onclusions paper addressed problem estimating tracking gaze visual focus attention group participants involved social interaction proposed bayesian dynamic formulation exploits correlation head movements eye gaze one side visual focus attention eye gaze side described detail proposed model particular showed entries matrix vfoa transition probabilities small number different possibilities provided formulae immediate consequence simplified transition matrix associated learning require large training dataset showed problem simultaneously inferring vfoas gaze directions time cast framework switching model yields tractable learning inference algorithms applied proposed method two datasets vernissage laeo vernissage contains several recordings humanrobot interaction scenario experimented motion capture data gathered vicon system rgb data gathered camera mounted onto robot head also experimented laeo dataset contains several hundreds video shots extracted shows quite remarkable result parameters obtained training model vernissage data successfully used testing method laeo data explained fact social interactions even different contexts share lot characteristics compared method three methods based hmms hmms geometric model interest methods including resides fact eye detection unlike many existing gaze estimation methods needed feature makes methods practical effective large number situations social interaction note gaze inference head orientation problem indeed correlation gaze head movements person dependent well context figure results obtained proposed method rgb data gaze directions shown green arrows head reference directions arrows observed head directions red arrows vfoa shown black circle top row displays image camera top views room show results obtained left person middle row right person bottom row figure figure shows results obtained laeo dataset top row shows results obtained coarse head orientation bottom row shows results obtained fine head orientation head orientations shown red arrows algorithm infers gaze directions green arrows vfoas blue circles people looking others shown dashed blue line dependent however important detect gaze whenever eyes reliably extracted images properly analyzed proposed solve problem based fact alignments often occur gaze directions finite number targets sensible assumption practice contextual information could considerably improve results indeed additional information speaker recognition speaker localization speech recognition detection may used learn vfoa transitions multimodal dialog systems future plan investigate discriminative methods based neural network architectures inferring gaze tions head orientations contextual information example one could train deep learning network pairs head pose visual focus attention purpose one combine system accurately detect eyes several participants estimate head poses associated algorithms infer speaker speech information ppendix vfoa ransition robabilities using notations introduced section let active target section showed practice probability transition matrix different entries completeness entries listed vfoa neither active passive target vti vtq vti vfoa passive target vti vti vti appendix provides formulae allowing estimate transitions probabilities explained section vtq vtq vtq vtq ppendix vfoa earning vfoa active target vti vti vti vti vti vti vti vti vti vti vtq vtq vtq vtq vtq vtq vtq vtq vtq vtq acknowledgments authors would like thank vincent drouard valuable expertise head pose estimation tracking eferences freedman sparks coordination headunrestrained gaze shifts rhesus monkeys journal neurophysiology freedman coordination eyes head visual orienting experimental brain research vol jayagopi vernissage corpus multimodal dataset idiap tech zisserman eichner ferrari detecting people looking videos international journal computer vision vol eizenman new methodology determining eye tracking systems ieee transactions biomedical engineering vol oct toyama kieninger shafait dengel gaze guided object recognition using eye tracker proceedings etra symposium hong pelz cockburn lightweight sidemounted mobile eye tracking system ieee wnyipw kurzhals hlawatsch seeger weiskopf visual analytics mobile eye tracking ieee transactions visualization computer graphics vol jan smith shah vitoria lobo determining driver visual attention one camera ieee transactions intelligent transportation systems vol krafka khosla kellnhofer kannan bhandarkar matusik torralba eye tracking everyone ieee cvpr june matsumoto ogasawara zelinsky behavior recognition based head pose gaze direction measurement ieee iros vol ohno mukawa simple calibration gaze tracking system enables interaction proceedings etra symposium acm sugano okabe sato adaptive linear regression gaze estimation ieee transactions pattern analysis machine intelligence vol oct okabe sugano sato learning gaze biases head motion head gaze estimation image vision computing vol trivedi head pose estimation computer vision survey ieee transactions pattern analysis machine intelligence vol zabulis sarmis argyros head pose estimation multiple distant views bmvc chamveha sugano sugimura siriteerakul okabe sato sugimoto head direction estimation low resolution images scene adaptation computer vision image understanding vol rajagopal subramanian ricci vieriu lanz sebe exploring transfer learning approaches head pose classification surveillance images international journal computer vision vol yan ricci subramanian liu lanz sebe learning framework head pose estimation target motion ieee transactions pattern analysis machine intelligence vol qin shelton social grouping tracking head pose estimation video ieee transactions pattern analysis machine intelligence vol stahl amplitude human head movements associated horizontal saccades experimental brain research vol goossens van opstal human coordination two dimensions different sensorimotor conditions experimental brain research vol stiefelhagen zhu head orientation gaze direction meetings human factors computing systems lanillos ferreira dias bayesian hierarchy robust gaze estimation interaction international journal approximate reasoning vol asteriadis karpouzis kollias visual focus attention environments using gaze estimation international journal computer vision vol odobez recognizing visual focus attention head pose natural meetings ieee transactions system men cybernetics part sheikhi odobez recognizing visual focus attention human robot interaction human behavior understanding workshop yucel salah mericli mericli valenti gevers joint attention gaze interpolation saliency ieee transactions system men cybernetics part otsuka yamato takemae conversation scene analysis dynamic bayesian network based visual head tracking ieee icme duffner garcia visual focus attention estimation unsupervised incremental learning ieee transactions circuits systems video technology sheikhi odobez combining dynamic head mapping robot conversational state attention recognition interactions pattern recognition letters vol horaud simultaneous estimation gaze direction visual focus attention interaction ieee icme seattle jul murphy switching kalman filters berkeley tech simon kalman filtering state constraints survey linear nonlinear algorithms control theory applications iet bishop pattern recognition machine learning springerverlag viola jones rapid object detection using boosted cascade simple features ieee cvpr vol bae yoon robust online tracking based tracklet confidence online discriminative appearance learning ieee cvpr drouard horaud deleforge evangelidis robust estimation based mixture linear regressions ieee transactions image processing vol zisserman reid high five recognising human interactions shows british machine vision conference girin horaud gannot localization based features likelihood maximization spatial sparsity regularization transactions audio speech language processing vol oct gebru horaud speaker diarization based spatiotemporal bayesian fusion ieee transactions pattern analysis machine intelligence benoit received degree applied mathematics computer science ensimag institut national polytechnique grenoble france degree graphics vision robotics joseph fourier grenoble france currently phd student perception team inria grenoble rhonealpes research interests include scene understanding machine learning computer vision special emphasis attention recognition interaction received applied mathematics signal processing university dakar dakar senegal mathematics computer vision machine learning ecole normale cachan paris france phd student researcher idiap research institute martigny switzerland worked probabilistic models object tracking human activity recognition researcher telecom bretagne brest france working variational models geophysical data processing worked innovation lab marseille france research engineer used computer vision machine learning principles methods develop interaction software tools researcher perception team inria grenoble working machine learning computer vision models interaction since may computer vision scientist videostitch paris radu horaud received degree electrical engineering degree control engineering degree computer science institut national polytechnique grenoble france fellow artificial intelligence center sri international menlo park currently holds position director research inria grenoble founder head perception team research interests include computer vision machine learning audio signal processing audiovisual analysis robotics radu horaud collaborators received numerous best paper awards area editor elsevier computer vision image understanding member advisory board sage international journal robotics research associate editor kluwer international journal computer vision program ieee iccv acm icmi radu horaud awarded erc advanced grant project vision hearing action vhia awarded erc proof concept grant project vhialab
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atomistic fingerprint algorithm learning initio molecular force fields dec tang dongkun zhang george karniadakis division applied mathematics brown university rhode island usa abstract molecular fingerprints feature vectors describing atomistic neighborhood configurations important abstraction key ingredient modeling potential energy surface interatomic force paper present canonically aligned fingerprint decaf fingerprint algorithm robust efficient fitting scalar vector quantities fingerprint essentially continuous density field formed superimposition smoothing kernels centered atoms rotational invariance fingerprint achieved aligning fingerprint instance neighboring atoms onto local canonical coordinate frame computed kernel minisum optimization procedure show approach superior methods especially atomistic neighborhood sparse contains symmetry propose distance density fields measured using volume integral pointwise difference efficiently computed using optimal quadrature rules require discrete sampling small number grid points also experiment choice weight functions constructing density fields characterize performance fitting interatomic potentials applicability fingerprint demonstrated set benchmark problems keywords active learning gaussian process regression quantum mechanics molecular dynamics next generation force fields introduction molecular dynamics simulations widely used studying atomistic systems proteins catalysts due ability precisely capture transient events predict macroscopic properties microscopic details prevalent implementation trajectory atomistic system integrated time according newton law motion using forces calculated negative gradient hamiltonian whose functional form parameters collectively referred force field traditionally pairwise terms comprise force field derived empirically fitting quantum mechanical calculations experimental data three properties directly relate applicability force field accuracy transferrability complexity years large number force fields developed carrying tang molecular fingerprints figure pipeline machine molecular computations atomistic neighborhood configurations transformed feature vectors called fingerprints used train regression models particular emphasis three properties however combinatorial complexity atomistic systems easily outpace force field development efforts difficulty explodes following curse dimensionality deceptively simple system demonstrate situation water triatomic molecule molecular structure fact common water models succeeded reproducing small number structural dynamical properties water due difficulty modeling strong intermolecular effects hydrogen bonding polarization lieu force field quantum mechanical calculations employed straightforwardly drive molecular dynamics simulations method achieves significantly better accuracy transferrability solving electronic structure system however computational complexity methods least cubic number electrons consequently time length scales accessible molecular dynamics severely constrained assuming smoothness potential energy surface atomistic system one possible strategy accelerate molecular dynamics use calculations subset time steps interpolate similar atomic configurations schematic overview process given figure enabled recent development nonlinear statistical learning regression techniques gaussian process regression artificial neural networks paper focuses particular aspect molecular computation pipeline fingerprint algorithms whose importance arises naturally aforementioned regression protocol fingerprint encoding atomistic configuration facilitate regression tasks similarity comparison across structures consisting variable numbers atoms elements pointed previously good fingerprint possess following properties encoded vector facilitate regression particularly artificial neural networks complete different atomistic neighborhood configurations lead different fingerprints tang molecular fingerprints vice versa distance fingerprints proportional intrinsic difference atomistic neighborhood configurations continuous regard atomistic coordinates change fingerprint approximately proportional structural variation characterized example internal coordinates invariant permutation rotation translation computationally feasible straightforward implement proceed details work first briefly review several fingerprints closely related work smooth overlap atomic positions soap kernel coulomb matrix graph approximated energy grape kernel smooth overlap atomic positions soap soap kernel built idea representing atomistic neighborhoods smoothed density fields using gaussian kernels centered neighbor atom similarity measured inner product density fields rotational invariance achieved integrating possible rotations performed analytically using power spectrum density field fact fingerprint algorithm inspired idea treating atoms smoothed density fields however take different approach endorse fingerprint rotational invariance use euclidean distance instead inner product distance metric coulomb matrix practice using graphs represent atomistic neighbor configurations first implied coulomb matrix later formulated grape kernel diffusion distance proposed similarity measure different local chemical environments idea construct undirected unlabeled graph atoms serving vertices pairwise interactions weighting edges example coulomb matrix treated laplacian matrix dij aij degree matrix encodes polynomial fit atomic energies nuclear charge adjacency matrix corresponds coulombic interactions pairs atoms due use relative positions atoms adjacency matrix coulomb matrix automatically invariant translation rotation however matrix invariant permutation swapping order two atoms result exchange corresponding columns rows address sorted list eigenvalues coulomb matrix used instead feature vector norm used distance metric practice due fact number tang molecular fingerprints neighbor atoms may change shorter eigenvalue list padded zeros distance computation graph approximated energy grape grape kernel evaluates simultaneous random walks direct product two graphs representing two atomistic neighborhood configurations permutational invariance achieved choosing uniform starting stopping distribution across nodes graphs however cost distance computation two graphs scales diagonalization cost sections present new fingerprint algorithm namely canonically aligned fingerprint decaf paper organized follows section introduce robust algorithm determine canonical coordinate frames obtaining projections section present numerical recipes use smoothed atomistic density fields fingerprint molecular configuration section demonstrate capability fingerprint via examples involving regression atomistic potential energy surfaces section discuss connection algorithm previously proposed ones conclude discussion section localized canonical coordinate frame rotationally invariant description atomistic neighborhood kernel minisum approach improve model generalization minimizing data redundancy fingerprint algorithm able recognize atomistic structures differ transformation permutation atoms element extract feature vectors invariant transformations summarized table variety strategies successfully employed common fingerprint algorithms achieve rotational invariance however approaches provide means acquisition quantities rotational invariant form one approach acquire interpolate potential energy scalar quantity take derivative regression model approach however triggers need methods decompose total energy among atoms property entire system rather individual atoms another approach proposed project vector quantities onto potentially overcomplete set basis vectors obtained weighted sum atomic coordinate vectors kxi exp however approach may suffer robustness issues example generated different point direction radial distance atoms equal tang molecular fingerprints configuration atoms cos sin leads cos sin cos sin thus gets close zero always point toward either even vector quantity interest may point directions table comparison strategies used fingerprint algorithms obtain feature vectors invariant translation permutation rotation invariance fingerprint coulomb matrix behler soap grape translation permutation rotation relative distance relative distance relative distance relative distance sorting eigenvalues summation summation uniform distribution graph ignoring angular information integrating rotations uniform distribution present robust kernel algorithm explicit determination canonical coordinate frame within projection atomistic neighborhood invariant rigidbody rotation furthermore canonical coordinate frame directly used capture quantities form given atoms position first formulate pca algorithm optimization problem seek unit vector maximizes sum projections kxi distance origin atom unit vector pointing toward atom respectively optimization process uniquely determine orientation projection vector line consequence heuristics needed identify specific direction pca vectors overcome difficulty generalize term weight function radial distance term bivariate kernel function two vectors attempt seek unit vector minimizes kernel summation particular found square angle kernel exponentiated cosine kernel tang molecular fingerprints perform well practice exp shown figure kernels minimal parallel monotonically reach maximum antiparallel intuitively optimizing minisum objective function generated kernel yield vector loosely speaking bisects sector occupied atoms kernel exhibits similar behavior leads smoother objective function shown figure allows determination projection vector without ambiguity even atom configuration contains perfect symmetry square angle exponentiated cosine pca figure shown illustration minisum algorithm determines projection vectors rotational invariant description atomistic neighbor configurations black dots represent atoms carry equal importance vectors point origin atoms used input bivariate kernels compute minisum objective function drawn solid lines reference negated values pca objective function drawn dashed lines projection vectors obtained finding unit vector minimizes objective function major advantage kernel minisum approach versus pca lies robustness presence structural symmetry continuity resulting principal axes respect angular movement input data shown figure kernel minisum particularly suitable atomistic systems strong symmetries common continuity angular movement desired minisum framework also used customized kernels suit characteristics specific application scenarios tang molecular fingerprints rotational invariance angular continuity cluster angular continuity pca pca kernel minisum figure comparison orthogonal bases obtained using principal components analysis pca kernel minisum msa kernel pca algorithm used conjunction norm fails extract principal axis rotates system exhibits planar symmetry msa pca norm accommodate scenario principal axes change orientation abruptly system undergoes slight angular motion contrast msa output continuous regard movement always bisects angle formed atoms origin loosely speaking msa axis points majority direction atoms single cluster present within cutoff distance bisects angle two atom clusters different results deliver robust rotational invariance well continuity arrows drawn different lengths improve visual clarity situations overlapping understood unit vectors performed radial distances atoms carry physically significant information solving kernel minisum optimization problems optimization problem solved efficiently using gradient descent algorithm detailed square angle objective function minisum problem using square angle kernel asa tang molecular fingerprints gradient asa respect asa arccos note arccos singular treated numerically replacing removable arccos singularities limwt truncating gradient finite threshold near poles local minimum iteratively searched gradient descent renormalizing iteration moreover due locally quadratic nature objective function found algorithm significantly accelerate convergence minimal cost algorithm presented alg algorithm gradient descent solving square angle minisum problem function minsquareangle repeat compute gradient using obtain tangential component gradient step small initial step size bootstrapping else adaptive subsequent steps end save save update normalize unit length return end function exponentiated cosine kernel objective function minisum problem using exponentiated cosine aec exp exp gradient aec respect aec tang molecular fingerprints gradient contains singularity however always locally quadratic convex cause algorithm generate negative step sizes consequently divert search towards maximum luckily easily overcome always using absolute value step size generated algorithm enforcement prevents minimization algorithm going uphill complete algorithm given alg algorithm gradient descent solving exponentiated cosine minisum problem function minexpcosine repeat compute gradient using obtain tangential component gradient step small initial step size bootstrapping else adaptive subsequent steps end save save update normalize unit length return end function shown table alg alg converge quickly consistently across wide range representative point configurations commonly found molecular systems however gradient descent method find local optima thus multiple trials performed using different initial guesses ensure global minimum located complete set orthogonal projection vectors canonical coordinate frame complete set orthogonal bases found greedily using protocol described alg specifically use globally optimal solution minisum optimization problem first basis constrained optimal solution plane orthogonal second basis special care must taken determining third basis degree freedom sign due orthogonality constraint straightforward approach choosing direction gives smaller objective function value may fail example system contains improper rotational symmetry case interchangeable perpendicular axis result two candidates align axis thus indistinguishable kernel minisum however projection atoms two seemingly equivalent coordinate frames identical rather mirror images fortunately addressed choosing direction created plane yields smaller kernel objective function bisector versus points lies rule also handle general situations symmetry tang molecular fingerprints table listed number iterations initial guesses used gradient descent algorithm find local optimum kernel minisum problems numbers averaged repetitions convergence criterion cases iterative algorithm converge within iterations optimization restarted new guess kernel square angle exponentiated cosine configuration guesses guesses single point two points angle two points angle two points angle planar planar tetrahedra octahedra improper improper random points random points algorithm procedure determining canonical coordinate frame using kernel minisum function global minimum minisum using multiple runs alg alg contains point construct arbitrary else global minimum minisum using multiple runs alg alg subjecting constraint probe vector gradient else end end return end function tang molecular fingerprints difficult prove global uniqueness kernel minisum solution given nature exponentiated cosine square angle kernels fact seems kernel allows analytical proof solution uniqueness whose solution simply corresponds weighted center mass neighbor atoms unfortunately simple kernel robust reflectional rotational symmetry luckily rare cases two global optimal solutions coexist safely captured repeated searching procedure starting different seeds thus fingerprint extracted using resulting coordidate frame may mildly increase size training set could even advantagenous training data scarce canonically aligned fingerprint density field approximation volume integral atoms density figure two density profiles generated two different atomistic configurations using smoothing kernel functions distance measured norm pointwise difference corresponds orange area middle plot shown density field using smoothing kernels whose widths depend distances atoms origin darker shades indicate higher density local density field around point formulated superimposition smoothing kernel functions centered neighbor atom relative displacement regard within cutoff distance kxi density field pointed previously may used fingerprint local atomistic environment assume smoothing kernel takes form stationary gaussian exp also assume density scaling function ensures continuity density field atoms enter exit cutoff distance function compact support discussion found section section respectively tang molecular fingerprints achieve rotational invariance project atom coordinates canonical coordinate frame determined kernel minisum algorithm generating density field kxi depending specific application may necessarily overlap scalar properties acquired directly target atom properties force acquired interpolated local orthogonal coordinates define distance two density fields weighted volume integral pointwise difference weight function provides additional flexibility emphasizing particular regions atomistic neighborhood could play important role fitting properties steep gradients repulsive part potential introduce optimal quadrature rule approximate integral computationally tractable manner quadrature rule numerical recipe form numerically approximates definite integral using discrete evaluations integrand determine quadrature nodes weights decompose volume integral surface integral spherical shells integral along radial direction sin surface integral optimally approximated using lebedev quadrature rule sin weights positional unit vectors number lebedev nodes respectively radial integral fits well generalized quadrature formula weight function ern weights coordinates number laguerre nodes respectively combining composite quadrature rule generated consisting several spherical layers tang molecular fingerprints nodes shown figure radial coordinates quadrature nodes determined laguerre quadrature nodes angular positions determined lebedev quadrature nodes respectively composite quadrature formula translates volume integral summation discrete grid points ern using right hand side replace integral use notation enumerate quadrature nodes located weights ern obtain final discretized formula computing distance fingerprints quick reference tabulated appendix values laguerre quadrature points values lebedev quadrature points addition quadrature nodes could radially scaled outer nodes lie radius within cutoff distance allows fit laguerre quadrature order within arbitrary cutoff distance scaled quadrature rule given max since scaling simply constant among nodes safely omitted many regression tasks relative distance fingerprints significance radial weight functions section examine two radial weight functions used density field fingerprint density scaling function appears weight integral appears driven interest reducing computational cost would like use cutoff distance select atoms involved constructing density field however important ensure atoms enter exit neighborhood smoothly naturally requests contribution atom zero outside cutoff increase continuously smoothly atom approaching entrance correspondingly become unity origin smoothly approach zero cutoff distance ensure differentiability regression models based tang molecular fingerprints laguerre lebedev fingerprint quadrature shell shell shell figure laguerre quadrature nodes order normalized reciprocal largest node onto unit interval class lebedev grid points unit sphere decaf molecular fingerprint essentially comprises grid quadrature nodes optimally samples density field induced neighbor atoms shown example one composite quadrature grid combines laguerre quadrature rule three layers lebedev quadrature nodes order respectively fingerprint candidates satisfying conditions include example kernel polynomial kernel compact support detailed appendix approximation radial integral using laguerre quadrature requires integrand pointwise difference atomistic density fields decays sufficiently fast beyond outermost quadrature nodes order achieve acceptable convergence addition steeply repulsive yet flat attractive interatomic interactions prompt sensitivity fingerprint adjusted correspondingly order avoid numerical difficulties training regression model weight integral provides convenient means achieving purpose different instead satisfy following conditions normalized decays sufficiently fast necessarily beyond outer quadrature node sufficiently smooth beyond outermost quadrature node candidates includes kernel kernel albeit different normalization factor laplacian kernel exp properly sized length scale also appears candidate due similarity part tang molecular fingerprints bell weight integral unity fingerprint distance density scaling function tent unity figure shown examples distance matrices fingerprints sampled biatomic system manifested difference weight integral helps emphasize near field meanwhile obvious discontinuities second row matrices demonstrate importance density scaling function fingerprint algorithm uses atoms within finite cutoff distance weight function laguerre quadrature note constant kernel may also choice long density field already decays fast enough due density scaling function figure demonstrate effect density scaling function weight integral distance matrices fingerprint obtained pair atoms comparison panel shows integration weight allows distance fingerprints change tang molecular fingerprints rapidly atoms closer slowly atoms farther apart visible discontinuity second row clearly demonstrates importance damping function atoms within finite cutoff distance used compute fingerprint examine impact weight functions performance gaussian process regression gpr using fingerprint interatomic force minimal system containing two nitrogen atoms despite simplicity system case fundamental importance ubiquity repulsive regime potential could cause difficulty small change system configuration trigger large changes regression target function figure compare performance among combination four weights integral two density scaling functions initial training set consists two samples collected regression refined using greedy strategy consecutively learns point largest posterior variance largest uncertainty defined twice posterior standard deviation less active learning scheme able delivery gpr model every combination weight functions closely fits target function however numbers refinement iterations achieved accuracy vary therefore important evaluate choose weight functions context specific application scenarios quadrature resolution density kernel despite formal convergence composite quadrature decaf cost distance calculations kernel evaluations needed sample density field generated atoms using quadrature nodes less prominent cost associated distance calculation comes cost floating point operations thus practice often desirable use nodes possible capture information density field within certain band limit accordingly integral cutoff number quadrature nodes width density kernel need tuned obtain optimal balance resolution computational speed designing composite quadrature rule chose laguerre quadrature radial direction nodes denser near origin sparser farther away consistent physical intuition near field generally stronger influence far field atomistic neighborhood example van waals potential grows rapidly atoms direct contact flattens first coordinate shell accordingly may possible use sparser outerlayer grids reduce total number quadrature nodes still keeping enough nodes inner layers maintain sensitivity quadrature toward close neighbors cooperatively also use gaussian density kernels whose width dependent distance atom origin way sparser nodes still sufficiently sample smoother far field wider kernels remote atoms also reduce total difference far fields two fingerprints statistical sense thus contribution far field integral effectively tuned even though weights quadrature nodes remain figure demonstrate quadrature combined widening smoothing density kernel simultaneously reduce computational complexity preserve quality tang molecular fingerprints gpr truth scaling density seed refine bell tent points rmse points rmse bell points rmse points rmse tent points rmse points rmse laplace weight integral points rmse points rmse unity distance distance figure gaussian process regression force two nitrogen atoms function interatomic distance using different combinations radial weight functions inset figures plots regression function using distances feature space tang molecular fingerprints fingerprint column dense grid used sample density fields generated wide smoothing length examining distance matrices fingerprints sampled bond stretching angular stretching movements note radial similarity decreases monotonically angular similarity changes nearly constantly column number quadrature nodes kept smoothing length reduced attempt increase fingerprint sensitivity better response near field radial direction obtained linearity far field angular direction compromised column fingerprint performs even worse due combination sparser quadrature grid small smoothing length column performance recovered let smoothing length parameter gaussian density kernels depend distance origin atom simultaneously adjust quadrature node density according pattern nodes layers nodes layers nodes layers nodes radial angular fingerprint distance figure comparison fingerprint distance matrices corresponding bond stretching angular stretching movements dense grid large smoothing length radial similarity decreases monotonically angular similarity changes nearly constantly dense grid smaller smoothing length better fingerprint sensitivity near field bond stretching compromised linearity far field angular stretching sparser grid smaller smoothing length compromised performance bond angular movements grid radially dependent smoothing length good resolution linearity near far fields tang molecular fingerprints demonstration method regression tasks throughout work performed using gaussian process regression gpr nonlinear kernel method treats training data points discrete observations instantiation gaussian process predictions made using posterior mean variance joint distribution test data training data one particular interesting property gaussian process posterior variance may interpreted measure prediction uncertainty exploited design active learning algorithms sampling expensive functions actual computation used software implementation made publicly available zenodo use square exponential covariance kernel compute covariance correlation samples kse exp decaf fingerprints distance norm computed kernel stationary meaning covariance depends relative distance two samples absolute position training process searches hyperparameters output variance length scale maximizes likelihood training data detailed tutorial gpr found ref illustration complete workflow using density field fingerprint perform regression tasks given figure potential energy surface first attempt fit potential energy surface protonated water dimer system configuration function distance dihedral angle two planes formed water molecule shown figure system contains improperly rotational symmetry wish capture kernel minisum algorithm gpr model seeded training points corresponding combinations subsequently active learning protocol used greedily absorb points highest uncertainty training set despite restricted training data within subdomain shown figure able accurately reproduce target function entire parameter space active learning steps decaf fingerprint used constructed spherical layers within cutoff distance consisting lebedev quadrature nodes respectively weight integral chosen polynomial defined appendix density scaling function vector atom fingerprint center kernel defined density kernel sits oxygen atoms assumes form gaussian discussed section wdo exp vector atom fingerprint center vector atom quadrature node tang molecular fingerprints regression sample prediction uncertainty kernel minisum aligned prediction neural network gaussian process matrix distance computation density field generation density field quadrature fingerprint feature vector figure shown workflow regression using density field fingerprint key stages kernel minisum optimization density field generation distance computation covered detail sec sec fingerprints also readily used train artificial neural networks tang molecular fingerprints density kernel hydrogen atoms different weight width ensure discriminability wdh exp oxygen hydrogen figure gaussian process regression carried fit potential energy surface protonated water dimer function two internal variables distance dihedral angle plane determined two water molecules system contains improperly rotational symmetry correctly recognized kernel algorithm given alg quarter domain used train yet model accurately predict energy entire parameter space thanks symmetry detection geometry optimization vibrational analysis next demonstrate usability fingerprint fitting quantities performing geometry optimization vibrational analysis single water molecule process involves simultaneous regression energy molecular scalar quantity force vector quantity dipole molecular vector quantity correspondingly performed gpr energy dipole using fingerprints extracted center mass molecule gpr force using fingerprints extracted atom component vector properties modeled independently scalar gaussian process training set consists configurations uniformly covering range shown table gpr model successful drive calculations infrared spectrum molecule randomly perturbed initial structures arbitrary orientation fingerprint configuration previous section molecular dyanmics trajectory shown figure attempt fit forces felt atoms benzene molecule along trajectories obtained sibling database density kernel carbon atoms assumes functional form oxygen atoms uses different smoothing length function rest parameters inherited previous examples training configurations chosen adaptively iterative process using sum gpr posterior variance prediction error acquisition function tang molecular fingerprints table geometry optimization vibrational analysis single water molecule using gpr proposed fingerprint algorithm independent trials performed using coordinates water perturbed equilibrium gaussian noise followed randomly chosen rotation energy static dipole residual force mode gpr dft frequency intensity gpr dft gpr dft time mae time mae time mae fit truth truth truth figure force exerted carbon atoms benzene molecule fit using configurations molecular dynamics trajectory time step size color coding corresponds carbon atom crosses indicate atomistic configurations used gpr training tang molecular fingerprints connection fingerprint algorithms figure compare ability distinguish atomistic configurations fingerprint well soap coulomb matrix work inspired soap descriptor proposes use smoothed densities represent atomistic neighborhoods however instead converting density field frequency domain using spherical harmonics perform density field sampling comparison directly real space enabled thanks available canonical coordinate frame computed kernel minisum optimization mainly used norm compute distance atomistic neighborhoods however fingerprint exhibits similar behavior soap used together inner product formula demonstrated figure thus fingerprint could used conjunction wide variety covariance functions based either euclidean distance inner product similarity soap coulomb random rotation radial stretch soap decaf dot coulomb interatomic distance random rotation noise perturbation figure comparison distance measure used decaf soap coulomb matrix fingerprint distances shown plots measured randomly chosen initial state first sight decaf different coulomb matrix fingerprint grape algorithms however instead trying capture overall density field measure contribution individual atom quadrature nodes row vector stacked results yield matrix regarded incidence matrix atoms quadrature nodes similar abstraction seen coulomb matrix grape kernel however cases vertices graph represent atoms edges represent pairwise interatomic interactions incidence matrix adopts opposite pattern constructs graph quadrature nodes vertices atoms edges adjacency matrix case tang molecular fingerprints molecule graph coulomb matrix sensors adjacency matrix incidence matrix grape figure comparison molecular fingerprints coulomb matrix grape kernel construct graphs nodes corresponds atoms weights edges determined pairwise interactions contrast incidence matrix construct graph set quadrature nodes whose connectivity weighted sum contributions individual atoms computed inner product weight edges represented elements adjacency matrix interpreted total flux contributed paths bridged atom numerically found smallest eigenvalues except eigenvalue symmetric normalized laplacian dii invariant rotation certain noise level even quadrature nodes rotate atoms nonetheless detour appears represent pure theoretical interest rather practical value conclusion paper presented canonically aligned fingerprint decaf exploring idea using smoothed density fields represent compare atomistic neighborhoods one tang molecular fingerprints key enabling technique decaf kernel minisum algorithm allows unambiguous identification canonically aligned coordinate frame used rotationally invariant projection density field well associated vector quantities performed detailed analysis study behavior fingerprint changing various parameter resolution smoothing length choice weight functions demonstrate fingerprint algorithm used implement highly accurate regressions scalar vector properties atomistic systems including energy force dipole moment could useful building block constructing next generation force fields accelerate molecular mechanics calculations accuracy comparable driven quantum mechanical theories calculators acknowledgment work supported department energy doe collaboratory mathematics mesoscopic modeling materials work also supported army research laboratory cooperative agreement number references zhao perilla yufenyuy meng chen ning ahn gronenborn schulten aiken others mature capsid structure microscopy molecular dynamics nature maragakis piana shaw picosecond millisecond structural dynamics human ubiquitin journal physical chemistry rappe casewit colwell goddard skiff uff full periodic table force field molecular mechanics molecular dynamics simulations journal american chemical society cornell cieplak bayly gould merz ferguson spellmeyer fox caldwell kollman second generation force field simulation proteins nucleic acids organic molecules journal american chemical society jorgensen maxwell development testing opls force field conformational energetics properties organic liquids journal american chemical society frenkel smit understanding molecular simulation algorithms applications volume academic press leach molecular modelling principles applications pearson education tang molecular fingerprints cheng goddard sun adaptive accelerated reaxff reactive dynamics validation simulating hydrogen combustion journal american chemical society braun boresch steinhauser transport dielectric properties water influence comparing bmw models journal chemical physics boonstra onck van der giessen charmm water model suppresses peptide folding solvating unfolded state journal physical chemistry behler symmetry functions constructing neural network potentials journal chemical physics kondor representing chemical environments physical review kermode vita molecular dynamics machine learning forces physical review letters khorshidi peterson amp modular approach machine learning atomistic simulations computer physics communications rasmussen williams gaussian processes machine learning specht general regression neural network ieee transactions neural networks rupp tkatchenko muller von lilienfeld anatole von lilienfeld von lilienfeld fast accurate modeling molecular atomization energies machine learning physical review letters haut barros learning molecular energies using localized graph kernels journal chemical physics sun learning molecules representations kernels phd thesis harvard university coifman lafon lee maggioni nadler warner zucker geometric diffusions tool harmonic analysis structure definition data diffusion maps proceedings national academy sciences united states america payne kondor gaussian approximation potentials accuracy quantum mechanics without electrons physical review letters botu ramprasad learning scheme predict atomic forces accelerate materials simulations physical review tang molecular fingerprints barzilai borwein step size gradient methods ima journal numerical analysis lebedev laikov quadrature formula sphere algebraic order accuracy doklady mathematics rabinowitz weiss tables abscissas weights numerical evaluation integrals form mathematical tables aids computation leistedt mcewen exact wavelets ball ieee transactions signal processing tang reference implementation algorithms presented atomistic fingerprint algorithm learning initio molecular force fields doi arbabzadah chmiela tkatchenko insights deep tensor neural networks nature communications chmiela tkatchenko sauceda poltavsky machine learning accurate molecular force fields science advances gross yellen graph theory applications chapman liu liu lam constructing smoothing functions smoothed particle hydrodynamics applications journal computational applied mathematics lucy numerical approach testing fission hypothesis astronomical journal appendix polynomial smoothing functions compact support candidates weight integral density scaling functions section class compact polynomials satisfy criteria compactly supported strictly positive within cutoff distance decreases monotonically least twice continuously differentiable minimal number terms tang molecular fingerprints normalized complementary coordinate within span kernel optional normalization factor ensure integral kernel ball radius unity parameters free parameters used adjust smoothness width kernel take real numbers satisfying condition note kernel equivalent lucy kernel commonly used smoothed particle hydrodynamics simulations kernel evaluated efficiently using multiplication addition integers kernel second derivative first derivative gaussian cosine figure visualization polynomial kernels given unit support radius kernels derivative origin first second derivatives kernels transition smoothly support radius contrast gaussian kernel derivatives decay zero finite distance second derivative cosine kernel mentioned previous work zero cutoff distance table quadrature nodes weights table list nodes weights laguerre quadrature rules using notations table list nodes weights lebedev quadrature rules using notations laguerre lebedev quadrature nodes combined using composite grids sampling atomistic density field tang molecular fingerprints table laguerre quadrature nodes weights points tang molecular fingerprints table lebedev quadrature nodes weights points
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nov incorporation fuzzy sets system declarative semantics implementation applications clemente dep information systems university chile centre argument technology university dundee departamento universidade santiago compostela spain abstract paper analyse benefits incorporating fuzzy sets system syntax declarative semantics implementation extension presented formalised show using potential applications fuzzy logic programming frameworks enhanced correctly work together lexical resources ontologies order improve capabilities knowledge representation reasoning keywords fuzzy sets approximate reasoning lexical knowledge resources fuzzy logic programming fuzzy prolog introduction motivation nowadays lexical knowledge resources well ontologies concepts widely employed modelling domain independent knowledge email address clrubio clemente preprint submitted summited studies computational intelligent seriesnovember automated reasoners case approximate reasoning makes possible incorporate general knowledge system independent programmer background inside former current frameworks fuzzy logic programming argue lexical reasoning might appropriate way tackling challenge type knowledge usually expressed linguistically however computational point view source information involves vagueness uncertainty consequently must specifically addressed fuzzy set theory good candidate shows particular limitations aim sometimes words mean different things different people generates additional layer uncertainty adequately handled definition membership functions word meaning also debatable question therefore achieving agreement means standard fuzzy set difficult iii respect semantic similarity measures used proposal dominant one therefore two given words different degrees resemblance obtained resulting additional level uncertainty specific field fuzzy logic programming fuzzy prolog systems little attention paid impact type high degree uncertainty vagueness inherent lexical knowledge used definition knowledge bases inference processes next simple example introduced order illustrate building prolog knowledge base example suppose extract internet two people opinions particular football player first one says normal player second one says bad player consider label qualifying highest quality good basic component lexical knowledge could modelled using two annotated facts football player good football player good respectively case use football player good given infimum usually employed however observed information first person lost case iii deserves special attention given involves use independent linguistic resources wordnet similarity said tool provide different measures according alternative criteria assessing degree similarity two words example illustrate situation means simple case example suppose fact loves extract closeness loves desires using two different semantics measures obtaining therefore order represent semantic knowledge could employ two facts either desires desires order address examples inside frame propose enhance system fuzzy sets ivfss since allow capture uncertainty associated lexical knowledge better several advantages pointed dealing environments high uncertainty imprecision using ivfss authors also shown ivfss generate better results standard fss additionally use intervals describing uncertain information successfully applied realms decision making risk analysis engineering design scheduling example example easily modelled means ivfss using interval combining information different sources single fact football player good desires respectively main contribution paper design implement intervalvalued fuzzy logic language incorporate system task involves different challenges theoretical implementation points view former entails adding ivfss arithmetic warren abstract machine based similarity swam latter means establish declarative semantics language classical way formalising notion least interval valued fuzzy herbrand model fuzzy definite programs paper divided following sections section introduces concepts support approach section describes details syntax semantics implementation proposed language section analyses different realms programming language applied section main differences proposal others described literature discussed finally section summarizes main conclusions ideas future work preliminary concepts fuzzy sets ivfss fuzzy formalism based two membership mappings instead single one like standard fss one membership functions called lower membership function upper membership function established universe discourse map element real number interval elements belongs according interval definition fuzzy set crisp set ordered triples lower upper membership functions respectively satisfying following condition observed definition intervals included closed ends hand arithmetic operations recalled since useful operating cardinalities ivfss let intervals arithmetic operations power defined follows min max operations union intersection ivfss defined triangular norms let ivfss union fuzzy set membership function intersection ivfss thus morgan laws ivfss let lattice intervals satisfies also definition hence smallest greatest elements approximate deductive reasoning consider collection imprecise premises possible imprecise conclusion inferred prolog program applying process approximate deductive reasoning set statements interpreted two different frames prolog program conditional interpretations former assumed imprecise premise assertion qualified degree truth john tall means degree truthfulness sentence using ivfs hand latter adopted interval qualifies sentence means degree membership element specific set john tall means membership john set tall people conclusion inferred imprecise premise must also qualified type degree john good player order preserver coherence classical prolog adopt propositional interpretation interval indicates degree truth assertion consequently approximate deductive reasoning based modus ponens deduce defined lattice simple fuzzy prolog syntax semantics implementation design programming language involves three main steps firstly definition syntax secondly elaboration formal study semantics thirdly implementation system order address tasks related syntax semantics follow guidelines established implementation task follow guidelines detailed sintax fuzzy program conveys classical prolog knowledge base set ivfss used annotating facts means fuzzy degree definition fuzzy definite clause horn clause form called head denote conjunction called body variables clause assumed universally quantified definition fuzzy definite program finite set fuzzy clauses example let intervalvalued fuzzy definite program generates first order language whose alphabet comprised set variable symbols constant symbols function symbols predicate symbols appear clauses assume first order language least one constant symbol assertion constants available alphabet artificial constant must added first order language generated assume familiarity theory practice logic programming declarative semantics logic programming declarative semantics program traditionally formulated basis least herbrand model conceived infimum set interpretations section formally introduce semantic notions herbrand interpretation herbrand model least herbrand model fuzzy program order characterise framework facts modelled terms degrees fuzzy interpretation pair domain interpretation mapping assigns meaning symbols specifically relation symbols interpreted mappings order evaluate open formulas introduce notion variable assignment variable assignment interpretation mapping set variables elements interpretation domain notion extended set terms structural induction usual following definition formalises notion valuation formula framework definition given fuzzy interpretation variable assignment valuation formula inf inf predicate symbol atomic formulas body clause defined assignment variables definition let first order language herbrand universe set ground terms formed constants function symbols appearing definition let first order language herbrand base set ground atoms formed using predicate symbols ground terms herbrand universe arguments example let consider language generated program example herbrand universe herbrand base classical case possible identify herbrand interpretation subset herbrand base therefore convenient generalization notion herbrand interpretation fuzzy case consists establishing fuzzy herbrand interpretation fuzzy subset herbrand base definition fuzzy interpretation given first order language fuzzy herbrand interpretation mapping hence truth value ground atom sometimes represent fuzzy herbrand interpretation extensively set pairs introduce notion fuzzy herbrand model formalised definitions employ declarative semantics based threshold intuitively threshold delimiting truth degrees equal greater true therefore going speak fuzzy herbrand model level simply definition fuzzy herbrand interpretation model fuzzy clause definition fuzzy herbrand interpretation model fuzzy program clause theorem let fuzzy program suppose herbrand proof suppose let intervalvalued fuzzy herbrand interpretation going prove interpretation clauses let clause initial supposition definition fuzzy program iff let implies definition let fuzzy program let fuzzy clause logical consequence level fuzzy interpretation proposition logical consequence fuzzy program level every fuzzy herbrand interpretation fuzzy herbrand proof first let suppose logical consequence level definition fuzzy interpretation moreover theorem must exist fuzzy herbrand model level establishes first side argument every interpretation herbrand model level herbrand let interpretation necessarily herbrand theorem model ground instances result hence establishes side argument ordering lattice extended set fuzzy interpretation follows iff fuzzy atom important note pair complete lattice comes equipped fuzzy interpretation fuzzy interpretation therefore top element lattice bottom element fuzzy interpretations important property allow characterize semantics fuzzy program definition model level model level contains fuzzy atom degree min proposition intersection property models let fuzzy program let set model levels respectively min proof prove proposition induction number interpretations base case let models levels fuzzy clause min model inductive case let models levels fuzzy clause min properties minimum definition let fuzzy program least model defined follows call fuzzy degree min theorem let fuzzy program let least model let ground atom fuzzy herbrand base min logical consequence level min proof first definition hence model min logical consequence level min establishes first side argument logical consequence definition model min min therefore definition least model implies min establishes another side argument fixpoint semantics section give deeper characterisation least herbrand model fuzzy program using fixpoint concepts possible fuzzy program associated complete lattice fuzzy herbrand interpretations define continuous operator lattice allows provide constructive vision meaning program defining immediate consequences operator construct least herbrand model means successive applications definition fixpoint characterization least herbrand model let fuzzy program mapping defined follows let fuzzy herbrand interpretation ground instance clause inf case classical logic programming fuzzy herbrand interpretations models characterised terms operator theorem let fuzzy program let fuzzy herbrand interpretation proof clause therefore fulfilled every variable assignment therefore supposing without loss generality properties minimun min implies hence properties minimun inf implies ready demonstrate main theorem subsection first recall following results fixpoint theory theorem fixpoint theorem let complete lattice monotonic mapping least fixpoint inf inf proposition let complete lattice continuous mapping proof see theorem let fuzzy definite program proof least model intersection lattice fuzzy herbrand models complete one use theorem proposition theorem applying continuity establishes theorem example given program example least herbrand model therefore fixpoint reached next item operational semantics begin providing definitions fuzzy used later showing soundness completeness system definition let either derived using mgu following conditions hold resolvent atom called selected atom mgu iii fuzzy goal min definition fuzzy successful empty clause last goal derivation empty clause say derivation length empty clause derived min definition let fuzzy program fuzzy goal fuzzy computed answer substitution obtained restricting composition variables sequence mgu employed finite fuzzy approximation degree definition let fuzzy program intervalvalued fuzzy goal answer say fuzzy correct answer logical consequence level min min implementation section briefly explain fuzzy sets incorporated describe structure main features abstract machine created extension swam execution programs appropriately modified compiler machine instructions swam structures beta version http founded url order trigger fuzzy resolution worth noting best knowledge first swam implementation supports fuzzy resolution mandatory step achieve result include new data structure architecture computing fuzzy sets data structure implemented using class called intervalfs formed two private attributes double type upper limit lower limit define public method constructor intervalfs double double four methods sets gets double getupperlimit double getlowerlimit void setupperlimit double void setlowerlimit double additionally overwrite tostring equals ods usual way finally methods adding substracting computing minimum interval valued fuzzy set implemented intervalfs add intervalfs intervalfs intervalfs substract intervalfs intervalfs intervalfs min intervalfs intervalfs following example illustrates new features swam enhanced ivfss example let suppose want represent following knowledge football player good fast tall coordinated know particular player fast quite tall coordinated thus good player answering question scenario linguistic expression coordinate could represented fact coordinate linguistic term fast could represented fact fast quite tall could represented fact tall possible solution employing program described follows facts coordinate fast tall rules fast coordinate swam enhanced ivfss allows obtain answer swam code generated program follows allocate call coordinate call fast call tall deallocate proceed coordinate proceed fast proceed tall proceed query call halt first instruction executed one labelled key query hence execution starts position degree fixed instruction trust line line query launched variable created create variable instruction line line first subgoal coordinate launched execution goes line unification term coordinate produced line put value get constant instructions new approximation degree established min trust instruction terms unify following subgoal ast line line line launched approximation degree min terms unify following subgoal tall line line launched approximation degree min finally assignation produced implemented limit expansion search space computation called ivfss lambdacutivfs flag set value different weak unification process fails computed approximation degree goes stored lambdacutivfs value therefore computation also fails possible branches starting choice point discarded default lambdacutivfs value however lambda cut flag set different value means directive interval example could established using following directive lambdacutivfs applications main realms application ivfss programming language described paper involve natural language semantics processing section discuss two linguistic knowledge modelling logic programming using linguistic resources linguistic knowledge modelling linguistic knowledge modelling handles computational representation knowledge embedded natural language framework enhanced combining multiadjoint paradigm fuzzy sets example define annotated atoms let assume definition suitable journal given journal high impact factor medium immediacy index relatively big halflife bad position listing category introduce program following inference rule let suppose ieee transactions fuzzy system journal following properties high impact factor small immediacy index relatively small cited half life best position regarding linguistic variables high medium relatively big bad related following respectively considering variables medium bad similar meaning knowledge could model fuzzy logic language follows high impact factor small immediacy index relatively small best position query suitable journal launched system answers ieee logic programming based wordnet logic programming framework provides capability enriching semantically classical logic programming languages using proximity equations pes limitation approach pes mostly defined specific domain designer manually fixes values equations fact makes harder use plp systems real applications possible solution consists obtaining proximity equations wordnet requires employ fuzzy sets order deal high uncertainty generated possibility using several different semantic similarity metrics let assume fragment deductive database stores information people preferences proximity equations generated wordnet put see detail loves mountaineering loves mary mountaineering likes football likes john football peter plays basketball plays peter basketball person practises sports healthy person healthy practices sport automatically generated wordnet related work literature proposals address goal found one relevant ones reason analyse detail differences order clarify reinforce novelty proposal point view implementation ivfss included means constrains hence translator must implemented result programmer must code variables order manage truth values get answers system based constraints hand ivfss included different way compiler warren abstract machine enhanced using ivfss data structure created adapted architecture result intervals work standard data structure code program instead particular set variables defined hoc programmer feature allows include ivfss fuzzy unification see fuzzy resolution addition framework also allows possible extensions incorporation reasoning module using wordnet see point view syntax bousiprolog although prolog languages well differentiated syntax former allows annotation facts rules annotated allow use aggregator operator computing annotated ivfss latter hand allows user annotation fact rules means ivfss addition focus inference engine extends resolution mechanism uses proximity equations young extends resolution unification process point view semantics bousiprolog relevant differences semantic levels well firstly implements concept allows user imposes threshold system according precise want answer substantial change due introduction threshold operational semantics therefore operational mechanism behaves much one prolog system obtaining correct answers one one option available ciao semantics mentioned section ivfs approximation degrees implemented concepts interpretation least model semantics model presented defined different way operational semantics based extension sld resolution type resolution based classical sld resolution prolog systems conclusions future work formally defined efficiently implemented simple intervalvalued fuzzy programming language using fuzzy sets modelling uncertainty imprecision knowledge associated lexical resources future work propose extend language provide results soundness completeness additionally want develop fully integrated framework fuzzy sets intervalvalued fuzzy relations combined framework acknowledgements authors gratefully acknowledges comments made reviewers work partially supported feder state research agency aei spanish ministry economy competition grants cultura universitaria postdoctoral training grants centro singular galicia accreditation european regional development fund erdf work done collaboration research group somos funded research agency graduate school management university references miller wordnet lexical database english communications acm liu singh commonsense reasoning natural language negoita howlett jain eds proceedings international conference intelligent information engineering systems kes wellington new zealand september berlin heidelberg springer bobillo fuzzy ontology extension wordnet eurowordnet specialized knowledge proceedings tke towards fuzzy lexical reasoning journal intelligent fuzzy systems feb medina logic programming proceedings international conference information processing managment uncertainty systems ipmu moreno penabad declarative semantics fuzzy logic language managing similarities truth degrees springer international publishing cham moreno penabad fuzzy logic programming environment managing similarity truth degrees strass rfuzzy syntax semantics implementation details simple expressive fuzzy tool prolog information sciences special issue information engineering applications based lattices straccia managing uncertainty vagueness description logics logic programs description logic programs springer berlin heidelberg berlin heidelberg liu tran fuzzy linguistic logic programming applications theory practice logic programming may pedersen patwardhan michelizzi measuring relatedness concepts wordnet demonstration papers stroudsburg usa association computational linguistics turksen four methods approximate reasoning valued fuzzy sets international journal approximate reasoning mar bustince fuzzy sets soft computing international journal computational intelligence systems medina adjoint pairs fuzzy sets proceedings international conference information processing managment uncertainty systems ipmu springer berlin heidelberg fuzzy linguistic prolog applications journal intelligent fuzzy systems wam springer berlin heidelberg berlin heidelberg reflections interpretations statements syllogistic reasoning archives philosophy history soft computing hajek metamathematics fuzzy logic springer science business media lloyd foundations logic programming berlin declarative semantics bousi prolog proceedings acm sigplan conference principles practice declarative programming ppdp new york usa acm ebrahim fuzzy logic programming fuzzy sets systems medina formal concept analysis via concept lattices fuzzy sets systems declarative semantics clp qualification proximity theory practice logic programming atanassov georgiev intuitionistic fuzzy prolog fuzzy sets systems guadarrama muoz vaucheret fuzzy prolog new approach using soft constraints propagation fuzzy sets systems
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tropical limits probability spaces part jun intrinsic distance asymptotic equipartition property configurations matveev portegies abstract entropy finite probability space measures observable cardinality large independent products probability space two probability spaces entropy almost bijection large parts way asymptotically equivalent turns challenging generalize notion asymptotic equivalence configurations probability spaces collections probability spaces maps article introduce intrinsic distance space configurations probability spaces concentrating geometry pass asymptotic distance induces asymptotic equivalence relation sequences configurations probability spaces call equivalence classes tropical probability spaces context prove asymptotic equipartition property configurations states tropical configurations always approximated homogeneous configurations addition show solutions certain problems respect asymptotic distance follows two statements order solve informationoptimization problem suffices consider homogeneous configurations finally show spaces trajectories length certain stochastic processes particular stationary markov chains tropical limit introduction aim present article develop theory tropical probability spaces asymptotic classes finite probability spaces together accompanying techniques expect relevant problems arising information theory causal inference artificial intelligence neuroscience matter introduction motivation research presented article start considering simple examples single probability spaces consider finite probability space finite set probability measure simplicity assume measure full support next introduction consider bernoulli sequence probability spaces denotes cartesian product product measure situation arises several contexts example physics would encode state system comprised many identical weakly interacting subsystems state space information theory would describe output random source dynamical systems stochastic processes setting corresponds bernoulli shifts bernoulli processes entropy exponential growth rate observable cardinality tensor powers observable cardinality loosely speaking cardinality set biggest possible set small measure elements negligible measure removed turns observable cardinality might much smaller cardinality whole following sense asymptotic equipartition property states every suf ficiently large one find typical subset takes almost mass probability distribution almost uniform normalized logarithmic scale holds cardinality may much smaller still grow exponentially even though many choices set exponential growth rate respect upto fact exists number choice typical subset holds limit growth rate called entropy explained detail section ent lim lim law large numbers ent formula shannon entropy usually introduced entropy especially easy evaluate space uniform since finite probability space uniform distribution holds ent point view entropy goes back original idea boltzmann according entropy logarithm number equiprobable introduction states system comprised many identical weakly interacting subsystems may take asymptotic equivalence asymptotic equipartion property implies sequence asymptotically equivalent sequence uniform spaces following sense let denote probability distribution supported uniform support sequence independent samples according hard discriminate sequence independent samples similarly probability spaces entropy sets asymptotically equivalent sense bijection typical sets seen change code essentially content shannon source coding theorem gromov proposed existence basis asymptotic equivalence relation sequences probability spaces even though greatly influenced ideas found gromov definition extend easily configurations probability spaces configuration probability spaces mean collection probability spaces maps give precise definition section consider particular examples formalizing studying notion asymptotic equivalence configurations probability spaces main topic present article configurations probability spaces suppose instead single probability space consider pair probability spaces joint distribution probability measure pushes forward coordinate projections words consider triple probability spaces pair maps later call minimal probability spaces particular instance configuration probability spaces three examples three examples object shown figure interpreted following way spaces cardinality six uniform distribution weight atom spaces cardinality distribution also uniform weights support measure colored grey pictures maps coordinate projections introduction figure examples pairs probability spaces together joint distribution view equation ent ent ent would like ask following question possible find sufficiently high powers arbitrary given precision shannon coding theorem described end previous subsection generally proper generalization asymptotic equivalence relation discussed previous subsection sequences tensor powers would like argue even though entropies constituent spaces pairwise three examples pairwise asymptotically different establish examples figure different isomorphic see also section relatively easy since symmetry groups however present different argument lends generalization prove examples hand asymptotically equivalent also gives quantitative difference distinguish example one could argue along following lines could try add third space pair provide joint distribution projection first two factors third factor could evaluate entropies various denote corresponding coordinate projections introduction probability spaces fit commutative diagram arrow reduction simply map probability spaces consider set possible extensions form denote ext extension ext four new entropies ent ent ent ent addition known entropies vector ent ent ent ent ent ent ent entropy vector extension set possible values entropy vector extensions extension call unstabilized relative entropic set unstabilized relative entropic sets examples turns unstabilized relative entropic sets different see let calculate particular points unstabilized relative entropic sets examples consider constrained informationoptimization problem finding extension space reduction ent ent spaces independent conditioned ent ent ent ent iii sum ent ent maximal subject conditions introduction easy read solutions optimization problem examples right pictures figure indeed condition says must partition condition says set partition must rectangular must cartesian product subset subset quantity maximized average sets partition optima easy find hand example partition three squares examples one solutions partition resp rectangles thus optimal values example examples stabilized relative entropic set seen examples told apart determining unstabilized relative entropic set however really interested rather wonder whether high tensor powers distinguished way relates used adjective unstabilized relative entropic set usually grows stable taking tensor powers every holds general set side strictly larger set left view inclusion may define stabilized relative entropic set closure lim set turns convex stabilized relative entropic set examples fact stabilized relative entropic set also differentiates examples proof fact follows lines section stabilization makes argument much technical expect stabilized relative entropic set differentiate examples however types relative entropic sets problems differentiate examples relative entropic sets discussed section problems relative entropic sets section used problem find particular points unstabilized relative entropic set coincidence link stabilized problems stable relative entropic set made explicit stable relative entropic set convex completely characterized informationoptimization problems vice versa introduction problems play important role information theory causal inference artificial intelligence information decomposition robotics neuroscience techniques developed article allow one address type problems easily efficiently intrinsic distance mentioned end section one tempted define asymptotically equivalent configurations along lines shannon source coding theorem following two configurations would asymptotically equivalent almost bijection subspaces almost full measure high tensor powers however found approach inconvenient instead finding almost bijection large parts two spaces consider stochastic coupling transportation plan joint distribution pair spaces measure deviation isomorphism probability spaces bijection measure deviation isomorphism leads notion intrinsic kolmogorovsinai distance stable version asymptotic distance explained section case single probability spaces define intrinsic kolmogorovsinai distance two probability spaces inf ent ent ent ent infimum taken choices joint distribution note summands nonnegative vanishes corresponding marginalization isomorphism probability spaces sense distance measures deviation existence bijection furthermore define asymptotic distance two probability spaces lim definition could generalized configurations probability spaces say two configurations asymptotically equivalent asymptotic distance vanishes asymptotic equipartition property examples property symmetry group acts transitively support measure particular instances call homogeneous configurations section show asymptotic equipartion property configurations theorem states every sequence tensor powers configuration approximated asymptotic distance sequence homogeneous configurations introduction asymptotic equipartition property allows one substitute configurations probability spaces homogeneous approximations homogeneous probability spaces uniform probability spaces first simple consequence asymptotic equipartition property asymptotic distance probability spaces computed equals ent ent homogeneous configurations unlike homogeneous probability spaces rather complex objects nonetheless seem simpler arbitrary configurations probability spaces types problems would like address specifically show section optimal values stabilized problems depend asymptotic class configuration continuous respect asymptotic distance many cases optimizers continuous well asymptotic equipartition property implies purposes calculating optimal values approximate optimizers one needs consider homogeneous configurations greatly simplify computations summarizing asymptotic equipartition property continuity problems important justifications choice asymptotic equivalence relation introduction intrinsic asymptotic distances article article following structure section devoted basic setup used throughout text section explain mean configurations probability spaces give examples describe simple properties operations section generalize notion probability distribution configurations discuss theory types configurations section intrinsic distance asymptotic distance introduced technical tools estimation kolmogorov distance developed section contains estimates distances types use estimates proof asymptotic equipartition property configurations section section deals extensions configurations prove extension lemma used show continuity extensions implies particular solutions constrained optimization problem entropies extensions respect asymptotic distance thus depend asymptotic classes configurations section briefly discuss special type configurations called mixtures play important role construction tropical probability spaces finally section introduce notion tropical probability spaces configurations thereof list properties continue study tropical probability spaces configurations subsequent articles category technical illuminating proofs deferred section technical proofs electronic version one move proof technical section statement main text following link arrow acknowledgments would like thank tobias fritz misha movshev johannes rauh inspiring discussions grateful participants wednesday morning session casa group eindhoven university technology valuable feedback introduction article finally thank max planck institute mathematics sciences leipzig hospitality category probability spaces configurations section devoted basic setup used throughout present article introduce category probability spaces reductions similar categories introduced define configurations probability spaces corresponding category last subsection recalls notion entropy elementary properties probability spaces reductions consider probability spaces support probability measure finite space contains subspace isomorphic finite space thus call objects finite probability spaces probability space denote supp support measure cardinality slightly abusing language call quantity cardinality pair probability spaces reduction class measurepreserving maps two maps equivalent coincide set full measure composition two reductions reduction two probability spaces isomorphic bijection supports probability measures bijection defines invertible reduction one space another clearly cardinality isomorphism invariant automorphism group aut group probability space called homogeneous automorphism group aut acts transitively support measure property homogeneous isomorphism invariant isomorphism class homogeneous space representative uniform measure finite probability spaces reductions form category denote prob subcategory homogeneous spaces denoted probh isomorphism category coincides notion isomorphism category prob small category however small full subcategory contains object every isomorphism class prob every pair objects contains available morphisms category imagine subcategory chosen fixed replaces prob considerations product prob given cartesian product probability spaces denote canonical reductions given projections factors pair reductions tensor product reduction equal class cartesian product maps representing tensor product however categorical product product leaves subcategory homogeneous spaces invariant probability measure usually denoted simply risk confusion low configurations probability spaces essentially configuration fij commutative diagram consisting finite number probability spaces reductions transitively closed morphism two configurations fij gij combinatorial type collection reductions corresponding individual objects commute reductions within configuration fij gij need keep track combinatorial structure collection reductions within configuration several possibilities reductions form directed transitively closed graph without loops spaces configuration form poset underlying combinatorial structure could recorded finite category last option seems convenient since many operations necessary analysis already diagram category finite category pair objects morphism space homg homg contains one element diagram category configuration probability spaces modeled functor prob collection configurations probability spaces modeled fixed diagram category forms category functors prob prob objects prob configurations functors prob morphisms prob natural transformations configuration prob diagram category called combinatorial type diagram category configuration prob denote number objects category object diagram category called initial target morphism except identity likewise terminal object source morphism except identity morphism note category terminology somewhat unconventional point view category theory diagram category called complete unique initial object thus configuration modeled complete category includes space reduces spaces configuration terminology transfers configurations modeled initial space prob one target space reduction within configuration terminal space source reduction complete unique initial space tensor product probability spaces extends tensor product configurations prob fij gij define fij gij occasionally also talk configuration sets denote set category finite sets surjective maps constructions could repeated sets instead probability spaces thus could talk category configurations sets set given reduction two probability spaces restriction surjective map given configuration fij probability spaces underlying configuration sets obtained taking supports measures level restricting reductions supports denote fij supp pxi thus forgetful functor prob set consider two important examples diagram categories configurations modeled give examples section configuration modeled category three objects one initial two terminal special significance since simplest nontrivial configurations see later essentially triple probability spaces pair reductions reduction another triple reductions commute reductions within fan following diagram commutative category isomorphisms automorphism group aut defined accordingly note terminal spaces labeled reductions preserve labeling called minimal unique reduces given always reduction minimal minimal reduction unique isomorphism explicitly take set consider probability distribution induced map cartesian product reductions original notion minimal fact categorical notion could equivalently defined saying minimal reduction holds isomorphisms also isomorphism consequently one specifies reductions terminal spaces minimal another exists one extension reduction whole fan inclusion pair probability spaces terminal vertices minimal equivalent specifying joint distribution arbitrary configuration called minimal every also contains minimal terminal spaces denote space minimal configurations modelled diagram category prob given terminal spaces point one may construct conditional probability distribution denote corresponding space usual bar normally used conditioning interferes notations cardinality spaces give details section diamond configuration diamond configuration modeled diamond category consists course also morphism lies transitive closure given four morphisms rule skip writing morphisms implied transitive closure diamond configuration minimal top minimal entropy working definition entropy based following version asymptotic equipartition theorem bernoulli process see configurations theorem suppose finite probability space exists subset holds moreover two subsets satisfying two conditions cardinalities satisfy rrr rrr rrrrr rrr rrrr define clearly view property theorem limit independent choice typical subsets entropy satisfies shannon inequality see example namely minimal diamond configuration ent lim lim following inequality holds ent ent ent ent furthermore entropy additive respect tensor product pair probability spaces prob holds ent ent ent pair probability spaces included minimal define conditional entropy ent ent ent quantity always view shannon inequality moreover following identity holds see ent ent configurations section look configurations detail start considering important examples configurations examples configurations singleton denote diagram category single object clearly configurations modeled probability spaces prob prob chains chain length category objects morphisms whenever configuration prob chain reductions category three objects two morphisms see also section simplest configurations asymptotic equivalence classes contain information entropies entries recall fan called minimal pair points positive weights exists one reduces equivalently reduction must isomorphism full configuration full category objects category objects indexed subsets morphism whenever collection random variables one may construct minimal full configuration prob considering joint distributions marginalization reductions denote configuration hand terminal vertices full configuration viewed random variables domain definition given unique initial space suppose prob minimal full configuration terminal vertices convenient view distribution cartesian product underlying sets terminal vertices underlying sets terminal spaces fixed correspondence full minimal configurations distributions configurations configuration category consists five objects two initial three terminal morphisms follows thus typical configuration consists five probability spaces reduction probability spaces initial terminal configuration previous example could generalized category fence configuration fence category consists six objects morphisms category three objects morphisms diagram diamond configurations diamond configuration modeled diamond category consists fan see also section examples complete examples contain tropical limits configurations simple essentially tropical limits correspond tuple numbers corresponding entropies constituent spaces therefore call configurations containing simple configurations constant configurations suppose probability space diagram category one may form constant considering functor maps objects morphisms identity morphism denote constant configuration simply clear context constant configuration automatically minimal fij another reduction write sometimes simply collection reductions fij configurations configurations course operation configuration could iterated given pair diagram categories could form could speak example configurations type prob prob prob cartesian product graphs every diagram category could considered transitively closed directed graph operation commutative thus example rarely need anything beyond configurations called minimal extension form reduction must isomorphism recall could also viewed gconfiguration probability spaces following lemma show order verify minimality configurations sufficient check minimality constituent lemma let diagram category minimal constituent probability spaces minimal minimal reduction exists exists minimal configurations included following diagram even though lemma rather elementary many similar statements true thus compelled provide proof found section page similarly full configuration called minimal every minimal terminal configurations lemma following corollary counterpart full configurations corollary let diagram category full configuration minimal constituent full configurations probability spaces minimal full configuration minimal reduction exists restrictions extensions suppose functor two diagram categories configuration prob pullback configuration prob defined composition called extension functor injective call write likewise called restriction operation functorial sense given two configurations prob reduction canonical reduction thus considered functor prob prob important examples restrictions extensions restriction full configuration smaller full configuration recall explained section terminal vertices full configuration could considered random variables collection random variables generates full configuration full configuration subset denote restriction full configuration generated configurations restriction operator also make use notation prob prob restriction given full configuration prob may forget part data example forget top space relation pair three terminal spaces end configuration operation corresponds inclusion functor preserves show section corresponding restriction operator prob prob surjective objects morphisms thus map collections objects right inverse however natural right inverse exists restriction starting full configuration might choose forget initial space reductions domain remaining configuration combinatorial type fence operation corresponds functor corresponding operator prob prob surjective find extendable interesting problem see example references therein hope methods developed article might useful address questions configurations doubling first example interesting functor diagram categories injective situation term restriction really reflect operation well however come better terminology doubling operation restriction configuration consider category category define functor setting note extends uniquely spaces morphisms since morphism space either empty set thus left right isomorphic operation along particular problem related copy operation used find many nonshannon information inequalities described example adhesion given minimal configuration prob one could always construct extension full configuration prob following way explained section construct minimal full configuration terminal vertices sufficient provide distribution correct marginals setting straightforward check appropriate restriction full configuration defined manner indeed original configuration essentially extend need provide relationship coupling spaces declaring independent relative instance operation called adhesion see call top vertex full configuration entropies achieve equality shannon inequality ent ent ent ent adhesion provides right inverse restriction functor described section prob prob configurations figure examples homogeneous configurations important note though map functorial fact functorial inverse exists homogeneous configurations configuration prob modeled diagram category called homogeneous automorphism group aut acts transitively every probability space three examples homogeneous configurations given introduction examples homogeneous configurations combinatorial type shown figure subcategory homogeneous configurations modeled denoted prob fact homogeneous sufficient aut acts transitively every initial space thus complete initial space check homogeneity sufficient check transitivity action symmetries functoriality restriction operator restriction homogeneous configuration also homogeneous words functor configurations prob prob associated restriction operator prob prob particular individual spaces homogeneous configuration homogeneous prob probh however homogeneity whole configuration stronger property homogeneity individual spaces configuration thus general prob probh single probability space homogeneous representative isomorphism class uniform measure holds true chain configurations configuration contain however complex configurations example simple description available universal construction homogeneous configurations examples homogeneous configurations could constructed following manner suppose finite group collection subgroups consider collection sets consider natural surjection fij whenever subgroup equipping uniform distribution one turn configuration sets fij homogeneous configuration complete smallest subgroup inclusion among configuration complete minimal together pair groups collection intersection also belongs collection fact homogeneous configuration arises way suppose configuration fij homogeneous set aut choose collection points fij denote stab one applies construction previous paragraph collection subgroups one recovers original configuration conditioning suppose configuration contains fan given point weight one may consider conditional probability distributions distribution supported given distributions types distribution pushforward recall minimal underlying set assumed product case denote corresponding space discussed end section assumptions possible condition whole specifically configuration contains probability space satisfying condition every fan terminal vertices may condition whole given positive weight denote configuration spaces conditioned configuration combinatorial type called slice note space may may belong conditioning may depend choice fan however complete conditioning independent choice fans suppose two subconfiguration addition constant configuration diagram category let well defined independent choice space element considered homogeneous also homogeneous isomorphism class depend choice entropy fij define entropy function prob fij ent convenient equip target thus ent complete initial space shannon inequality obvious estimate ent ent distributions types section recall elementary inequalities relative entropies total variation distance distributions finite sets furthermore generalize notion probability distribution set distribution distributions types configuration sets finally give perspective theory types also introduce types context complete configurations distributions single probability spaces finite set denote collection probability distributions unit simplex real vector space often use fact compact convex set whose interior points correspond fully supported probability measures denote total variation signed measure define entropy distribution addition lies interior define relative entropy entropy probability space often defined formula standard fact verified help lemma holds ent justifies name entropy function define divergence ball radius centered interior fixed ball also lies interior lemma let finite set pinsker inequality holds interior exists positive constant holds iii suppose point interior also lies interior exist constant holds max first claim lemma pinsker inequality inequality instance information theory proof found second claim follows fact fixed interior relative entropy function first argument bounded smooth interior simplex minimum distributions types prove last claim note entropy function smooth interior simplex last claim follows first claim distributions configurations map two finite sets induces affine map configuration sets fij define space distributions configuration fij essentially element collection distributions sets consistent respect maps fij consistency conditions fij form collection linear equations integer coefficients respect standard convex coordinates thus rational affine subspace product simplices particular convex structure complete initial set specifying distribution uniquely determines distributions setting situation complete collection initial sets isomorphic affine subspace product cut linear equations integer coefficients corresponding initial sets among simplify notation probability space configuration write discuss briefly theory types types special subspaces tensor powers consist seqences empirical distribution explained details detailed discussion reader referred generalize theory types complete configurations sets complete configurations probability spaces theory types configurations complete complex addressed subsequent article types single probability spaces let finite set denote also collection rational points denominator say rational number denominator distributions types define empirical distribution map sends empirical distribution given clearly image lies space equipped uniform measure called type symmetric group acts permuting coordinates action leaves empirical distribution invariant therefore could restricted type acts transitively thus probability space uniform distribution homogeneous space suppose probability space let pushforward empirical distribution map clearly supp thus finite probability space therefore reduction call empirical reduction particular follows side depend probability long compatible following lemma records standard facts types checked elementary combinatorics found distributions types lemma let probability space iii probability space rational probability distribution denominator type called true type tpx corollary lemma equation obtain following corollary finite set holds ent particular finite probability space rational distribution denominator holds ent ent ent following important theorem known sanov theorem derived lemma found theorem sanov theorem let finite probability space let empirical reduction every divergence ball relative entropy ball defined types complete configurations subsection generalize theory types configurations modeled complete category theory configurations complex addressed future work give three equivalent definitions type complete configuration useful way describe three approaches need preparatory material distributions types lemma given diamond configuration probability spaces following two conditions equivalent minimal reduction diamond isomorphic adhesion equivalently following independence condition holds holds suppose diamond lemma top row subconfiguration consider conditioning element satisfies two conditions lemma thus constructed reduction suppose reduction pair probability spaces induced reduction included following diamond configuration satisfies conditions lemma means reduction ready give definitions types let prob complete configuration fij initial space let type configuration configuration types define type whose individual spaces types individual spaces corresponding fij distributions types types tensor power section mean consistent collection points fij whenever fij defined define projection symmetric group acts collection sections tensor power permuting coordinates let orbit action suppose pair fij defined since map turn call type fij value impirical distribution since initial space coincides initial space reductions coincide conclude type conditionining tensor power extend configuration adding empirical reduction let recall may define section define type definition holds let using lemma discussion thereafter conclude therefore empirical construct terminal vertices constant every probability space every reduction identity distance help lemma construct minimal reduction let within holds every type homogeneous configuration suppose complete configuration probability distribution initial set rational denominator call true type denote distance turn space configurations space introducing intrinsic distance asymptotic distance brevity usually call kolmogorov distance asymptotic kolmogorov distance intrinsic distance obtained taking infimum shared information distance possible joint distributions two probability spaces name justified fact shared information distance name appears proof theorem generating partitions ergodic systems kolmogorov sinai see example note kolmogorov distance statistics refers different notion kolmogorov distance asymptotic kolmogorov distance kolmogorov distance case single probability spaces twofan define distance probability spaces respect ent ent ent ent ent essentially measures deviation statistical map defined deterministic bijection minimal reduction satisfies minimal distance also calculated distance respectively entropy relative entropy functions defined pair probability spaces define intrinsic distance inf optimization takes place terminal spaces view inequality one could well optimize space minimal also refer couplings tensor product trivially provides coupling set couplings compact therefore optimum always achieved finite bivariate function prob prob defines notion pseudodistance vanishes exactly pairs isomorphic probability spaces follows directly shannon inequality general statement proven proposition kolmogorov distance complete configurations definition kolmogorov distance complete configurations repeats almost literally definition single spaces fix complete diagram category considering configurations prob consider three configurations fij gij hij prob recall define ent ent ent quantity vanishes fan provides isomorphisms individual spaces commute inner structure configurations provides isomorphism prob intrinsic distance configurations defined analogy case single probability spaces inf infimum terminal vertices following proposition records intrinsic kolmogorov distance fact prob provided complete diagram category unique initial space proposition let complete diagram category bivariate function prob prob distance prob moreover two configurations prob satisfy isomorphic prob idea proof simple case single probability spaces coupling constructed coupling coupling adhesion see section triangle inequality follows shannon inequality however since dealing configurations combinatorial structure requires careful treatment therefore provide detailed proof page important note proof uses fact complete fact even though definition could easily extended bivariate function space configurations fixed combinatorial type fails satisfy triangle inequality general composition couplings requires completeness asymptotic distance let complete diagram category define asymptotic distance two configurations prob lim show corollary sequence subadditive therefore limit definition always exists holds corollary proposition definition immediately obtain also asymptotic distance prob corollary let complete diagram category bivariate function prob prob prob satisfying following homogeneity property pair configurations prob holds show later section however probability spaces isomorphic rest section derive elementary properties intrinsic kolmogorov distance asymptotic kolmogorov distance distance lipschitz property operations section show certain natural operations configurations namely tensor product entropy function restriction operator lipschitz continuous section show lipschitz continuity certain extension operations tensor product show tensor product space configurations later allow give simple description tropical configurations points asymptotic cone prob limits certain sequences classical configurations proposition let complete diagram category respect kolmogorov distance prob tensor product prob prob variable every triple prob following bound holds statement direct consequence additivity entropy respect tensor product details found page follows directly definition proposition asymptotic kolmogorov distance enjoys similar property corollary let complete diagram category respect kolmogorov distance prob tensor product prob prob variable another corollary obtain subadditivity properties intrinsic kolmogorov distance asymptotic kolmogorov distance corollary let complete diagram category let prob implies particular shifts maps prob prob corollary let complete diagram category either kolmogorov distance asymptotic kolmogorov distance prob let prob shift map prob prob distance map respect either kolmogorov distance asymptotic kolmogorov distance entropy recall defined entropy function prob evaluating entropy individual spaces target space endowed respect natural coordinate system choice entropy function respect kolmogorov distance prob proposition suppose complete diagram category either kolmogorov distance asymptotic kolmogorov distance prob entropy function prob fij entxi proof proposition application shannon inequality see page details restrictions restriction operators also lipschitz shown next proposition proposition suppose functor two complete diagram categories stands either kolmogorov asymptotic kolmogorov distance restriction operator prob prob lipschitz seen proof page lipschitz constant proposition bounded fact careful analysis provides better bound maximal number objects mapped single object slicing lemma slicing lemma proposition allows estimate kolmogorov distance two configurations integrated kolmogorov distance slices configurations obtained conditioning another probability space slicing lemma along local estimate section turned powerful tool estimation kolmogorov distance used many occasions described section reduction configuration fij single space mean collection reductions individual spaces commute reductions within fij distance alternatively whenever single probability space appears together gconfiguration commutative diagram replaced constant proposition slicing lemma suppose complete diagram category given prob four prob probability spaces included following threetents configuration minimal following estimate holds ent ent idea proof slicing lemma page follows every pair consider optimal guv coupling fans underlying configuration sets construct coupling convex combination distributions guv weighted estimates resulting imply proposition various implications slicing lemma summarized next corollary corollary let complete diagram category prob prob given configuration following inequality holds ent given fan following inequality holds ent distance iii let reduction ent holds local estimate fix complete diagram category consider gconfiguration sets set initial set discussed section space distributions could identified space distributions initial set therefore probability spaces underlying configuration sets equal correspondence interior points set interior consists fully supported measures set carries total variation distance respect convex coordinates simplex task presently compare total variation distance kolmogorov distance space configurations fixed underlying configuration sets upper bound kolmogorov distance derive two summands one linear total variation distance slope proportional second one total variation distance depend following interesting observation course summand always dominates linear one locally however cardinality becomes large linear summand starts playing main role estimate suppose given configuration sets fij set modeled complete diagram category initial set use isomorphism sends component initial space inverse given pair distributions denote total variation difference consider binary probability space weight one atoms equal distance types proposition let fij set configuration sets modeled complete diagram category initial set let two probability distributions denote ent prove local estimate decompose convex combination common part rests coupling common parts gives contribution distance worst possible estimate parts still enough get bound lemma using corollary part details proof found page fact lower bound also holds specifically given complete gconfiguration sets constant holds ent use fact therefore include proof sets fixed map prob seen discussion even though map continuous lipschitz distance types explained section given complete sets rational distribution construct homogeneous configuration called type goal section estimate kolmogorov distance two types two different distributions terms total variation distance purpose use lagging technique explained lagging trick let binary probability space let fij gij two configurations modeled complete diagram category included minimal recall left terminal vertex interpreted constant assume distribution rational denominator follows also rational denominator distance types construct lagging follows right leg induced right leg original left leg obtained erasing symbols reduce applying remaining symbols target space reduction true type lagging behind factor specifically reduction constructed follows let components reduction given define subset indexes define component equivariance reduction homogeneous spaces since inverse image point cardinality moreover reductions commute reductions explained section therefore reduction configurations next lemma uses lagging estimate kolmogorov distance terminal configurations lemma let prob two configurations modeled complete diagram category included minimal distribution rational denominator ent ent lemma proposition closely related local estimate proposition immediate consequence slicing lemma particular corollary part ent tacit ingredient proof local estimate subadditivity kolmogorov distance ent distance types bound almost estimate lemma except lemma estimates distance types rather tensor powers soon see tensor powers types close kolmogorov distance however purpose proof lemma suffices know entropies close estimate provided corollary proof lemma use lagging constructed equation namely coupling estimate kolmogorov distance recall corollary probability space rational distribution ent ent ent thus estimate follows ent ent ent ent ent ent ent ent minimality original shannon inequality bound ent ent ent second part sum estimated using relation follows ent ent ent ent ent ent ent ent ent ent combining obtain estimate conclusion lemma distance types section use lagging trick described estimate distance types two different distributions complete configuration sets distance types proposition suppose complete sets initial set suppose let ent ent local estimate idea proof write convex combination common distribution small amounts respectively use lagging trick estimate distances types well types present details proof proof proposition recall complete configuration initial set goal write convex combination three distributions could following way let proposition follows trivially constructing fan assume define three probability distributions setting every min denote distributions corresponding affine isomorphism thus construct pair aep configurations setting fij fij reductions given coordinate projections following isomorphisms estimate distance types apply lemma fans ent reason similarity local estimate distance estimate types become clear next section establish asymptotic equivalence bernoulli sequence probability spaces sequence types rational distributions approximating true distribution asymptotic equipartition property configurations prove bernoulli sequence approximated sequence homogeneous configurations essentially asymptotic equipartition theorem configurations theorem suppose prob complete configuration probab ility spaces exists sequence homogeneous configurations ombinatorial type aep configurations precisely sequence may chosen constant depending proof denote underlying configuration sets true distribution construct approximating homogeneous sequence taking types rational approximations converge sufficiently fast true distribution specifically select rational distributions homogeneous spaces set tpn show kolmogorov distance satisfies required estimate first apply slicing along empirical defined section equation page estimate use fact ent slicing see corollary along empirical tpn ent tpn tpn estimate integral split domain small divergence ball around true distribution complement tpn tpn tpn set radius equal estimate first integral side equality note distance two types configuration sets always crudely estimated extensions moreover sanov theorem theorem estimate empirical measure complement divergence ball used definition conclude last inequality therefore obtain tpn define side smaller set otherwise every satisfies pinsker inequality lemma triangle inequality consequently estimate distance types proposition tpn ent using definition find tpn hence combining estimates tpn precise check shows constants appearing depend worth noting type considered subspace tensor power takes small probability fact probability converges zero growing calculation shows terms probability configuration consists polynomially many types relative exponential growth sizes parts polynomially many good one difference setup used gromov extensions extensions introduction already emphasized close relationship relative entropic sets problems definitions restricted extensions full configurations corresponding three random variables generalize definitions make relationship relative entropic sets informationoptimization problems explicit prove extension lemma use show relative entropic set associated full configuration depends continuously configuration relative entropic set section introduced restriction operator prob prob follows minimal full configuration denote restriction minimal full configuration generated call minimal configuration prob configuration denote class extl recall full configuration prob record entropies probability spaces vector denote ent entries vector nonnegative simplify notations set denote dual introduction introduce unstabilized relative entropic set extl additivity property entropy respect tensor powers inclusion sum left hand side minkowski sum allows define limit lim define stabilized relative entropic set closure lim extensions closed convex subset property vector define problem ioc inf inf ent coordinates vector respect basis dual standard basis note equation implies sequence ioc subadditive hence limit lim ioc always exists may equal ioc ioc call optimization problem associated stable general define stabilized optimization problem iosc lim ioc stabilized relative entropic set convex intersection defined linear inequalities entropies iosc words stabilized information optimization problems occur often practice identify supporting convex set solution linear problems determine shape relative entropic set vice versa entropic set entropic cone definitions relative entropic sets motivated classical notion entropic cone briefly discuss entropic set defined prob minimal closure usually referred entropic cone closure indeed entropic cone closed convex cone entropic cone polyhedral completely described shannon inequalities however situation much complicated known polyhedral shape entropic cone known time writing article important open problem information theory find tight bounds entropic cone hope techniques developed article eventually lend finding useful characterization extensions fact entropic cone considered relative entropic set prob corresponds empty diagram empty configuration category diagram category let denote constant configuration spaces given prob restriction last terminal spaces induces linear isomorphism extension lemma lipschitz continuity relative entropic sets follow following important proposition refer extension lemma proposition extension lemma let let prob minimal full configurations every extl exists extl key behind proof extension lemma full configuration extends optimal coupling configuration chosen restriction full configuration generated terminal spaces estimate directly follows shannon inequalities present details page follows immediately extension lemma lipschitz property entropy function asymptotically equivalent configurations solutions problems consequently stabilized relative entropic set fact much stronger statement unstabilized stabilized relative entropic sets lipschitz dependence configuration distance sets measured hausdorff distance let endow collection subsets hausdorff metric respect two subsets define hausdorff distance inf size around origin fact point hausdorff distance extended pseudometric sense may take infinite values may vanish pairs points suppose given two minimal full configurations prob suppose point lies unstabilized relative entropic set mixtures means extension extl extension lemma exists configuration extl property entropy function point close point thus obtained following corollary extension lemma corollary let prob hausdorff distance unstabilized relative entropic sets satisfies following lipschitz estimate note particular distance unstabilized relative entropic sets always finite let denote metric space closed convex sets endowed hausdorff distance theorem let prob hausdorff distance stabilized relative entropic sets satisfies lipschitz estimate words map minimal full configurations prob finally primer section note set sequence converges hausdorff distance set viewed metric space restricting convergence expressed saying asymptotic cone equals isomorphic mixtures mixtures mixtures provide technical tools use section input data mixture operation family parametrized probability space result one obtains another conditionals one particular instance mixture one mixes two configurations latter constant probability spaces operation used substitute taking radicals section definition elementary properties let complete diagram category probability space let family parametrized mixture family reduction mix mixture exists uniquely defined property isomorphism identity denote top configuration mixture also call mixture family binary space write simply mixture configuration subindexed always first summand entropy mixture evaluated following formula mixtures satisfy distributive law respect tensor product mix mix mix tropical probability distance estimates recall diagram category deg note constant spaces mixture may serve ersatz taking radicals configuration following lemma provides justification distance estimates related mixtures used section lemma let complete diagram category prob ent ent iii ent proof found page note distance estimates lemma respect asymptotic kolmogorov distance essential since perspective intrinsic kolmogorov distance mixtures badly behaved tropical probability spaces configurations section introduce notion tropical probability spaces configurations configurations tropical probability spaces points asymptotic cone space prob limits certain divergent sequences normal configurations first give construction asymptotic cone abstract context next apply construction particular case configurations probability spaces background asymptotic cones see instance asymptotic cones metric spaces asymptotic cone captures geometry metric space abstractly asymptotic cone pointed metric space pointed limit sequence spaces obtained given one scaling metric course convergence general means assured sometimes weaker type convergence using ultrafilters considered since case asymptotic cone evaluated relatively explicitly give definition convergence convergence respect ultrafilter instead give construction would like understand asymptotic cones space configurations probability spaces considered metric space fixed complete diagram category space prob monoid operation additional property shifts maps simplifies construction analysis asymptotic cone fact see later metric already asymptotic relative tropical probability application asymptotic cone construction metric allows obtain complete metric space simple description points note even though monoid prob abelian property prob one thus metric perspective good abelian metrics versus set bivariate function satisfying axioms distance function except nonnegative definite rather positive definite function allowed vanish pairs points set equipped called space isometry spaces map point target space point image distance zero given space one could always construct isometric metric space identifying pairs points distance zero apart property formulated terms holds simultaneously space metric quotient convenient construct spaces instead passing quotient spaces asymptotic cone metric abelian monoid let monoid satisfies following properties shifts holds call monoid satisfies conditions metric abelian monoid follows property holds quadruple holds particular monoid operation respect argument direct consequence every also holds sequence define defect respect distance function sup tropical probability sequence called denote sets linear respectively sequences respect distance two elements qld define asymptotic distance lim lemma pair limit lim exists finite provide proof section page bivariate function set call two sequences asymptotically equivalent write call sequence weakly asymptotically equivalent sequence note space weakly sequences also endowed asymptotic distance isometric space sequences see later natural operations consider therefore coincide asymptotically equivalent sequences thus given weakly sequence could always replace equivalent sequence without visible effect thus take liberty omit adverb weakly whenever say sequence mean weakly sequence silently replaced asymptotically equivalent genuine sequence necessary validity following constructions easy verify omit proofs set admits action multiplicative semigroup defined following way let define action action asymptotic equivalence similarly constructions follow tacitly assuming valid asymptotic equivalence action continuous respect moreover homothety dilation tropical probability group operation induces fact group operation multiplying sequences semigroup structure distributive respect particular path called convex interpolation conditions completeness would like call asymptotic cone however clear general whether complete space simply consider metric completion call asymptotic cone feel however adds another level obscurity points circumstances however completeness space quasilinear sequences comes free subject proposition suppose metric abelian monoid additional property exists constant sequence exists asymptotically equivalent sequence defect bounded case say metric monoid uniformly bounded defect property proposition suppose metric abelian monoid distance function homogeneous uniformly bounded defect property space complete proof proposition found page tropical probability density linear sequences section shown bernoulli sequences configurations approximated sequences homogeneous configurations proposition allow extend statement wider class sequences gives sufficient condition linear sequences dense sequences proposition suppose bounded defect property every dense see page proof asymptotic metric original semigroup starting element one construct linear sequence view inequality map contraction inclusions induced metric satisfying following condition gained note moreover begin coincides iteration construction may iterate constructions may apply instead one may wonder purpose however already observed satisfies scaleinvariance condition one conditions going proof completeness proposition moreover later apply theory section particular case prob see prob show latter space bounded defect property every virtue bound sequences respect also respect since associated asymptotic distance coincides show lemma also corresponds sequences order organize statements precise let include spaces following commutative diagram tropical probability maps isometries maps isometric embeddings next lemmas show also isometric embedding dense image lemma natural inclusion isometric embedding lemma image isometric embedding dense proofs two lemmas found page tropical probability spaces configurations apply construction space complete configurations fixed combinatorial type fix complete diagram category consider space prob configurations modeled carries following structures tensor product iii entropy function prob tensor product configurations commutative metric perspective recall corollary subadditivity established namely prob holds space prob metric abelian monoid note also prob along lines section however metric moreover unclear whether metric semigroup prob uniformly bounded defect property iterate construction announced section consider space sequences instead tropical probability lemma complete diagram category every space prob bounded defect property sequence prob exists asymptotically equivalent sequence defect exceeding applying general setup previous section metric semigroups prob prob corollary lemma obtain following theorem theorem consider commutative diagram prob prob qlk prob prob prob following statements hold maps isometries maps isometric embeddings map dense image corresponding target space iii space corner prob complete may finally define space tropical space corner diagram prob prob theorem space complete entropy function prob extends linear functional prob norm one defined lim sequences homogeneous configurations dense let prob stand weakly linear sequences homogeneous configurations sequences asymptotically equivalent linear sequence necessarily homogeneous spaces sequence homogeneous spaces prob define aep sequence asymptotically equivalent tropical probability extend commutative diagram follows prob prob aep prob qlk prob prob prob asymptotic equipartition property configurations theorem map aep isometry hence following theorem theorem map aep prob prob isometric embedding dense image let prob prob denote space weakly sequences configurations prob every configuration homogeneous refer prob space homogeneous tropical configurations denote aep embedding aep prob prob theorem asymptotic equipartition theorem tropical configurations let complete diagram category map aep prob prob isometry proof need show every tropical configuration prob exists homogeneous tropical configuration prob lemma proposition every exists sequence prob prob asymptotic equipartition property configurations theorem sequences homogeneous configurations define tropical probability moreover function chosen monotonically increasing every unique define follows since divergent sequence theorem follows thus shown arrows diagram isometric embeddings dense images would like conjecture fact isometries case difference metrics small defined dense subset domain definition coincide whenever defined write hat anymore use notation metric prob tropical probability spaces tropical chains section evaluate spaces prob prob chain diagram category introduced page recall finite probability space homogeneous aut acts transitively support measure property homogeneous invariant isomorphism every homogeneous space isomorphic probability space uniform distribution homogeneous chains also simple description chain reductions homogeneous individual spaces homogeneous simple description allows evaluate explicitly kolmogorov distance spaces weakly linear sequences homogeneous chains consequently space tropical chains theorem prob prob rrrrrr rrrr rrrrrr rrrrrr side cone tropical probability prove theorem evaluate first isometry class space weakly linear sequences homogeneous spaces chains present argument single spaces since argument chains similar lemma probh note side complete metric space thus asymptotic equipartition theorem tropical configurations theorem together lemma imply theorem prove lemma need evaluate kolmogorov distance two homogeneous spaces chains homogeneous spaces subject next lemma lemma follows immediately lemma denote finite uniform probability space cardinality ent ent stochastic processes often stochastic processes naturally give rise sequences include last subsection indication statements together construction tropical cone much larger reach sequences independent random variables brief come back topic subsequent article minimal diamond configuration define conditional mutual information given ent ent ent ent shannon inequality says conditional mutual information always minimal completed diamond probability space terminal vertex mutual information defined ent ent ent technical proofs let stationary stochastic process finite state space thus jointly distributed random variables generate full configuration explained section collection full configurations consistent sense canonical isomorphisms restriction operator introduced section property stationary means canonical isomorphisms finite subset call initial space configuration space trajectories process denote xkl note stationarity every moreover side increasing function make following important observation defect sequence equal sup sup ent ent sup sup therefore sequence lim condition satisfied stochastic process defines tropical probability space prob note condition satisfied stationary markov chains technical proofs section contains proofs make main text technical proofs statements section configurations lemma let diagram category minimal constituent probability spaces minimal minimal reduction exists exists minimal included following diagram proof need following lemma lemma suppose given two probability spaces minimal let minimal reduction reduction exists reduction proof define terminal spaces coincide prove lemma need provide dashed arrow makes following diagram commutative reduction constructed simple diagram chasing using minimality suppose commutativity solid arrows diagram technical proofs similarly thus minimality follows hence constructed setting proceed prove claim lemma let mij diagram category prob three recall also considered gconfiguration fij minimizing reduction induces reductions index set follows minimal prove implication direction suppose minimal show minimal well suppose exist nonminimal fan among index let homg homg choose index minimal minimal consider minimal reduction construct gij setting otherwise fij gij otherwise reduction provided lemma applied diagram thus constructed reduction identity terminal contradicts minimality technical proofs address second assertion lemma observe argument gives algorithm construction minimal reduction statements section kolmogorov distance proposition let complete diagram category bivariate function prob prob prob moreover two configurations prob satisfy isomorphic prob proof symmetry immediate follows fact entropy target space reduction greater entropy domain particular instance shannon inequality proceed prove triangle inequality make use following lemma lemma minimal full configuration probability spaces holds proof shannon inequality page ent ent ent ent similarly ent ent ent therefore continue proof proposition let arbitrary complete reduction category suppose fij gij hij initial spaces respectively let technical proofs two optimal minimal satisfying recall individual spaces construct coupling following manner starting configuration initial spaces use adhesion extend full configuration thus constructing coupling full configuration could pushed provides full extensions lower levels thus could compose couplings use shannon inequality establish triangle inequality kolmogorov distance details follows consider configuration extend adhesion described section together reductions gives rise note minimal reductions subconfigurations respectively lemma technical proofs consider minimization configuration could also viewed configurations minimal corollary apply lemma level conclude finally follows proposition let complete diagram category respect kolmogorov distance prob tensor product prob prob variable every triple prob following bound holds proof claim follows easily additivity entropy equation suppose fij gij three optimal fan ent ent ent consider fan technical proofs additivity entropy equation ent ent ent ent ent ent therefore thus tensor product probability spaces respect argument proposition suppose complete diagram category either kolmogorov distance asymptotic kolmogorov distance prob entropy function prob fij entxi proof let prob let optimal fan components fixed index estimate difference entropies ent ent ent ent symmetry ent ent adding inequalities additivity entropy also obtain property entropy function respect asymptotic kolmogorov distance proposition suppose functor two complete diagram categories stands either kolmogorov asymptotic kolmogorov distance restriction operator prob prob lipschitz technical proofs proof claim follows functoriality restriction operator argue follows suppose functor corresponding restriction operator prob let optimal fan fan terminal vertices restrictions considered individual spaces also appears fan thus obtain rough estimate since restriction operator commutes tensor powers estimate also holds asymptotic kolmogorov distance proposition slicing lemma suppose complete diagram category given prob four prob probability spaces included following threetents configuration minimal following estimate holds ent ent proof since minimal probability space could considered underlying set subset cartesian product underlying sets pair positive weight consider optimal guv zuv zuv zuv let puv probability distributions zuv individual spaces configuration zuv next step take convex combination distributions puv weighted construct coupling technical proofs first extend configuration full gconfigurations top vertex distribution puv described section integrate obtain puv use holds puv pxi therefore pxi way pyi note exactly corresponds adhesion described section follows ent ent ent ent extended configuration contains configurations terminal vertices call initial vertex xyi fij following estimates conclude proof slicing lemma first use definitions intrinsic kolmogorov distance estimate ent xyi ent ent next apply definition conditional entropy rewrite righthand side ent xyi ent ent xyi ent ent ent ent ent ent use rearrange terms obtain ent xyi ent ent ent ent ent ent xyi ent ent technical proofs integral formula conditional entropy applied first three terms get ent xyi ent ent however simplifies therefore ent ent proposition let fij set configuration sets modeled complete diagram category initial set let two probability distributions denote ent proof need following obvious rough estimate kolmogorov distance holds obtained taking tensor product coupling goal write convex combination three distributions smallest possible could following way let proposition follows rough estimate assume define three probability distributions setting technical proofs every min denote distributions corresponding isomorphism thus construct configuration setting fij fij reductions given coordinate projections note following isomorphisms hold apply part corollary obtain desired inequality ent ent ent technical proofs statements section extensions proposition extension lemma let let prob minimal full configurations every extl exists extl proof denote initial spaces configurations respectively let initial space full generated let initial space optimal coupling recall could considered cartesian product underlying sets spaces generating distribution similar view holds thus particular define full minimal configuration prob providing distribution explained section distribution defined clear contains coupling configuration restrictions also contains minimal full configuration coupling pair spaces denote initial space minimal fan terminal spaces considered gij gij using notation estimate gij ent gij ent ent ent ent ent ent ent ent technical proofs theorem let prob hausdorff distance stabilized relative entropic sets satisfies lipschitz estimate words map minimal full configurations prob proof note corollary directly proposition hence scaling properties hausdorff distance convenience introduce notation closure closure recall definition closure closure note superadditivity property unstabilized relative tropic sets see inclusion sequences monotonically increasing sequences sets select large radius let denote ball radius around origin compactness definition stabilized relative entropic set therefore also inequality holds every estimate lemma follows technical proofs statements section mixtures lemma let complete diagram category prob ent ent iii ent proof recall empirical reduction quantity counts number black squares sequence binomially distributed random variable mean variance first claim proven following calculation lim lim ent lim ent lim ent ent lim second claim proven similarly third follows second property tensor product finally fourth follows corollary slicing arguments along statements section tropical probability lemma suppose sequence real numbers bounded constant holds limit lim technical proofs exists finite proof lemma standard sometimes refered fekete subadditive lemma include proof convenience reader assume first sequence satisfies particular let lim inf choose find let quotient reminder integer division last inequality holds sufficiently large specifically max therefore lim sequence subadditive converges previous argument thus lim lim lim lemma pair limit lim exists finite proof suppose two sequences elements thus sequence lemma limit lim exists finite proposition suppose metric abelian monoid technical proofs distance function homogeneous uniformly bounded defect property space complete proof given cauchy sequence elements need find limiting element version diagonal process define value sufficiently large depending would follow fact fixed sufficiently large set uniformly bounded give detailed argument first replace element sequence asymptotically equivalent element defect bounded constant according assumption lemma still call new sequence cauchy sequence satisfies sup assumption lemma holds dividing obtain pass limit sending infinity keeping fixed given let number holds may assume nondecreasing function following bound ready define limiting sequence setting technical proofs first verify convergence shown follows let quotient remainder division fix let lim lim lim lim since arbitrary lim proposition suppose bounded defect property every dense proof let sequence need approximate linear sequences let sequence asymptotically equivalent satisfying provided bounded defect property define sequence technical proofs lim lim lim thus lim lemma natural inclusion isometric embedding proof let two sequences sequences show two numbers lim lim equal since shifts maps follows immediately left show opposite inequality follows fix lim lim passing limit respect gives required inequality lemma image isometric embedding dense technical proofs proof given element find sequence define show follow lim lim lim lim max worth noting defect need uniformly bounded respect lemma complete diagram category every space prob bounded defect property sequence prob exists asymptotically equivalent sequence defect exceeding proof let sequence let find asymptotically equivalent sequence defect less define new sequence number chosen later first verify sequences asymptotically equivalent lim estimate asymptotic distance individual members sequences using lemma follows ent technical proofs thus two sequences asymptotically equivalent next show sequence evaluate defect using lemma let defect ent thus choosing solution inequality ent make sure lemma denote finite uniform probability space cardinality ent ent proof consider unm construct specific reductions identify unm cyclic groups corresponding order unm znm consider short exact sequences mod mod znm znm choose left splitting first exact sequence left splitting second exact sequence technical proofs constructed unm let minimal reduction estimate implies ent ent ent max prove second assertion note entropy function therefore ent ent ent ent substituting definition asymptotic kolmogorov distance obtain required equality references nihat nils bertschinger ralf der frank eckehard olbrich predictive information explorative behavior autonomous robots european physical journal samson abramsky rui soares barbosa kohei kishida raymond lal shane mansfield contextuality cohomology paradox arxiv preprint dmitri burago yuri burago sergei ivanov course metric geometry volume graduate studies mathematics american mathematical society providence john baez tobias fritz tom leinster characterization entropy terms information loss entropy nils bertschinger johannes rauh eckehard olbrich jost nihat quantifying unique information entropy imre method types ieee trans inform theory information theory thomas cover joy thomas elements information theory wiley series telecommunications john wiley sons new york publication randall dougherty chris freiling kenneth zeger information inequalities four random variables arxiv preprint karl friston principle rough guide brain trends cognitive sciences misha gromov search structure part entropy preprint available http frantisek matus infinitely many information inequalities information theory isit ieee international symposium pages ieee bastian steudel nihat inference common ancestors entropy sinai introduction ergodic theory princeton university press princeton translated scheffer mathematical notes technical proofs sander van dijk daniel polani informational organization behavior advances complex systems raymond yeung first course information theory springer science business media
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consideration publication theory practice logic programming may default rules sergio antoy computer science portland state university oregon antoy michael hanus institut informatik cau kiel kiel germany submitted january revised april accepted may abstract functional logic programs rules applicable independently textual order rule potentially used evaluate expression similar logic languages contrary functional languages haskell enforces strict sequential interpretation rules however situations convenient express alternatives means compact default rules although default rules often used functional programs nature functional logic programs allow directly transfer concept functional functional logic languages meaningful way paper propose new concept default rules curry supports programming style similar functional programming preserving core properties functional logic programming completeness use functions discuss basic concept propose implementation exploits advanced features functional logic languages appear theory practice logic programming tplp keywords functional logic programming semantics program transformation motivation functional logic languages combine important features functional logic programming single language see antoy hanus hanus recent surveys particular functional logic language curry hanus conceptually extends haskell common features logic programming nondeterminism free variables constraint solving moreover amalgamated features curry support new programming techniques like deep pattern matching use functional patterns evaluable functions pattern positions antoy hanus extended version paper presented international symposium practical aspects declarative languages padl invited rapid communication tplp authors acknowledge assistance conference program chairs marco gavanelli john reppy sergio antoy michael hanus example suppose want compute two elements list property distance two elements elements use condition program discussed later course may many pairs elements list satisfying given condition denotes concatenation lists dist length defining functions case distinction pattern matching useful feature functional patterns make feature even convenient however functional logic languages feature slightly delicate possibility functional patterns typically stand infinite number standard patterns textual order among rules defining operation variables functional pattern bound like variables ordinary patterns simple example consider operation isset intended check whether given list represents set contain duplicates curry might think implement follows isset false isset true first rule uses functional pattern returns false argument matches list two identical elements occur intent second rule return true identical elements occur argument however according semantics curry ensures completeness finding solutions values rules tried evaluate expression therefore second rule always applicable calls isset expression isset evaluated false true unintended application second rule avoided additional requirement rule applied rule applicable call rule default rule mark adding suffix default function name order avoid syntactic extension base language thus define isset rules isset false isset default true isset evaluates false isset true paper propose concept default rules curry define precise semantics discuss implementation options next section review main concepts functional logic programming curry intended concept default rules informally introduced sect examples showing convenience default rules programming presented sect order avoid introduction new semantics specific default rules define precise meaning default rules transforming already known concepts sect options implement default rules efficiently discussed evaluated default rules curry sect benchmarking alternative implementations default rules shown sect relate proposal work conclude functional logic programming curry presenting concept implementation default rules detail briefly review elements functional logic languages curry necessary understand contents paper details found recent surveys functional logic programming antoy hanus hanus language report hanus curry declarative language combining seamless way features functional logic concurrent programming concurrency irrelevant work goes hence ignored paper syntax curry close haskell peyton jones type variables names defined operations usually start lowercase letters names type data constructors start uppercase letter denotes type functions mapping elements type elements type also functional type functional types curried application operation argument denoted juxtaposition addition haskell curry allows free logic variables conditions sides rules expressions evaluated interpreter moreover patterns defining rule nonlinear might contain multiple occurrences variable abbreviation equalities occurrences example following simple program shows functional logic features curry defines operation concatenate two lists identical haskell encoding second operation dup returns list element least two dup dup free operation applications contain free variables evaluated lazily free variables demanded arguments instantiated hence condition rule defining dup solved instantiating anonymous free variables evaluation method corresponds narrowing slagle reddy curry narrows possibly unifiers ensure optimality computations antoy note curry requires explicit declaration free variables rule dup ensure checkable redundancy sergio antoy michael hanus note dup operation since might deliver one result given argument evaluation dup yields values operations interpreted mappings values sets values important feature contemporary functional logic languages hence also predefined choice operation thus expression evaluates value chosen operations defined easily directly using functional patterns antoy hanus functional pattern pattern occurring argument side rule containing defined operations data constructors variables pattern abbreviates set standard patterns functional pattern evaluated narrowing instance rewrite definition dup dup functional patterns powerful feature express arbitrary selections tree structures xml documents hanus details semantics constructive implementation functional patterns unification procedure found antoy hanus set functions antoy hanus allow encapsulation computations manner defined operation denotes corresponding set function encapsulates caused evaluating except caused evaluating arguments applied instance consider operation decorinc defined decorinc decorincs evaluates abstract representation set caused decorinc encapsulated set however decorincs evaluates two different sets due nondeterministic argument caused argument encapsulated property desirable essential define implement default rules transformational approach shown sect following section discuss default rules intended semantics default rules concept informal semantics default rules often used functional logic programming languages rules applied textual order haskell prolog loosely speaking every rule default rule preceding rules instance following standard haskell function takes two lists returns list corresponding pairs excess elements longer list discarded default rules curry zip zip zip second rule applied first rule applicable one argument lists empty avoid consideration rule orderings replacing second rule rules patterns matching first rule zip zip zip zip general coding cumbersome since number additional rules increases patterns first rule complex need three additional rules operation combining three lists moreover coding might impossible conjunction functional patterns first rule isset functional patterns conceptually denote infinite set standard patterns complement set infinite prolog one often uses cut operator implement behavior default rules instance zip defined prolog predicate follows zip zip zip although definition behaves intended instantiated lists completeness logic programming destroyed cut operator instance goal zip provable prolog compute answer goal zip examples show neither functional style logic style default rules suitable functional logic programming functional style based textual order curtails logic style based cut operator destroys completeness computations thus new concept default rules required functional logic programming want keep strong properties base language particular completeness evaluations presenting exact definition default rules introduce informally discuss intended semantics intend extend standard operation definition one default rule hence operation definition default rule following form denotes sequence objects tkn default consider conditional rules since unconditional rule regarded conditional rule condition true sergio antoy michael hanus call first rules standard rules final rule default rule informally default rule applied standard rule applicable rule applicable pattern matches condition satisfied hence expression expressions evaluated follows arguments evaluated enough determine whether standard rule applicable whether exists standard rule whose side matches evaluated condition satisfied evaluable true standard rule applicable applied otherwise default rule applied argument previous points apply independently choice combination arguments particular argument free variable instantiated every potentially applicable rule used usual language like curry arguments operation application evaluated demanded operation pattern matching condition however failure argument evaluation passed inside condition evaluation precise definition inside antoy hanus def behavior quite similar set functions encapsulate internal therefore exploit set functions implement default rules discussing advantages implementation default rules explain motivate intended semantics proposal first noted concept distinguishes outside inside rule application difference irrelevant purely functional programming essential functional logic programming example consider operation zip defined default rule zip zip zip default since standard rule applicable zip default rule ignored expression solely reduced zip since standard rule applicable zip default rule applied yields value altogether value zip however argument one value use evaluation principle combination thus call zip yields two values considerations even relevant evaluation condition might following example shows example consider operation look values keys association list lookup key assoc assoc key val val default rules curry val free lookup default nothing note condition standard rule evaluated various ways particular evaluated true false fixed association list key therefore using otherwise branch haskell instead default rule might lead unintended results evaluate lookup condition standard rule satisfiable default rule ignored since condition two solutions val val yield values evaluate lookup condition standard rule satisfiable default rule applicable obtain result nothing hand arguments might trigger different rules applied consider expression lookup since arguments leads independent evaluations expressions lookup lookup obtain results nothing similarly free variables arguments might lead independent results since free variables equivalent values antoy hanus instance expression lookup yields value binding also value nothing binding well many solutions latter desirable property also implications handling failures occurring arguments evaluated instance consider expression lookup failed failed predefined operation always fails whenever evaluated evaluation condition standard rule demands evaluation failed subsequent failure comes outside condition entire expression evaluation fails instead returning value nothing motivated fact need value association list order check satisfiability condition thus decide applicability standard rule value available example see design decision reasonable consider following contrived definition operation checks whether argument unit value value unit type isunit true isunit default false proposal evaluation isunit failed fails alternative design like prolog construct one might skip failure condition checking proceed next rule case would return value false expression isunit failed quite disturbing since deterministic operation isunit one possible input value could return two values true call isunit false call isunit failed moreover call operation free variable like isunit obtain single binding value true since free variables never bound failures thus sergio antoy michael hanus either semantics would incomplete logic computations compute many values order get consistent behavior require failures arguments demanded condition checking lead failures evaluations examples show applicability convenience default rules functional logic programming sketch examples section example default rules important combination functional patterns since functional patterns denote infinite set standard patterns often finite complement consider operation lookup introduced example functional patterns default rules operation conveniently defined lookup key key val val lookup default nothing example functional patterns also useful check deep structure arguments case default rules useful express easy manner check successful instance consider operation checks whether string contains float number without exponent optional minus sign functional patterns default rules definition predicate easy isfloat isdigit isdigit true isfloat default false example classical puzzle one must place queens chess board queen attack another queen solved computing permutation list element denotes row queen placed column check whether permutation safe placement queen attack another diagonal latter property easily expressed functional patterns default rules rule fails placement safediag abs length failed safediag default hence solution obtained computing safe permutation queens safediag permute example shows default rules convenient way express logic programming default rules curry example programming pattern also applied solve map coloring problem map consists states pacific northwest list adjacent states data state adjacent furthermore define available colors operation associates nondeterministically color state data color red green blue color red green blue map coloring computed operation solve takes information potential colorings adjacent states arguments compute correct colorings evaluating initial expression solve map color adjacent operation solve fails coloring two states identical color adjacent otherwise returns coloring solve failed solve default note compact definition standard rule solve exploits ordering definition adjacent arbitrarily ordered adjacency lists extend standard rule follows solve failed transformational semantics order define precise semantics default rules one could extend existing logic foundation functional logic programming include meaning default rules approach partially done without considering different sources inside outside important intended semantics discussed sect fortunately semantic aspects issues already discussed context encapsulated search antoy hanus christiansen put proposal foundations hence develop new logic foundation functional logic programming default rules provide transformational semantics specify meaning default rules transformation existing constructs functional logic programming start description transformational approach explaining translation default rule zip default rule applied standard rule applicable rule pattern match argument rule condition satisfiable hence translate default rule regular sergio antoy michael hanus rule adding condition rule applicable purpose generate original standard rules set test applicability rules side replaced constant unit value thus single standard rule zip produces following new rule zip test add default rule condition zip test applicable since interested failure attempts apply zip test actual argument check application value furthermore failures evaluation actual arguments must distinguished similar outcomes caused evaluation condition requirements call encapsulation search values zip test inside outside distinguished handled differently fortunately set functions antoy hanus sketched sect provide appropriate solution problem since set functions strategyindependent denotational semantics christiansen use specify implement default rules using set functions one could translate default rule zip isempty zip test hence rule applied attempts apply standard rule fail complete example add translated default rule alternative standard rule obtain transformed program zip test zip zip zip isempty zip test thanks logic features curry one also use definition generate appropriate argument values zip instance evaluate equation zip curry implementation search space finite computes among others solution unfortunately scheme yield best code ensure optimal computations understand potential problem consider following operation intuitively best strategy evaluate call starts case distinction second argument since value determines rule apply value case strategy checks first argument since value determines whether apply first rule formal characterization operations allow strategy antoy discussion strategy presented sect example pattern matching strategy follows evaluate second argument head normal form value apply second rule value evaluate first argument try apply first rule default rules curry otherwise rule applicable particular loop denotes operation call loop evaluates contrast haskell peyton jones performs pattern matching left right haskell loops call strategy optimal class programs referred inductively sequential antoy intended extended functional logic computations needed narrowing antoy overlapping rules antoy order cover general functional logic programs consider following default rule default apply transformation scheme sketched obtain following curry program test test isempty test result definition longer inductively sequential since sides first third rule overlap since argument demanded rules rules could applied independently fact curry implementation loops call loop since tries evaluate first argument order apply first rule whereas yields result without default rule avoid undesirable behavior adding default rules could try use strategy standard rules test default rule done translating original standard rules auxiliary operation redefining original operation one either applies standard rules default rules example transform definition default rule following functions test test init init dflt isempty test init dflt test init inductively sequential optimal needed narrowing strategy applied simply denotes choice without argument evaluation two expressions evaluated optimally observe one expressions reducible result curry implementation evaluates loop run loop overall transformation default rules described following sergio antoy michael hanus scheme simplicity advantageous obtain comprehensible definition semantics default rules operation definition tkn default transformed test init dflt new operation identifiers test test tkn init init tkn dflt isempty test init dflt note patterns conditions original rules changed hence transformation also compatible advanced features curry like functional patterns patterns patterns local declarations etc furthermore efficient strategy exists original standard rules strategy applied presence default rules property formally stated follows proposition let program without default rules program except default rules added operations overlapping inductively sequential proof let operation interesting case default rule operation produces four different operations dflt init test first two overlapping inductively sequential since defined single rule last two overlapping inductively sequential overlapping inductively sequential since definitional tree modulo renaming symbols proposition could tightened little operation case three four operations produced transformation well prop important efficiency computations overlapping inductively sequential systems needed redexes exist easily efficiently computed antoy original system strategy reduces needed redexes transformed system strategy reduces default rules curry needed redexes ensures optimal computations preserved transformation regardless result contrast haskell prolog concept default rules based sequential testing rules might inhibit optimal evaluation prevent limit hence concept default rules powerful existing concepts functional logic programming see also sect relate values computed original system computed transformed system vice versa expected extending operation default rule preserves values computed without default rule proposition let program without default rules program except default rules added operations expression evaluates value evaluates proof let expression step evaluation interesting case default rule definitions init init trivial induction length evaluation completes proof converse prop hold typically computes values reason default rules following statement relates values computed values computed proposition let program without default rules program except default rules added operations expression evaluates value either evaluates default rule applied proof let denote evaluation never applies default rules operation steps two kinds init init expressions remove steps kind replace init obtain evaluation curry design textual order rules irrelevant default rule constructive alternative certain kind failure reasons single default rule opposed multiple default rules without order conceptually simpler adequate practical situations nevertheless default rule operation may invoke auxiliary operation multiple ordinary rules thus producing behavior multiple default rules sergio antoy michael hanus implementation implementation default rules curry based transformational approach available preprocessor preprocessor integrated compilation chain curry systems pakcs hanus future version curry one could also add specific syntax default rules transform front end curry system transformation scheme shown previous section mainly intended specify precise meaning default rules similarly specification meaning guards haskell peyton jones although transformation scheme leads reasonably efficient implementation actual implementation improved various ways following present two approaches improve implementation default rules avoiding duplicated condition checking transformation scheme default rules generates set standard rules auxiliary operations test init test used condition translated default rule check applicability standard rule whereas init actually applies standard rule since alternatives standard rules default rule eventually tried application pattern matching condition checking standard rule might duplicated instance standard rule applicable call call matches pattern default rule might tried twice standard rule applied init pattern condition tested test order test emptiness set results although amount duplicated work difficult assess accurately due curry lazy evaluation strategy check condition dflt suffices compute one element set risk operationally complex conditions patterns functional patterns kind duplicated work avoided sophisticated transformation scheme common parts definitions test init joined single operation operation first tests application standard rule case successful test returns continuation proceed corresponding rule instance consider rules zip presented example operations zip test zip init generated first transformation scheme joined single operation zip testc following transformation zip testc zip zip dflt zip let zip testc isempty zip dflt else choosevalue standard rule translated rule new operation zip testc rule side encapsulated lambda abstraction avoid default rules curry immediate evaluation rule applied actual implementation zip first checks whether set lambda abstractions empty case standard rule applicable default rule applied otherwise continue sides applicable standard rules collected lambda abstractions set general transformation scheme obtain behavior defined follows operation definition form tkn default transformed testc testc tkn dflt let testc isempty dflt else choosevalue obviously modified scheme avoids potentially duplicated condition checking standard rules sophisticated since requires handling sets continuations depending implementation set functions might impossible values operations results computed set functions actually sets scheme applied since sets require equality operation elements order eliminate duplicated elements fortunately scheme applicable pakcs hanus computes results set functions require equality elements thus compare run times schemes operations shown contain complex applicability conditions functional patterns benchmarks executed linux machine debian jessie intel core processor memory figure shows run times seconds evaluate operations schemes benchmarks indicate new scheme might yield reasonable performance gain although clearly depends particular example alternative transformation scheme discussed following section operation choosevalue chooses value given set sergio antoy michael hanus system pakcs hanus operation arguments isset isset lookup lookup queens sect sect fig performance comparison different transformation schemes transforming default rules standard rules situations behavior default rule provided set standard rules almost universally standard rules efficient example situation provided operation zip example operation defined default rule definition using standard rules shown beginning sect relations two definitions identical section introduce concepts describe obtain sufficient conditions set standard rules behave default rule programs considered section donnell extension functional patterns discussed later thus disjoint sets operation symbols denoted constructor symbols denoted pattern expression form operation symbol arity expression consisting variables constructor symbols linear repeated occurrences variable pattern pattern operation pattern ground contain variable program rule form side pattern extension conditional rules discussed later given redex step called contractum although curry allows patterns convenience programmer transformed linear ones simple syntactic transformation following first consider specific class programs called inductively sequential rules operation organized definitional tree antoy definition definitional tree symbols rule exempt branch appearing uninterpreted functions classifying nodes tree partial definitional tree pattern either rule node rule exempt node exempt branch node branch variable also called inductive variable set constructors type substitution maps xai xai fresh variables arity partial definitional tree pattern definitional tree operation finite partial definitional tree pattern default rules curry arity pairwise different variables contains rules defining variable renaming case call inductively sequential definitional trees comprehensible graphical representation instance definitional tree operation defined example shown fig graphical representation pattern node shown root node branch children rule nodes inductive variable branch left operand referring def maps variable rule nodes side rule shown arrow exempt nodes marked keyword exempt shown figures sss fig definitional tree operation sake completeness sketch definitional trees used evaluation strategy details found antoy discuss compute rewrite expression rooted operation general cases reduced shown step computed needed thus let expression rooted operation definitional tree traversal finds deepest node whose pattern matches node pattern exist every rule node redex reduced exempt node computation aborted value head head empty list branch node match inductive variable expression rooted operation strategy recursively seeks compute step presenting transformation state important property definitional trees definition mutually exclusive exhaustive patterns let operation symbol set patterns say patterns mutually exclusive iff ground pattern two distinct patterns match say patterns exhaustive iff ground pattern exists pattern matches lemma uniqueness let operation defined set standard rules definitional tree rules patterns leaves exhaustive mutually exclusive sergio antoy michael hanus proof let ground pattern pattern node suppose matches initially show leaf exactly one child pattern matches let inductive variable subexpression matched since ground proper subexpression rooted constructor symbol let set constructors type defining let arity appropriate def children patterns fresh variable appropriate hence exactly one patterns matches since matches iff going back proposition claim since pattern root matches induction depth exactly one leaf whose pattern matches inductive sequentiality sufficient necessary set exhaustive mutually exclusive patterns later show sequential operation exhaustive mutually exclusive patterns nevertheless inductive sequentiality supports constructive method transform default rules since every definitional tree useful define transformation first restrict set definitional trees definition minimal definitional tree definitional tree minimal iff rule node branch node tree example consider operation isempty defined single rule isempty true fig shows tree rules defining isempty right child root branch node rule node minimal tree rules defining isempty right child would exempt node isempty isempty isempty true isempty exempt isempty exempt fig definitional tree operation isempty investigate sufficient conditions equivalence operation defined default rule operation defined standard rules definition replacement default rule let operation defined set standard rules default rule pairwise different variables expression let minimal definitional tree standard rules let exempt default rules curry nodes tki pattern node substitution tki following set standard rules called replacement default rule fig shows minimal definitional tree single standard rule operation zip defined beginning sect leaf tree holds rule since leaf branch nodes definitional tree minimal according def remaining two leaves hold patterns match combinations arguments default rule would applicable patterns instantiated default rule see expression reduced rules need additional evaluation respect default rule zip exempt zip ppp ppp zip qqq zip exempt ppp ppp zip zip fig definitional tree standard rule operation zip defined sect lemma correctness let operation defined set standard rules default rule form variable appropriate expression let replacement ground pattern reduced root default rule iff reduced root rule proof proof done two steps first prove reduced iff reduced rule prove contracta two rules lemma patterns rules exhaustive mutually exclusive therefore reduced reduced rule reduced rule prove equality contracta remainder proof substitutions restricted argument variables reduced match pattern also reduced rule def form substitution consequently match since ground thus contractum rule sergio antoy michael hanus def replacement default rule constructed minimal definitional tree hypothesis minimality used proof lemma reason lemma claims property ground patterns execution program default rule may applied expression may neither pattern ground hypothesis minimality ensures case additional evaluation required replacement rule applied instead default rule fact counter intuitive since pattern default rule matches expression whereas patterns replacement rules except degenerate case set standard rules empty however default rule applicable standard rule applicable therefore expression must evaluated enough determine standard rule applicable following lemma shows evaluation right application replacement rule lemma evaluation let expression reduced default rule according transformational semantics sect let minimal definitional tree standard rules exists exempt node whose pattern matches proof first note standard rules rules test defined sect identical sides hence also minimal definitional tree rules test used check applicability default rule prove claim construct path definitional tree following invariant properties pattern unifies last node leaf exempt node establishing invariant root node definition pattern fresh distinct variables hence unifies invariant holds furthermore leaf rule node otherwise would never reduced default rule hence exempt node invariant holds maintaining invariant assume invariant holds node leaf base case must exempt node hence assume branch show invariant extended child since branch node unify child let pattern let inductive variable branch node definition xac constructor symbol arity fresh variable appropriate let match variable child satisfies invariant otherwise must rooted constructor symbol say following reasons minimal one rule nodes pattern rules instance constructor symbol position matched hence unless would impossible tell rules reduces hence would impossible say whether must reduced standard rule default rule hence child mapped xad invariant also holds default rules curry define replacement default rule set standard rules four assumptions assess significance assumptions inductive sequentiality standard rules inductively sequential mild requirement practice instance every operation curry prelude except choice operator shown sect inductively sequential sequential operations problematic evaluate efficiently following operation adapted berry prop defined rules admit definitional tree false true false true true false apply evaluation constructor normal form two three arguments necessary sufficient practical way known determine two arguments without evaluating three furthermore since evaluation argument may terminate three arguments must evaluated concurrently see antoy middeldorp general pattern assumed transformation pattern default rule general arguments operation variables choosing general pattern keeps statement lemma simple direct assumption extra evaluation arguments needed application replacement rule relax assumption modify def follows side default rule look general unifier say tki rule replacement default rule iff exists unconditional rules standard rules default rule unconditional adding condition default rule straightforward similar transformation shown sect condition default rule directly transferred replacement rule extending display def condition contrast conditions standard rules require care modest loss generality assume standard rules definitional tree leaf node conditional rule form boolean expression expression lemma proves ground pattern matched standard rule matches hence reduced root default rule iff satisfied therefore need following rule replacement default rule denotes negation condition satisfied patterns matched satisfy spirit functional logic programming evaluated example consider operation takes list colors say red green blue removes red occurrences list sergio antoy michael hanus data color red green blue remred red remred free remred default first rule applied exist satisfy condition list red green red blue two combinations thus negation condition must negate existence automatically done according transformational semantics presented sec applied single rule example replacement default rule shown remred isempty remred test remred test red constructor patterns standard rules defining operation constructor patterns curry also provides functional patterns presented sec rules defined functional patterns transformed ordinary rules antoy hanus def moving functional pattern matching condition rule hence absence functional patterns discussion intrinsic limitation since functional patterns quite expressive operations defined functional patterns often consist single program rule default rule examples shown sect instance previous operation remred defined functional pattern follows remred red remred remred default hence improved transformation scheme presented sect still useful applied combination transformation shown section benchmarking show practical advantage transformation described previous section evaluated simple operations defined typical functional programming style default rules instance boolean conjunction defined default rule true true true default false replacement default rule consists two rules transformation yields following standard rules true true true true false false false false similarly computation last element list defined default rule default rules curry last last default last final example extracts values list optional maybe values catmaybes catmaybes catmaybes catmaybes default catmaybes introduction default rules order evaluation may become arbitrary even though needed steps executed example first definition operation arguments must evaluated order application standard rule evaluation one argument terminate one evaluates false order two arguments evaluated becomes observable situation directly related presence default rule two natural inductive definitions operation one evaluates first argument first second definition another evaluates second argument first single standard rule say two definitions intended default rule operation replaced set standard rules per sec resulting definition inductively sequential explicitly arbitrarily encode two arguments evaluated first discussed earlier functional logic computations execute narrowing steps steps variable expression instantiated rule reducing expression depends instantiation variable example consider operation simplicity evaluation true free variable narrows true apply standard rule narrows false apply default rule narrowing step variable instantiated unification expression evaluated side rule work default rule since arguments side variables particular transformational semantics rule unify false obtain intended behavior narrowing steps variables instantiated generators antoy hanus example discussed boolean generator true false figure shows run times seconds evaluate operations discussed section different transformation schemes scheme sect replacement default rules presented section different curry implementations call size denotes number calls lengths input lists examples benchmarks executed machine benchmarks sect results clearly indicate advantage replacing default rules standard rules particular pakcs less sophisticated implementation set functions related work section compare proposal default rules curry existing proposals languages sergio antoy michael hanus system operation call size pakcs hanus zip last catmaybes sect sect system zip last catmaybes sect sect operation call size fig performance comparison different schemes different compilers operations discussed section functional programming language haskell peyton jones explicit concept default rules since haskell applies rules defining function sequentially top bottom common practice haskell write catch rule final rule avoid writing several nearly identical rules see example zip beginning sect thus proposal default rules increases similarities curry haskell however approach general since also supports computations powerful since ensures optimal evaluation inductively sequential standard rules contrast haskell shown sect since haskell applies rules sequential manner also possible define one default rule function rule different specificity directly expressed default rules one default rule allowed however one obtain behavior introducing sequence auxiliary operations operation one default rule logic programming language prolog deransart based backtracking rules defining predicate sequentially applied similarly haskell one also define catch rules final rules predicate definitions order avoid unintended application rules one put cut operators preceding standard rules already discussed sect cuts meaningful instantiated arguments otherwise completeness logic programming might destroyed hence kind default rules used predicate called particular mode contrast approach completeness arbitrary modes might require addition concepts curry prolog like instantiation free variables various encapsulation operators proposed functional logic programs default rules curry encapsulate computations data structure set functions antoy hanus proposed strategyindependent notion encapsulating deal interactions laziness encapsulation see details one also use set functions distinguish successful computations similarly logic programming exploiting possibility check result sets emptiness encapsulated computations nested performed lazily turns one track encapsulation level order obtain intended results discussed christiansen thus surprising set functions related operators fit quite well proposal actually many explicit uses set functions functional logic programming implement implicitly tersely encoded concept default rules shown examples default rules also explored functional logic programs works operator fails introduced check whether every reduction expression form successful proposes use operator define default rules functional logic programming however authors propose scheme default rule applied standard rule able compute head normal form quite unusual contrast functional programming proposal default rules applied pattern matching conditions standard rules fail computations rules sides taken account decide whether default rule applied applies early proposal default rules eager functional logic language since treatment different sources interaction explored time nested computations failures considered works consequence operator fails might yield unintended results used nested expressions instance use fails instead set functions implement operation isunit defined example evaluation isunit failed yields value false contrast intended semantics finally proposed antoy hanus change curry rule selection strategy sequential one however turned change drawbacks evaluation strategy since formerly optimal reductions longer possible particular cases instance consider operation defined sect call loop sequential rule selection strategy one starts testing whether first rule applicable since arguments demanded rule one might evaluate left right done implementation antoy hanus evaluation terminate problem avoided proposal returns even presence default rule moreover examples presented antoy hanus expressed default rules similar way sergio antoy michael hanus conclusions proposed new concept default rules curry default rules available many languages sensible inclusion functional logic language demanding therefore used advanced features encapsulating search define implement default rules thanks approach typical logic programming features like evaluating operations unknown arguments still applicable new semantics distinguishes approach similar concepts logic programming simply cut alternatives approach lead elegant comprehensible declarative programs shown several examples paper moreover many uses often implemented functional logic programs complex applications encapsulation operators easily expressed default rules since encapsulated search costly simple pattern matching also shown opportunities implement default rules efficiently particular standard rules inductively sequential unconditional one replace default rules set standard rules usage encapsulated search completely avoided acknowledgments authors grateful sandra dylus anonymous reviewers suggestions improve previous version paper material based part upon work supported national science foundation grant references antoy definitional trees proc international conference algebraic logic programming springer lncs antoy optimal functional logic computations int conf algebraic logic programming alp vol springer lncs southampton antoy echahed hanus needed narrowing strategy journal acm antoy hanus declarative programming function patterns proceedings international symposium program synthesis transformation lopstr springer lncs antoy hanus overlapping rules logic variables functional logic programs proceedings international conference logic programming iclp springer lncs antoy hanus set functions functional logic programming proceedings acm sigplan international conference principles practice declarative programming ppdp acm press antoy hanus functional logic programming communications acm antoy hanus curry without success proc international workshop functional constraint logic programming wflp ceur workshop proceedings vol default rules curry antoy middeldorp sequential strategy theoretical computer science berry computation recursive programs hanus huch encapsulating functional logic computations journal functional logic programming hanus reck new compiler curry haskell proc international workshop functional constraint logic programming wflp springer lncs christiansen hanus reck seidel semantics weakly encapsulated search functional logic programs proc international symposium principle practice declarative programming ppdp acm press deransart cervoni prolog standard reference manual springer approach declarative programming based rewriting logic journal logic programming hanus declarative processing semistructured web data technical communications international conference logic programming vol leibniz international proceedings informatics lipics hanus functional logic programming theory curry programming logics essays memory harald ganzinger springer lncs hanus antoy engelke koj niederau sadre steiner pakcs portland aachen kiel curry system available http hanus curry integrated functional logic language vers available http proof theoretic approach failure functional logic programming theory practice logic programming default rules extension constructive negation languages proc eleventh international conference logic programming mit press donnell computing systems described equations springer lncs peyton jones haskell language revised report cambridge university press reddy narrowing operational semantics functional languages proc ieee internat symposium logic programming boston constructive failure programming theory implementation journal universal computer science slagle automated theories simplifiers commutativity associativity journal acm
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sep conjugacy search problem conjecture dmitry panteleev alexander ushakov abstract develop new computational methods studying potential counterexamples conjecture particular examples devise number algorithms attempt disprove interesting counterexample improve metric properties search space set balanced presentations introduce new transformation called acmmove generalizes original transformations discuss details practical implementation reduce growth search space introduce strong equivalence relation balanced presentations study space modulo automorphisms underlying free group finally prove applied unfortunately despite lot effort unable trivialize keywords conjecture presentations trivial group conjugacy search problem computations mathematics subject classification introduction conjecture acc open problem topology combinatorial group theory proposed andrews curtis categorizing possible counterexamples conjecture later wright formulated equivalent conjecture associated finitely presented groups thus showing zeeman conjecture implies known zemman conjecture also implies conjecture implied cases despite recent progress solution conjecture validity remains open although motivating examples come topology conjecture usually formulated language combinatorial group theory question equivalence presentations trivial group paper use language combinatorial group theory omitting topological aspects problem date september second author partially supported nsf grant panteleev ushakov balanced presentations trivial group let free group finite subset normal closure denoted ncl smallest normal subgroup containing pair defines quotient group ncl denoted called presentation sum called total length presentation denoted say symmetrized contains cyclically reduced words closed taking inverses cyclic permutations denote minimal symmetrized set containing words cyclically reduced presentation symmetrized finite presentation efficiently symmetrized symmetrization change computational properties fundamental problems say group presentation balanced group presentations define trivial group trivial presentation generators course called canonical presentation define set fnn balanced trivial group ncl use vector notation tuples fnn problem deciding defines trivial group undecidable see open problem true balanced presentations see magnus problem problem transformations group presentations several types transformations general group presentation produce new presentation set generators group transformations simply type transformations recognized nielsen transformations tuple transformation conjugation element tuple since invertible say write exists sequence transforming generally transformation named replaces single element element satisfying produces isomorphic presentation easy see particular types also recognized slightly generalized easy see conjecture transformed done therefore check one use yet another transformation group presentation change group automorphism move application aut every component known system equivalent system section conjecture denote set tuples obtained canonical tuple sequence generally denote set tuples conjecture states every balanced presentation trivial group converted canonical presentation sequence despite nearly years research conjecture still open widely believed andrewscurtis conjecture false theoretic works attempting disprove common approach fix group homomorphism investigate exists sequence taking clearly answer negative choice original conjecture hold several classes groups investigated way solvable groups finite groups grigorchuk group negative answer found potential counterexamples big obstacle towards solution problem algorithm test particular balanced presentation trivial group satisfies conjecture number particular balanced presentations known satisfy conjecture examples xyx yxy examples exponent sum neumann example examples referred potential counterexamples acc examples balanced presentations known canonical presentation found shown means computer experiment counterexamples total length less later shown every balanced presentation total length either canonical presentation xyx yxyi makes shortest potential counterexample panteleev ushakov computational approach disproving counterexample check given tuple canonical presentation one enumerate equivalent presentations applying canonical presentation found see several general computational problems associated approach would like mention infinite terminating condition allows enumeration procedure stop negative answer enumeration procedure terminate positive answer finds canonical presentation lengths tuples unbounded exponential growth alleviate problems one bound lengths words tuples constant process tuple acequivalent given contains cyclic word length greater approach used allows use fixed memory slots words makes search space finite also good heuristic process shorter tuples first paper consider case use compact memory representation balanced pairs represent letter number thus packing letters approach saves memory allows implement operations cyclic shift processor instructions compared usual approach includes several memory writes work paper develop new efficient techniques enhance algorithmic search work similar previous computational investigations acc goes much presentation main object study algorithms tested big goal prove counterexample satisfies acc unfortunately able achieve goal list key features work section show used practice notice requires enumerating short conjugates one relator group given element later problem efficient solution even known conjugacy problem decidable one relator groups based techniques described design heuristic procedure enumerating short conjugates discuss details implementation prove section used presentations regardless whether conjecture holds section introduce equivalence relation pairs define normal forms equivalence classes show practice equivalence two pairs checked normal forms conjecture computed allows work quotient space elements infinite equivalence classes working work large blocks elements thus say space much smaller even though sets infinite countable section use heuristics investigate certain properties groups described could reason unsuccessful search trivialization section present results experiments section describe implementation algorithm given pair constructs subset set fixed parameter value ideally algorithm construct whole set algorithm based weighted xdigraphs weighted formally weighted tuple defines directed graph labeling function weight function often use following notation edge origin terminus label weight say edge inverse denote say weighted xdigraph folded every exists one edge origin labeled inverse every edge graph contains rooted comes designed vertex called root path sequence adjacent edges label weight circuit path origin terminus inverse weighted labeled digraph root number called pseudo conjugacy graph following conditions satisfied circuit simplest nontrivial example pseudo conjugacy graph loop shown fig implement generate sufficiently large graph harvest circuits weight large pseudoconjugacy graph generated starting loop applying panteleev ushakov figure pseudo conjugacy graph loop procedure times variation coset enumeration first described reviewed precise complexity bounds harvest shortly discussed section operations weighted shortly describe several operations graphs used later sequel given folded distinct define graph obtained follows add new vertex edge add edge edge add edge remove general graph folded define graph shift obtained changing weights edges incident follows weight increased weight decreased easy see preserves weights circuits hence preserves property graph number arithmetic operations required operation clearly bounded number edges incident folding folded contains two edges consider several cases mod remove one edges mod replace modulus number gcd remove one edges apply shift shift achieve identify remove conjecture straightforward check described operation produces pseudoconjugacy graph graph see detail since folding decreases number edges sequence folds eventually stops folded graph final result folding unique shifts weights denote fold recall finite set called symmetrized every contains cyclic permutations complete given weighted symmetrized relators means add circuit labeled weight every easy check graph result graph well clearly requires linear time given graph general result folded complexity weighted folding let weighted follows description folding fold forget weight function result folding difference weighted folding folding weightprocessing applications shift shift section idea modify procedure take weights account folding recall folding done nearly linear time function see sense folding describes following equivalence relation exists path results graph effectively represent sets identified vertices equivalence classes one use compressed tree representation vertex contains pointer parent root points vertex parent always belong equivalence class thus tree represents equivalence class presentation allows compare merge two classes efficiently results complexity bound weighted folding achieve similar complexity bound weighted folding need take account shifts weight since middle folding process work equivalence classes section requires shifting weights whole class avoid shifting weights many times vertex keep number called shift value defines total shift vertex hence perform identifying case folding procedure section set instead panteleev ushakov shif immediately set comparison merge two vertex equivalence classes easily extended tree presentations shifts therefore following proposition holds proposition number additions performed folding procedure would like point values weight function grow exponentially fast linearly binary nevertheless experiments never encountered values greater corollary let symmetrized presentation total number additions required apply weighted graph harvest describe procedure weighted folded xdigraph finds circuits weight length since number circuits expected grow exponentially one expect efficient implementation nevertheless certain heuristics allow speed enumeration significantly every vertex find reduced paths length distribute bins ends consider pairs compatible bins set circuits constructed finally vertex removed process applied another vertex note may skip vertices adjacent edges weight number operations bounded much smaller practice implementation efficiency implementation described constructs subset set conjugates given bounded length general proper subset result depends value number completion steps used construct graphs denote section shortly prove define parameter called depth conjugate responsible complexity set let boundary closure let finite connected planar set vertices set conjecture edges let set cells connected simply connected bounded components unbounded component called outer cell denoted cout edge free belong denote label boundary cell traversed counterclockwise direction starting vertex makes closed path giving word called boundary label depending starting vertex get cyclic permutation word rest subsection let finite connected planar base vertex graph van kampen diagram every boundary label boundary label read starting counterclockwise direction note need also specify first edge read starting boundary position important considerations omit issue let exclude one cells call inner cell cin denote vout pick vertex vin call annular schupp diagram see two boundary labels cin cout read counterclockwise direction vin vout correspondingly called inner outer labels exists annular diagram measure diagram complexity using notion depth introduced van kampen annular diagram define dual graph undirected graph cout annular diagrams add cin denote graph distance depth generalized van kampen diagram defined max cout depth annular diagram max min cout cin similar notion diagram radii see define conjugate depth two words min annular diagram otherwise next theorem shows relation complexity conjugacy search problem conjugacy depth theorem theorem exists algorithm given finite symmetrized presentation words panteleev ushakov terminates affirmative answer furthermore complexity bounded purposes useful define another characteristic annular diagrams inner conjugacy depth max cin conjugacy depth min annular diagram otherwise theorem assume conjugate number applied loop implementation applied produces pair proof proof theorem easily follows corollary general iterations rcompletion procedure require exponential time fortunately experiments observed value sufficient application two produce additional conjugates change highlighted figures table section nielsen automorphisms section discuss namely applications automorphism aut known adding transformations results equivalent system transformations even presentations nevertheless following true lemma proposition iii lemma transformed using automorphisms transformed using associate end defined way treat monoid lemma naturally acts lemma assume conjecture proof clearly sufficient prove result case obtained single may assume hence lemma immediately implies lemma since every aut lemma let end following holds proof may assume hence associate monoid endac end usual composition lemma implies endac endac whenever aut endac endac acc holds prove aut endac every examples lemmas show aut proofs obtained using computer program brevity use lemma aut defined proof pair xyxy yxyxy modified follows xyxy xyxy xyx xyxy appendix provides detail used lemma aut defined panteleev ushakov proof pair xyxy xyxy modified follows xyxy xyxyxy xyxyxy xyxy xyxyxy xyxy xyxyxy xyc xyxy xyxy xyxyxy xyxy xyxyxy xyxyxy xyxy xyxy xyxyxy xyxy xyyxy xyxy xyyxy xyxy xyyxy xyxy xyxy xyxy xyxy xyxy lemma aut defined proof pair xyxy xyxy xxy modified follows xyxy xxy xyxy xxyxy xxyxy yxy xxyxy yxy xyxy yxy yxy xyx yxy yxy yxy xyxy yxy xyxy proposition aut proof automorphisms considered lemmas generate aut hence proposition holds aut next corollary implies adding increase orbits presentations conjecture corollary every aut natural raise question similar result holds balanced presentations performed experiments several randomly generated presentations results always positive presentations unable prove automorphic images conjecture true every aut note conjecture immediately implies negative answer acc table reader find particular balanced presentations suspected satisfy conjecture canonical forms presentations given relation tuple search space infinite computer procedure exhaust elements reduce search space one introduce equivalence relation fnn define efficiently computable representatives equivalence classes study quotient space way one achieve compression search space single element representing infinite equivalence class clearly coarser relations fnn give better compression consider two equivalence relations results hold first one referred cyclic relation used casson series unpublished work according followed casson second relation new significantly stronger cyclic relation let transitive closure following pairs arbitrary words call cyclic relation define canonical representatives cyclic relation lowing fix order generators say denote corresponding shortlex order corresponding lexicographic order let easy see taking least cyclic permutation least cyclic permutation sorting obtained words produces least representative equivalence class denoted clearly normal form efficiently computable panteleev ushakov easily follows definition hence naturally defined problem approach computing cyclic normal form negates applications result component broken subcomponents become disconnected particular implementation found http take normal form pair obtained completely negates advantage using normal forms explain solve problem cyclic relation automorphisms define equivalence relation taking closure following pairs arbitrary words arbitrary automorphism aut note defined relation makes equivalent general known hence possible equivalence class contains element nevertheless following true proposition every proof follows corollary proposition allows replace original component cak factor much smaller problem taking normal form pair negates still relevant use original really help follows theorem choosing sufficiently large value parameter produce conjugate section normal form pair defined minimal pair equivalence class show normal forms computed efficiently main tool following classic result theorem whitehead theorem see proposition let cyclic words free group conjecture aut minimal among aut arepwhitehead automorphisms forp plength strict inequality unless recall section whitehead automorphisms automorphisms two types automorphisms permute letters automorphisms fixed multiplier carry elements one exactly whitehead automorphisms free group rank according whitehead theorem total length given tuple cyclic words decreased application automorphism decreased application single whitehead automorphism hence compute normal form pair following first minimize total length applying whitehead automorphisms total length decreases construct set equivalent pairs least total length applying automorphisms finally choose least cyclic normal form among pairs least length procedure described efficient except maybe second step construct set pairs least total length currently theoretical polynomial bounds size set nevertheless computations maximal size observed equivalent presentations bound average size set pairs least total length groups high dehn function one potential challenge computer enumeration techniques described bridson lishak papers use similar idea based properties following group yxy introduced baumslag satisfying inequality dehn first observed lishak constructs particular sequence balanced presentations parametrized satisfying following conditions canonical presentation panteleev ushakov number steps required obtain canonical presentation later property comes consequence inequality curious possibility reason program fails find tested words obtained experiments word attempted bound dehn function group purpose used holt package identify automatic groups automatic groups quadratic dehn functions left perhaps groups among classified type presentations presentations relation none type presentations satisfied condition presentations identified presentations clearly heuristic approach guarantee list presentations contain baumslag groups isomorphism problem groups known also unable classify remaining presentations case someone would like investigate published obtained lists https light heuristic results seems interesting computational problem classify short groups find precise upper bounds dehn functions conjecture baumslag group yxy highest dehn function among groups results described algorithms tested several known potential counterexamples attention mainly focused schupp presentations test performance compare experimental results also ran programs presentations known canonical presentation already mentioned section set bound length conjugates obtained harvest phase also set limit total length pairs notice need taking normal form described section increase length one words beyond allowed implementation experiments run machine two ghz intel xeon cpu ram enumeration presentations shown section used together applied presentations particular one use normal forms section compress array stored presentations table shows dynamics growth component cak constructed conjecture program different values cell table corresponds value value presents number pairs total length equal constructed program table cell shows number pairs total length obtained program run length bound highlighted cells increase increased took program days finish enumeration bound consuming days cpu time running time bound expected days decided proceed beyond value memory usage experiments moderate never exceeded cpu time main obstacle however notice numbers rows table stabilize least values instance conjecture number normal forms equivalent presentations total length less canonical one among old also tested program balanced presentations eliminated list potential counterexamples program trivializes almost immediately less seconds single computational core less panteleev ushakov xyx yxy yxy gordon presentation also considered type presentations analyzed several randomly generated presentations exponent sum attempted trivialize show automorphic images tasks dealt different success table contains pairs program failed prove equivalence aut defined particular could trivialize corresponding presentations table contains type presentations program proved automorphic equivalence aut able trivialize table contains trivializable presentations purpose tables provide reference future experiments xyyyxyy xxxyxyxy xyyyxyy xxxyxxyxxy xyyyxyy xxxyxyxxxy xyyyxyy xxxxyxyxxy xyyyxyy xxxyxyxxxy xyyyxyy xxxyxyxxxy xyyyxyy xxxxyxyxxy xyyyxyy xxxxyxxyxy xyyyxyy xxxyxyxxxy xyyyxyy xxxyxxyxxy xyyyxyy xxxyxyxy xyyyxyy xxxxyxyxxy xyyyxyy xxxxyxyxxy xyyyxyy xxxxyxxyxy xyyyxyy xxxxyxxyxy xyyyxyy xxxxyxxyxy xyyyxyy xxxyxyxxxy xyyyxyy xxxxyxyxxy xyyyxyy xxxxyxxyxy xyyyxyy xxxxyxyxxy xyyyxyy xxxyxyxxxy xyyyxyy xxxxyxxyxy xyyyxyy xxxxyxyxxy xyyyxyy xxxyxyxxxy xyyyxyy xxxxyxxyxy xyyyxyy xxxxyxxyxy xyyyxyy xxxxyxxyxy xyyyxyy xxxyxyxxxy xyyyxyy xxxxyxyxxy table pairs unknown equivalence aut xxxyxxy xyyyyxyyy xxxyxxy xxyyyxyy xxxyxxy xyyyxyyyy xxxyxxy xxyyxyyy xyyyxyy xxxyxyxy xyyyxyy xxxyxyxy xxxyxxy xyyyxyyyy xxxyxxy xxyyyxyy xyyyxyy xxxxyxxxy xyyyxyy xxxyyxxy xxxyxxy xxyyxyyy xyyyxyy xxxyyxxy xyyyxyy xxxyxyxy xxxyxxy xyyyyxyyy xyyyxyy xxxxyxxxy xyyyxyy xxxyxyxy table pairs equivalence aut known trivializable xyyxyyy xxyyyxyxyxyy xyyxyyy xxyxyxyyxyy xyyxyyy xyxyxyy xyyxyyy xyxyyxyyyyyy xyyxyyy xxyyyxyyxyxy xyyxyyy xxyyyxyxyxyy xyyxyyy xxyyxyxyxyy xyyxyyy xyxyyxyxyyxy xyyxyyy xxyxyxyyyxy xyyxyyy xyxyyyyyyxy xyyxyyy xyyxyyyxyyy xyyxyyy xyyyxyyxyy table trivializable presentations conjecture conclusion despite lot effort unable disprove new kurby type presentations fact numbers table rows stabilize value parameter increases suggesting class contain canonical presentation thus supporting common opinion acc hold appendix used justification section prove groups identities used lemmas acm moves every proof demonstrates used lemma xyx xyxy xyx xyx xyx xxyx xyxy used lemma xyxy xyxyxy xyxy yxy yxy yxxy xyxyxy xyxy xyxy xyxyxy yxy xyxyxy xyxy yxy xyxy yxy xyxy xyxyxy xyxy xyxy xyxy xyxy xyxyxy yxy xyx yxy yxy yyx panteleev ushakov xyxy xyyxy xyyx xyxy xyxy shift yxy xyxy xyx xyx yxy used lemma yxy xxyxy yxy yxy yxy yxy xyx yxy xyxy xyxy xyx yxy yxy xyx yxy xyx yxy xyxy yxy yxy yxy xyxy yxy yxy yxy conjecture references andrews curtis free groups handlebodies proceedings american mathematical society andrews curtis extended nielsen operations free groups amer math baumslag group whose finite factor groups cyclic australian math boone word problem proc natl acad borovik lubotzky myasnikov finitary conjecture infinite groups geometric combinatorial dynamical aspects volume progress mathematics pages birkhuser basel bowman mccaul fast searching trivializations experiment brady riley short geometry word problem finitely generated groups advanced courses mathematics crm barcelona birkhauser bridson complexity balanced presentations conjecture arxiv preprint burns macedonska balanced presentations trivial group bull london math gersten riley filling length finitely presentable groups geometriae dedicata gersten dehn functions finite presentations algorithms classification combinatorial group theory pages berlin springer gillman rolfsen zeeman conjecture standard spines equivalent conjecture topology havas ramsay search conjecture int algebra holt version downloaded http december ivanov balanced presentations trivial group invent lishak balanced finite presentations trivial group preprint available http lyndon schupp combinatorial group theory springer mazurov khukhro unsolved problems group theory kourovka notebook available http miasnikov genetic algorithms conjecture international journal algebra computation morar ushakov search problems groups branching processes int algebra myasnikov extended nielsen transformations trivial group mat zametki myasnikov shpilrain ushakov cryptography complexity problems chapter mathematical surveys monographs american mathematical society myropolska nielsen equivalence relations infinite groups preprint available http novikov algorithmic unsolvability word problem group theory proc steklov schupp dehn algorithm conjugacy problem mathematische annalen panteleev ushakov tarjan van leeuwen analysis set union algorithms acm march touikan fast algorithm stallings folding process international journal algebra computation ushakov fundamental search problems groups phd thesis center wright group presentations formal deformations transactions american mathematical society zeeman dunce hat topology department mathematics stevens institute technology hoboken usa address aushakov
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normalized volume singularity lower semicontinuous feb harold blum yuchen liu abstract show flat family klt singularities normalized volumes lower semicontinuous respect zariski topology quick consequence smooth points largest normalized volume among klt singularities using alternative characterization developed liu show generic empty condition flat family log fano pairs introduction given complex klt singularity chi introduced normalized volume function space valx real valuations centered precisely valuation normalized volume defined vol log discrepancy respect vol according vol volume according define normalized volume klt singularity min vol vol shown recently also denote existence minimizer vol vol vol normalized volume klt singularity carries interesting information geometry topology shown second author vol equality holds supp smooth see theorem theorem local algebraic fundamental group klt singularity always finite moreover assuming conjectural finite degree formula normalized volumes conjecture size bounded see remark limit einstein fano manifolds showed vol volume density closed point see background materials article shown normalized volume singularity lower semicontinuous families theorem let together section flat family complex klt singularities normal variety function vol lower semicontinuous respect zariski topology date february harold blum yuchen liu one quick consequence theorem smooth points largest normalized volumes among klt singularities see theorem theorem another natural consequence limit fano manifolds volume density function lower semicontinuous zariski topology stronger lower semicontinuous euclidean topology mentioned see corollary also state following natural conjecture constructibility normalized volumes klt singularities see also conjecture conjecture let together section flat family complex klt singularities normal variety function vol constructible verifying zariski openness important step construction algebraic moduli space varieties smooth family fano manifolds odaka donaldson showed locus fibers admitting metrics equivalently discrete automorphism groups zariski open generalized wang proved zariski openness flat families smoothable varieties construction proper moduli space smoothable varieties see related results common feature analytic methods used essentially proving results using alternative characterization affine cone construction developed second author apply theorem prove following result weak openness unlike results described previous paragraph result proved using purely algebraic methods hence applied families fibers generally families log fano pairs theorem let flat family complex log fano pairs normal base log closed point following statements hold exists intersection countably many zariski open neighborhoods log closed point particular log general closed point denote generic point geometric generic fiber log assume conjecture true chosen genuine zariski open neighborhood following corollary generalizes theorem follows easily theorem corollary suppose complex log fano pair specially degenerates log ksemistable log fano pair also log strategy prove theorem study invariants ideals instead invariants valuations liu characterization normalized volume normalized multiplicities normalized volume singularity lower semicontinuous ideals see theorem theorem know vol inf lct infimum taken ideals oxt cosupported ideals parametrized relative hilbert scheme countably many components clearly lct lower semicontinuous hilbert scheme may upper semicontinuous thus unclear semicontinuity properties lct may fix issue introduce normalized colength singularities taking infimum lct ideals satisfying normalized colength function behaves better families since colength function always locally constant hilbert scheme lct constructibly lower semicontinuous hilbert scheme thus properness hilbert schemes implies constructibly lower semicontinuous prove key equality asymptotic normalized colength normalized volume vol small see theorem using local bodies following see lemma convex geometry see appendix establishing uniform approximation volumes colengths see theorem generalizing izumi properness estimates families see theorems show normalized colengh functions uniformly approximate normalized volume function see theorem putting ingredients together get proof theorem paper organized follows section give preliminaries including notations normalized volumes singularities flat families klt pairs section introduce concept normalized colengths singularities show theorem normalized volume klt singularity asymptotic normalized colength proof theorem uses comparison colengths multiplicities established lemma section study normalized volumes normalized colength algebraically closed field extensions section establish uniform approximation volume valuation colengths valuation ideals section generalize izumi properness estimates families results sections enable prove uniform approximation normalized volumes normalized colengths families see section proofs main theorems presented section give applications main theorems section theorem generalizes inequality part theorem show volume density function limit manifolds lower semicontinuous zariski topology see corollary give effective upper bound degree finite maps klt singularities limits fano manifolds see theorem appendix provide certain convex geometric results lattice points counting needed proving lemma appendix provide results constructbility funtions needed proving uniform approximation results section harold blum yuchen liu acknowledgements first author would like thank advisor mircea numerous useful discussions constant support addition would like thank mattias jonsson ilya smirnov tommaso fernex many useful conversations second author would like thank chi chenyang fruitful discussions wish thank linquan sam payne xiaowei wang ziquan zhuang helpful comments also grateful ruixiang zhang help proposition preliminaries notations paper varieties assumed irreducible reduced defined necessarily algebraically closed field characteristic variety denote residue field point given morphism varieties write spec scheme theoretic fiber also denote geometric fiber spec suppose variety point field extension denote spec let normal variety effective say kawamata log terminal klt pair coefficients log resolution klt pair called log fano pair addition proper ample klt pair together closed point called klt singularity let klt pair ideal sheaf define log canonical threshold respect lct inf orde orde infimum taken prime divisors log resolution often use notation lct abbreviate lct klt pair specified single closed point define multiplicity lim dim denotes length valuations let variety defined field closed point valuation function field mean valuation trivial convention set valuation center maximal ideal write valx set valuations center valuation valx associated valuation ideal defined locally note ideal valx set min normalized volume singularity lower semicontinuous normalized volumes singularities let algebraically closed field characteristic klt singularity valx recall introduced normalized volume function vol valx vol vol vol denote log discrepancy volume defined volume given vol lim sup log discrepancy denoted defined case klt pairs normalized volume singularity given vol lim vol inf vol uncountable infimum minimum following characterization normalized volumes using log canonical thresholds multiplicities ideals crucial study note right hand side studied fernex ein smooth theorem theorem notation vol inf lct following theorem provides alternative characterization using affine cone construction state general form special cases found theorem proposition let log fano pair dimension algebraically closed field characteristic satisfying cartier affine cone defined spec let corresponding denote cone vertex vol equality holds log flat families klt pairs section field assumed algebraically closed definition harold blum yuchen liu given normal variety flat family klt pairs consists surjective flat morphism variety effective avoiding codimension singular points following conditions hold fibers connected normal contained supp klt pair flat family klt pairs together section called flat family klt singularities denote unique closed point lying proposition let flat family klt pairs normal variety following hold exists closed subset codimension least codimension least every smooth morphism normal morphism normal variety base change flat family klt pairs kxt base change proof assume relative dimension let clear zariski closed since characteristic singular locus hence codimxt normal know codimension least smooth thus tsing regular tsing codimension least satisfies property since flat point depth depth depth corollary hence easy see satisfies property since normal hence normal let note smooth since fibers irreducible also irreducible thus argument implies satisfies means normal since smooth know kxt equivalent restriting since codimension least equivalence extends thus finish proof definition let normal projective variety let effective qdivisor say log fano pair klt pair ample say variety log fano pair let normal variety family klt pairs called flat family log fano pairs proper normalized volume singularity lower semicontinuous following proposition states well known result behaviour log canonical threshold families see corollary similar statement proof omitted follows arguments similar proposition let flat family klt pairs normal variety let ideal sheaf function lct constructible addition proper function lct lower semicontinuous respect zariski topology comparison normalized volumes normalized colengths normalized colengths klt singularities definition let klt singularity algebraically closed field characteristic denote local ring given constants define normalized colength respect lct inf note assumption implies ideal given constant define asymptotic normalized colength function respect lim inf clear increasing function main result section following theorem theorem klt singularity algebraically closed field characteristic exists whenever vol proof first show direction let take sequence valuations vol may rescale since vol bounded theorem know exists vol izumi type inequality theorem exists ordm ordm result thus let take therefore lim inf lct lct vol vol last inequality use lct proof theorem thus vol vol finishes proof direction harold blum yuchen liu direction show vol logarithmic version izumi type estimate theorem exists constant ordm valuation valx function ideal exists divisorial valuation valx computing lct lemma hence following skoda type estimate lct ordm ordm let positive number lct choose sufficiently small vol ideal satisfying thus suffices show lct vol vol lim inf inf lctn lemma know exists inf lctn hence proof finished inf lct vol following result comparison colengths multiplicities crucial proof theorem note lemma special case lech inequality theorem regular local ring lemma let analytically irreducible noetherian local domain assume residue field algebraically closed positive numbers exists ideal proof section admits good valuation total order let closed convex hull know strongly convex cone exists linear functional exists ordm ordm normalized volume singularity lower semicontinuous suppose ideal satisfying know similarly positive integer let define semigroup follows denote easy see satisfies thus proposition implies lim vol convex body defined section easy see let define know ideal satisfying define clear also satisfies since algebraically closed onedimensional leaves proposition lim vol vol lim since know denote clear hand denote since vol vol vol exists positive numbers depending vol vol vol vol let pick vol vol second inequality guaranteed applying proposition sub convex body fixed convex body thus vol vol vol vol vol vol vol harold blum yuchen liu last inequality follows hence finish proof normalized volumes field extensions rest section use hilbert schemes describe normalized volumes singularities field extension let klt pair point let spec denote thickening consider hilbert scheme hilbd field extension know parametrizes ideal sheaves satisfying mkxk oxk particular point corresponds ideal satisfying two conditions denote proposition let field characteristic let klt pair let point field extension algebraically closed inf lct assumption vol vol proof first prove direction definition infimum lct oxk ideal satisfying mkxk mxk oxk hence represents point suppose lying point clear spec hence lct lct direction proved next prove direction proposition know function lct constructible lower semicontinuous denote set closed points since set closed points dense stratum respect lct function following equality inf lct inf lct satisfies algebraic extension since algebraically closed embedded subfield hence exists point lying thus similar arguments implies lct lct direction proved know hence follows theorem following corollary experts present proof using normalized volumes corollary let log fano pair field characteristic following equivalent log log field extension normalized volume singularity lower semicontinuous iii log field extension say geometrically log one conditions holds proof let take affine cone cartier let corresponding denote byx cone vertex let field extension theorem implies log hence corollary vol consequence proposition finish section natural speculation suppose klt singularity field characteristic zero necessarily algebraically closed definition normalized volume singularities extend verbatimly expect vol vol also denote vol normalized volumes stable base change algebraic closures speculation consequence stable degeneration conjecture sdc stated conjecture conjecture roughly says valuation vmin unique vmin invariant action gal hence normalized volume restriction uniform approximation volumes colengths section prove following result gives approximation volume valuation colengths valuation ideals result consequence arguments section turn relies ideas properties function theorem let together section flat family klt singularities set dim dim every exists positive integer following holds satisfies axt vol positive integers divisible begin approximating volume valuation multiplicity valuation ideals proposition let klt singularity defined algebraically closed field positive integer cartier fix valx satisfying xsing supp jacx vol harold blum yuchen liu xsing supp vol proof fix valx satisfying simplify notation set theorem jacx since see proof proposition follows previous inclusion combined jacx apply teissier minkwoski inequality example previous inclusion find jacx dividing sides taking limit gives jacx vol since vol desired inequality follows case xsing supp stronger inequality follows similar argument observation jacx trivial neighborhood proceeding recall basic properties jacobian ideal let flat finite type morphism noetherian rings assume induced map spec spec constant relative dimension relative jacobian ideal defined fitting ideal means take free resolution view matrix elements given previous definition gives local description flat finite type morphism noetherian schemes lemma assumptions theorem exists nonempty open set jacxt jacx proof sufficient consider case affine choose free resolution generic flatness may choose nonempty open set restriction free resolution thus jacxt jacx proposition assumptions theorem fix positive integer cartier every exists following holds satisfies jacxt supp jacxt normalized volume singularity lower semicontinuous proof simplify notation set jacxt supp noetherianity suffices prove following claim exists jacxt nonempty open set positive integer proceed prove claim fix apply lemma choose nonempty open affine set jacxt jacx applying chevalley theorem see constructible thus may shrink either holds case claim complete therefore assume holds shrink accordingly next choose nonzero element jacx restriction denoted nonzero set note dimensional jacxt first inequality follows inclusion jacxt second precisely lech inequality theorem thus jacxt proposition may shrink constant since dim therefore exists integer completes claim following proposition consequence results appendix proposition keep assumptions notation theorem fix integer exists following holds point ideal satisfying proof set max consider union hilbert schemes hilbm spec let denote morphism point corresponds ideal universal ideal sheaf applying proposition irreducible components endowed reduced scheme structure see set functions hbh finite next fix previous paragraph exists hbh harold blum yuchen liu consider point ideal oxt satisfying since map spec therefore implies deduce theorem propositions proof theorem simplify notation set axt order prove theorem suffices prove following claim every exists integer vol divisible indeed vol since set bounded proposition claim implies conclusion theorem fix proceed bound latter two terms proposition first apply proposition find positive integer next apply proposition find positive integer following holds jacxt supp jacxt jacxt set max proposition implies vol next note therefore may apply proposition find integer thus vol normalized volume singularity lower semicontinuous first inequality follows combined inclusion second inequality therefore setting completes claim izumi properness estimates families section generalize results families klt singularities results used prove theorem theorem estimate let together section flat family klt singularities variety exists constant following holds satisfies axt axt theorem properness estimate let together section flat family klt singularities variety exists constant following holds satisfies axt axt axt vol proofs theorems rely primarily result techniques found main new ingredient found proposition proved using arguments appendix order functions let normal variety defined algebraically closed field closed point order vanishing defined ordx max mjx smooth ordx valuation function field singular case ordx may fail valuation example inequality ordx ordx ordx may strict following consider alternative function ord defined lim ordx sup ordx ord let denote normalized blowup write prime divisors following statement terms proved theorem gives interpretation ord exceptional divisors harold blum yuchen liu proposition function ordei ord ord building upon results section show comparison ordx ordx proposition exists klt pair ordx ordx ord dim supremum proof first inequality follows definition ord second inequality assume ordx note since first inclusion follows fact klt second therefore ord skoda theorem see claim complete izumi type estimates propositions section concern following setup arise proof theorem let affine klt singularity algebraically closed field fix projective compactification resolution singularities assume exists ample line bundle restriction denoted log resolution proposition exists constant following holds closed point ordy ordx furthermore write prime divisor formula constant given terms coefficients intersection numbers dimension proposition refined version theorem proof relies ideas appendix proof fix closed point element let denote blowup exceptional divisor write strict transform consider divisor given closure write lies support note ordy ordei since factors normalized blowup along normalized volume singularity lower semicontinuous proposition implies ordei min ord simplify notation set max goal find constant finding ordx ordy ord min last inequality follows proposition thus desired inequality hold proceed find constant set note ample example consider last equality follows fact principal divisor neighborhood set cij see cij cii note cij case cij computing cij terms intersection numbers find cij otherwise additionally otherwise set cij note cij set similarly note set max cij choice distinct zariski main theorem implies connected therefore set conclude harold blum yuchen liu proposition exists constant following holds valx satisfies ordy furthermore coefficients condition holds proof proof statement found proof theorem case general statement follows similar argument proofs theorems proof theorem sufficient prove theorem case affine show exists nonempty open set constant conclusion theorem holds induction dimension proof complete fix relative projective compactification denote ideal sheaf fix projective resolution singularities restriction denoted log resolution set write prime divisor order prime divisors dominates positive integer generic smoothness exists nonempty open set log resolution smooth shrinking may assume let assume rest proof kyt kxt note divisors may multiple irreducible components next apply spa tag find morphism irreducible irreducible components generic fiber geometrically irreducible denote generic points write decomposition irreducible components set equal closure applying spa tag may find open subset divisor geometrically irreducible shrinking may assume divisors distinct choose nonempty open subset contained image seek find constant ordy normalized volume singularity lower semicontinuous since image suffices establish inequality form singularities let line bundle ample write pullback fix fixed function sends constant lemma choice know irreducible therefore may apply proposition find constant desired inequality holds next choose set proposition axt ordy cyt combining see desired inequality holds proof theorem theorem follows immediately theorem theorem proof theorem proofs applications convergence result normalized colengths theorem let together section flat family klt singularities every exists constant integer following holds vol divisible beginning proof previous theorem record following statement proposition let together section qgorenstein flat family klt singularities exists constant inf vol vol axt proof first note exists real number vol indeed vol lct lct function sends lct takes finitely many values propositions thus vol inf vol vol next fix constant satisfying conclusion theorem axt therefore proposition satisfies vol holds harold blum yuchen liu proof theorem fix constant satisfying conclusion previous proposition simplify notation set proceed proving following two claims claim exist constants following holds inf lct proposition implies exist constants consider since therefore claim follows definition claim exists following holds lct vol integers divisible theorem exists following holds vol integers divisible note lct lct axt therefore multiplying lct yields desired result return proof corollary fix constants satisfying conclusions claims set postive integer divisible satisfies inf lct inf vol vol first equality follows claim second claim third choice normalized volume singularity lower semicontinuous proofs following theorem stronger result implies theorem theorem let together section flat family klt singularities field characteristic function vol lower semicontinuous respect zariski topology proof let thickening section spect let denote hilbd since proper know also proper let normalization denote pulling back universal ideal sheaf obtain ideal sheaf denote projection provides flat family klt pairs following notation proposition assume point lying denote construction ideal sheaf spec pull back flat base change spec spec hence lct spec lct spec simplicity abbreviate equation lct lct applying propon sition family ideal implies function defined lct constructible lower semicontinuous respect zariski topology since lct lct descends function lct since proper know function defined nmin constructible lower semicontinuous respect zariski topology proposition implies thus conclude constructible lower semicontinuous respect zariski topology let fix point theorem exist vol divisible since constructibly lower semicontinuous exists zariski open neighborhood theorem exist vol let choose min combining yields vol vol harold blum yuchen liu proof finished following theorem stronger result implies theorem theorem let flat family log fano pairs field characteristic assume geometric fiber log point exists intersection countably many zariski open neighborhoods log point addition uncountable log general closed point geometrically log locus log stable generalization proof satisfying cartier define relative affine cone spect assume sufficiently large easy see locally free thus spec let qdivisor corresponding section projection together section cone vertices flat family klt singularities since theorem implies vol theorem exists intersection countably many zariski open vol since neighborhoods vol global volumes log fano pairs constant flat families vol vol theorem implies let point exists countably many zariski open neighborhoods generalization belongs zariski open neighborhoods proof theorem clear follows theorem constructibility normalized volumes implies set proof theorem chosen zariski open neighborhood argument proof theorem works following corollary stronger result implies corollary corollary let family complex log fano pairs assume isotrivial zariski open subset log ksemistable closed point log normalized volume singularity lower semicontinuous proof since log theorem implies log ksemistable general closed point hence exists hence log remark acc normalized volumes bounded families true conjecture follows applying theorem moreover suspect much stronger result discreteness normalized volumes away see also question might true much evidence yet applications section present applications theorem following theorem generalizes inequality part theorem theorem let complex klt singularity dimension let largest coefficient components containing vol proof suppose component containing coefficient let din normalization applying theorem din together natural diagonal section din din vol vol general closed point may pick smooth point vol coordinate hyperplane let vol take local coordinates monomial valuation weights satisfies aan ordva vol hence aan aordv vol vol vol proof finished theorem let klt pair function vol lower semicontinuous respect zariski topology let irreducible subvariety general closed point sup vol vol particular exists countable intersection zariski open constant subsets vol proof part follows quickly applying theorem together diagonal section part denote normalization proof follows quickly applying theorem together natural diagonal section next study case limit fano lower semicontinuous manifolds note function vol respect euclidean topology following corollary improves result follows easily part theorem corollary let limit fano manifolds lower semicontinuous function vol respect zariski topology harold blum yuchen liu following theorem partially generalizes lemma proposition theorem let limit fano manifolds let closed point finite morphism singularities deg particular weil divisor ind ind denotes cartier index proof theorem finite degree formula holds vol since vol theorem theorem deg vol corollary deg vol remark finite degree formula conjecture true klt holds finite morphism singularity clearly deg klt singularities particular would get loc known finite effective upper bound see theorem partial result dimension theorem let complex variety dimension let largest integer exists weil divisor satisfying proof consider orbifold cone spec cone vertex let specv partial resolution exceptional divisor klt singularity component hence ordv theorem implies minimizes vol vol hence ordv vol ordv vol theorem theorem proof finished since vol appendix asymptotic lattice points counting convex bodies appendix prove following proposition proposition positive number exists closed convex body integer vol proof induction dimensions closed interval length kvol hence know kvol kvol normalized volume singularity lower semicontinuous holds next assume proposition true dimension denote coordinates let sectional convex body let image projection onto last coordinate know vol vol induction hypothesis exists vol vol clear vol next know function vol concave brunnminkowski theorem particular find vol reaches maximum hence increasing decreasing applying proposition respectively yields vol vol since vol vol vol vol vol therefore setting max inequality follows easily combining proposition monotonic function proof may assume increasing function denote since whenever harold blum yuchen liu similarly clear result appendix families ideals function following proposition concerns behavior function along family ideals statement new proof give follows arguments found found definition local ring ideal samuel function denoted given note dim proposition let morphism finite type assume integral section ideal filtration every closed function hat constant proof prove result sufficient show exists nonempty open set proceed find set hat constant hat therefore consider finitely generated gra generic flatness may choose nonempty open set gra flat function constant since flat zero dimensional support since furthermore lemma proved implies ait therefore ait constant proof complete stating following lemma introduce notation let ring ideal set gri normalized volume singularity lower semicontinuous lemma let morphism rings ideal mod gri flat mod gri gri proof follow argument given consider surjective map claim injective flat order prove claim induct claim holds since clearly isomorphism flat assumption next consider exact sequence assume claim holds positive integer since flat flatness may tensor get exact sequence exact sequence injectivity implies injectivity claim proven lemma follows applying claim previous short exact sequence references ambro variation log canonical thresholds linear systems int math res bhatt gabber olsson finiteness fundamental groups reduction modulo preprint available blum existence valuations smallest normalized volume appear compos available boucksom fernex favre urbinati valuation spaces multiplier ideals singular varieties recent advances algebraic geometry london math soc lecture note cambridge univ press cambridge boucksom favre jonsson refinement izumi theorem valuation theory interaction ems ser congr eur math cutkosky multiplicities associated graded families ideals algebra number theory fernex ein multiplicities log canonical threshold algebraic geom donaldson algebraic families constant scalar curvature metrics surveys differential geometry regularity evolution nonlinear equations surv differ int press somerville ein lazarsfeld smith uniform approximation abhyankar valuation ideals smooth function fields amer math flenner manaresi equimultiplicity equidimensionality normal cones recent progress intersection theory bologna trends fujita optimal bounds volumes fano manifolds appear amer available hein sun manifolds isolated conical singularities appear publ math inst hautes available jonsson valuations asymptotic invariants sequences ideals ann inst fourier grenoble harold blum yuchen liu kaveh khovanskii convex bodies multiplicities ideals proc steklov inst math rational curves algebraic varieties ergebnisse der mathematik und ihrer grenzgebiete berlin seifert preprint available arxiv singularities minimal model program volume cambridge tracts mathematics cambridge university press cambridge collaboration lazarsfeld positivity algebraic geometry ergebnisse der mathematik und ihrer grenzgebiete berlin lazarsfeld convex bodies associated linear series ann sci norm lech note multiplicities ideals ark mat minimizing normalized volumes valuations appear math available equivariant volume minimization appear duke math available correspondence fano manifolds reine angew math liu metrics volume minimization appear adv available wang proper moduli space smoothable fano varieties preprint available stability valuations components preprint available stability valuations higher rational rank preprint available liu volume singular fano varieties appear compos available liu cubic threefolds preprint available matsumura commutative algebra second edition mathematics lecture note series publishing reading multiplicities graded sequences ideals algebra odaka moduli fano manifolds proceeding kinosaki algebraic geometry symposium odaka compact moduli spaces fano varieties publ res inst math sci spotti sun explicit compactifications moduli spaces fano manifolds preprint available spotti sun yao existence deformations metrics smoothable varieties duke math spa stacks project authors stracks project available http finiteness algebraic fundamental groups compos math interaction singularity theory minimal model program submitted icm algebraic complex geometry session available department mathematics university michigan ann arbor usa address blum department mathematics yale university new usa address
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small clique number graphs three trivial critical ideals nov carlos alfaro carlos valencia abstract critical ideals graph determinantal ideals generalized laplacian matrix associated graph article provide set minimal forbidden graphs set graphs three trivial critical ideals use forbidden graphs characterize graphs three trivial critical ideals clique number equal introduction given connected graph set indeterminates generalized laplacian matrix matrix rows columns indexed vertices given otherwise muv number edges vertices definition critical ideal determinantal ideal given det submatrix say critical ideal trivial equal critical ideals firstly defined studied general framework algebraic number trivial critical ideals algebraic allows separate set simple connected graphs following graph families simple connected graph proven induced subgraph thus implies set closed induced subgraphs therefore say graph forbidden moreover define set minimal forbidden follows definition let forb set minimal induced subgraphs property forbidden graphs graph called forb equal set graphs given family graphs graph called induced subgraph isomorphic member thus forb therefore characterizing forb leads characterization mathematics subject classification primary secondary key words phrases critical ideal critical group laplacian matrix forbidden induced subgraph authors partially supported conacyt grant first author partially supported conacyt second author partially supported sni carlos alfaro carlos valencia ideas used obtain characterization precisely found forb thus consists complete graphs hand forb consists graphs cricket dart consists graphs isomorphic induced subgraph one following graphs tripartite complete graph denote disjoint union graphs denote join main goal paper provide set minimal forbidden graphs partial description specifically characterize graphs clique number equal therefore prove graph clique number induced subgraph one graph family see figure converse stronger induced subgraph graph belongs graphs clique number less graph family graphs iii family graphs figure family graphs black vertex represents clique cardinality white vertex represents stable set cardinality gray vertex represents single vertex article divided follows section show characterization leads characterization graphs critical group invariant factors equal section construct graph replacing vertices cliques stable sets show novel method verify whether critical ideal trivial method applied prove induced subgraph graph family belongs section give family minimal forbidden graphs used prove graph clique number induced subgraph graph see figure section devoted prove graph clique number induced subgraph one graph family applications critical group laplacian matrix evaluation degree vector considering linear map cokernel quotient module torsion part module critical group critical group studied intensively several contexts last years group components picard group jacobian group sandpile group game laplacian unimodular equivalence known theorem critical group connected graph vertices described follows positive integers integers called invariant factors laplacian matrix besides greatest common divisor small clique number graphs three trivial critical ideals laplacian matrix invariant factor equal details see theorem definition given integer number let number invariant factors critical group equal also let simple connected graph study characterization great interest particular results conjectures graphs cyclic critical group found section conjectures hand easy see consists complete graphs besides several people see posed interest characterization sense attempts done characterized graphs whose third invariant factor equal later characterizations graphs cut vertex graphs number independent cycles equal given recently complete characterization given however nothing known crucial result linking critical groups critical ideals theorem states degree vector invariant factors thus critical ideal trivial equal equivalently equal critical ideal trivial critical ideals behave better critical group induced subgraph property difficult see unlike closed induced subgraphs instance cone claw graph belongs claw graph belongs also belongs meanwhile belongs moreover induced subgraph always true example hand consequence theorem therefore analysis invariant factor laplacian matrix graphs characterization obtained see instance characterizations cliques stable sets critical ideals let simple graph suppose subset induced subgraph subgraph whose vertex set whose edge set set edges ends subset subgraph subgraph whose edge set whose vertex set consists ends edges let two subsets denote set edges one end end clique subset mutually adjacent vertices maximum size clique clique number subset called independent set stable set two vertices adjacent graph vertices whose vertex set induces stable set called trivial graph denoted cardinality maximum stable set called stability number denoted given simple graph vector graph constructed follows vertex associated new vertex set clique cardinality negative stable set cardinality positive vertex adjacent vertex adjacent graph called underlying graph carlos alfaro carlos valencia convenient way visualize means drawing vertex colored black negative colored white positive indicate cardinality writing inside drawing vertex may color gray avoid writing cardinality see figure hand useful avoid writing cardinality see figure general computation bases critical ideals complicated however rest section show novel method developed decide whether critical ideal trivial define follows theorem theorem let graph vertices critical ideal xgd trivial evaluation trivial thus procedure verify whether family graphs belongs becomes evaluation critical ideal underlying graph family let underlying graph family graphs see figure vertex set let positive integers negative integers thus using computer algebra system check since evaluation equal theorem critical ideal xgd therefore graph family graphs algebraic let underlying graph family graphs see figure vertex set let positive integer using computer algebra system check since evaluation theorem critical ideal xgd therefore graph algebraic hand verified equal graphs algebraic induced subgraph graph algebraic proposition graph belongs description possible compute algebraic connected graphs vertices using software nauty computation algebraic connected graphs vertices required hours macbookpro ghz intel quad core processor ram besides computation algebraic connected graphs vertices required weeks computation computer let family graphs shown figure family represents graphs forb vertices since exists minimal forbidden graph vertices likely forb small clique number graphs three trivial critical ideals figure family graphs black vertex represents clique cardinality white vertex represents stable set cardinality gray vertex represent single vertex proposition graph belongs forb proof easily checked using computer algebra system graph algebraic equal one main results article following theorem graph clique number induced subgraph graph divide proof two characterizations graphs clique number equal converse theorem stronger proposition induced subgraph graph belongs however difficult recognize graphs clique number greater equal theorem let simple connected graph isomorphic induced subgraph graph see figure iii figure family graphs white vertex represents stable set cardinality gray vertex represents single vertex carlos alfaro carlos valencia since graph isomorphic induced subgraph graph proposition implies graph belongs note graphs induced subgraph graph see figure theorem let simple connected graph isomorphic induced subgraph graph clique number section devoted proof theorem give proof theorem proof theorem since graph belongs graph thus get one implication suppose let since vertex adjacent time vertex neighbor set set vertices adjacent let clearly sets induces trivial graph let define thus two possible cases empty empty first consider empty case following statements claim one sets empty cardinality one proof suppose empty let vertex set induces graph isomorphic impossible suppose cardinality take induces graph isomorphic contradiction thus cardinality one claim edge set induces either complete bipartite graph complete bipartite graph minus edge proof first note vertex incident vertex except one vertex induced subgraph isomorphic impossible similar way vertex incident vertex except one vertex thus edge set must equal edge set minus matching fact cardinality matching must one otherwise induced subgraph isomorphic contradiction claim possible time edge set induces complete bipartite graph minus edge proof suppose situations occur time let let vertex set induces graph isomorphic contradiction thus three possible cases induces bipartite complete graph minus edge small clique number graphs three trivial critical ideals let denote set vertices adjacent follows describe vertex set case let note possible vertex adjacent vertex vertex time get moreover claim exist two vertices one adjacent vertex one adjacent vertex proof suppose two cases either impossible contradiction without loss generality suppose adjacent thus claim vertex set cardinality proof two possible cases either vertex adjacent common vertex suppose exists adjacent otherwise induce thus vertex set induces graph isomorphic forbidden case possible suppose adjacent common vertex following possible cases yields contradiction since case vertex set induces graph isomorphic case vertex set induces graph isomorphic case vertex set induces graph isomorphic two possibilities either adjacent vertex adjacent one vertex adjacent either vertex one vertex adjacent adjacent vertex set induces graph isomorphic note adjacent vertex graph isomorphic induced subgraph graph meanwhile adjacent one vertex graph isomorphic induced subgraph graph finally graph complete bipartite graph case suppose without loss generality claims vertex set cardinality one let two cases adjacent either vertex vertex let consider adjacent vertex two possibilities either however none two cases allowed since first case get vertex set induces graph isomorphic second case vertex set induces graph isomorphic thus remaining case adjacent vertex case must adjacent vertex otherwise graph appears induced subgraph note graph isomorphic induced subgraph graph case let note adjacent adjacent since would obtain induced subgraph possible case adjacent graph isomorphic induced subgraph consider carlos alfaro carlos valencia adjacent vertex without loss generality suppose adjacent vertex adjacent otherwise clique cardinality obtained hand adjacent otherwise would adjacent vertices thus adjacent vertex set induces graph isomorphic contradiction thus empty graph isomorphic induced subgraph consider case empty one vertex sets cardinality one otherwise would contain induced subgraph thus let assume let denote vertex set whose vertices adjacent claim empty adjacent vertex adjacent vertex proof let suppose one edges exist induces graph isomorphic contradiction note vertex set induces stable set otherwise thus empty graph isomorphic induced subgraph graph consider case claim vertex adjacent either unique vertex vertex proof result easy check suppose cardinality greater equal let adjacent adjacent since vertex set induces graph isomorphic get contradiction result follows claim let adjacent cardinality greater vertex adjacent either vertex proof suppose exists adjacent let adjacent four possible cases cases vertex set induces graph isomorphic cases occur hand induces graph isomorphic case occur finally vertex set induces graph isomorphic contradiction thus pair vertices adjacent vertices result follows thus two cases vertex adjacent unique vertex vertex adjacent vertex note induces stable set otherwise first case either vertex set empty otherwise would induced subgraph therefore case isomorphic graph small clique number graphs three trivial critical ideals proof theorem one implication easy proposition implies graph implication much complex suppose let clique cardinality let note denote vertex set whose vertices adjacent vertex since vertex set empty pair vertex set induces stable set furthermore claim induced subgraph isomorphic either proof first consider induced subgraphs since forbidden graphs forbidden thus exist clearly vertex set induces trivial subgraph since nvx nvx therefore component complete bipartite subgraph component cardinality least three another component existence another component makes appears induced subgraph impossible component cardinality least two another component since forbidden thus result turns claim induced subgraph least components isomorphic complete bipartite least three vertices vertex set empty proof let two vertices adjacent suppose empty three possibilities case vertex set induces graph isomorphic respectively since graphs forbidden obtain contradiction empty claim least components isomorphic complete bipartite least three vertices vertex set empty proof let two vertices adjacent suppose let three possibilities carlos alfaro carlos valencia case isomorphic respectively since graphs forbidden empty remark claims imply connected components complete bipartite graph least vertices vertex sets different elements moreover claim vertex set one following vertex sets complete bipartite graph cardinality least next result describes induced subgraph set connected cardinality claim suppose vertex set connected cardinality set empty isomorphic different elements one following sets isomorphic one following sets isomorphic proof difficult prove either empty induces complete bipartite graph result follows checking possibilities computer algebra system rest proof case obtained remark claim analyze remaining edges sets vertex set may refer different elements case consider subsection describe induced subgraph claim vertex sets empty edge set empty proof suppose take result follows since vertex set induces graph isomorphic forbidden graph claim empty proof let suppose exists since induced subgraph isomorphic get contradiction empty claim induces either complete bipartite graph complete bipartite graph minus edge small clique number graphs three trivial critical ideals proof let suppose exist since induced subgraph isomorphic forbidden vertex adjacent least one vertices similar way vertex adjacent least one vertices therefore edge set induces complete bipartite graph minus matching suppose matching cardinality greater equal exist since isomorphic get contradiction matching cardinality therefore vxy induces complete bipartite graph complete bipartite graph minus edge claim one following two statements holds exists proof since edge joining vertex vertex claim implies let claim edge set induces either complete bipartite graph complete bipartite graph minus edge sets induce either complete bipartite graph done remains check two cases one edge set induces complete bipartite graph one induces complete bipartite graph minus edge edge sets induce complete bipartite graph minus edge first consider former case suppose exists since isomorphic case possible consider second case let two possible cases either suppose since isomorphic case impossible therefore claim edge set empty induced subgraph isomorphic one following graphs complete tripartite graph complete tripartite graph minus edge complete tripartite graph minus edges proof claim induced subgraph complete tripartite graph minus three edges analyze cases edges removed suppose induces complete tripartite graph minus edges since induces complete bipartite graph complete bipartite graph minus edge edges removed unique edge set two possibilities exist exist vxy first case graph induced set isomorphic forbidden graph impossible second case induced subgraph carlos alfaro carlos valencia isomorphic impossible thus case edges removed possible suppose complete tripartite graph minus edges since induces complete bipartite graph complete bipartite graph minus edge edges removed unique edge set thus four possible cases cases impossible argument following case impossible induced subgraph isomorphic forbidden graph case induced subgraph isomorphic forbidden graph case induced subgraph isomorphic forbidden graph thus edges removed possible case previous claims following cases set empty set vxy vyz vxy vyz induces bipartite complete graph induces bipartite complete graph minus edge vab vac vbc vab vac vab vbc vac vbc vxz vyz vxz vyz edge sets vxz vyz induce complete bipartite graph vxz vyz vxz vyz exists vxy vxz vxz vxy vyz vyz vxy isomorphic complete tripartite graph isomorphic complete tripartite graph minus edge isomorphic complete tripartite graph describe vertex set remark let suppose adjacent vertex vertex adjacent vertex otherwise shortest path would contains graph isomorphic thus vertex adjacent vertex claim adjacent either adjacent vertex adjacent vertex moreover vertex adjacent vertex either exists vertex vertex adjacent vertex adjacent vertex proof since first statement easy cardinality assume cardinality least let suppose adjacent adjacent since isomorphic get contradiction adjacent let suppose vertex adjacent otherwise remark get contradiction let adjacent thus two possible cases small clique number graphs three trivial critical ideals since first case isomorphic second case isomorphic get contradiction thus vertex adjacent vertex adjacent vertex claim let vertex adjacent induces either trivial graph furthermore exists vertex adjacent clique cardinality proof first note forbidden induced subgraph vertices induce graph isomorphic see component clique let component suppose clique two vertices adjacent say let smallest path contained length greater equal induced subgraph hence contradiction therefore clique hand graph forbidden induced subgraph therefore one component component cardinality one let vertex adjacent vertex adjacent suppose induces stable set cardinality least take get contradiction since induced graph isomorphic thus clique cardinality thus claims case following possible cases clique cardinality vertex adjacent one vertex trivial graph induces complete bipartite graph note graphs isomorphic induced subgraph graph claim clique cardinality vertex adjacent vertex proof let vxy vyz vxy vyz easy see adjacent vxy vyz adjacent vertices otherwise induced subgraph isomorphic thus vertex adjacent vertex suppose adjacent since induced subgraph vxy vyz isomorphic get contradiction induces clique cardinality previous claim get case clique cardinality vertex adjacent vertex graph isomorphic induced subgraph graph claim edge set induces complete bipartite graph clique cardinality vertex adjacent one vertex carlos alfaro carlos valencia proof first prove vertex adjacent vertex one sets suppose exists vertex adjacent since induced subgraph isomorphic get contradiction vertex adjacent vertices one vertex sets suppose two vertices adjacent adjacent since isomorphic impossible vertices adjacent vertices one vertex sets either suppose adjacent two vertices say take adjacent since induced subgraph isomorphic occur thus vertex adjacent one vertex finally suppose induces trivial graph cardinality least let adjacent take adjacent since induced subgraph isomorphic get contradiction result follows previous claim case vertex set induces clique cardinality vertex adjacent one vertex case graph isomorphic induced subgraph graph claim let adjacent adjacent proof suppose adjacent note adjacent adjacent thus two cases either adjacent adjacent cases impossible former case isomorphic meanwhile second case isomorphic contradiction consider case let adjacent claim vertex adjacent difficult see fact vertex adjacent otherwise would contain induced subgraph applying claim induced subgraph get clique cardinality graph isomorphic induced subgraph graph consider case claim get clique cardinality vertex adjacent vertex graph isomorphic induced subgraph graph claim let adjacent adjacent proof suppose adjacent note adjacent adjacent thus two cases either adjacent adjacent cases impossible former case isomorphic meanwhile second case isomorphic contradiction consider cases let vxz vyz vxz adjacent vyz claims vertex adjacent vxz vyz claim get clique cardinality graphs isomorphic induced subgraph graph consider case let vxy vxz vyz vxy adjacent vxz vyz vxz adjacent vyz claims vertex small clique number graphs three trivial critical ideals adjacent vxy vxz vyz claim get clique cardinality graphs isomorphic induced subgraph graph finally case claim vertex set clique cardinality vertex adjacent one vertex case corresponds graph isomorphic induced subgraph graph case first obtain satisfies one following statements induces complete bipartite graph exists vertex called apex exists vertex called apex prove following claims claim let either proof suppose three possible cases first two cases possible since induced subgraph would isomorphic respectively last case induced subgraph isomorphic impossible thus result follows last claim implies incident one vertex apex suppose vertex vertices claim vertex adjacent induced subgraph impossible implies vertex adjacent vertex adjacent vertices similar argument yields apex vertex adjacent vertex adjacent vertices thus three cases complete bipartite minus edges vertex apex vertices complete bipartite minus edges vertex apex vertices complete bipartite claim either induces complete bipartite graph empty proof let suppose one two following possibilities happen first case induced subgraph isomorphic meanwhile second case induced subgraph isomorphic cases occur get result carlos alfaro carlos valencia thus case occur follows describe vertex set vertex set whose vertices edge common vertex set remark let vertex adjacent vertex otherwise shortest path would contains graph induced subgraph let adjacent adjacent vertices otherwise induced subgraph would isomorphic forbidden case see vertex adjacent vertex let supppose vertex adjacent vertex adjacent one vertices must adjacent vertex otherwise let induced subgraph would isomorphic remark adjacent vertex claim vertex set induces stable set proof suppose adjacent since adjacent induced subgraph isomorphic forbidden thus adjacent therefore stable set thus case corresponds graph isomorphic induced subgraph graph consider case let adjacent claim adjacent vertex vertex adjacent vertex proof let suppose adjacent adjacent vertices get induced subgraph isomorphic thus adjacent least one vertex remark adjacent vertices exist adjacent vertex set would induce graph isomorphic hand adjacent vertex set induces subgraph therefore adjacent vertex suppose another vertex argument adjacent vertex must adjacent vertex also adjacent vertex must adjacent vertex suppose adjacent vertex adjacent vertex two possibilities either let first case second case since graphs forbidden get contradiction thus adjacent therefore adjacent vertex claim either vertex adjacent vertex vertex adjacent vertex proof let clearly adjacent vertex adjacent vertex otherwise would induced subgraph claim vertex adjacent vertex different apex vertex adjacent vertex thus get result small clique number graphs three trivial critical ideals claim vertex set induces either clique cardinality trivial graph proof first note forbidden induced subgraph induce get component clique let component suppose clique two vertices adjacent say let smallest path thus length greater equal induced subgraph hence therefore complete graph hand graph forbidden happen therefore one component component cardinality one first case claim either adjacent otherwise possibilities claim allowed second case claim adjacent two vertices get forbidden subgraph trivial since possibilities claim allowed much effort reader see cases corresponds graph isomorphic induced subgraph graph case claim two possible cases either vertex adjacent vertex vertex adjacent vertex vertex adjacent vertex first case vertex set empty adjacent vertices induced subgraph forbidden vertex set empty otherwise vertex set must stable set exist two adjacent vertices taking vertex vertex get forbidden finally case vertex adjacent vertex vertex adjacent vertex get clique cardinality get forbidden graphs isomorphic induced subgraph graph case complete bipartite graph cardinality least assume complete bipartite graph cardinality lest three bipartition claim proof suppose exist get contradiction since induced subgraph forbidden claim let adjacent adjacent vertex part containing proof suppose adjacent vertex result follows may assume since connected cardinality least exists adjacent two possibilities either first case carlos alfaro carlos valencia induced subgraph isomorphic second case induced subgraph isomorphic since cases forbidden get contradiction previous claims suggest divide vertex set three subsets vertices adjacent vertex vertices adjacent vertex vertices adjacent vertex follows assume claim cardinality sets let since proof suppose exist similar isomorphic forbidden get contradiction case empty claim one sets let thus proof suppose impossible empty since applying previous claim one sets thus possible cases following describe set vertices adjacent vertex let vertex adjacent vertex otherwise shortest path would contains graph induced subgraph claim let adjacent vertex adjacent vertex parts partition cardinality greater equal proof let vertex adjacent suppose prove two things adjacent vertex adjacent vertex let consider case see adjacent vertex suppose vertex adjacent take thus two possibilities either adjacent case impossible forbidden meanwhile case impossible forbidden thus adjacent therefore adjacent vertex see adjacent vertex note case may equal suppose adjacent vertex let since induced subgraph isomorphic forbidden graph get contradiction thus adjacent vertex applying previous case get adjacent vertex small clique number graphs three trivial critical ideals next claim show happens case claim one edges sets induces complete bipartite graph proof let claim vertices adjacent vertex suppose adjacent adjacent two possibilities either adjacent induced subgraph isomorphic impossible hand isomorphic forbidden get contradiction result follows claim let adjacent adjacent vertex parts partition cardinality greater equal moreover adjacent unique vertex proof let adjacent claim therefore three cases taking forbidden induced subgraph suppose would appear get contradiction thus claim adjacent vertex parts cardinality greater equal take would isomorphic forbidden graph therefore similar way previous case get suppose adjacent vertex parts partition cardinality greater equal case suppose take know since get contradiction thus easy see adjacent vertex otherwise suppose would appear therefore claim adjacent vertex parts partition cardinality greater equal moreover claim suppose adjacent take since proof let adjacent isomorphic hand claims adjacent vertex thus induced subgraph isomorphic forbidden therefore adjacent suppose exists vertex since let adjacent adjacent isomorphic possible claim let adjacent vertex adjacent proof suppose adjacent claims vertices adjacent vertex let four cases obtained combinations following possible cases either either induced subgraph isomorphic induced subgraph possible induced subgraph isomorphic possible carlos alfaro carlos valencia isomorphic possible finally induced subgraph isomorphic possible thus adjacent adjacent adjacent claim let proof first suppose adjacent let claim adjacent vertex thus isomorphic contradiction therefore adjacent suppose adjacent let adjacent obtained similar way previous case thus assume adjacent isomorphic impossible therefore adjacent claim claim adjacent proof let suppose adjacent claim vertices adjacent thus isomorphic contradiction therefore thus applying previous claims cases obtain following possibilities induces complete bipartite graph induces complete bipartite graph induces complete bipartite graph induces complete bipartite graph induces complete bipartite graph induces complete bipartite graph induces complete bipartite graph induces complete bipartite graph similar argument claim obtain either trivial cases since taking hand since isomorphic cases difficult see case isomorphic induced subgraph graph case induces case assume let following discussion also applies one vertex set cardinality claim moreover adjacent vertex one following sets proof suppose edge joining vertex since forbidden induced subgraph contradiction obtained adjacent vertex suppose adjacent one vertices prove adjacent one vertex say thus suppose adjacent adjacent small clique number graphs three trivial critical ideals since adjacent vertex say adjacent done assume adjacent adjacent possible therefore either adjacent adjacent neither claim vertex adjacent vertex one following sets proof let consider following cases adjacent vertex adjacent vertex adjacent vertex adjacent vertex adjacent vertex adjacent vertex case impossible forbidden hand cases allowed forbidden thus result follows thus satisfies one following three cases induces complete bipartite graph induces complete bipartite graph induces complete bipartite graph describe set vertices adjacent vertex let vertex adjacent vertex otherwise shortest path would contains graph induced subgraph claim adjacent vertex adjacent vertex proof suppose adjacent since forbidden also must adjacent similar way get opposite case turns result consider cases following arguments works cases first note exists adjacent adjacent vertex reason following suppose adjacent vertex since adjacent vertex say vertices adjacent contradiction adjacent claim edge set induces complete bipartite graph hand prove similar way exists adjacent vertex adjacent vertex therefore vertex adjacent vertex furthermore set stable set adjacent taking adjacent induced subgraph would isomorphic impossible cases correspond graph isomorphic induced subgraph graph let consider case claim adjacent vertex vertex adjacent vertex proof let suppose adjacent vertex take since induced subgraph carlos alfaro carlos valencia isomorphic get contradiction adjacent vertex thus claim adjacent vertex suppose exists adjacent since vertex set would induce occur therefore adjacent vertex suppose another vertex argument adjacent vertex must adjacent vertex done hand claim adjacent vertex must adjacent vertex suppose adjacent vertex adjacent vertex two possibilities either first case second case since graphs forbidden adjacent therefore adjacent vertex claims obtain two possible cases either vertex adjacent vertex vertex adjacent vertex consider first case exists vertex adjacent vertices thus isomorphic allowed empty consider second case let thus adjacent since isomorphic get contradiction graph case isomorphic induced subgraph graph cases one following graphs sake clarity suppose induce complete bipartite graph empty going obtain claims describe edge sets joining claim let empty vxy vxy empty proof let vxy suppose vxy adjacent vxy contradiction suppose vxy adjacent case vxy impossible therefore result turns claim implies claim let edge set induces complete bipartite graph proof let vxy suppose vxy adjacent vxy contradiction finally suppose vxy adjacent since forbidden get contradiction result turns claim implies vertex adjacent vertex claim empty proof let vbc suppose vbc adjacent vbc isomorphic forbidden therefore edge joining vertex vertex small clique number graphs three trivial critical ideals claim let induces complete bipartite graph proof let vax suppose vax adjacent since vax isomorphic forbidden vertex adjacent vertex claim edge set induces complete bipartite graph proof let vab vac know adjacent vab vac adjacent vac adjacent vab suppose vab vac adjacent vab vac impossible therefore vertex adjacent vertex claim let edge set induces complete bipartite graph proof let vax vbc know vbc vbc vab vac vax suppose vax vbc adjacent vax vbc impossible therefore vertex adjacent vertex analyze claim adjacent vertex proof let suppose consider following cases adjacent adjacent adjacent cases impossible would induced subgraph obtained respectively finally case induced subgraph isomorphic impossible claim exists vertex adjacent vertex proof let vbc suppose adjacent vbc contradiction obtained since vbc thus adjacent vertex claim exists vertex adjacent vertex proof let vax suppose adjacent vax since induced subgraph vax isomorphic get contradiction result follows claims imply vertex adjacent vertex implies thus graph isomorphic induced subgraph graph case vertex set let claim edge sets empty claim edge set empty let vxy since isomorphic forbidden vxy empty pair hand claim edge set empty graph isomorphic see figure carlos alfaro carlos valencia case without loss generality suppose empty claim vertex adjacent vertex claim let proof let vxc suppose vxc adjacent two possible cases either vxc adjacent since first case vxc second case vxc isomorphic get contradiction thus vxc adjacent claim vertex adjacent vertex vertex adjacent vertex claim vertex adjacent vertex proof let vab suppose exists vxc vxc vab vxc since vab vxc isomorphic get contradiction vertices vab vxc adjacent result turns claim vertex adjacent vertex proof suppose vac vbc vbc vac let since vac vbc get contradiction describe vertex set set vertices adjacent vertex claim adjacent vertex proof let suppose consider following cases adjacent adjacent cases impossible would induced subgraph obtained respectively thus adjacent vertex claim vertex adjacent vertex proof suppose adjacent vab let since isomorphic get contradiction vab adjacent claim vertex adjacent vertex proof let vxc suppose adjacent vxc since vxc isomorphic get contradiction adjacent vxc thus edge therefore empty therefore graph isomorphic induced subgraph graph small clique number graphs three trivial critical ideals case without loss generality suppose complete claim claim let either induces complete bipartite graph induces complete bipartite graph proof let vxc first note vxc adjacent time otherwise vxc would isomorphic also vxc adjacent time otherwise vxc would isomorphic thus vxc adjacent one vertex either suppose vxc adjacent let vxc adjacent vxc isomorphic thus vxc induces complete bipartite graphs adjacent isomorphic hand vxc induces complete bipartite graph suppose vxc adjacent let vxc adjacent vxc isomorphic thus vxc induces complete bipartite graph adjacent isomorphic hand vxc thus induces complete bipartite graph claim empty vertex adjacent vertex either proof let vac vbc suppose vac adjacent vbc adjacent two cases either vac vbc adjacent first case vac vbc isomorphic case impossible second case vac vbc isomorphic case possible suppose vac adjacent vbc adjacent two cases either vac vbc adjacent first case vac vbc isomorphic case impossible second case vac vbc isomorphic case possible thus vac vbc adjacent vertex either result follows claim empty set vac vbc induces complete bipartite graph proof let vac vbc suppose vxc vyc adjacent vxc vyc adjacent since get contradiction vac vbc induces complete bipartite graph claim let induces complete bipartite graph proof let vxc vab suppose vxc vab adjacent two cases either vxc adjacent vxc adjacent vxc adjacent induced subgraph vab vxc isomorphic contradiction case impossible hand vxc adjacent induced subgraph vac vxc isomorphic contradiction thus case also impossible therefore induces complete bipartite graph carlos alfaro carlos valencia let describe claim exists vertex adjacent vertex claim vertex adjacent vertex proof let vab suppose vab adjacent since isomorphic get contradiction therefore vab adjacent claim let vertex adjacent vertex proof let vxc vab suppose vertex adjacent vxc two cases either vxc adjacent vxc adjacent first case impossible since vxc isomorphic forbidden second case occur vxc follows adjacent vertex thus previous claims vertex set empty vertex adjacent vertex isomorphic induced subgraph graph case without loss generality suppose claim let either induces complete bipartite graph empty proof let suppose exist since induced subgraph isomorphic get contradiction thus case possible therefore adjacent either vertex vertex going discard possibility induces complete bipartite graph suppose vertex since induced subgraph isomorphic get contradiction thus either induces complete bipartite graph empty claim let satisfies one following induces complete bipartite graph empty edge set induces complete bipartite graph minus edge proof suppose exist since induced subgraph isomorphic get contradiction thus case possible result follows claim satisfies one following induces complete bipartite graph empty edge set induces complete bipartite graph minus edge induces perfect matching induces complete bipartite graph minus two edges proof since cases checked easily computer algebra system similar arguments rest proof asume claim small clique number graphs three trivial critical ideals check possibilities edge sets possible cases discard following induces complete bipartite graph induces complete bipartite graph minus edge edge set induces complete bipartite graph minus edge two removed edges share common vertex let suppose first case thus adjacent vertex adjacent vertex since induced subgraph isomorphic get contradiction case possible suppose second case thus adjacent vertex adjacent vertex since induced subgraph isomorphic get contradiction case possible finally suppose third case thus adjacent vertex adjacent vertex since induced subgraph isomorphic get contradiction case possible result turns claim empty induces complete bipartite graph proof let vab vac suppose vab vac adjacent since induced subgraph vab vac isomorphic get contradiction thus vertex adjacent vertex claim edge sets induces complete bipartite graph proof first prove let vab vac suppose vab vac adjacent since induced subgraph vab vac isomorphic get contradiction thus let suppose cardinality least let vax vax vax vax since induced subgraph vax vax isomorphic get contradiction finally suppose let vab vac since vab vac isomorphic get contradiction thus cardinality claim let induces complete bipartite graph induces complete bipartite graph proof let vab vac suppose vab vac adjacent since induced subgraph vay vax isomorphic get contradiction thus induces complete bipartite graph claim let one following statements true induces complete graph edge set induce complete bipartite graph proof analyze following four cases carlos alfaro carlos valencia induces complete bipartite graph exist exists consider first case let vax vbc suppose vax vbc adjacent since induced subgraph vax vbc isomorphic get contradiction thus induces complete bipartite graph consider second case let vax vbc suppose vax vbc adjacent since induced subgraph vax vbc isomorphic get contradiction thus induces complete bipartite graph follows prove last two cases impossible implies rest possibilities claim impossible consider third case let vax suppose exist four possible cases vax vax vax vax vax vax vax vax since first case vax second case vax third case vax fourth case vax get contradiction last case get contradiction otherwise first case second case thus case impossible claim let induces complete bipartite graph edge sets induce complete bipartite graphs proof let vax suppose two cases either vvax vvax since induced subgraph vax isomorphic first case second case respectively get contradiction thus therefore edge set induces complete bipartite graph suppose vvax previous result since induced subgraph vax isomorphic get contradiction thus vvax therefore induces complete bipartite graph previous claims following cases vac edge set induces complete bipartite graph vac edge set induces complete bipartite graph edge set induces complete bipartite graph edge set induces complete bipartite graph vac induces complete bipartite graph induces complete bipartite graph small clique number graphs three trivial critical ideals induce complete bipartite graph induces complete bipartite graph induces complete bipartite graph induce complete bipartite graph induces complete bipartite graph induces complete bipartite graph minus edge induces complete bipartite graph minus two edges induces perfect matching induces complete bipartite graph describe set vertices adjacent vertex let vertex adjacent vertex otherwise shortest path would contains graph induced subgraph claim proof let vax suppose adjacent vax three possible cases adjacent adjacent vax adjacent vertices vax first consider case case two subcases either adjacent vax adjacent vax isomorphic since forbidden adjacent vax hand adjacent vax isomorphic therefore case impossible consider case two possible cases either adjacent vax adjacent vax isomorphic forbidden thus adjacent vax adjacent vax isomorphic thus case occur finally consider case subcases either adjacent vax adjacent vax isomorphic forbidden adjacent vax adjacent induced subgraph vax isomorphic forbidden thus get case impossible therefore adjacent vax result follows claim let moreover proof let vax vbc suppose adjacent vbc claim adjacent vertex claims vertex vax adjacent vbc one following three cases occur adjacent vax vbc adjacent vbc carlos alfaro carlos valencia adjacent vax vbc case possible induced subgraph vax vbc would isomorphic impossible case vax vbc isomorphic since forbidden case possible finally case possible since vbc isomorphic forbidden thus adjacent vbc vertex adjacent vertex therefore empty thus previous claim cases vertex set empty cases cases correspond graph isomorphic induced subgraph rest cases correspond graph isomorphic induced subgraph claim let adjacent adjacent vertex let wbc wbc adjacent wbc adjacent vertex let adjacent induces complete bipartite graph two vertices wbc exist time proof suppose wbc exist time two possible cases either wbc adjacent wbc adjacent wbc isomorphic impossible wbc adjacent wbc isomorphic impossible wbc exist time suppose exist time two possible cases either suppose two possible cases either adjacent adjacent isomorphic impossible adjacent isomorphic impossible thus suppose note induced subgraph vertex adjacent adjacent applying previous wbc case induced subgraph get exist time suppose wbc exist time two possible cases either suppose two possible cases either wbc adjacent wbc adjacent wbc isomorphic impossible wbc adjacent wbc isomorphic impossible suppose note induced subgraph vertex adjacent wbc adjacent thus applying first case induced subgraph get wbc exist time claim let induces complete bipartite graph adjacent induces complete graph proof first see adjacent suppose adjacent since induced subgraph isomorphic forbidden graph get contradiction thus adjacent vertex see adjacent suppose adjacent since induced subgraph isomorphic get contradiction therefore adjacent vertex claim either vertex adjacent one vertex clique vertex adjacent trivial proof let proving following cases possible follows possible cases either equal wui small clique number graphs three trivial critical ideals equal wui wuj implies result wuj wui wuj wui wuj wui wui wuj wui wuj wui wuj since cases induced subgraph isomorphic respectively cases impossible hand case induced graph isomorphic impossible also case induced subgraph isomorphic possible finally case induced subgraph isomorphic possible claim induces complete bipartite graph vertex adjacent one vertex vertex adjacent vertex proof let suppose adjacent adjacent since induced subgraph isomorphic get contradiction result follows claim let induces complete bipartite graph vertex adjacent one vertex clique cardinality proof let suppose adjacent since isomorphic get contradiction adjacent one vertex let suppose two cases either adjacent adjacent induced subgraph isomorphic get contradiction adjacent induced subgraph isomorphic get contradiction thus finally let adjacent suppose adjacent since induced subgraph isomorphic get contradiction case claims following possible cases vertex adjacent vertex trivial graph vertex adjacent vertex clique cardinality vertex adjacent vertex clique cardinality cases corresponds graph isomorphic induced subgraph remark let adjacent adjacent vertices case claim remark vertex adjacent vertex carlos alfaro carlos valencia clique trivial graph clique difficult see graphs isomorphic induced subgraph claim let suppose adjacent vertex one following statements holds vertex adjacent induces complete bipartite graph proof first case follows remark check adjacent vertex let adjacent suppose adjacent since induced subgraph isomorphic contradiction adjacent also thus remark adjacent finally applying claim induced subgraph get also adjacent vertex thus adjacent vertex suppose cardinality least take thus adjacent since induced subgraph isomorphic get contradiction thus cardinality claim let suppose adjacent vertex adjacent vertex either adjacent induces complete bipartite graph proof first case follows remark prove adjacent adjacent suppose adjacent since induced subgraph isomorphic contradiction adjacent also thus remark adjacent finally suppose let applying claim induced subgraph get also adjacent vertex thus adjacent vertex since isomorphic get contradiction let therefore case claims one following cases holds clique cardinality vertex adjacent trivial vertex adjacent vertex trivial vertex adjacent checked graphs isomorphic induced subgraph graph claim let adjacent suppose induces complete bipartite graph adjacent proof remark adjacent one vertex adjacent see adjacent vertex suppose adjacent first consider case adjacent since isomorphic get contradiction thus adjacent adjacent isomorphic impossible adjacent vertex small clique number graphs three trivial critical ideals suppose exist another vertex previous discussion adjacent since isomorphic get contradiction thus cardinality let therefore case claims adjacent isomorphic induced subgraph claim let adjacent proof since adjacent vertex adjacent adjacent suppose adjacent adjacent since isomorphic get contradiction therefore also adjacent let thus case claims trivial vertex adjacent therefore isomorphic induced subgraph references alfaro valencia graphs two trivial critical ideals appear discrete appl math alfaro corrales valencia critical ideals signed graphs twin vertices preprint bacher harpe nagnibeda lattice integral flows lattice integral cuts finite graph bull soc math france biggs critical group graph alg combin bondy murty graph theory grad texts vol springer chan hou shiu graphs whose critical groups larger rank acta math sinica cori rossin sandpile group dual graphs european combin corrales valencia critical ideals appear grayson stillman software system research algebraic geometry available grone merris watkins laplacian unimodular equivalence graphs brualdi friedland klee eds combinatorial problems linear algebra godsil royle algebraic graph theory grad texts vol new york jacobson basic algebra second edition freeman company new york lorenzini arithmetical graphs math ann lorenzini finite group attached laplacian graph discrete math lorenzini smith normal form laplacians combin theory mckay nauty users guide version available http merino game discrete math merris unimodular equivalence graphs linear algebra appl pan wang note third invariant factor laplacian matrix graph preprint wagner critical group directed graph preprint arxiv departamento centro estudios avanzados del ipn apartado postal mexico city address alfaromontufar cvalencia
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connectivity natura forest sites europe christine estreguil giovanni caudullo daniele rigo european commission joint research centre institute environment sustainability via fermi ispra italy jun politecnico milano dipartimento elettronica informazione bioingegneria via ponzio milano italy context european biodiversity policy green infrastructure strategy one supporting tool mitigate fragmentation increase spatial functional connectivity protected unprotected areas joint research centre developed integrated model provide set indices evaluate connectivity natura network forms backbone green infrastructure europe model allows wide assessment comparison performed across countries terms structural spatially connected isolated sites functional connectivity distances sites influenced distribution distance land cover main conclusion natura network europe shows differences among countries terms sizes numbers sites distribution well distances sites connectivity assessed basis average distance roads intensive land use barrier effects well presence green corridors countries natura network mostly made sites physically connected highest functional connectivity values found spain slovakia romania bulgaria natural landscape sweden finland result high network connectivity due large distances distribution subnets respect roads explains higher share isolated subnets portugal belgium cite estreguil caudullo rigo connectivity natura forest sites europe doi arxiv corresponding author europe green infrastructure strategy holistic policy initiative integrating nature biodiversity sustainable development one supporting tool implement european biodiversity strategy achieve targets one aims mitigate fragmentation render protection effective one condition achieve increase spatial functional connectivity natural protected unprotected areas protected areas natura sites form backbone green infrastructure europe besides size quality connectivity contributes movement dispersal animals plants estreguil caudullo rigo connectivity natura forest sites europe figure illustration site rasterization method five sites left resulting three subnets right perspective influencing factors related site distribution grey infrastructure distance connectivity indices isolated sites unprotected landscape plays role enhancing reducing conservation resilience protected habitats example grey infrastructure artificial lands roads intensive land use often pose biggest threats disturbances biodiversity conservation need conservation tools accounting connectivity tandem landscape planning connectivity measures necessarily link individual habitat patches physical structures corridors similar habitat ensure existence required functional connections protected sites distances landscape permeability identify potentially isolated sites joint research centre developed integrated spatially explicit model provide european vision connectivity sites pattern natura site network network natura sites designated habitat birds directives covers circa european union sites include forest polygons representing extracted site areas converted raster layer order generate subnet components formed one natura sites cases overlap sites physically connected figure differences among countries terms sizes numbers sites distribution well distances sites figure small frames figure ireland frame network portugal frame large isolated subnets spain frame large connected sites germany frame small densely distributed sites sweden frame mainly small distant subnets bulgaria frame large subnets site network pattern terms size number network structural connectivity morphological criteria complex simple subnets network functional connectivity figure natura forest site network six zoom frames showing network differences estreguil caudullo rigo connectivity natura forest sites europe extract sitestwithtfocalthabitat rasterize merge landtuse artificial agriculture natural landscapehmatrix sharehofhroadhnetwork landtcovertshares numberhofhsubnets subnethsizehcmax hmedianl reclassify structuralhsubnethmap simplehsubnets tnodeb complexhsubnets tnodeb tsimpletlnodesb tcomplextllinksmtnodesb reclassify frictiontvaluestllogtscaleb forteachtlandtuse frictionhmap txttfiles rpc sitetareatweighetdtconnectivity rapc unweightedtconnectivity sharetin isolatedhsubnets functionalhconnectivity conefor inter sitetnetworktfunctionaltconnectivity mspahmap coremtedgemtbridge loopmtbranchmtislet guidostmspa intra sitetnetworktstructuraltconnectivity figure data input information flow compute analysis natura network connectivity intermediate steps generate raster layers input data categorical variables transformed friction maps functional distance cost estimates paths computed derived indices dimensionless values exception indices summarising number sites subnets size notation workflow follows semantic array programming paradigm applied semantics based details less obvious modules may found simplifiedhlandhusehmap pintensiveptagriculturem natural semi natural mainhroadhnetwork highwaysmttrunksmtprimary roadstwithouttbridges tunnels reclassify extract rasterize clc osm pythontprocessing withtesritspatialtlibraries pythontprocessing indextfamily derivedtindices specializedtsoftware derivedtmap intermediatetstep inputtdata nvk site networktpattern estreguil caudullo rigo connectivity natura forest sites europe estreguil caudullo rigo connectivity natura forest sites europe figure structural connectivity morphological analysis subnets based guidos software left reclassification subnets right landscape resistance species dispersal identify green corridors new european land use based friction map created corine land cover map year spatial resolution openstreetmap layer model connectivity protected areas model based two available software packages guidos free software conefor integrated gis python programming tools automated processing figure approach harmonized applies structural functional criteria represents compromise biological species models commonly used connectedness measures figure illustration functional connectivity parameters refer subnet areas probability connectivity area landscape unit network structural connectivity spatial configuration sites characterized terms simple subnets made one node complex subnets made several interconnected nodes links figure country two structural connectivity indices proposed share complex subnets share simple subnets probability connectivity measured function landscape resistance costi subnets applies probability connectivity average distance dist also accounts presence green corridors subnets figure network functional connectivity model based probabilistic power weighted dispersal function proxy costi estreguil caudullo rigo connectivity natura forest sites europe figure functional connectivity illustrated two subnets tunnel along main road left accounted identifying structural connectivity site network pattern country name natura number number subnet size sites subnets median max coverage simple complex subnet subnet functional connectivity land cover shares road rpc rapc isolated austria belgium bulgaria cyprus czech germany denmark estonia spain finland france greece hungary ireland italy lithuania luxembourg latvia malta netherlands urban agricult natural poland portugal romania sweden slovakia kindom table country based table indices highlighting highest value orange lowest purple estreguil caudullo rigo connectivity natura forest sites europe country three functional connectivity indices applied site area weighted root probability connectivity rpc sensitive size subnets rpc country based results natura network connectivity countries network mostly made simple subnets share complex subnets physically connected sites range bulgaria latvia large numbers subnets similar small median sizes found germany france connectivity emphasis sites sizes rather low rpc approx higher intersite distances landscape focus rapc approx highest rpc found spain slovakia romania bulgaria natural landscape sweden finland result high network connectivity due large distances distribution subnets respect roads explains higher share isolated subnets portugal belgium root average probability connectivity rapc sensitive unprotected landscape resistance functional distances subnets rapc share functionally isolated subnets natura network figure european map root average probability connectivity rapc share subnets functionally isolated rpc rapc special cases general family indices power weighted probability dispersal pwpd also belong simplified formulation instances parameters respectively rpc rapc estreguil caudullo rigo connectivity natura forest sites europe figure national profile site area weighted root probability connectivity rpc chart includes cover percentage per country proxy gap connectivity computed difference datails please refer estreguil caudullo connectivity natura forest sites eur luxemburg publications office european union jrc estreguil rigo caudullo proposal integrated modelling framework characterise habitat pattern environmental modelling software references bennett openstreetmap packt publishing isbn page bennett bento pais berry didicescu fichter hoellen miko onida smith wakenhut green infrastructure implementation proceedings european commission conference november karhu european commission http page rigo applying semantic constraints array programming module mastrave modelling library semantic array programming mastrave introduction semantic computational modelling http page environment env multifunctionality green infrastructure science environment policy http page estreguil caudullo rigo connectivity natura forest sites europe environment env natura data european network protected sites temporal coverage european environment agency web portal http page european commission council directive may conservation natural habitats wild fauna flora official journal european union pages european commission directive european parliament council november conservation wild birds official journal european union page european commission life insurance natural capital biodiversity strategy brussel com final http page european commission green infrastructure enhancing europe natural capital brussel com final http fin pdf page european environment agency corine land cover raster data version european environment agency web portal http page estreguil caudullo rigo whitmore reporting european forest fragmentation standardized indices web map services ieee earthzine http quarter theme forest resource information pages estreguil caudullo connectivity natura forest sites eur luxemburg publications office european union jrc pages estreguil rigo caudullo proposal integrated modelling framework characterise habitat pattern environmental modelling software pages estreguil rigo caudullo supplementary materials proposal integrated modelling framework characterise habitat pattern http extended version supplementary materials published environmental modelling software page haklay weber openstreetmap street maps pervasive computing doi page mchugh thompson rapid ecological network assessment tool use locating habitat extension areas changing landscape journal nature conservation page saura conefor sensinode software package quantifying importance habitat patches landscape connectivity environmental modelling software page soille vogt morphological segmentation binary patterns pattern recogn lett page van rossum drake python language reference manual version network theory limited isbn page
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sep stochastic gradient descent going fast possible faster alice schoenauer sebag altschuler lab dpt pharm chem ucsf san francisco marc schoenauer tau orsay sebag tau orsay abstract applied training deep neural networks stochastic gradient descent sgd often incurs steady progression phases interrupted catastrophic episodes loss gradient norm explode possible mitigation events slow learning process paper presents novel approach control sgd learning rate uses two statistical tests first one aimed fast learning compares momentum normalized gradient vectors random unit vectors accordingly gracefully increases decreases learning rate second one change point detection test aimed detection catastrophic learning episodes upon triggering learning rate instantly halved abilities speeding slowing learning rate allows proposed approach called sal learn fast possible faster experiments standard benchmarks show sal performs well practice compares favorably state art machine learning algorithms require efficient optimization techniques whether solve convex problems svms ones neural networks convex setting main focus order convergence rate nesterov defazio case still experimental science significant efforts devoted devising optimization algorithms robust default values associated tailored typical regime models problem instances deep convolutional neural networks mnist cun imagenet deng duchi zeiler schaul kingma tieleman hinton data size model dimensionality increase mainstream convex optimization methods adversely affected approaches optimally handle convex optimization problems however scale approximations required martens overall stochastic gradient descent sgd increasingly adopted convex settings good performances linear tractability bottou bousquet hardt within sgd framework one main issues know control learning rate objective reach satisfactory learning speed without triggering catastrophic event manifested sudden rocketing training loss gradient norm finding much much terms learning rate slippery game depends current state system weight vector current often eventual convergence sgd ensured decaying learning rate robbins monro defazio zinkevich number learning rate decay effectively prevents catastrophic events main cause days weeks computation behind many breakthroughs deep learning many diverse approaches thus designed achieve learning rate adaptation amari duchi schaul kingma tieleman hinton andrychowicz section paper proposes novel approach adaptive sgd called sal safe agnostic leraning rate adaptation sal based conjecture learning catastrophes well taken care learning process speed whenever successive gradient directions show general agreement direction frequent advent catastrophic episodes long observed neural net practitioners goodfellow chapter raises question best mitigate impact answer depends whether events could anticipated precision framing catastrophic episodes random adopt purely curative strategy opposed preventive one detecting instantly curing catastrophic episodes formally sequential cumulative sum change detection test test page hinkley adapted used monitor learning curve reporting minibatch losses change learning curve detected system undergoes instant cure halving learning rate backtracking former state instant cure thought terms dichotomic approximation line search see defazio risk catastrophic episodes well addressed learning rate adapted agile manner agnostic learning rate adaptation process increases resp decreases learning rate whenever correlation among successive gradient directions higher resp lower random comparing actual gradient momentum agnostic momentum built random unit vectors contribution paper twofold first proposes original efficient way control learning dynamics section secondly opens new approach handling catastrophic events salvaging significant part runs section experimental validation thereof compares favorably state art mnist benchmarks section related work sgd revived last decade effective method training deep neural networks linear computational complexity size dataset bottou bousquet hardt sgd faces two limitations depending learning rate large learning trajectory leads catastrophic episodes small convergence takes ages dynamic adjustment learning rate therefore acknowledged key issue since early sgd days robbins monro dealing catastrophic events deep learning exploding gradient problem described goodfellow chapter encounter steep cliff structures derivative landscape learning frequently met training neural networks even training recurrent neural networks bengio comes dealing events published work focuses creating conditions arise among possibilities use regularizations regularization pascanu regularization srivastava gradient clipping constraining gradient norm remain smaller constant pascanu another possibility introduction batch normalization ioffe szegedy also helps diminishing frequency events finally proper initialization glorot bengio sutskever unsupervised erhan initializing optimization trajectory good region parameter space also diminish frequency events learning rate adaptation addressing slow speed sgd learning rate adaptation acknowledged key issue since late see george powell review using far tractable computational resources involved prediction information contained correlation successive gradient directions already heart update rules proposed jacobs briefly rule states parameter current gradient relaxed sum past gradients sign learning rate incremented additively opposite sign learning rate decremented multiplicatively decrementing learning rates faster increasing already advocated author adapt faster case catastrophic events natural gradient descent ngd approach amari considers riemaniann geometry parameter space using fisher information matrix estimated gradient covariance matrix precondition gradient due quadratic complexity dimension parameter space ngd approximations designed deep networks pascanu bengio notably approaches martens interpreted ngd pascanu bengio dagrad duchi also uses information past gradients precondition update manner dividing learning rate sum squared past gradients several approaches proposed refine dagrad mitigate learning rate decay including dadelta zeiler rmsprop tieleman hinton dam kingma dam based estimating first second moments gradient parameter using ratio update parameters moment estimates maintained exponential moving averages different weight factors default inertia first moment higher two orders magnitude second seen sal also builds upon use gradient second moment difference compared fixed agnostic counterpart schaul learning rate computed approximately maximally decrease expected loss loss function locally approximated parabola finally andrychowicz address learning rate adaptation reinforcement learning problem exploiting evidence gathered current time steps infer would good decisions earlier accordingly optimizing adjustment policy remotely related momentum approaches classic polyak nesterov versions nesterov derived sutskever rely relaxed sum past gradients indicating robust descent direction current gradient sal sal involves two components learning rate adaptation scheme ensures learning system goes fast catastrophic event manager charge detecting undesirable behaviors getting system back track agnostic learning rate adaptation rationale basic idea proposed learning rate update compare current gradient descent random walk uniformly chosen gradient directions indeed sum successive normalized gradient vectors referred cumulative path following larger norm sum uniformly drawn unit vectors gradient directions positively correlated cases learning process global direction process afford speed opposite norm cumulative path smaller random equivalent gradient directions process alternating opposite directions bouncing sides narrow valley hovering around local optimum learning rate decreased scheme takes inspiration famed hansen ostermeier nes wierstra algorithms today considered among continuous optimization algorithms approaches facto implement natural gradient optimization amari instantiate optimization paradigm ollivier space normal distributions ird formally maintains normal distribution variance normal distribution aka updated basis comparison cumulative path algorithm moving exponential average successive steps random walk gaussian moves fixed mechanism said agnostic makes assumption whatsoever properties optimization objective algorithm partial adaptation scheme minimization loss function parameter space defined follows let solution time gradient current loss current learning rate sgd computes solution time let denote norm associated dot product definition exponential moving average normalized gradients weight random equivalent defined independent random unit vectors ird proposition expectation variance krt defined krt krt proof appendix let denote limits respectively time step scheme updates cumulative path comparing norm distribution agnostic momentum defined learning rate increased decreased depending normalized gap squared norm kpt exp algorithm approach implemented algorithm lines algorithm given well initial learning rate size iteration computes new exponential moving average normalized gradient line performs agnostic update learning rate line updating parameter usual way line learning rate controlled fashion independently maintaining exponentiated moving average updating learning rate layer neural network lines algorithm used experiments section learning rate adaptation noted kingma parameterwise control learning rate desirable contexts scheme extended achieve update learning rate follows let denote squared ith coordinate straightforward show expectation respectively standard deviation expectation divided respectively standard deviation divided given squared ith coordinate noted thus likewise adjusted comparison random counterpart update therefore becomes exp see appendix full derivation algorithm catastrophic event management said ability learn fast requires emergency procedure able detect emergency recover algorithm agnostic learning rate adaptation change detection input model loss function parameters memory rate factor parameters initial learning rate ratio run parameters initialize init initialization first lmin initialize stopping criterion met new perform forward pass empirical mean batch losses lmin min lmin lmin lmin cumulated deviations mean lower bound deviations triggered backtrack else exp end end save possible backtracks compute gradient backward pass exponential moving average normalized gradients agnostic learning rate update standard parameter update recovery healthy learning regime training error decrease along time noise due variance unless learning system abruptly meets cliff structure goodfellow usually blamed large learning rate uneven gradient landscape convex noiseless optimization setting computationally tractable best strategy compute approximation optimal learning rate line context thought experiment let used update resulting would yield performance yields worse performance continued optimization process likely diverge yields performance improvement yields performance improvement trajectory likely bounce back forth walls optimum valley overall safety zone learning rate caveat safety zone narrower optimization problems proposed safeguard strategy primarily aims detect steps outside safety zone see change point detection test apply correction get back upon change detection loss sal implements straightforward correction halving learning rate recovering last solution test triggering halving process iterated needed sending back exponentially fast except perturbations gradient due variance rationale halving trick based number successive dividing iterations could indeed made even smaller using larger dividing factor required standard iterations needed reach optimal learning rate reached safety zone choice dividing factor discussed appendix fact two time counters full sal algorithm global usual one one related test one used simplicity reasons see defazio handle noise case composite loss function detection sal applies change detection test signal given minibatch loss detection test page hinkley chosen provides optimal guarantees detection delay upon change affecting average standard deviation signal mean time false alarms maintains empirical mean signal cumulative empirical mean finally records empirical bounds lmin mint lmax maxt case stationary signal expectation construction change test thus triggered gap empirical bounds higher threshold controls alarm rate test implemented sal algorithm follows greyish lines algorithm set first one tenth empirical loss first minibatch experiments issue discussed section variables lmax maintained lines learning context decrease loss signal welcomed expected case increasing signal thus monitored upon test triggering lmax learning rate halved weight vector reset last solution line test reinitialized line experiments goal following experiments validate algorithmic ideas introduced section comparing application widely used optimization techniques see section straightforward architectures experimental setting datasets experiments performed mnist cun krizhevsky datasets respectively contain training examples contain test examples classified classes data normalized according mean standard deviation along coordinate training set algorithms adagrad nag adam used baselines agnostic adaptation rule change detection applied independently order separate effect original algorithms studied alera white lines algorithm implements agnostic learning rate adaptation without change detection uses agnostic adaptation learning rate top dam algorithm finally change detection mechanism implemented agnostic adaptation yielding salera described algorithm well version spalera algorithm appendix hyperparameters exploration space algorithms done grid possible values exception adagrad nag momentum adam alera see algorithm parameters salera spalera additional parameter part recommended values dam used dam part values used part initial learning rate ranges depending algorithm finally size set either training set size reported results based independent runs performed hyperparameter set unless otherwise specified network models experiments consider following network architectures softmax regression model loss hidden layers test takes account extreme value phenomenon considering upper cumulative deviation lower cumulative deviation defined adding resp subtracting margin sal set fully connected hidden layers relu activation top hidden layers respective sizes mnist identical model except batch normalization layers ioffe szegedy added hidden layer convolutional models cun models contain convolutional layers followed fully connected layers relu activation respective sizes mnist batch normalization used layer architectures specifically optimized task hand rather chosen compare performances past novel algorithms wide variety situations experimental conditions computations performed gpus titan pascal gtx tesla using torch library collobert double precision typical run titan pascal gpu size takes minutes algorithms metrics mnist classification problems therefore report classification accuracy test set epochs full passes training set epochs end runs well standard deviations independent runs experimental results mnist cifar nag adagrad dam dam sal spal table best performances test error tested parameter settings epochs nag dagrad dam dam sal average standard deviation runs line best results bold results less away italics learning performances experimental evidence table shows dam quite often slightly statistically significantly improves dam possible explanation dam flexible adjustment learning rate dam possibly increasing orders magnitude many cases sal yield similar results indeed whenever meet catastrophic episodes sal behaviors representative run sal undergo catastrophic episodes depicted fig runs random seed faces series catastrophic episodes training error reaches eventually stabilizes medium training loss test error meanwhile sal reacts upon first catastrophic episode around epoch halving learning rate layer faces catastrophic episode around epoch halves learning rates overall faces less frequent less severe terms train loss test error deteriorations accidents eventually sal recovers acceptable train test errors mnist cifar nag dam dam sal spal table best performance test error robust settings runparameters settings batch size momentum nag dam dam sal spal interesting note learning rates fig constantly increasing contradiction common knowledge practice learning rate behavior depends dataset neural architecture seed diverse constant decrease constant increase time increase followed decrease diverse learning rate behavior viewed original feature proposed approach made possible ability detect recover catastrophic explosions training loss actual behavior algorithms depicted model fig dam sal respectively getting first second third rank terms test error epochs terms optimization per sal respectively reaches training error close epoch resp epoch whereas dam reaches plateau epoch meanwhile test error decreases test loss increases three algorithms tentative interpretation fact neural net yields crisp output close change error increasing loss result suggests several perspectives work section order determine extent best results table depend settings define best configurations considered benchmarks sensitivity analysis performed comparing results models epochs setting choosing one lowest sum ranks results robust settings displayed table dagrad mentioned proposed sal found interestingly enough find optimal dam instead suggested original paper kingma proposed approaches still show advantage dam nag though seem sensitive parameter tuning left work derive precise recommendations depending model characteristics catastrophe management performances let define failed run run attains test error epochs observed failed runs parameter range defined catastrophe management scheme makes possible sal avoid approximately failures reaching rate failure parameter range would course possible diminish failure rate sal raby setting alarm threshold first rather used experiments reported however would potentially interfere learning rate adaptation triggering learning rate halvings even serious alert made thus preventing bold possible indeed setting causes decline half figure comparison representative run mnist random seed sal test error top row minibatch error middle row learning rates bottom row first catastrophic episode met around epoch sal reacts dividing learning rate three layers note catastrophic events met sal rare less severe sal eventually yields better training loss considerably better test error better seen color figure comparative behaviors model best configuration algorithm see text left test error middle test loss right error better seen color sal performances data shown even though manages approximately halve number failed runs furthermore setting diminish failure rate therefore harming sal learning performances hand setting significantly improve best performances higher failure rate one tenth initial loss therefore good balance aggressive learning rate adaptation scheme braking counterpart least datasets architectures discussion first proposed contribution relies comparison gradient momentum fixed reference meant estimate overall correlation among sequence gradients thought ratio process generated current solution objective successive depending ratio process accelerated slowed procedure implements idea proves significantly able increase decrease learning rate furthermore process plugged dam performance improvement average price pay flexibility increases risk catastrophic episodes instant rocketing training loss gradient norm proposed approach relies conjecture catastrophic episodes rigorously observed detected second conjecture neural net optimizer almost doomed face episodes along optimization process events mostly detrimental optimization run one often chooses small learning rates thus slow convergence prevent run mostly recovered based two conjectures second contribution paper agnostic principled way detect address episodes detection relies change point detection test soon event detected learning rates halved previous solution recovered perspective research apply proposed approach recurrent neural networks consider complex datasets another perspective replace halving trick approximating line search exploiting gaps actual momentum reference one several values momentum weight factor third perspective regards adaptation detection threshold learning runs test triggered resulting small learning rate preventing improvement goal adapt mechanism line algorithm current loss values another perspective apply sal acknowledgments heartfully thank steve altschuler lani making work possible supporting well insightful discussions also thank yann ollivier sigurd angenent insightful discussions anonymous reviewers preliminary version paper accurate constructive comments references amari natural gradient works efficiently learning neural issn doi url http andrychowicz denil colmenarejo hoffman pfau schaul freitas learning learn gradient descent gradient descent lee sugiyama von luxburg guyon garnett editors nips pages bengio simard frasconi learning dependencies gradient descent difficult trans neur mar issn doi url http bottou bousquet tradeoffs large scale learning platt koller singer roweis editors advances neural information processing systems volume pages nips foundation http url http collobert kavukcuoglu farabet environment machine learning biglearn nips workshop defazio bach saga fast incremental gradient method support convex composite objectives ghahramani welling cortes lawrence weinberger editors pages deng construction analysis large scale image ontology vision sciences society duchi hazan singer adaptive subgradient methods online learning stochastic optimization technical report eecs department university california berkeley mar url http erhan bengio courville manzagol vincent bengio unsupervised help deep learning mach learn mar issn url http george powell adaptive stepsizes recursive estimation applications approximate dynamic programming mach issn doi url http glorot bengio understanding difficulty training deep feedforward neural networks proceedings thirteenth international conference artificial intelligence statistics aistats chia laguna resort sardinia italy may pages url http goodfellow bengio courville deep learning mit press http hansen ostermeier completely derandomized evolution strategies evolutionary computation hardt recht singer train faster generalize better stability stochastic gradient descent corr url http hinkley inference change point cumulative biometrika ioffe szegedy batch normalization accelerating deep network training reducing internal covariate shift proceedings international conference machine learning icml lille france july pages url http jacobs increased rates convergence learning rate adaptation neural networks kingma adam method stochastic optimization corr url http krizhevsky learning multiple layers features tiny images technical report cun bottou bengio haffner learning applied document recognition proceedings ieee pages martens deep learning via optimization proceedings international conference machine learning june haifa israel pages url http martens sutskever swersky estimating hessian curvature icml omnipress nesterov method solving convex programming problem convergence rate soviet mathematics doklady ollivier arnold auger hansen optimization algorithms unifying picture via invariance principles journal machine learning research url http page continuous inspection schemes biometrika pascanu bengio revisiting natural gradient deep networks international conference learning representations conference track apr url http pascanu mikolov bengio difficulty training recurrent neural networks proceedings international conference international conference machine learning volume icml pages url http polyak methods speeding convergence iteration methods ussr computational mathematics mathematical physics robbins monro stochastic approximation method annals mathematical statistics schaul zhang cun pesky learning rates proceedings international conference international conference machine learning volume pages srivastava hinton krizhevsky sutskever salakhutdinov dropout simple way prevent neural networks overfitting mach learn issn url http sutskever martens dahl hinton importance initialization momentum deep learning proceedings international conference international conference machine learning volume icml pages url http tieleman hinton lecture rmsprop coursera neural networks machine learning technical report technical report wierstra schaul glasmachers sun peters schmidhuber natural evolution strategies mach learn issn zeiler adadelta adaptive learning rate method corr url http zinkevich online convex programming generalized infinitesimal gradient ascent proceedings twentieth international conference international conference machine learning icml pages aaai press isbn url http appendix derivation formula let define unit vector uniformly drawn ird recurrence thus krt kul dealing unit vectors uniformly drawn thus denoting kronecker delta kul taking expectation krt krt derivation formula similar appendix safe agnostic learning rate adaptation algorithm gives detailed implementation version white lines sal plus greyed lines briefly described section line denotes multiplication algorithm spal change detection input model parameter loss function parameters memory rate factor parameters initial learning rate ratio run parameters initialize init initialization lmin first initialize stopping criterion met new perform forward pass empirical mean batch losses lmin min lmin lmin lmin cumulated deviations mean lower bound deviations triggered backtrack else save possible backtracks compute gradient backward pass exponential moving average normalized gradients exp agnostic learning rate update end standard parameter update end end appendix analysis dividing factor use context notations section make even simpler assuming one dimension gradient direction change minimizing parabola straightforward show independently current solution optimal value learning rate value loss deteriorate let assume current learning rate recovery phase sal used prevent catastrophic event dividing factor let show best bringing back reaching optimal value discussed section test triggered first phase brings successive divisions number divisions log log second phase uses standard procedure reach value context let assume simplified procedure updates multiplying small let consider two cases depending whether smaller greater compute expectation number iteration sal needed reach hence standard update phase decreases multiplying becoming less close length update phase increases thus construction minimal meanwhile length division phase decreases increases thus optimal value let consider two intervals respective expectations number iterations reach expectation total number iteration sum weighted probability arriving respective intervals lengths intervals weights hence expected value independent counting number log multiplications needed get close number iterations log log integrating gives log log log log similarly expected value computed interval using approach counting number multiplications needed get close comes log log log log log let summarize different computational costs involved catastrophe detected grail reached cost dividing iterations involves forward pass current minibatch let denote cost hand standard iterations larger cost involving forward pass plus backward pass weights update let denote cost looking value minimize total cost reaching catastrophic event detected minimizes equivalently minimizes log log log log average cost salera zeta figure cost function equation different values constant plots smaller values indistinguishable log easy empirically check figure global minimum depends value constant assumed small increases asymptotic value however value initially chosen historical reasons reference famed doubling trick frequently used different areas machine learning light results simple case work investigate slightly larger values
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discrete mathematics theoretical computer science dmtcs vol jun universal partial words herman sergey torsten brian school science tianjin chengjian university china department computer information sciences university strathclyde glasgow institut mathematik berlin germany college mathematics system science xinjiang university urumqi xinjiang china received revised may accepted may universal word finite alphabet integer word every word appears exactly subword cyclically linearly easy prove universal words exist work initiate systematic study universal partial words words addition letters may contain arbitrary number occurrences special joker symbol substituted symbol example linear partial word binary alphabet first three letters yield subwords present results existence linear cyclic universal partial words different situations depending number positions including various explicit constructions also provide numerous examples universal partial words found help computer keywords universal word partial word bruijn graph eulerian cycle hamiltonian cycle introduction bruijn sequences topic combinatorics years found widespread use applications areas molecular biology computer security computer vision robotics psychology experiments recently also studied general context constructing universal cycles fundamental combinatorial structures permutations subsets fixed ground set see context words finite alphabet say word universal contains every word length exactly subword distinguish cyclic universal words linear universal words cyclic setting view cyclic word consider subwords length cyclically across boundaries linear setting hand view linear word consider subwords start end within index range letters last author supported scientific research program higher education institution xinjiang uygur autonomous region natural science foundation xinjiang university issn author distributed creative commons attribution international license herman chen sergey kitaev torsten brian sun definition follows length cyclic linear universal word must respectively example binary alphabet linear universal word observe cyclic universal word easily transformed linear universal word existence results cyclic setting imply existence results linear setting note also reversing universal word permuting letters alphabet yields universal word following classical result starting point work see theorem finite alphabet exists cyclic universal word standard proof theorem really beautiful concise using bruijn graph line graph eulerian cycles see section universal partial words paper consider universality partial words words addition letters may contain number occurrences additional special symbol idea every occurrence substituted symbol think joker wildcard symbol formally define say word appears factor word integer cyclic setting consider indices definition modulo example linear setting ternary alphabet word occurs twice factor subwords whereas appear factor partial words introduced applications see references therein combinatorics partial words appear context primitive words avoidability sets partial words also study number squares infinite partial words concept partial words extended permutations notion universality given extends straightforwardly partial words refer universal partial word upword short distinguish cyclic upwords linear upwords simplest example linear upword word consisting many call trivial let consider interesting examples linear upwords binary alphabet linear upword whereas linear upword replacing first two letters yields factor last three letters similarly linear upword word appear factor word appears twice factor results work initiate systematic study universal partial words turns words rather shy animals unlike ordinary counterparts universal words without joker symbols stark contrast theorem general existence results upwords also many results borderline two cases seems rather complicated makes subject even interesting true also alphabets constructions paper indicate addition size alphabet length factors also consider number positions upword problem parameters universal partial words thm thm thm thm thm thm thm thm thm thm thm thm thm thm thm thm thm thm tab examples linear upwords single position beginning end possible values upwords closer end word beginning obtained reversal dash indicates upword exists first focus linear upwords linear upwords containing single following results alphabets linear upword single theorem binary alphabet situation interesting see table denoting position linear upword theorem linear upwords following three cases theorem conjecture cases herman chen sergey kitaev torsten brian sun cor cor thm thm thm tab examples linear upwords two binary alphabet conjecture support conjecture performed search indeed found linear upwords values possible values ones excluded beforementioned results examples listed table remaining ones available third author website www also prove special cases conjecture theorems respectively linear upwords containing two following results first table shows examples linear upwords two binary alphabet establish sufficient condition binary linear upwords two theorem particular shows ways placing two among positions yield valid upword moreover conclude two binary linear upwords two adjacent corollary namely see table also construct infinite family binary linear upwords two theorem let discuss cyclic upwords note trivial solution cyclic upword universal partial words cyclic setting following rather general result gcd cyclic upword corollary particular binary alphabet odd cyclic upword fact know single cyclic upword binary alphabet namely cyclic shifts reversal letter permutations outline paper paper organized follows section introduce notation collect basic observations used throughout rest paper sections prove results linear upwords containing single two respectively section contains proofs cyclic upwords finally section discusses possible directions research including extensions results alphabets preliminaries rest paper assume alphabet denotes size alphabet often consider special case binary alphabet write complement moreover word let denote length mentioned reversing universal word permuting letters alphabet yields universal word thus assume upword letters appear increasing order left right first occurence symbol first occurence symbol whenever moreover factored xyz contain assume one standard approach prove existence universal words define suitable graph search hamiltonian graph another algebraic approach uses irreducible polynomials specifically bruijn graph gna vertices elements words length directed edge vertex vertex whenever last letters first letters call edge last letter equals figure shows graph gna respectively clearly linear universal word corresponds hamiltonian path gna cyclic universal word hamiltonian cycle graph observe furthermore gna line graph recall line graph directed graph directed graph vertex every edge directed edge end vertex equals starting vertex therefore problem finding hamiltonian gna equivalent finding eulerian existence eulerian follows fact bruijn graph connected vertex one euler famous theorems see also theorem proves theorem fact existence proof easily turned algorithm actually find many universal words using hierholzer algorithm fleury algorithm discuss standard approach proving existence universal words extended universal partial words specifically collect simple powerful observations used proofs later vertex gna let denote sets herman chen sergey kitaev torsten brian sun fig bruijn graphs spanning subgraph linear upword shown solid edges part figure shows schematic drawing graph line graph highlighted sequences edges respectively sets vertices gna mentioned clearly observation vertex gna set vertices different set given vertex gna set vertices different set given linear upword define spanning subgraph bruijn graph gna follows see figure let denote set words obtained subword length starting position replacing occurences letter alphabet clearly many subword different possibilities substitution note sets form partition vertex set gna directed edges given edges gna induced every pair consecutive sets example linear upword binary alphabet spanning subgraph shown figure give another example linear upword binary tree depth additional edge emanating root universal partial words solid edges fig illustration proof lemma following observation follows straightforwardly definitions observation let linear upword vertex vertex vertices vertices last observation graph determined positions intuitively lead branching graph due different possibilities substituting symbols particular spanning path gna hamiltonian path back setting theorem searching linear universal partial word particular number certain positions essentially search copy spanning subgraph gna exploit idea existence proofs constructions particularly useful computationally much efficient search copy gna directly rather search ing sequences edges seen generalizations eulerian paths used proof theorem see figure example search linear upword single position prescribe first letters letters particular choice symbols iterating possible choices search remains deleting edges correspond eulerian path subgraph prescribed prefix see proofs theorems idea generalized straightforwardly search upwords patterns see example proof theorem next lemma used repeatedly proofs existence upwords proof uses previous two observations derive dependencies letters upword lemma let linear upword require moreover proof observation vertex set vertex observation vertices different ones gna set herman chen sergey kitaev torsten brian sun many namely see figure every vertex gna exactly vertices except ones already least follows vertices must equal one vertices follows moreover least two vertices ending different symbols must equal one vertices impossible words set end symbol follows must linear upwords single results first result completely excludes existence linear upwords single alphabets theorem linear upword single proof suppose upword exists claim preceded followed least symbols would different factors strictly less assume followed least symbols lemma implies therefore means word appears twice factor starting positions words vertex identical vertex contradiction next result excludes several cases single binary alphabet theorem linear upword single position beginning end proof first consider case suppose upword assuming must otherwise word would appear twice factor word appear factor appears twice contradiction rest proof assume suppose upword note equivalently holds assumption followed least symbols applying lemma yields means word appears twice factor starting positions contradiction contrast theorem binary alphabet exclude following three small cases addition cases excluded theorem exceptions marked table theorem linear upword single position beginning end following three cases universal partial words proof suppose upword case applying lemma twice reverse obtain contradiction prove second case suppose upword form applying lemma twice reverse obtain form assume word must appear somewhere factor since possible starting positions however starting positions excluded immediately would cause appear twice factor hand starts positions neighboring letters must causing respectively appear twice factor contradiction proof third case proceeds similarly second case allows conclude upword must form assume word must appear somewhere factor since possible starting positions starting positions excluded immediately would cause appear twice factor hand starts positions neighboring letters must causing respectively appear twice factor contradiction existence results conjecture binary alphabet single cases discussed previous section ones conjecture linear upword containing single position every case covered theorem theorem recall numerical evidence conjecture discussed introduction remainder section prove cases general conjecture theorem linear upword single first position begins note lemma every linear upword single form satisfies conditions begins letter permutations proof consider word corresponding three edges bruijn graph denote graph tained removing three edges isolated vertex clearly edges form connected graph every vertex exactly two except vertex one vertex one therefore eulerian path starting ending eulerian path yields desired upword begins binary word write word given herman chen sergey kitaev torsten brian sun fig subgraph constructed proof theorem informally speaking obtained concatenating infinitely many copies truncating resulting word length complementing last symbol example using terminology starting segment linear upword theorem written next result considerable extension previous theorem theorem linear upword single position begins idea proof theorem straightforward generalization approach used prove theorem boils showing bruijn graph without edges given prescribed upword prefix still eulerian path proof words table show statement true rest proof assume consider word let denote two words obtained substituting respectively moreover let unique word unique word define proceed show two words defined coincide namely given first letters given first letters equal words corresponding set vertices size see figure verified directly considering number leading trailing vertices assume every word except possibly contains factor exactly uniquely identified position factor proving words uniquely identified number leading equals implying show disjoint prove use words except possibly contain factor moreover word contains factor however contain factor case starts ends unlike words case proving disjoint remains show words contain least one universal partial words word satisfies last letter one positions left complementary recall definition property hold words implying moreover case words contain factor exactly uniquely identified position factor might contain factor last two positions potential conflict could arise case ends however case still different conclude cases combining observations shows claimed consider set edges bruijn graph see figure span subgraph following pairs vertex vertex vertices vertex denote graph obtained removing edges isolated vertex clearly every vertex except vertex one vertex one complete proof theorem show contains eulerian path must start end suffices degree conditions show connected first consider case vertex follow either reach vertex vertex next could happen right beginning case follow reach vertex follow reach ever follow edges forward direction claim process never use edge shows connected see suppose encounter vertex next means trailing use following leads vertex trailing next step vertex trailing note vertices trailing trailing avoid edges way moreover way via use edges either vertex starts least two transitions vice versa reading left right using consider case vertex follow either reach vertex vertex next case follow single follow reach similarly need argue never use edge process way never use edges vertices path except first one last two contain factor vertices different word contains factor implying edges except possibly last one safe however since last edge safe arguments show connected eulerian path eulerian path yields desired upword begins completes proof herman chen sergey kitaev torsten brian sun linear upwords two section focus binary alphabets many conditions provided section generalized straightforwardly alphabets briefly discuss section results theorem linear upword two form table shows examples linear upwords two whenever conditions theorem violated put differently every upword table one numbers note already first condition choices placing two among positions excluded possible candidates upwords proof first assume applying lemma yields word appears twice factor contradiction assume case follows symmetry note number factors every subword ending first letter contains one giving rise two factors every subword ending second letter contains two giving rise four factors number strictly less therefore must case assume applying lemma yields implying word appears twice factor contradiction assume case must number factors strictly less assume let consider subword since position since applying lemma yields moreover lemma yields contradiction hand factors obtained subword appear twice starting position position contradiction corollary linear upwords containing two adjacent reversal letter permutations proof linear upwords two adjacent follows theorem upword subword two empty estimate third part proof theorem strengthened show number factors strictly less unless means continue argument leading contradiction exceptional case excluded follows applying lemma shows universal partial words becomes clear factor whatever position within placed would appear twice possible linear upwords two adjacent lemma leads covered must impossible appears twice factor starting positions possible linear upword two existence results next result provides infinite number binary linear upwords two see table theorem linear upword two begins proof consider word easy check yields different factors factors gives rise edge bruijn graph set edges end vertices form connected subgraph vertices except respectively denote graph obtained removing edges vertices clearly every vertex except vertex one outgoing edge vertex one incoming edge complete proof theorem show contains eulerian path must start end suffices degree conditions show connected consists edges connected graph rest proof assume consider vertex ends consider maximum number trailing note vertices correspond cases follow alternatingly starting either reach vertex vertex could happen right beginning follow vertex ends following follow single proceed vertex directly follow note ever follow edges forward direction claim process never use edge shows connected see first consider case start vertex trailing vertex reached via vertex segment consecutive surrounded also none next vertices reaching contain factor unlike word vertex reached following either stop vertex ends next vertex reached via could subsequent vertices including since contain factor unlike word shows none edges traversed moreover none vertices traversed contain factor unlike word indeed reach without using edges herman chen sergey kitaev torsten brian sun consider case start vertex ends interesting case two starting vertex ends namely edges starting however different part different assumption conclude follow arguments show connected eulerian path eulerian path yields desired upword begins completes proof cyclic upwords throughout section indices considered modulo size corresponding word notions introduced section extended straightforwardly cyclic upwords factors taken cyclically across word boundaries particular defining graph cyclic upword consider subsets words cyclically first two statements observation hold vertices next lemma analogue lemma cyclic upwords lemma let cyclic upword proof suppose observation vertex set vertex observation vertices different ones gna set many namely every vertex gna exactly vertices already least follows vertices part contradiction fact spanning subgraph gna lemma immediately yields following corollary captures various rather severe conditions cyclic upword must satisfy relating length size alphabet value parameter corollary let cyclic upword least one divides proof lemma two symbols distance must well thus indices partitioned gcd many residue classes modulo symbols positions residue class either letters let denote number among consecutive symbols least one letters starting position gives rise different factors implying furthermore many within consecutive letters partitioned gcd many blocks pattern gcd must divide condition equivalent dividing gcd dividing immediate corollary last result exclude existence cyclic upwords many combinations universal partial words corollary let gcd cyclic upword particular odd cyclic upword proof suppose upword exists corollary divides however gcd divide must divide impossible yielding contradiction corollaries binary alphabet remaining potential parameter values cyclic upwords etc case easily exluded word form leading factor appearing twice appear factor however cyclic upword already mentioned introduction cyclic upword binary alphabet know cyclic upwords even alphabet size constructed paper outlook paper initiated systematic study universal partial words hope results numerous examples upwords provided tables see also extensive data available website www generate substantial interest researchers continue exploration possibly one directions suggested concerning binary alphabet would interesting achieve complete classification linear upwords containing single suggested conjecture two task seems somewhat challenging recall table theorem see data www examples binary linear upwords three listed table deriving general existence results setting would certainly interest next step would consider situation three present linear upword following example direction communicated rachel kirsch theorem linear upword many complementing theorem prove following result direction possible obtain general results theorem linear upword begins proof theorem easy applying lemma first second leave details reader would also interesting find examples binary cyclic upwords mentioned finally natural direction would search linear cyclic upwords alphabets anticipate upwords exist cases recall theorem evidence following general result setting theorem large enough linear cyclic upword exactly many herman chen sergey kitaev torsten brian sun thm tab examples linear upwords three theorem shows particular fixed alphabet size fixed number diamonds finitely many possible candidates upwords diamonds principle could checked exhaustive search proof idea fixed large enough upword must contain symbol distance applying lemma lemma yields contradiction recall proof theorem omit details positive side upwords even alphabet sizes constructed upwords even cyclic question touched paper algorithmic problem efficiently generating upwords preliminary observation direction remark linear upwords constructed theorem also obtained straightforward modifications fkm bruijn sequences constructed efficient generation algorithms known acknowledgements authors thank martin gerlach assistance computer searches rachel kirsch collaborators well artem pyatkin providing particular examples small upwords second author grateful sergey avgustinovich helpful discussions universal partial words bill chen arthur yang hospitality author visit center combinatorics nankai university november work supported project pcsirt project ministry education national science foundation china also thank universal partial words anonymous referees paper several valuable suggestions references helped improving presentation references berstel boasson partial words theorem fine wilf theoret comput words rouen blakeley gunter rampersad complexity deciding avoidability sets partial words theoret comput gutin digraphs theory algorithms applications springer edition primitive partial words discrete applied mathematics open problems partial words editors new developments formal languages applications pages springer berlin heidelberg berlin heidelberg brownstein kalcic palumbo weyand unavoidable sets partial words theory comput chung diaconis graham universal cycles combinatorial structures discrete claesson kitaev pattern avoidance partial permutations electron paper compeau pevzner tesler apply bruijn graphs genome assembly nature biotechnology bruijn combinatorial problem koninklijke nederlandse akademie wetenschappen euler solutio problematis geometriam situs pertinentis comment academiae sci petropolitanae fredricksen kessler algorithm generating necklaces beads two colors discrete fleury deux problemes geometrie situation journal mathematiques elementaires pages fredricksen maiorana necklaces beads colors bruijn sequences discrete herman chen sergey kitaev torsten brian sun goeckner groothuis hettle kell kirkpatrick kirsch solava universal partial words alphabets nov halava harju partial words inform process halava harju infinite partial words theoret comput holroyd ruskey williams shorthand universal cycles permutations algorithmica hurlbert universal cycles beyond bruijn proquest llc ann arbor thesis state university new jersey new brunswick hierholzer wiener ueber die einen linienzug ohne wiederholung und ohne unterbrechung umfahren math lempel closed sequences combinatorial theory ser salvi collewet forest optimised bruijn patterns shape acquisition image vision computing ralston bruijn model example interaction discrete mathematics computer science math ruskey savage wang generating necklaces algorithms sohn bricker simon hsieh optimal sequences trials balancing practice repetition effects behavior research methods instruments computers scheinerman determining planar location via brujin sequences using discrete optical sensors ieee transactions robotics automation dec stevens williams coolest way generate binary strings theory comput www currently http yoeli binary ring sequences amer math monthly
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online stochastic matching new algorithms brian karthik aravind pan nov department computer science university maryland college park usa original version may version november abstract online matching received significant attention last years due close connection internet advertising seminal work karp vazirani vazirani optimal competitive ratio standard adversarial online model much effort gone developing useful online models incorporate stochasticity arrival process one popular model known model different arrive online known distribution develop algorithms improved competitive ratios basic variants model integral arrival rates including case general weighted edges improve ratio due haeupler mirrokni zadimoghaddam case improve ratio jaillet also consider extension stochastic rewards variant edge independent probability present setting stochastic rewards arrival rates present simple optimal algorithm ratio special case edge unweighted uniform constant probability present improve upon proposing strengthened benchmark one key ingredients improvement following offline approach bipartitematching polytopes additional constraints first add several valid constraints order get good fractional solution however give less control structure next remove additional constraints randomly move feasible point matching polytope coordinates set chosen integer structure solution inspired jaillet mathematics operations research tractable structure algorithm design analysis appropriate random move preserves many removed constraints approximately high probability exactly expectation underlies improvements could independent interest preliminary version appeared european symposium algorithms esa email bbrubach supported part nsf awards cns ccf email kabinav supported part nsf awards cns ccf email srin supported part nsf awards cns ccf research awards adobe email panxu supported part nsf awards cns ccf introduction applications internet advertising driven study online matching problems recent years problems consider bipartite graph set vertices available offline set vertices arrive online whenever vertex arrives must matched immediately irrevocably one vertex offline vertex matched one context internet advertising set advertisers set impressions edges define impressions interest particular advertiser impression arrives must choose available advertiser match wlog consider case matched since advertising forms key source revenue many large internet companies finding good matching algorithms obtaining even small performance gains high impact stochastic known model arrival given bipartite graph finite online time horizon assume round vertex sampled replacement known distribution sampling distributions independent identical online rounds captures fact often historical data impressions predict frequency type impression arrive matching general model context advertising every advertiser gains given revenue matched particular type impression type impression refers class users demographic group interested subset advertisements special case model matching weights associated advertisers words given advertiser revenue generated matching user types interested modern business models revenue generated upon matching advertisements user clicks advertisement model historical data one assign probability particular advertisement clicked type user works including capture notion stochastic rewards assigning probability edge one unifying theme approaches use benchmark additional valid constraints hold respective models use optimal solution guide online actions use various modifications dependent randomized rounding preliminaries technical challenges unweighted online known stochastic bipartite matching problem given bipartite graph set available offline vertices arrive online drawn replacement distribution given arrival rate expected number times arrive exception sections paper focus integral arrival rates setting reasons described assume wlog assumption integral arrival rates case total number online rounds variant every vertex weight seek maximum weight matching variant every edge weight seek maximum weight matching stochastic rewards variant edge weight probability present probe edge seek maximize expected weight matching asymptotic assumption notation always assume large analyze algorithms goes infinity write instead suppressed terms subtract competitive ratios another fact note competitive ratio defined slightly differently usual set problems similar notation used alg particular defined opt algorithms adaptive arrives adaptive algorithm modify online actions based realization online vertices thus far algorithm specify actions start online phase throughout use refer worst case instance various algorithms benchmark deterministic rewards prior work see use following upper bound optimal offline expected performance also use guide algorithm case rewards deterministic case stochastic rewards use slightly modified lps whose definitions defer sections first show unweighted variant describe changes settings usual variable edge let set edges adjacent vertex let maximize subject variants objective function maximize variant maximize variant refers constraint matching constraint vertices constraint valid vertex arrival rate constraint used captures fact expected number matches edge valid large probability given vertex arrive rounds constraint similar previous one pairs edges two neighbors given probability neither arrive therefore sum variables two distinct edges exceed notice constraints reduces gap optimal solution performance optimal online algorithm fact without constraint general achieve competitive ratio better edge realization process independent different edges step algorithm probes edge probability edge exists remaining probability realization edge determined affect random realizations rest edges note work use optimal value offline instance instead use simulations wherein simulate arrival sequence compute vector approximating via simulation probability matching edge offline optimal solution use similar approach problems reasons weighted variants namely edge versions number samples depends maximum value weight making expensive unweighted version running time sampling based algorithm hand show section based algorithm solved much faster time worst case even faster practice stochastic rewards setting offline problem known solvable paper shows assumption constant opt obtain approximation optimal solution however assumptions strong used setting finally stochastic rewards setting one might tempted use achieve property obtained simulation via adding extra constraints context uniform stochastic rewards edge associated uniform constant probability really need exp guarantee via straightforward approach add family constraints however number constraints exponential seems obvious separation oracle overcome challenge showing suffices ensure inequality holds constant thus resultant polynomial solvable overview algorithm contributions key challenge encountered special could lead length four cycles type shown figure fact used cycle show algorithm could perform better using mentioned tighter constraints section could avoid bottleneck propose technique use note solution produced essential component random list algorithm show randomized rounding algorithm construct similar simplified vector solution stricter benchmark allows inclusion additional constraints importantly constraint using rounding algorithm combined tighter constraints upper bound probability vertex appearing cycle figure see lemma additionally show deterministically break length four cycles type without creating new cycles type finally describe algorithm utilizes techniques improve previous results unweighted settings algorithm first solve section input graph section show use technique obtain sparse fractional vector present randomized online algorithm similar one uses sparse fractional vector guide achieve competitive ratio previously gap best unweighted algorithm ratio due negative result figure cycle source negative result described jaillet thick edges thin edges due take step towards closing gap showing algorithm achieve unweighted variants integral arrival rates make progess open questions book overview algorithm contributions challenge arises applying power two choices setting edge included matchings case copy offer benefit second arrival wasted use example related work haeupler choose two matchings following way attained solving constraints rounding integral solution constructed finding maximum weight matching removing edges already included key element proof showing probability edge removed approach paper construct two three matchings together correlated manner reduce probability edge included matchings show general technique construct ordered set matchings easily adjustable parameter show probability edge appearing algorithms presented first solve input graph round solution vector sparse integral vector use vector construct randomly ordered set matchings guide algorithm online phase begin section simple algorithm uses set two matchings guide achieve competitive ratio improving best known result problem follow slight variation improves ratio complex algorithm relies convex combination algorithm separate algorithm overview stochastic rewards contributions algorithm algorithm presented section believe known model stochastic rewards interesting new direction motivated work adversarial model introduce new general see specifically setting show simple algorithm using solution directly achieve competitive ratio proved optimal among algorithms shown randomized algorithm achieve ratio better adversarial open questions state following general close gap upper lower bounds sense ratio achieved integral case nice round number one may suspect correct table summary contributions problem previous work paper section section unweighted stochastic rewards section general version restricted version model hence achieving model shows lower bound extend model paper shows using one achieve ratio better discuss challenges relating techniques used prior work directly extend model finally consider restricted version problem edge unweighted uniform constant probability integral arrival rates proposing family valid constraints able show restricted setting one indeed beat summary contributions theorem online stochastic matching integral arrival rates online algorithm achieves competitive ratio least theorem online stochastic matching integral arrival rates exists algorithm achieves competitive ratio least algorithm achieves competitive ratio least theorem online stochastic matching arbitrary arrival rates stochastic rewards online algorithm achieves competitive ratio optimal among algorithms theorem unweighted online stochastic matching integral arrival rates uniform stochastic rewards exists adaptive algorithm achieves competitive ratio least runtime algorithm section discuss implementation details algorithm algorithms solve step dimension determined constraint matrix consists rows columns however note number entries matrix order recent work shows sparse programs solved time using interior point methods known perform well practice sparsity reason solve large instances problem second critical step perform randomized rounding note variables step randomized rounding due incur running time hence total running time obtain rounded solution order finally recall operations part step hence online phase algorithm incurs running time stochastic rewards case fact smarter implementation using binary search runs fast log algorithms section rounding technique algorithms presented first solve benchmark input instance get fractional solution vector round integral solution using two step process call first step multiply second step apply dependent rounding techniques gandhi khuller parthasarathy srinivasan new vector paper always choose vertex may appear probably two three times dependent rounding typically applied values useful properties extend naturally case kfe may greater edge understand process easiest imagine splitting kfe two edges integer value fractional value kfe former remain unchanged dependent rounding since already integer latter rounded probability otherwise final value would sum two rounded values two properties dependent rounding use marginal distribution every edge let kfe vertex let fractional degree kfw kfe integral degree random variable related work study online matching began seminal work karp vazirani vazirani gave optimal online algorithm version unweighted bipartite matching problem vertices arrive adversarial order following series works studied various related models book mehta gives detailed overview version introduced aggarwal goel karande give optimal ratio adversarial arrival model setting studied adversarial model feldman korula mirrokni muthukrishnan consider additional relaxation addition adversarial known models online matching also studied several variants random arrival order unknown distributions known adversarial distributions setting random arrival order arrival sequence assumed random permutation online vertices see case unknown distributions round item sampled fixed unknown distribution sampling distributions required round called unknown otherwise called adversarial stochastic input known adversarial distributions round item sampled known distribution allowed change time another variant problem edges stochastic rewards models stochastic rewards previously studied among others known model related work unweighted settings unweighted settings many results starting feldman mehta mirrokni muthukrishnan first beat competitive ratio unweighted problem improved manshadi gharan saberi adaptive algorithm addition showed even unweighted variant integral arrival rates algorithm achieve ratio better finally jaillet presented adaptive algorithm used clever achieve unweighted problems respectively related work setting model haeupler mirrokni zadimoghaddam first beat achieving competitive ratio use discounted tighter constraints basic matching similar seen employ power two choices constructing two matchings offline guide online algorithm related work devanur gave algorithm achieves ratio adwords problem unknown arrival model knowledge optimal budget utilization ratios alaei considered matching problem arrives distinct known distribution round gave competitive algorithm minimum capacity matching integral arrival rates simple algorithm describe simple algorithm achieves competitive ratio introduces key ideas approach begin solving get fractional solution vector applying described subsection get integral vector construct bipartite graph copies edge note max degree since thus decompose two matchings using hall theorem finally randomly permute two matchings ordered pair matchings matchings serve guide online phase algorithm similar entire algorithm model denoted summarized algorithm analysis algorithm show algorithm achieves competitive ratio let randomly ordered pair matchings note might exist edge appears algorithm construct solve benchmark input instance let optimal fractional solution vector call get integral vector create graph copies edge decompose two matchings randomly permute matchings get random ordered pair matchings say vertex arrives first time try assign arrives second time try assign vertex arrives third time nothing step matchings therefore consider three types edges say edge type denoted appears similarly appears finally appears let probabilities getting matched respectively according result haeupler lemma bounds probabilities lemma proof details section given worst case use lemma prove algorithm achieves ratio examining probability given edge becomes type proof analysis consider following two cases case marginal distribution property dependent rounding one copy probability including since edge appear either equal probability thus ratio case similarly marginal distribution follows thus ratio noting first term case second term case edge algorithm section describe improvement upon previous algorithm get competitive ratio start making observation performance algorithm solving let edges called large edges called small let sets large small edges respectively notice previous analysis small edges achieved much higher competitive ratio versus large edges primarily due fact may get two copies large edge case copy better chance matched since edge block edge offline neighbor gets matched first copy chance matched correct imbalance make additional modification values applying rest algorithm exactly let parameter optimized later large edges set small edges adjacent large edge let largest edge adjacent note two large neighbors care largest one set words increase values large edges ensuring reducing values neighboring small edges proportional original values note possible two large edges adjacent since must small edges adjacent large edges leave values unchanged apply new vector multiplying applying dependent rounding analysis prove theorem proof analysis consider large small edges separately two cases case adjacent large edges case analysis algorithm still get competitive ratio edges case adjacent large edge case let value largest neighboring edge original solution achieves ratio follows lemma particular first two terms result set algorithm two numbers probabilities matched respectively note decreasing function respect worst case due constraint ratio ratio edge follows fact decreasing function respect see decreasing function note rearranged substituting appropriate values value hence expression written show done optimal value choose turns hence choosing optimal value yields overall competitive ratio new algorithm need show value ensures less modification since note modified value always less equal original value decreasing range value less since algorithm next subsections describe final algorithm attenuation factors keep modular give following guide reader note definition large small edges given subsection different definition previous subsection section describes main algorithm internally invokes two algorithms described sections respectively theorem proves final competitive ratio proof depends performance guarantees given lemmas respectively proof lemma depends claims found appendix claims careful analysis intuitively refers case offline vertex incident one large edge one small edge analysis large edge refers case incident three small edges refers case incident small edge large edge analysis small edge proof lemma depends claims found appendix claims proven careful analysis since many cases given diagram cases prove algorithm section describe algorithm algorithm achieves competitive ratio algorithm first solves benchmark subsection obtains fractional optimal solution invoking obtains random integral solution notice constraint see therefore consider graph edge associated value say edge large small note differs definition large small subsection design two algorithms denoted take sparse graph input difference two algorithms favors small edges favors large edges final algorithm take convex combination run probability probability algorithm solve benchmark input let optimal solution vector invoke obtain vector independently run probabilities respectively details algorithm proof theorem presented following sections algorithm section describe randomized algorithm algorithm suppose view graph another way edge copies let refer process constructing graph copies edge decomposing three matchings randomly permuting matchings first invokes obtain random ordered triple matchings say notice constraint properties edge appear two three matchings small edge say type two neighbors say type exactly one neighbor wlog assume every otherwise add dummy node neighborhood note use terminology assign denote edge matched algorithm matched step algorithm invoke obtain random ordered triple matchings say vertex comes first time assign comes second time assign comes third time either large edge small edge type assign however small edge type probability assign otherwise nothing comes fourth time nothing step parameter fix end analysis let competitive ratio small edge large edge respectively lemma achieves competitive ratio large small edge respectively proof case large edge divide analysis three cases case corresponds one three matchings combine conditional probabilities using bayes theorem get final competitive ratio two types small edges similarly condition based matching appear combine using bayes theorem complete version proof found section appendix algorithm algorithm algorithm algorithm takes input performs well large edges recall previous algorithm first invokes obtain random ordered triple matchings contrast invoke routine denoted algorithm generate random ordered pair recall integral solution vector wlog assume every otherwise dummy vertex ensure case algorithm suppose two neighbors say large edge small edge add primary matching secondary matching suppose three neighbors incident edges take random permutation say add probability probability parameters fixed analysis algorithm describes algorithm invoke generate random ordered pair say vertex comes first time assign comes second time try assign vertex comes third time nothing step let competitive ratios small edges large edges respectively lemma achieves competitive ratio large small edge respectively proof analyze basis considering local neighborhood edge large edge two possible cases neighborhood small edge eight possible cases fact large edge small edges neighborhood small edge large small edges neighborhood choosing worst case among two large edge worst case among eight small edge prove claim complete details proof found section appendix convex combination section prove theorem proof let competitive ratios achieved large small edges respectively similarly let denote following two cases marginal distribution property know thus final ratio properties know thus final ratio competitive ratio convex combination maximized value stochastic matching integral arrival rates section consider online stochastic matching bipartite graph known model integral arrival rates present algorithm offline vertex competitive ratio least recall invoking obtain random integral vector define let graph induced edge thus takes value notice implies neighbors ignore edges section focus sparse graph main idea algorithm follows solve benchmark section let optimal solution vector invoke obtain integral vector fractional vector apply series modifications transform another solution see section run randomized list algorithm based denoted rla graph first briefly describe overcome bottleneck case algorithm explain algorithm full detail case shown figure happens node competitive ratio recall analyze algorithm considering cases various neighborhood structure given offline vertex analysis node figure competitive ratio least hence boost performance cost specifically increase value decrease value cases figure illustrate modification new cycle shown figures fact unweighted case well however lemma implies cycles avoided probability least helps improve ratio even unweighted case lemma describes formally lemma given appears cycle probability proof consider graph obtained notice vertex appear cycle must neighboring edge try bound probability event easy see hence thus edges possibly rounded note two edges since hence following two cases case contains one edge let know notice hence since increasing function constraint case contains two edges let probability either note two events mutually exclusive since using analysis case follows constraint know hence present details rla based given section discuss two modifications transforming section give formal statement algorithm section related analysis rla algorithm discuss apply rla based sparse graph let set neighboring nodes assume wlog thus least neighbors since satisfies better situation additionally see section two modifications sum edge values incident unchanged hence time vertex comes rla first generates random list permutation based follows say sample random list say sample permutation verify sampling distributions described equations valid since full details random list algorithm rla shown algorithm algorithm rla random list algorithm based vertex comes generate random list satisfying equation list matched drop vertex otherwise assign first unmatched list two kinds modifications stated earlier first modify running rla algorithm section describe modifications first modification figure left jaillet case right three possible types cycles length applying thin edges thick edges first modification break cycles deterministically three possible cycles length graph denoted give efficient way break shown figure cycle modified hence bottleneck unweighted case notice breaking cycles new cycles created graph since randomized construction solution gives control probability cycles occurring would like break controlled way create new cycles procedure summarized algorithm correctness proved lemma algorithm cycle breaking algorithm offline phase cycle type break cycles type break one cycle type return first step lemma applying algorithm value preserved cycle type exists new cycle type added proof lemma lemma follows following three claims claim breaking cycles change value claim breaking cycle type vertices never part length four cycle claim length four cycles type breaking exactly one cycle type create new cycle type proof claim shown figure increase decrease edge values way sums vertex preserved notice cycles freely broken without creating new cycles removing cycles type removing single cycle type create cycles type hence algorithm removes cycles without creating new cycles proof claim consider structure breaking cycle type note edge permanently removed hence four vertices together never part cycle length four vertices respectively edges therefore appear length four cycle vertices one additional edge since edge removed never part cycle length less six proof claim first note since edges added process create new cycle length four join cycle type therefore cycles could affected type however every cycle type falls one two cases case cycle breaking case become cycle type since remove two edges break cycle case cycle breaking case one edges converted edge let length four cycle breaking note differ least one vertex break two edges converted cover four vertices therefore one edges note breaking one cycle type could create cycles type cycles always broken next iteration breaking another cycle type second modification informally second modification decreases rates lists associated nodes increases rates lists associated nodes illustrate intuition following example figure example need second modification left competitive analysis shows case achieve high competitive ratio expense right example balancing strategy making slightly likely pick comes consider graph figure let thin thick edges represent respectively calculate competitive ratio applying rla let denote probability gets matched algorithm let denote event among random lists exists list starting gvu denote event among lists exists successive lists lists starts lists arrive order ensures matched algorithm lemma corollary following lemma follows lemma suppose part cycle length gvu node definition event among lists random list comes least twice notice list comes probability thus pois similarly get resultant observe let rla rla rla competitive ratio achieved rla respectively hence rla rla rla intuitively one improve worst case ratio increasing arrival rate reducing suppose one modifies arrival rate gets modified respectively resulting changes rla rla hence performance instance improves notice modifications figure describes various modifications applied vector values top edge denote new values cases help improve upon described figure algorithm analysis algorithm full details algorithm stated follows figure illustration second modification value assigned edge represents value second modification algorithm construct solve input instance invoke output apply two kinds modifications morph run rla graph algorithm consists two different random processes offline phase rla online phase consequently analysis consists two parts first given graph analyze ratio rla node analysis similar second analyze probability transforms fractional values three discrete cases seen first part combining results two parts get final ratio let first analyze competitive ratio rla given let probability gets matched rla notice value determined algorithm rla also modifications applied define competitive ratio vertex achieved rla modifications lemma gives respective ratio values proof found section appendix lemma consider given vertex respective ratios achieved rla modifications described competitive ratio rla first cycle rla otherwise competitive ratio rla competitive ratio rla essentials prove theorem proof lemmas know present cycle probability consider node let probability first modification first cycle respectively lemma get final ratio least minimizing expression subject get minimum value node know ratio least min value rla rla completes proof theorem arrival rates stochastic rewards setting strictly generalized previous sections following ways firstly allows arbitrary arrival rate say fractional stochastic vertex notice total number rounds secondly associated value captures probability edge present probe try assign assume process independent stochastic arrival show simple algorithm introduced extended general case achieves competitive ratio note manshadi show algorithm possibly achieve ratio better arrival rates even case thus algorithm optimal algorithm model use similar case arrival rates let expected number probes edge multiple copies count sum probes among copies offline optimal thus realizations greater max algorithm summarized algorithm notice last constraint ensures step algorithm valid let prove theorem algorithm construct solve wlog assume optimal solution vertex arrives assign neighbors probability proof let event safe beginning round event vertex matched round conditioned algorithm know follows consider edge graph notice probability gets matched matched arrives integral arrival rates uniform stochastic rewards section consider special case model studied section show indeed surpass barrier specialize model following two ways consider unweighted case uniform constant edge probabilities constant constant arbitrary independent problem parameters vertex comes online integral arrival rate usual wlog refer special model unweighted online stochastic matching integral arrival rates uniform stochastic rewards note even special case given offline instance sequence realizations online arrival unclear efficiently solve approximate exact offline optimal within without extra assumptions hence directly apply simulation technique approximate exact expected offline optimal within arbitrary desired accuracy present strengthened benchmark upper bound offline optimal max exp lemma valid upper bound expected offline optimal proof suffices show constraint valid correctness constraints follows previous section consider given let number copies edges offline instance probability edge included offline optimal straightforward calculation follows using linearity expectation get definition since assumed sufficiently large follows poisson distribution mean substituting fact exp equation prove lemma note impossible beat using benchmark without extra constraint see hardness instance shown main idea online phase primarily based offline phase first solve get optimal solution vertex arrives generate random list two choices based denoted online decision based follows safe available match else second choice safe match random list generated based satisfies following two properties max algorithm solve let optimal solution vertex arrives generate random list two choices based satisfies property safe available assign else second choice safe match several ways generate satisfying one simple way shown section verify calculations shown extended incorporate independent process present probability assign hence final ratio follows viewed counterpart equation page alg min opt observe verify given rhs expression inequality increasing function interval thus important step lower bound given following key lemma viewed counterpart lemma lemma given proof consider given define thus property max property thus lower bound essentially need maximize let set edges called contributing edge thus observe max constraint equation get pfe exp substituting inequality back exp easy verify expression maximum value thus let complete proof theorem proof need prove defined lower bound consider first case easy verify consider second case lemma simple calculations show conclusion future directions paper gave improved algorithms models previously gap best unweighted algorithm ratio due negative result due took step towards closing gap showing algorithm achieve unweighted vertexweighted variants integral arrival rates made progess open questions online matching allocation survey possible approach rounding simpler fractional solution allowed employ stricter edgeweighted variant showed one significantly improve power two choices approach generating two matchings solution variant edge weights nonintegral arrival rates stochastic rewards presented algorithm showed bound given adversarial model stochastic rewards extend known model natural next step setting use adaptive strategy vertexweighted problem one easily see stricter use still gap addition utilize fractional solutions however dependent rounding gives solutions allowing random lists length greater three stricter lps longer lists could yield improved results stochastic model rewards integral arrival rates open question either improve upon ratio general case work showed certain restrictions possible beat however serious limitation comes fact polynomial sized insufficient capture complexity problem acknowledgments authors would like thank aranyak mehta anonymous reviewers valuable comments significantly helped improve presentation paper references aggarwal goel karande mehta online bipartite matching budgeted allocations proceedings annual symposium discrete algorithms siam alaei hajiaghayi liaghat online matching applications allocation proceedings acm conference electronic commerce acm alaei hajiaghayi liaghat online stochastic generalized assignment problem approximation randomization combinatorial optimization algorithms techniques springer assadi khanna stochastic matching problem queries proceedings acm conference economics computation acm brubach sankararaman srinivasan attenuate locally win globally framework online stochastic matching timeouts proceedings conference autonomous agents multiagent systems richland aamas international 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foundations computer science focs ieee mehta waggoner zadimoghaddam online stochastic matching unequal probabilities proceedings annual symposium discrete algorithms siam appendix complementary materials section model proof lemma prove lemma using following three claims recall one kind large edge two kinds small edges hence following claim characterizes performance claim large edge parameter achieves competitive ratio claim small edge type achieves competitive ratio regardless value claim small edge type achieves competitive ratio setting two types small edges ratio get achieves thus proves lemma proof claim proof consider large edge graph let small edge incident edges appear following three ways notice random triple matchings generated invoking property know occur probability ignore second copy lemma matched matched matched hence matched matched proof claims proof consider small edge type let two small edges incident given triple matchings say type appears two remaining two matchings similarly define type case appears respectively notice probability type similar calculations proof claim therefore matched matched consider small edge type define type appears large edge incident appears remaining two matchings similarly hence ratio small edge type proof lemma prove lemma using following two claims claim large edge achieves competitive ratio min claim small edge achieves competitive ratio therefore setting get proves lemma proof claim proof figure shows two possible configurations large edge figure diagram configurations large edge thin thick lines represent small large edges respectively consider large edge configuration know always probability respectively following cases happens probability matched happens probability matched happens probability therefore matched matched matched matched consider configuration know always always thus matched matched hence completes proof claim proof claim proof figure shows possible configurations small edge figure diagram configurations small edge thin thick lines represent small large edges respectively similar proof claim analysis various configurations let event gets matched given denote observe thus observe two cases case happens probability conditional probability case happens probability conditional thus observe observe two cases case happens probability conditional case happens probability conditional thus observe three cases case happens probability conditional case happens probability conditional case happens probability conditional similarly observe six cases therefore observe following six cases either either either therefore observe configuration additional six cases ones discussed let cases defined notice multiplicative factor consider six new cases either either either hence setting get competitive ratio small edge bottleneck cases configurations supplemental materials section proof lemma unweighted cycle show competitive ratio hence remaining cases use following claims claim rla claim rla claim rla recall event among random lists exists list starting gvu event among lists exist successive lists start different neighbors ensure matched notice probability gets matched rla compute gvu possibilities using lemma get first discuss two different ways calculate gvu cases use direct calculation rest use approximation method figure case calculation gvu two ways compute value gvu case consider case two neighbors shown figure assume two neighbors modifications thus second certificate event gvu corresponds event list starting comes time list comes second time figure case calculation gvu note arrival rate list starting rate list therefore gvu case consider case three neighbors value gvu approximated using markov chain method similar let use following example illustrate method consider following case shown figure three neighbors recall modifications simulate process getting matched resulting several successive random lists starting either markov chain follows states three numbers triple correspond matched respectively state corresponds matched matched status irrelevant chain initially starts probability state steps gives approximation gvu transition probability matrix shown follows notice since modifications arrival rate list starting decreases correspondingly let prove three claims give explicit analysis case remaining cases similar methods applied hence omit analysis present related computational results leads conclusion proof claim notice cycle thus lemma used figure describes possible cases node ignore cases since appear figure cases value assigned edge represents value second modification value indicates modification let two neighbors total combinations chosen chosen need find worst combination among value minimized find using lemma type compute values contribute term gvu example assume type denoted contributes term gvu thus total value contributes similarly compute let arg maxi arg maxj combination resulting value rla follows rla list gvu gvu notice modifications hence use equation compute lower bound gvu gvu notice large edge incident node modification modification thus equation get gvu notice small edge incident node modification modification thus gvu gvu modifications thus get gvu modifications thus get gvu gvu gvu gvu gvu gvu using computed values let compute ratio node three neighbors configuration three neighbors type value largest resultant ratio two neighbors configuration one neighbor type type resultant ratio proof claim proof similar claim figure shows possible configurations node note hence omit neighbor one one list values gvu gvu gvu gvu gvu gvu gvu figure cases value assigned edge represents value second modification value indicates modification gvu gvu gvu gvu hence structure one neighbor type resultant ratio proof claim figure shows possible configurations node omit cases list values gvu gvu figure cases value assigned edge represents value second modification value indicates modification gvu gvu gvu gvu hence node one neighbor type resultant ratio figure available jpg format http
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tail behavior class multivariate conditionally heteroskedastic processes rasmus olivier dec october abstract conditions geometric ergodicity multivariate autoregressive conditional heteroskedasticity arch processes bekk baba engle kraft kroner parametrization considered show class processes invariant distribution regularly varying order account possibility different tail indices marginals consider notion vector scaling regular variation vsrv closely related regular variation characterization tail behavior processes used deriving asymptotic properties sample covariance matrices ams subject classifications keywords phrases stochastic recurrence equations markov processes regular variation multivariate arch asymptotic properties geometric ergodicity introduction aim paper investigate tail behavior class multivariate conditionally heteroskedastic processes specifically consider department economics university copenhagen farimagsgade copenhagen denmark department mathematical sciences university copenhagen universitetsparken copenhagen denmark sorbonne upmc univ paris lsta case place jussieu paris france grateful comments suggestions thomas mikosch associate editor two referees led much improved manuscript moreover thank sebastian mentemeier valuable comments pedersen greatly acknowledges funding carlsberg foundation financial support anr network ameriska anr gratefully acknowledged wintenberger correspondence rasmus pedersen rsp bekk process introduced engle kroner satisfying positive definite matrix set real matrices initial value due assumption gaussian holds written stochastic recurrence equation sre mit mit process mutually independent mjt mit moreover process mutually independent mit knowledge representation process new moreover representation crucial studying stochastic properties process firstly find new sufficient condition terms matrices order geometrically ergodic particular case derive condition directly related eigenvalues line strict stationarity condition found nelson univariate arch process condition milder compared conditions found existing body literature processes secondly representation used characterize tails stationary solution whereas tail behavior univariate garch processes see basrak results tail behavior multivariate garch processes exist exceptions multivariate constant conditional correlation ccc garch processes see pedersen matsui mikosch class factor garch processes see basrak segers existing body literature relies rewriting transformed process companion form obeys multivariate sre characterization tails processes follows application kesten theorem kesten sres approach feasible ing processes stated terms sre special cases process apply existing results sres order show stationary distribution bekkarch process multivariate regularly varying specifically matrix invertible almost surely law absolutely continuous respect lebesgue measure denoted argue classical results kesten theorem see also alsmeyer mentemeier apply moreover product positive scalar random orthogonal matrix denoted similarity show results buraczewski apply importantly also argue results alsmeyer mentemeier rely rather restrictive conditions shown hold certain types processes particular much applied process diagonal denoted diagonal bekkarch specifically ruled alsmeyer mentemeier show diagonal process exhibits different marginal tail indices constant denoted condition order analyze class processes tail indices allowed differ among elements introduce new notion vector scaling regular variation vsrv distributions based scaling instead scaling arbitrary norm emphasize notion vsrv similar notion regular variation see resnick chapter additional condition addition spirit basrak segers introduce notion vsrv processes particular attention markov chains characterize extremal behavior argue stationary distribution diagonal process expected vsrv supported simulation study proving vsrv property applies requires new multivariate renewal theory developed leave task future research rest paper organized follows section state sufficient conditions geometric ergodicity process introduce notion regular varying vsrv distributions show distribution satisfies type suitable conditions section introduce notion vsrv processes state certain bekkarch processes satisfy property moreover consider extremal behavior process terms asymptotic behavior maxima extremal indices lastly consider convergence point processes based vsrv processes section consider limiting distribution sample covariance matrix relies point process convergence section contains concluding remarks future research directions notation let denote set invertible real matrices set real matrices let denote spectral radius denoting kronecker product real matrix let factors two matrices dimension denotes elementwise product unless stated otherwise denotes arbitrary matrix norm moreover kxk two matrices dimensions means aij bij two positive functions let denote distribution default mode convergence distributions weak convergence stationary solution model existence geometric ergodicity start stating following theorem provides sufficient condition geometric ergodicity process knowledge result together proposition new theorem let satisfy defined suppose inf log geometrically ergodic associated stationary solution kxt proof theorem follows alsmeyer theorems example theorem hence omitted remark sufficient condition existence finite moments obtained theorem feigin tweedie particular strictly stationary solution kxt example implies kxt result complements theorem pedersen rahbek contains conditions finite moments case case contains one term condition simplifies condition geometric ergodicity stated explicitly terms eigenvalues matrix proposition let satisfy let necessary sufficient condition exp log digamma function proof condition holds exists log let holds log log log kan log log log hence satisfied log kan log result follows observing kan remark holds hence condition equivalent exp log exp similar strict stationary condition found arch coefficient univariate arch process gaussian innovations see nelson boussama derive sufficient conditions geometric ergodicity bekk process specifically show sufficient condition setting condition simplifies stronger condition derived provide examples processes geometrically ergodic studied detail throughout paper example following alsmeyer mentemeier consider bekk processes corresponding sre satisfying certain irreducibility density conditions conditions section appendix specifically consider bivariate process writing sre obtain mit mutually independent processes mit assuming top lyapunov exponent strictly negative process geometrically ergodic notice one could consider general process structure containing terms lebesgue density clarified example moreover one could include additional terms say term containing full matrix autoregressive term presented remark focus simple bivariate process emphasize results apply general processes example similarity consider bekk process positive scalar orthogonal matrix implies sre amt definition similarity probability one recall matrix similarity written product positive scalar orthogonal matrix proposition exp log process geometrically ergodic important process satisfying similarity property scalar process aid identity matrix example diagonal consider process diagonal refer process diagonal process relying proposition process geometrically ergodic diagonal element less exp log modulus discussed bauwens diagonal bekk models typically used practice within empirical finance due relatively simple parametrization shown even though parametrization simple tail behavior rather rich sense marginal different tail indices general remark extension one may consider autoregressive process process recently studied applied nielsen rahbek modelling term structure interest rates notice process sre representation mit following arguments used proving theorem holds bekkarch process geometrically ergodic condition satisfied interestingly verified simulations nielsen rahbek lyapunov condition may hold even autoregressive polynomial unit roots reduced rank multivariate regularly varying distributions stationary solution process see theorem written even random matrices gaussian assumption maximum products may exhibit heavy tails precisely tails stationary distribution suspected extremal behavior power law function cases referred kesten cases seminal paper kesten subject monograph buraczewski class multivariate distributions satisfying property class multivariate regularly varying distributions haan resnick definition let borel random variable constant scalar define kxk distribution multivariate regularly varying exists radon measure satisfies vaguely set refer index regular variation refer haan resnick notion vague convergence additional details provide two examples multivariate regularly varying bekk processes example continued consider process example verifying conditions theorem alsmeyer mentemeier stated section appendix establish process multivariate regularly varying since gaussian hold moreover invertible probability one ensures satisfied also notice distribution lebesgue density strictly positive neighborhood ensures irreducibility density conditions satisfied fact independent implies condition holds lastly condition holds fact gaussian theorem alsmeyer mentemeier established following proposition proposition let satisfy top lyapunov exponent strictly negative stationary solution exists lim finite positive continuous function proposition implies marginal distribution regularly varying order theorem basrak conclude multivariate regularly varying whenever moreover since symmetric multivariate regular variation also hold odd integer see remark buraczewski proposition also apply seen observing strictly positive density sufficiently large sufficient establishing conditions example similarity continued similarity bekkarch introduced example fits setting buraczewski see also section buraczewski specifically using representation log exp log iii log distribution due theorem buraczewski following proposition proposition let satisfy orthogonal matrix exp log process unique strictly stationary solution multivariate regularly varying index satisfying following example clarify diagonal process introduced example satisfy conditions theorem alsmeyer mentemeier moreover argue marginals may different tail indices motivates notion vector scaling regular variation introduced next section example diagonal continued consider diagonal process example diagonal process distribution restricted apply results alsmeyer mentemeier example specifically irreducibility condition appendix shown hold clarified next holds sign hence always find open max sign example choose conclude condition hold diagonal process note element diagonal process written sre aii theorem goldie stationary solution marginal equation exists log case exists unique log hence marginal may general different tail indices precisely tail indices different diagonal elements aii heaviest marginal tail index corresponds largest diagonal coefficient unique except distribution considered multivariate regularly varying index limit measure degenerate marginals vector scaling regularly varying distributions previous example shows diagonal process fits case cases attracted much attention existing body literature however recent empirical studies matsui mikosch see also damek may suggest realistic consider different marginal tail behaviors modelling multidimensional financial observations idea use vector scaling instead scaling kxk definition reduced regular variation properties vector regular variation properties norm kxk precisely let stationary process let denote also framework consider distributions satisfying following condition condition marginal regularly varying order slowly varying functions indeed diagonal process introduced example satisfies condition moreover regularly varying distribution satisfying kesten property satisfies condition particular similarity bekkarch processes introduced examples respectively satisfy condition introduce notion vector scaling regular variation nonstandard regular variation book resnick condition extended negative components resnick sections definition distribution vector vector scaling regularly varying vsrv satisfies condition regularly varying exists normalizing sequence radon measure marginals vaguely usual way analyzing regularly varying vectors consider componentwise normalization standard regularly varying sense definition specifically satisfies definition satisfies definition index one throughout find helpful focus vector order preserve multiplicative structure tail chain introduced section used analyzing extremal properties vsrv processes following proposition state vsrv vector polar decomposition case condition satisfied note polar decomposition holds transformation original process condition natural radius notion max notice homogeneity due condition essential proof proposition suppose vector satisfies condition vsrv exists tail vector marginals defined moreover standard pareto distributed notice similar vector scaling argument introduced lindskog proof adapting theorem haan resnick definition vector scaling regularly varying distribution implies conversely condition regularly varying order slowly varying functions moreover regularly varying weak convergence applied borel sets thus regularly varying order slowly varying function one rewrite using slowly varying property obtain marginal homogeneity notice tail exists change variable obtain existence enough characterize entirely choosing arbitrarily small spectral properties vsrv expressed terms tail vector notice exists satisfying lim max consider continuous mapping argument satisfies regularly varying index homogeneity limiting measure multivariate regular variation may decompose limit product limiting distribution called simple distribution supported positive orthant called spectral measure see haan resnick details identification two expressions limit obtain following proposition proposition defined proposition distribution spectral measure moreover independent standard pareto distributed proof standard pareto distributed follows convergence associated regularly varying property ensuring homogeneity limiting measure using homogeneity follows independent example diagonal continued able establish existence satisfying except case scalar bekkarch diagonal elements identical case process special case similarity see example even case characterization spectral distribution easy task diagonality ruling theorem buraczewski section appendix included estimates spectral measure bivariate case plots suggest tails process indeed dependent emphasize new multivariate renewal theory developed order prove model vsrv leave task future regularly varying time series extremal behavior existence tail vector proposition allows extend asymptotic results perfekt vsrv vectors taking possibly negative values order use notion tail chain basrak segers adapted vsrv stationary sequences eventually different tail indices vector scaling regularly varying time series introduce new notion multivariate regularly varying time series based vsrv definition stationary process vsrv exists process marginals sequence called tail process following basrak segers extend notion spectral measure one spectral processes vsrv stationary process definition vsrv stationary process admits spectral process arguments similar ones proof proposition follows vsrv properties also characterize spectral process following stationary distribution distribution following proposition proposition vsrv stationary process marginals standard pareto distributed spectral process nondegenerate distributed independent moreover standard pareto distributed tail chain following focus dynamics tail process definition given existence restrict case markov chain implies also markov chain called tail chain see perfekt following proposition proposition let satisfy vsrv stationary process defined tail process admits multiplicative form proof following approach janssen segers one first notices existence kernel tail chain depend marginal distribution thus characterization kernel extends automatically usual multivariate regular variation setting vector scaling regular variation one straightforward check condition janssen segers conclude tail chain multiplicative structure tail chain vsrv process satisfying matter values marginal tail indices multivariate regularly varying case common tail indices coincides tail chain janssen segers condition notice extend tail chain backward time using corollary janssen segers asymptotic behavior maxima previous section tail chain quantifies extremal behavior let consider asymptotic behavior maxima max max let assume positive recall suitably scaled maxima converge distribution see haan resnick defining max exp vector scaling regularly varying case due condition expression max let assume following condition slightly stronger exists lim theorem let satisfy defined suppose condition holds suppose stationary distribution vsrv assuming existence definition max exp admits expression max max proof verify conditions theorem perfekt condition perfekt satisfied tractable condition janssen segers indeed tail chain depends markov kernel one apply lemma janssen segers extends immediately vector scaling regularly varying setting condition perfekt holds geometric ergodicity markov chain sequence log sufficiently large lastly finite clustering condition lim lim max log holds using reasoning proof theorem mikosch wintenberger drift condition dcp min also standard regularly varying actually drift condition holds thanks condition sufficiently large iterations markov kernel finally special case condition log perfekt obtain desired result characterization given theorem perfekt tail chain standardized markov chain restricted distribution assume identify distribution max constraint thus max obtain expression valid exploit homogeneity property obtain max max max max standard pareto distributed independent spectral process expression homogeneous extends possible homogeneity extremal indices random coefficients may large consecutive values large univariate case one says extremal values appear clusters indicator average length cluster inverse extremal index indicator extremal dependence see leadbetter thus natural extension extremal index function defined respectively notice reason depend explicit expression terms spectral process however extremal index marginal index still depends complete dependence structure multivariate markov chain thanks following proposition proposition let satisfy defined satisfying assuming existence definition extremal index defined positive continuous function bounded extended extremal indices marginals proof except positivity extremal index result follows proposition perfekt positivity ensured applying corollary segers example diagonal continued suppose vsrv conjectured example follows tail chain approach janssen segers stationary markov chain regularly varying thanks diagonal structure matrices amk one factorize expression provided proposition since independent recover similar expression remarks theorem haan max manage provide link extremal index multivariate stationary solution diagonal due different normalising sequences asymptotic extremal result given theorem extremal index depends constants expression gets simple standard pareto distributed supported one check satisfies aii marginal smallest tail extremal indices thus inverse extremal index multidimensional diagonal larger largest average length marginals clusters interpreted fact largest clusters concentrated along axis following interpretation multivariate extremal index given beirlant convergence point processes let consider vector scaling point process want characterize asymptotic distribution point process refer resnick details convergence distribution random measures order characterize limit adapt approach davis hsing multivariate vsrv case similar davis mikosch limit distribution cluster point process admitting expression arrival times standard poisson process mutually independent cluster processes following basrak tafro use back forth tail chain describe cluster process consider process satisfying sup kyt well defined condition satisfied kzt notice use fact standard pareto crucial mimic arguments basrak tafro limiting distribution point process coincides one theorem let satisfy defined suppose holds assume definition exists defined defined proof let denote sign operator sign applied coordinatewise vectors apply theorem davis mikosch transformed process sign standard regularly varying order order one check condition satisfied cluster index positive follows arguments developed proof theorem mixing condition davis mikosch implied geometric ergodicity thus limiting distribution point process sign cluscoincides one cluster point process ter process continuous mapping argument yields convergence sign limiting cluster process coincide distribution thanks definition vsrv processes sample covariances section derive limiting distribution sample covariances certain processes consider sample covariance matrix let vech denote operator matrix aij vech add derivation limiting distribution sample covariance matrix relies using multidimensional regularly varying properties stationary process vech let denote normalization matrix using theorem adapting continuous mapping argument proposition davis mikosch yield following result proposition let satisfy defined satisfying assuming existence definition vech vech let define assume note candidate tail index cross product actually case extra assumptions ensuring product non null see proposition resnick line theorem davis mikosch get main result asymptotic behavior empirical covariance matrix theorem let satisfy defined suppose holds assume definition exists moreover suppose lim lim sup var random variable theorem applies widest confidence interval covariance estimates supported marginal satisfying order apply theorem main difficulty show condition holds however notice theorem applies simultaneously crossproducts extra assumption next apply theorem ongoing examples example diagonal continued consider diagonal bekkarch process cross products inequality turns equality case ajj thus function markov chain satisfies drift condition dcp mikosch wintenberger one show satisfied using reasoning proof theorem mikosch wintenberger example similarity continued limiting distribution sample covariance matrix similarity follows directly theorem additional condition checked relying arguments example one would verify condition dcp mikosch wintenberger holds similarity process appears difficult task requires find suitable multivariate lyapunov function leave task future investigation consider special case scalar process introduced example aid identity matrix diagonal case least pair limiting distribution sample covariance derived along lines example specifically relies assuming exp log stationary solution exists noting index regular variation marginal given satisfying amt example continued whenever limiting distribution sample covariance matrix follows directly theorem similar example leave future investigation show whether condition holds previous examples important relation variance targeting estimation model considered pedersen rahbek univariate garch process vaynman beare shown limiting distribution suitably scaled variance targeting estimator follows singular stable distribution tail index process lies expect similar result hold process concluding remarks found mild sufficient condition geometric ergodicity class processes exploiting processes written multivaraite stochastic recurrence equation sre investigated tail behavior invariant distribution different processes specifically demonstrated existing results apply certain cases implying marginal invariant distribution tail index moreover shown certain empirically relevant processes existing renewal theory applicable particular show diagonal processes may different tail indices light property introduce notion vector scaling regular varying vsrv distributions processes study extremal behavior processes provide results convergence point processes based vsrv processes conjectured supported simulations diagonal process vsrv however remains open task verify formally property holds task require development new multivariate renewal theory results expected important future research related statistical analysis diagonal model recently shown avarucci suitably scaled maximum likelihood estimator general model gaussian limiting distribution moments finite order obtain limiting distribution presence heavy tails kxt believe arguments needed particular knowledge expected crucial analysis leave additional considerations direction future research references alsmeyer harris recurrence iterated random lipschitz functions related convergence rate results journal theoretical probability alsmeyer mentemeier tail behaviour stationary solutions random difference equations case regular matrices journal difference equations applications avarucci beutner zaffaroni moment conditions likelihood estimation multivariate arch models econometric theory basrak davis mikosch characterization multivariate regular variation annals applied probability regular variation garch processes stochastic processes applications basrak segers regularly varying multivariate time series stochastic processes applications basrak tafro complete convergence theorem stationary regularly varying multivariate time series extremes bauwens laurent rombouts multivariate garch models survey journal applied econometrics beirlant goegebeur segers teugels statistics extremes theory applications john wiley sons boussama fuchs stelzer stationarity geometric ergodicity bekk multivariate garch models stochastic processes applications buraczewski damek guivarc hulanicki urban stationary measures multidimensional stochastic recursions probability theory related fields buraczewski damek mikosch stochastic models tails equation springer series operations research financial engineering springer international publishing damek matsui componentwise different tail solutions bivariate stochastic recurrence equations https davis hsing point process partial sum convergence weakly dependent random variables infinite variance annals probability davis mikosch sample autocorrelations heavytailed processes applications arch annals statistics haan resnick limit theory multivariate sample extremes zeitschrift wahrscheinlichkeitstheorie und verwandte gebiete haan resnick vries extremal behaviour solutions stochastic difference equation applications arch processes stochastic processes applications einmahl haan piterbarg nonparametric estimation spectral measure extreme value distribution annals statistics engle kroner multivariate simultaneous generalized arch econometric theory feigin tweedie random coefficient autoregressive processes markov chain analysis stationarity finiteness moments journal time series analysis goldie implicit renewal theory tails solutions random equations annals applied probability janssen segers markov tail chains journal applied probability kesten random difference equations renewal theory products random matrices acta mathematica kulik soulier wintenberger tail empirical process regularly varying functions geometrically ergodic markov chains https leadbetter lindgren extremes related properties random sequences processes springer series statistics lindskog resnick roy regularly varying measures metric spaces hidden regular variation hidden jumps probability surveys matsui mikosch extremogram crossextremogram bivariate garch process advances applied probability mikosch wintenberger precise large deviations dependent regularly varying sequences probability theory related fields nelson stationarity persistence garch model econometric theory nielsen rahbek unit root vector autoregression volatility induced stationarity journal empirical finance pedersen targeting estimation models infinite fourth moments econometric theory pedersen rahbek multivariate variance targeting model econometrics journal perfekt extreme value theory class markov chains values advances applied probability resnick phenomena probabilistic statistical modeling springer science business media segers generalized pickands estimators extreme value index journal statistical planning inference multivariate extremes models constant conditional correlations journal empirical finance vaynman beare stable limit theory variance targeting estimator essays honor peter phillips chang fomby park emerald group publishing limited vol advances econometrics chap appendix theorem alsmeyer mentemeier consider general sre sequence random variables generic copy real matrix takes values consider following conditions alsmeyer mentemeier kak denotes operator norm kbk open subset let denote open let leb denote lebesgue measure holds borel set leb exists theorem alsmeyer mentemeier theorem consider sre suppose log hold exists unique lim log moreover sre strictly stationary solution satisfying lim finite positive continuous function estimation spectral measure bivariate diagonal process section consider estimation spectral measure diagonal process presented example specifically consider special case process process independent following approach sequences vectors given einmahl consider following estimator spectral measure arctan denotes rank among sequence satisfying einmahl showed estimator consistent series expect similar result hold geometrically ergodic processes reason asymptotic behavior empirical tail process used einmahl extended cases kulik consider estimation spectral measure different values particular matrix values determined according choices tail indices respectively satisfy determined analytical integration specifically pdf standard normal distribution aii aii aii aii figure contains plots estimates spectral measure estimates based one realization process period observations figure nonparametric estimates various choices
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feb state art gomes casper thule david broman dbro peter gorm larsen pgl hans vangheluwe february contents introduction motivation need survey outline modeling simulation dynamical systems models real systems simulators computing behavior trace simulation units reality compositional discrete event based simulation units orchestration challenges causality determinism confluence dynamic structure distribution continuous time based simulation units orchestration challenges modular composition algebraic constraints algebraic loops consistent initialization simulators compositional convergence error control compositional stability compositional continuity constraints hybrid approach hybrid scenarios challenges semantic adaptation predictive step sizes event location discontinuity identification discontinuity handling algebraic loops legitimacy zeno behavior stability theory approximated states standards hybrid classification methodology taxonomy state art discussion concluding remarks historical perspective one formalism dynamic iteration two formalisms digital analog many simulation units large scale state art frameworks scenario categorization requirements fault tolerance configuration reusability performance protection parallelism distribution hierarchy scalability platform independence extensibility accuracy open source simulator requirements information exposed causality time constraints rollback support availability framework requirements standard coupling number simulation units domain dynamic structure rate communication step size strong coupling support results visualization communication approach list acronyms abstract essential find new ways enabling experts different disciplines collaborate efficient development ever complex systems increasing market pressures one possible solution challenge use heterogeneous approach different teams produce conventional models carry usual analysis addition different models coupled simulation allowing study global behavior system due potential studied many different disciplines limited sharing findings aim work summarize bridge enhance future research multidisciplinary area provide overview approaches research challenges research opportunities together detailed taxonomy different aspects state art classification past five years main research needs identified finding generic approaches modular stable accurate coupling simulation units expressing adaptations required ensure coupling correct introduction motivation truly complex engineered systems integrate physical software network aspects emerging due external pressure development systems concurrent distributed divided different teams external suppliers domain tools participant develops partial solution constituent system needs integrated partial solutions later process integration done less optimal innovative truly optimal solutions achieved holistic development process partial solutions developed independently integrated sooner frequently furthermore traditional activities carried partial solution level requirements compliance check design space repeated global level salient properties spanning multiple constituent systems studied modeling simulation improve development partial solutions see friedman ghidella falls short fostering holistic development process understand one observe models partial solution exchanged integrated easily likely developed different specialized tools externally supplied models intellectual property hence cheaply disclosed system integrators proposed solution overcome important challenges consists theory techniques enable global simulation coupled system via composition simulators simulator black box constituent system developed provided team responsible system allows work part problem tools without coupled system mind eases relationship system integrators suppliers latter provide virtual solutions selling without revealing former perform early conformance checks evaluate different designs multiple competing suppliers alternative models described unified language simulated advantages approach domain particularities comes simulation see making impractical find language simulation algorithm fits part systematic review led current document see section details took note approaches publications applications approaches shaped taxonomy section applications shows last five years applied many different engineering domains fig shows concrete publications automotive electricity production distribution hvac soc design maritime robotics closer look publications shows however average reported scenario includes two simulators constituent system different domain gives evidence enhances development systems yet scale systems cpss unexplored potential recognized number completed ongoing projects address one reasons functional interface fmi standard created publications application domain automotive electricity production distribution hvac soc design maritime robotics year figure research publications applications past five years fmi defines interfaces enable modelling simulation tools cooperate simulation keeping models protected december tools implement many others planning support standard originally created coupling continuous systems used abused simulate hybrid systems extensions proposed literature facilitate standardize simulation zero step size transitions step size prediction complement adding understanding challenges arising hybrid system new concept many approaches state art particular discrete event based approaches studied may shed light hybrid system performed contribution present survey taxonomy attempt bridge relate classify many approaches state art https http http https http https number includes tools passed crosscheck https need survey despite growing interest benefits scientific challenges best knowledge existing survey attempts cover heterogeneous communities studied lack survey means techniques proposed independently limited sharing findings give example use models well known technique discrete event recently used continuous time approach hypothesis bridging different approaches means solutions techniques exchanged deeper understanding hybrid system attained scientifically mixes following fields research numerical analysis accuracy stability coupled system studied differential algebraic system simulation composition units general sense made algebraic constraints hybrid systems scenarios general sense hybrid systems optimization heterogeneous capabilities units pose interesting tradeoffs hierarchy systems systems hierarchical corresponding scenarios hierarchical well compositionality properties becomes interesting research challenge formal verification orchestration algorithm also known master certified correct certain assumptions scenarios system testing used exhaustively testing set constituent systems environment dynamic structure systems subsystems different dependencies depending level abstraction interact modeling subsystems different models different levels abstraction relationships multiple levels known correct dynamic switching levels abstraction made outline help structure characteristics simulators interact distinguish two main approaches discrete event described section continuous time described section used continuous discrete hybrid coupled systems call hybrid described section approach mixes approaches section categorizes features provided frameworks classifies state art taxonomy finally section concludes publication section provides terminology used rest survey modeling simulation dynamical systems models real systems section present informal manner concepts used throughout document dynamical system model real system instance physical system computer system characterized state notion evolution rules state set point values state space evolution rules describe state evolves independent variable usually time example traffic light real system modeled dynamical system time one four possible states red yellow green evolution rules may dictate changes red green time seconds another example modeled set first order ordinary differential equations odes equations describe state velocity changes continuously simulated time contrast traffic light system state take infinite number different values finite duration simulated time state behavior trace set trajectories followed state outputs dynamical system example state trajectory defined mapping time base set reals fig shows possible behavior trace example systems described example time base figure examples behavior traces refer time variable simulated time simply time ambiguity defined time base typical real numbers opposed time wct time passes real world computing behavior trace dynamical system interval simulated time computer takes units note survey focusing timed formalisms also called models computation interact hybrid environment formalisms logical notion time dataflow synchronous reactive discussed survey overview formalisms models computation please see book ptolemy following survey time depend therefore used measure performance simulators fig highlights different kinds simulation based relationship simulation relationship given cases required making sure obeyed simulation algorithm one main challenges simulation extension analytical simulation relationship restricted simulation tools offer interactive visualization allow user pause simulation set relationship knowing dynamical system used predict behavior real system crucial experimental frame describes abstract way set assumptions behavior trace dynamical system compared one real system real system mean either existing physical system fictitious one validity difference behavior trace dynamical system behavior trace real system measured assumptions specified experimental frame conveys predictive power dynamical systems example hooke law system used predict reaction force spring small deformations traffic light dynamical system experimental frame includes assumption transition red light green light instantaneous valid assumption provided executing platform controller software runs enough computing power model invalid within experimental frame assumptions behavior trace different one real system used predict properties real system order practical behavior trace dynamical system highlight features interest real system relevant tasks hand traffic light model precise amount time transition red green takes unknown deemed small enough neglected hooke law chosen maximum displacement mass large context system used taken account finally consider dynamical systems possible obtain meaning behavior trace even approximation simulators computing behavior trace two generally accepted ways obtaining behavior trace dynamical system translational translate dynamical system another model readily used obtain behavior trace obtaining analytical solution equations example approach instance traffic light model expressed statechart formalism translated devs model done borland used obtain behavior trace operational use solver algorithm takes dynamical system input outputs behavior trace example numerical solver used obtain approximation behavior trace focus latter simulator algorithm takes dynamical system computes behavior trace running digital computer often case simulator able approximate trace two aspects contribute error approximations inability calculate solution approx analytical approx analytical classification time constraints simulation based time approximate behavior trace system computed forward euler solver parameters displacement velocity dotted lines approximated behaviour traces solid lines analytical behaviour traces figure trajectory continuum finite representation infinitely small quantities simulators discrete dynamical systems may also tolerate inaccuracies behavior traces well brings performance benefit fig shows example approximation behavior trace system computed forward euler inaccuracies clear compared analytical trace order define accurate simulator even able talk error need postulate every dynamical system analytical behavior trace error defined absolute difference behavior trace computed simulator analytical trace accurate simulator one produces traces analytical behaviour even possible obtain analytical behavior every dynamical system theoretical results allow simulators control error make techniques applied section linear odes general analytical trace follows known structure traffic light timed statemachine models general analytical behavior trace obtained sequential solver respects causality events short validity property dynamical system whereas accuracy property simulator perfectly possible accurate behaviour trace model invalid vice versa continuous time systems choice appropriate solver important made domain experts meaning close depends numerical tolerance observer simulation units reality strict terms simulator readily executable needs dynamical system input trajectories dynamical system able compute behavior trace use term simulation unit composition simulator dynamical system simulation unit replacement real system ready take inputs produce behavior trace fmi standard analogous term functional unit fmu simulation behavior trace obtained simulation unit correctness simulation unit dictated correctness simulation depends accuracy simulator validity dynamical system compositional described section useful obtain correct simulations complex yet existing systems since constituent systems developed independently specialized simulation units produced simulation units coupled via produce behavior trace coupled system special kind simulation set simulations computed coupled simulation units simulation units independent black boxes possibly running different computers hence orchestrator necessary couple orchestrator controls simulated time progresses simulation unit moves data outputs inputs according cosimulation scenario scenario information necessary ensure correct obtained includes inputs simulation unit computed outputs experimental frames etc fmi standard orchestrator called master algorithm analogously simulator simulation unit concepts composition specific orchestrator scenario yields unit special kind simulation unit substitute real coupled system follows simulation trace computed unit characterization enables hierarchical scenarios units coupled main purpose orchestrator obtain correct behavior trace coupled system assuming simulation unit correctly defined make assumption simulation units often well known parts system developed experienced specialized enables design decisions tried model analysis cheaply early process possibly automatically survey focus coupling black box simulation units limited knowledge models simulators available however become clear sections black box restriction relaxed certain properties related correctness ensured understanding kind information revealed still protected active area research challenges related compositionality every simulation unit scenario satisfies property unit suitable orchestrator must also satisfy correctness property compositional properties include validity accuracy open research question ensure another aspect consider balance insights gained resources spent compositional given set properties following three sections provide overview information techniques used throughout state art divided three main approaches discrete event section continuous time section hybrid section discrete event based discrete event based approach describes family orchestrators characteristics simulation units borrowed discrete event system simulation domain start description systems extract main concepts characterize based traffic light good example system one possible modes red yellow green mode often used police countries characterized blinking yellow initially traffic light red seconds changes green alternatively seconds pass external entity police officer may trigger change red output system event signaling change new color example captures essential characteristics dynamical system reactivity instant reaction external stimuli turning external entity transiency system change state multiple times simulated time point receive simultaneous stimuli traffic light transiency would happen light changes always instead police officer would turn traffic light instant characteristics embraced based orchestrator acknowledges simulation units evolve internal state exchange values despite fact simulated time stopped simulation units simulation unit black box exhibits characteristics dynamical system dynamical system stands need one furthermore typical assume simulation units communicate environment via events opposed signals means outputs simulation units absent times event produced adapt definition originally proposed formally define original devs definition initial state absent value output function left implicit make explicit consistent section note also many variants discreteevent formalisms instance hardware description languages vhdl verilog actor based systems instance director ptolemy simulation unit denotes reference unit set possible discrete states input events output events respectively external transition function computes new total state based current total state input event internal transition function computes new total state current total state denotes elapsed units time since last transition internal external output event function invoked right internal transition takes place encodes absent value time advance function indicates much time passes next state change occurs assuming external events arrive initial state execution devs simulation unit described informally follows suppose simulation unit time marks current discrete state elapsed units time since total state let input event happens time output event computed new discrete state computed hand event time absent time solver changes state instead description two events happen time processed simulated time progresses due transiency reactivity properties state output trajectories simulation unit well identified time base traditionally positive real numbers includes way order simultaneous events simultaneous state changes example time base notion superdense time time point pair typically positive real numbers called index natural numbers time base state trajectory function vxi vxi set values state trajectory vui simultaneous states events formally represented incrementing indexes see introduction equations show examples simulation units simulation unit passive expects external coordinator set inputs call transition functions passivity enables easier composition simulation units cosimulation means coordination algorithm shown later section algorithm shows trivial orchestrator computes behavior trace single simulation unit specified inputs remarks holds time last transition initial elapsed time satisfies algorithm used coordinate execution traffic light simulation unit resulting behavior trace piecewise constant traffic light state together output events latter represented trajectory mostly undefined absent except single points output produced according algorithm single autonomous simulation unit orchestration based originally proposed data simulation unit initial discrete state account initial elapsed time true compute time next transition output take internal transition end orchestration devs simulation units communicate environment exclusively inputs outputs scenarios comprised multiple units coupled output input connections map output events one unit external events unit consider following simulation units traffic light police office respectively red yellow green working idle toauto tooff towork toidle idle working int working idle toidle working towork idle working idle yellow red yellow red red tooff toauto red green yellow red green yellow idle red green yellow red following remarks current state model definition elapsed time since last transition output event function executed immediately internal transition takes place must publish next state instead current model scenario police officer interacts traffic light output events mapped external events traffic light simulation unit example toauto tooff external input events handled traffic light simulation unit mapping defined toauto toidle tooff towork way police officer changes working state time output signal towork translated input event tooff traffic light simulation unit based idea abstract simulation units formalize illustrate idea scenario reference follows hucs ycs selecti ucs set possible input events external scenario ycs set possible output events scenario environment ordered set simulation unit references denotes unit defined set simulation units influence simulation unit possibly including environment external scenario excluding specifies mapping events select used deterministically select one simulation unit among multiple simulation units ready produce output events simultaneously time set simulation units imm one simulation unit reference function restricted select one among set imm select imm imm following scenario couples traffic light simulation unit police officer simulation unit ycs ics selecti ycs ics traffic light simulation unit police officer simulation unit output defined omitted functions map anything absent select function particularly important ensure trace unique example consider scenario suppose time simulation units ready output event perform internal transition traffic light output event perform internal transition first police office first general order actions performed matters reason way one simulation unit reacts simulation unit output may different depending internal state former example scenario end result always general case algorithm illustrates orchestrator autonomous without inputs scenario assumes scenario expect external events events affect simulation units produced simulation units scenario external output events possible though remarks tcs holds recent time last transition scenario elapsed time current state simulation unit time next transition scenario denotes chosen imminent simulation unit ics set simulation units produce output events environment ycs output event signal scenario environment holds simulation units influence algorithm autonomous scenario orchestration based data scenario ycs selecti tcs store initial discrete state unit true time next internal transition tcs time next internal transition select imm get next unit execute store new discrete state reset elapsed time executed unit ics ycs compute output scenario end trigger internal units influenced unit end update elapsed time remaining units end tcs advance time end fig shows behavior trace scenario algorithm similar algorithm time advance scenario corresponds time advance single simulation unit output produced state transition analogous function single simulation unit output state transition child together external transitions simulation units influenced analogous internal transition single simulation unit natural scenario specified made behave single simulation unit scs intuitively state scs set product total states child unit minimum time one units executes internal transition internal transition scs gets output event imminent unit executes external transitions affected units updates elapsed time unaffected units computes next state imminent unit external transition scs gets event environment executes external transition affected units updates elapsed time unaffected units formally ext int scs xcs ucs ycs qcs xcs qcs min min select imm xcs ics otherwise int xcs xcs int xcs xcs otherwise ext xcs ecs ucs xcs ecs zcs ucs ecs otherwise remarks cartesian product total state child simulation unit makes discrete state unit elapsed times child simulation unit managed solely unit whenever transition internal external external transition functions child executed mapping events produced current state imminent child next one computed internal transition child simulation unit may cause output event environment unit child connected output unit internal transition causes change child discrete state also due output event may cause external transitions child simulation units recursive iterative process one external transition occur affected child simulation units affected simulation units becomes ready internal transition waits next internal transition invoked coordinator unit resulting unit scs behaves exactly unit specified thus executed algorithm case inputs composed units hierarchical scenarios hierarchical scenarios elegantly correspond real hierarchical systems natural way deal complexity summary based exhibits following characteristics figure example trace traffic light police officer scenario reactivity simulation unit analogously unit process event moment occurs transiency algorithm unit time advance next imminent child internal transition zero successive iterations orchestrator able tolerate fact simulated time may advance several iterations predictable step sizes scenario without inputs orchestrator shown algorithm always predict next simulated time step scenario inputs environment provides time next event next simulated time step predicted possible black box simulation units able inform orchestrator time advance trivial task units simulate continuous systems whose future behavior trace especially reacting future inputs easily predicted without actually computing next main challenges based requirements capabilities solutions impose simulation units made explicit challenges causality sake simplicity algorithm sequential hierarchical unit imminent simulation unit closest performing internal transition one execute thus inducing global order events exchanged global order avoids causality violations pessimistic event causes another event changing internal state simulation unit turn changes next output however converse true necessarily imply caused means simulation unit could execute wallclock time without violating causality least within small window simulated time see suppose influence scenario would happen anyway regardless occurring moreover scenario holds information dependencies used determine influences parallel optimistic orchestrator takes account general faster wall clock time sense pessimistic sequential one however algorithm well known example require rollback capabilities simulation units simulation units proceed advance time optimistically assuming simulation units affect proven wrong receiving event occurs internal time happens simulation unit rollback state prior time timestamp event arrived may turn cause cascade rollbacks affected simulation units moreover parallel optimistic units scenario needs theoretically support multiple rollbacks enough memory arbitrary distant point past point past limited global virtual time gvt gvt represents minimum internal time simulation units definition event yet produced time timestamp smaller gvt make distinction multiple rollback single rollback capabilities support single rollback simulation unit needs store last committed state thereby saving memory causality compositionality property child simulation unit violate causality orchestrator ensure causality violated units coupled optimistic orchestration algorithms requiring rollback capabilities child simulation units whereas pessimistic algorithms cost performance determinism confluence determinism also compositional property select function scenario definition paramount ensure compositionality deterministic behavior function used ensure unique behavior trace obtained scenario executed algorithm turned unit alternative select function ensure possible interleavings executions always lead behavior trace known confluence intuitively unit compositional respect confluence also compositional respect determinism proving confluence hard general black box depends child simulation units react external events potentially valuable approach leaves confluence property satisfied modeler dynamic structure dependencies assumed fixed time performance perspective static sequence dependencies may conservative especially used ensure causality optimistic parallel see consider large scale simulation simulation unit may influence simulation unit specific set conditions may verified large amount simulated time passed pessimistic unit assumes may always affect hence tries ensure simulated time always smaller minimize possible rollbacks incurs unnecessary performance toll overall affect time making dynamic improve performance cosimulation since unit know time affect dynamic structure allows change time depending behavior trace simulation units used study systems distribution units whose child simulation units geographically distributed common interesting solutions like computation allocation bridging hierarchical encapsulation use models proposed mitigate additional communication cost moreover security becomes important solutions address continuous time based continuous time based approach orchestrators simulation units behavior assumptions borrowed continuous time system simulation domain describe simulation units continuous time simulation unit assumed state evolves continuously time easier get intuitive idea considering simulation unit continuous time dynamical system depicted left hand side fig state given displacement velocity mass evolution denotes time derivative spring stiffness constant damping coefficient mass initial position velocity denotes external input force acting mass time solutions satisfy constitute behavior trace dynamical system fig shows example trace generalized state space form state vector input output vectors initial state solution obeys behavior trace system linear analytical form obtained analytical solution obtained application mathematical identities example behavior trace obtained via translational approach described section alternatively behavior trace computed sufficiently differentiable approximated truncated taylor series max lim const denotes order truncated residual term size basis family numerical solvers iteratively compute approximated behavior trace example forward euler method given simulation unit assumed behavior similar one numerical solver computing set differential equations reinforce restrict simulation units mockups systems even though easier introduced fmi standard simulation unit analogous functional unit fmu cosimulation example simulation unit using forward euler solver written embedding solver size input output orchestration consider second system depicted right hand side fig governed differential equations denote stiffness damping coefficients spring damper respectively denotes displacement left end combining forward euler solver yields following simulation unit size inputs output suppose coupled setting resulting scenario represents system depicted fig figure system comprised two subsystems approach models equations would combined get following coupled model written state space form behavior trace obtained either analytically forward euler solver based overcome fact simulation unit sizes independent communication step size also known size communication grid size defined marks times simulation units exchange values suppose simulation unit time natural asked orchestrator execute time gets inputs valued extrapolation must used associate value inputs internal simulation unit words time hhi size extrapolation function built input values known previous communication time points used approximate value notice hhi allowed even though theoretically value obtained environment reason becomes clear section analogously interpolation techniques used orchestrator makes input value available time simulation unit still time example input simulation unit described defined similarly inputs simulation unit described defined simplest case extrapolations constant coupled example state art input extrapolation approaches classified increasing degree complexity constant linear polynomial contextaware estimated model often combined practical use cases see andersson arnold busch schweizer overview linear higher order extrapolation techniques affect accuracy trace orchestrator scenario time gets outputs simulation units computes inputs simulation unit instructed compute behavior trace next communication step size making use extrapolating functions get inputs micro steps equations ready formally define behavior simulation unit hxi state set typically input set typically output set typically function instructs simulation unit compute behavior trace making use input extrapolation interpolation function output function initial state instance simulation unit described follows obtained iterative application simulation unit finite number making use extrapolation defined continuous time scenario reference includes least following hucs ycs ycs ucs ordered set simulation unit references defined ucs space inputs external scenario ycs space outputs scenario set input approximation functions induces simulation unit coupling constraints coupling ycs ucs example scenario representing system fig simulation unit constituent system left simulation unit remaining constituent system inputs input outputs output algorithm summarizes generic way tasks orchestrator computing cosimulation scenario external inputs represents jacobi communication approach simulation units exchange values time independently compute trace next communication time way system solved depends simulation units coupled definition trivial case system reduces assignment output input orchestrator gets output simulation unit copies onto input simulation unit appropriate order concrete examples algorithm described please note formalization related formalization proposed broman main differences designed formalize subset fmi standard accommodates algebraic coupling conditions iii explicitly define port variables alternative jacobi communication approach sequential approach order simulation units function forced ensure time get inputs simulation unit already time approach allows interpolations inputs accurate hinders parallelization potential examples described algorithm generic jacobi based orchestrator autonomous scenarios data autonomous scenario ycs communication step size result trace true solve following system unknowns ycs instruct simulation unit advance next communication step advance time end similarly based scenario together orchestrator behave simulation unit form thus coupled simulation units forming hierarchical scenarios state unit set product states internal units inputs given ucs outputs ycs transition output functions implemented orchestrator communication step size used orchestrator analogous simulation unit sizes input extrapolation function algorithm makes clear simulation units coupled limited information internal details concrete output state transition functions need executable internal details remain hidden inputs need accessible state variables hidden represented merely illustrate internal state simulation unit changes executing however blind coupling lead compositionality problems discussed sections common trait addressing require individual simulation units either capabilities information internal hidden dynamical system figure system coupled link based example provided schweizer challenges modular composition algebraic constraints scenario described coupling condition translates set assignments outputs inputs inputs simulation unit system left hand side fig outputs simulation unit system represented right hand side picture connected directly vice versa practice simulation units models created specific coupling pattern mind complex example consider system coupled massless rigid link depicted fig first subsystem one left hand side fig simulation unit second constituent system governed following differential equations following simulation unit input simulation unit coupling force output state mass input simulation unit external force outputs state mass recall clearly mismatch outputs first simulation unit coupled directly input second simulation unit vice versa however massless link restricts states inputs two units whatever input forces may equal opposite sign hence orchestration algorithm find inputs ensure coupling constraints satisfied problem addressed arnold arnold asada schweizer schweizer sicklinger approach taken asada worth mentioning defines boundary condition coordinator bcc behaves extra simulation unit whose inputs outputs original two simulation units whose outputs show initial scenario constraint translated trivial constraint adding extra simulation unit illustrated fig figure transforming scenario constraint simpler scenario adding extra simulation unit induces trivial constraint promotes separation concerns transforming scenario make simpler marks important step separating concerns orchestrator fact newly created simulation unit run smaller internal size required meet stability accuracy criteria shown asada many solutions proposed information rate change sensitivity outputs states simulation unit respect changes inputs required solve coupling condition information either provided directly jacobian matrix system output functions estimated finite differences provided simulation units rolled back previous states frequent characteristic availability certain capabilities simulation units mitigate lack capabilities show sensitivity information useful one tasks bcc ensure close zero possible finding appropriate inputs possible since functions inputs constraint written one communication step next expanded taylor series known input equations combined obtain input next communication step simple orchestration algorithm perform following steps step let current position outputs two simulation units perform step known obtaining new outputs rollback simulation units state perform step obtaining approximate finite differences obtain corrected fec rollback simulation units state perform final step fec commit states advance time seen fig coupling carried without errors constraint accurately forced zero first try furthermore finding initial conditions initial inputs satisfy equations important usually requires fixed point iteration algorithm could changed perform arbitrary coupling solution trajectory time figure algebraically coupled masses parameters notice small disturbance initial conditions number iterations repeating steps close enough zero would increase accuracy also increase amount computation examples show rollback capabilities important simulation unit black box rollback capability provided simulation unit little orchestrator make lack feature see broman orchestrator takes account existence rollback feature hand simulation unit provides access state allows state set blockwitz orchestrator implement rollback keeping track state simulation unit rollback also plays key role dealing algebraic loops scenario finally explain refers modular composition simulation units example fig makes explicit one problems rigid protected nature simulation units make coupled simulation difficult contrast white box approach equations constituent systems available whole system simplified two masses lumped together coupling forces canceling simplified system lumped easily solvable approach common modeling languages modelica concrete coupled system obtained combining equations simplifying fig compares behavior trace produced algorithm applied scenario described analytical solution obtained coupled model obvious error due extrapolation functions large communication step size solution cosim analytical cosim analytical time figure comparison sample coupled system parameters modular equations made available constituent system coupled systems many different contexts without changes shows possible get around modularity aspect cost algebraic loops algebraic loops occur whenever variable indirectly depends see algebraic loops arise scenarios recall see state evolution output simulation unit written simplify things assume simulation units coupled set assignments outputs inputs input simulation unit output simulation unit scenario definitions easy see depending coupling assignments scenario output simulation unit may depend distinguish two kinds algebraic loops ones spanning input variables ones include state variables well first kind avoided using input extrapolations parameters output functions second kind arises implicit numerical solvers used input approximating functions interpolations instead extrapolations previous example first kind removed replacing corresponding extrapolation depend thus breaking algebraic loop methods ignore algebraic loop though shown arnold schiehlen empirically bastian neglecting algebraic loop lead prohibitively high error better way use fixed point iteration technique algebraic loops involving state variables step repeated convergence algebraic loop involve state variable iteration output functions see algebraic loops involving state variables arise suppose example constructed order imposed evaluation simulation units ensures computed indirectly depend approach improve accuracy shown arnold arnold arnold busch stanciulescu obviously execution simulation unit start simulation unit finished output evaluated input depends indirectly algebraic loop exists output function depends state simulation unit turn obtained executing simulation unit time using extrapolation input improvement input means whole step repeated get improved consequence improved output fixed point iteration technique makes use rollback repeat step corrected inputs called dynamic iteration waveform iteration strong onion coupling simulation units expose outputs every internal waveform iteration improved strong coupling approaches typically best terms accuracy worst terms performance approaches perform correction steps best terms performance worst accuracy variant attempts obtain method fixed limited number correction steps performed see schweizer schweizer examples approach current fmi standard possible perform fixed point iteration output variables step mode rollback simulation units repeat step effectively treating algebraic loop involving state variables assumed full knowledge models simulated simulation unit explain identify deal algebraic loops practice general simulation units extra information required identify algebraic loops according arnold benveniste broman binary flag denoting whether output depends directly input sufficient structural analysis example tarjan strong component algorithm performed identify loops consistent initialization simulators definition simulation unit assumes initial condition part simulation unit however seen example section initial states simulation units coupled algebraic constraints output functions implies initial states simulation units set independently used example constraint satisfied initial states general scenario defined extra coupling function time satisfied example ycs ucs denotes initial state simulation unit represents initial constraint necessarily equal may infinite number solutions case example provided section algebraic loops initialization problem identified blockwitz addressed galtier fmi standard dedicated mode possible fixed point iteration based search consistent initial state simulation units compositional convergence error control accuracy trace degree conforms real trace described section obtaining real trace challenge error difference cosimulation trace real measure accuracy context continuous accurate trace analytical solution coupled model underlies scenario example coupled model corresponding system fig implicitly created scenario described fortunately analytical solution obtained coupled model forms linear time invariant system practice analytical solution coupled model found easily calculating error precisely impossible cases getting estimate error grows well understood procedure numerical analysis simulation factors influence error model solver size naturally size time interval simulated extrapolation functions introduce error inputs simulation units translated error causing feedback error increase time intuitively larger step size larger error made extrapolation functions example forward euler solver used compute approximated behavior trace dynamical system single micro step making error order forward euler infinite taylor series obviously order error made one step commonly called local error depends unbounded derivatives see observe derivative infinite residual term bounded constant multiplied fortunately since continuous time dynamic systems model real system assumption satisfied solver used solvers midpoint method derived truncating higher order terms taylor series midpoint method local truncation error naturally larger micro step size larger local error local error assumes solver made one step starting accurate point compute approximate behavior trace accurate point solver starts initial value rest trace approximate error gets compounded multiple steps forward euler method limit reacts deviations parameter true parameter const const order total accumulation error defined terms size condition called global lipschitz continuity forward euler solver total global error solver useful must convergent computed trace must coincide accurate trace means error controlled adjusting micro step size concept convergence applies intuition suggests decreasing communication step size lead accurate trace answered yet general behavior coupled model induced coupling simulation units may satisfy lipschitz continuity context according arnold arnold busch schweizer hafner schiehlen simulation units convergent coupled model induced scenario coupling conditions written state space form unit induced jacobi gaussseidel strong coupling methods convergent polynomial extrapolation technique inputs presence algebraic loops complex coupling constraints one shown section factors may make impossible write coupled model state space form see arnold examples local error vector defined deviation true trace one step starting accurate point true state vectors outputs respectively simulation unit convergent unit techniques used traditionally simulation estimate error applied richardson extrapolation technique compatible simulation units long provide rollback state capabilities essential idea get estimate local error comparing less accurate point less accurate point computed orchestrator using larger communication step size seen larger communication step sizes affect accuracy two points far apart means communication step need changed importance notice less accurate point computed accurate starting point input extrapolation outputs two different order input approximation methods compared milne device similar previous ones extrapolation inputs compared actual value end step iterative approaches ones studied arnold arnold schweizer schweizer readily benefit technique parallel embedded method technique runs traditional adaptive step size numerical method parallel purpose piggy back auxiliary method decisions step size derivatives integrated simulation unit either provided estimated conservation laws local error estimated based deviation known conservation law extra domain knowledge coupling simulation units required example couplings form power bonds energy conserved across step practice always error due usual factors magnitude energy residual start end step serves estimate local error technique implemented studied sadjina advantage may require rollback functionalities embedded solver method individual simulation units support adaptive step size decisions made internally made public help orchestrator decide communication step size best knowledge orchestrator proposed performs fmi standard allows simulation units reject large communication step sizes error deemed large one methods correction applied pessimistically rollback repeating step optimistically adapt next step mitigate overhead pessimistic approach corrections may applied sensitive simulation units done verhoeven finally traditional simulation techniques applied chose right communication step size see busch schweizer approach gustafsson gustafsson techniques potentially applied compositional stability previous section presented conditions orchestration engine reduce communication step size arbitrarily small value order meet arbitrary accuracy theoretically useful tells orchestrator reducing local error also reduces global error practice communication step size reduced arbitrarily small value without facing performance roundoff error problems performance smaller communication step sizes takes steps compute behavior trace given interval time accuracy digital computer real numbers represented approximately computations involving small real numbers incur error means practice convergence imply arbitrary accuracy achieved better question analyze happens global error trace computed communication step size suppose analytical solution coupled model induced scenario eventually goes zero case coupled system fig described provided least one constants positive intuitively means system lose energy time eventually comes rest let denote analytical solution position mass system let solution computed unit exi denotes global error time made unit exi unit convergent arbitrarily small exi arbitrarily small since practice take arbitrarily small want know whether thus driving exi zero well case means assuming system eventually come rest unit property called numerical stability contrarily convergence numerical stability property depends characteristics system numerical stability always studied assuming system stable makes sense show trace grow unbounded provided system comparison two infinities one ways numerical stability studied calculating spectral radius error unit written autonomous linear discrete system give example recall coupled model induced scenario scribed written coupling conditions order write model autonomous linear discrete system write happens single step executed orchestrator presented algorithm since purpose analyze stability unit stability simulation units common assume simulation units compute analytical trace system enables study stability properties unit starting stable simulation units time simulation unit computing behavior trace following initial value problem ordinary differential equation initial conditions given previous step term denotes fact assuming constant extrapolation input interval linear time invariant value given analytically replacing integration variable matrix exponential rewriting discrete time system gives computation performed simulation unit single step state transition function end step output first simulation unit output function given plugging output repeating procedure second simulation unit yields state transition output functions since coupling conditions combine equations single discrete time system system stable behavior traces remain bounded going zero checked observing whether spectral radius parameters communication step size means unit stable damping constant unit would unstable stable shown fig different coupling methods different approximation functions yield different stability properties see busch busch schweizer stability analysis multiple coupling approaches approximating functions stability various units also studied arnold asada stanciulescu schiehlen schweizer rules thumb drawn papers summarized solution time figure behavior trace described parameters employ fixed point iteration techniques typically better stability properties coupling approach slightly better stability properties order simulation units compute appropriate example simulation unit highest mass computed first main problem applied industrial problems solvers models may coupled black box protect little knowledge kind solver model used stability properties best always use iterative techniques shown better stability properties however techniques require rollback functionalities difficult support certain simulation units even functionalities available cost computing trace prohibitively high compared approaches creates paradox industrial units make use iterative techniques performance toll may high compositional continuity let assume simulation unit continuous system prerequisite physical laws continuity obeyed using extrapolation inputs constant extrapolation laws may obeyed discussed busch sadjina consider point view simulation unit throughout step input kept constant next step input may change radically far away discontinuities inputs may wreak havoc performance simulation unit causing reduce inappropriately micro step size reinitialize solver discard useful information past solvers produce inaccurate values input extrapolation furthermore discontinuity may propagated simulation units aggravating problem numerical methods assume input discretized version continuous trace means discontinuity occurs simulation unit distinguish steep change continuous trace way traditional solvers deal behavior reduce micro step size change steep works continuous signal steep change work discontinuity even size reduced difference still depends communication step size micro step size solver reduce micro step size minimum reached point gives finally advantages micro step times gives acceptable results huge performance toll solver repeatedly retrying small size advance simulated time means huge computational effort goes waste solver finally gives defer discussion correct ways deal discontinuities scenario discontinuities welcome section continuous scenarios discontinuities occur solution avoid discontinuities input approximations use extrapolated interpolation methods instead constant extrapolation methods ensure least give example derive one possible linear extrapolated interpolation method interval since linear constants let avoid discontinuities require want putting constraints together gives solving system gives constraints introduced section major challenge simulation ensure simulation unit satisfy timing constraint challenge gets aggravated due presence multiple simulation units different capabilities order enable every simulation unit fast enough furthermore often needed one simulation unit actually original system wrapped simulation unit means measurements performed state system means noise signals therefore extrapolation functions used simulation units properly protected noise signal using statistical techniques kalman filtering finally quality network important simulation units needs receive inputs timely manner mitigate orchestration algorithm compensate delays receiving data provide inputs simulation unit hybrid approach sections described essential characteristics assumptions simulation units kind approach compared unit whose state evolves continuously time whose output may obey physical laws continuity unit state assume multiple values time transiency output dramatically change short period time orchestrator unit flexibility safe algebraic loops ugly coupling conditions computing contrast simulation unit get inputs produce outputs precise time event occurs due potentially drastic change outputs lipschitz continuous condition allows predicting delay output unit affect overall trace example simulation unit system constant extrapolation function running orchestrator algorithm change input affect output least units time continuous time solvers general seen explicit solver delayed response inputs normal differences units heart many challenges hybrid scenarios mixing two hybrid scenarios give formal definition hybrid scenarios related finding appropriate standard hybrid non trivial challenge see section instead define broadly mixing characteristics assumptions kinds simulation units scenarios together adequate orchestrator used mockups hybrid systems thermostat regulating temperature room classical example continuous constituent system represents temperature dynamics room accounting source heat radiator discrete event part controller turns radiator depending temperature continuous time simulation unit simulates following dynamics output temperature room denotes fast room heated cooled control input turns radiator discrete event simulation unit simulates statemachine shown fig one think input event toohot happening toocold output events assign appropriate value input therefore temperature kept within comfort region clearly two units coupled together via input output assignments orchestrator scenario reconcile different assumptions inputs output simulation unit figure statemachine model controller constituent system simulation unit expects continuous input whereas output simulation unit event signal output simulation unit continuous signal whereas simulation units expects event signal input coupling continuous time discrete event black box simulation units studied state art essence two approaches known based creating wrapper component around simulation unit adapt behavior hybrid wrap every unit simulation unit use based orchestration hybrid wrap every unit become unit use based orchestrator according formalization proposed simulation units hybrid approach applied thermostat example may involve wrapping simulation unit time advance matches size step keeping track output order produce output event whenever crosses thresholds conversely output event converted continuous signal input examples hybrid described awais bolduc vangheluwe camus fey kofman junco kounev kuhr neema nutaro quesnel vangheluwe widl zeigler hybrid example followed wrapping unit unit takes input temperature continuous signal internally reacts event caused crossing threshold conversely output event converted continuous signal examples hybrid include denil feldman garro falcone lawrence quaglia tavella tripakis regardless approach taken properties constituent systems retained fact otherwise discontinuous signal becomes continuous result linear higher order extrapolation may respect properties coupled system knowledge domain simulation units paramount third alternative compared using hybrid hybrid different mechanisms orchestrating simulation units depending semantic domain instance actor modeling language ptolemy actor many similarities simulation unit instead using either hybrid hybrid called director block used particular set connected actors context notion superdense time fundamental also discussed subsection different issues arise hybrid described read light hierarchical hybrid scenarios ality important challenges semantic adaptation generic wrapper based underlying model computation simulation unit used done realization approaches hybrid hybrid depends concrete scenario features simulation units shown thermostat example simply best choice wrappers scenarios even technical level manner events signals sent obtained unit may need adapted concrete simulation unit assume events communicated encoding single string signal opposed different signal signal denote different events account variability common adaptations captured configuration language done denil meyers specialization model computation done kuhr muller widl pedersen approaches require person domain knowledge describes simulation units adapted choice wrapper hybrid approach meant highlight another problem adaptations simulation units wrapper incorporates information ultimately encoded software controller argue need sophisticated semantic adaptations smaller later stages development components refined models thermostat decision turn radiator made wrapper predictive step sizes hybrid approach time advance defined recall setting whatever step size orchestrator decides work adapted simulation unit may produce many absent output events better adaptations proposed thermostat example propose time advance coincides moment leave comfort region thereby always simulated relevant times naturally approaches rely information may expose simulation units others try adaptively guess right time advance monitoring conditions interest set dynamics adapted simulation unit common approach quantization output space capability predict time advance also useful enhance based shown broman event location locating exact time continuous signal crosses threshold well known problem intimately related guessing right time advance predicting step size address solutions typically require derivative information signal causes event capability perform rollbacks thermostat example shows output controller changing time temperature room actually crossed confort zone may accurate large note consequence decisions made orchestrator discontinuity identification based discussion knowledge kind simulation units comprise general hierarchical simulation unit output may event signal coming wrapper unit case runtime signal often represented set points observing sequence points alone make possible discern steep change continuous signal true discontinuity occurs event signal extra information currently used formalization time include notion absent signal proposed broman lee zheng tavella extra signal used discern discontinuity occurs done fmi model exchange even facilitating location exact time discontinuity symbolic information dirac impulses characterize discontinuity included done nilsson discontinuity handling discontinuity located handled depends nature simulation units capabilities simulation unit continuous system traditionally discontinuities inputs handled reinitializing simulation unit step incur high performance cost especially numerical methods andersson andersson proposes improvement solvers furthermore discontinuity cause discontinuities producing cascade process may finish care must taken ensure physically meaningful properties energy distribution respected algebraic loops legitimacy zeno behavior algebraic loops dependencies simulation units detected using feedthrough information explained section based solution algebraic loops attained fixed point iteration technique covered section possibility solution algebraic loop fail converge result left unchecked orchestrator would move infinite number input output values simulation units point time based related property legitimacy undesirable version transiency property explained section illegitimate scenario cause orchestrator move infinite number events timestamp units never advancing time distance matrices used optimize parallel optimistic approaches explained fujimoto used ghosh leveraged detect statically presence classes illegitimacy similar behavior difficult detect zeno behavior occurs increasingly small interval time two consecutive events point sum intervals finite however illegitimate behaviors desired pure least theoretical sense zenoness desired feature hybrid scenarios say theoretical sense purposes scenarios zenoness still recognized appropriate measures regularization taken stability hybrid represents hybrid switched system possible particular sequence events cases system become unstable even individual continuous modes operation stable new analyses required identify whether units yield unstable trajectories result events wrapped simulation units keeping hidden theory approximated states pure based errors neglected computed trajectories essentially exact best knowledge zeigler addresses theoretically error discrete event system propagated based however error accepted well studied techniques exist control hybrid need analysis techniques provide bounds error propagation simulation units coupled sources error addition based analyzes possible simulation unit recognize error exceeded given tolerance measures taken reduce error techniques place allows hybrid orchestrator take appropriate measures adapt communication step size etc keep error bounded every simulation unit standards hybrid functional interface fmi standard high level architecture hla standard time writing standards limitations hybrid bogomolov garro falcone tavella extensions fmi standard awais proposes techniques perform simulation conforming hla recognizing hybrid far well studied broman proposes set idealized test cases hybrid unit underlying standard pass particular important correct handling representation time achieve sound approach simultaneity finally even standardized interface simulation units equal fact makes coding orchestration algorithm real challenge possible approach deal heterogeneity proposed gomes assume units implement set features code orchestration algorithm features delegate wrappers responsibility leveraging extra features mitigating lack section features classified classification described multiple facets section summarizes classification methodology requirements platform independence configuration reusability scalability performance extensibility protection accuracy hierarchy requirements simulator requirements framework requirements fault tolerance optional legend feature mandatory figure distribution feature optional parallelism figure requirements methodology find initial set papers related used google scholar keywords cosimulation coupled simulation collected first pages papers every paper filtered abstract read detail references collected guide reading influential papers gave higher priority cited papers collected read approximately papers create initial version taxonomy read new papers constantly revised taxonomy classified new references cause revisions taxonomy prompted classify collected papers systematic fashion papers collected inclusive including classified two main reasons justify last years interval limited time papers refer based prior work consequence classification would similar many related references prior papers classified report case studies noted create fig taxonomy taxonomy represented feature model structured three main categories shown fig requirements nfrs groups concerns performance accuracy protection reference addresses simulation unit requirements srs features simulation units orchestrator described paper examples information exposed causality availability rollback support framework requirements frs features provided orchestrator examples dynamic structure adaptive communication step size strong coupling support main group detailed figures abstract features denote concepts easily detailed chose sake brevity mandatory features required activity optional state art give example taxonomy used consider work van acker fmi based orchestration algorithm generated description scenario paper description language introduced reused manner orchestration code generator analyzes scenario identifies algebraic loops using feedthrough information separates fast moving simulation units slow moving ones using preferred step size information provides interpolation fast ones finds largest communication step size divides step sizes suggested simulation units uses throughout whole argue generated orchestrator fast decisions made code generation stage algebraic loops solved via successive substitution inputs storing restoring state simulation units based facts van acker classified follows requirements performance protection configuration reusability simulation unit requirements causal simulation units locally available simulation units preferred step size information solver feedthrough causality information state values time constraints rollback support framework requirements continuous time domain simulation fixed communication step size coupling fully implicit strong coupling support postmortem visualization results postmortem visualization results fmi underlying standard communication approach support three simulation units similar reasoning fmi standard version classified according assumptions makes participating simulation units highlighted fig remaining state art classified figures raw data available discussion observing fig accuracy observed nfr reports followed protection performance least observed nfrs fault tolerance hierarchy extensibility fault tolerance especially important long running one industrial partners project running takes minimum two weeks complete argue extensibility ability easily accomodate new features given importance heterogeneous set simulation units participate scenario combination capabilities provided see fig huge thus orchestrator either assume common homogeneous set capabilities common approach leverage capabilities provided one later approach lead extremely complex orchestration algorithm case extensibility key address new semantic adaptations recall section fig suggests could find approaches make use nominal values state output variables even though capabilities supported fmi standard see fig useful detect valid approaches important modularity explained section scarce http framework requirements fig least observed features dynamic structure interactive visualization algebraic coupling strong coupling support explained fact features depend upon capabilities simulation units may mature figures tell full story isolate feature feature interaction common phenomenon among many possible interactions highlight accuracy concern domain number simulation units supported protection seen fig one approach based number simulation units note mean work addresses challenges identified section accommodating different domains means approach assumes simulation units behave unit concern protection evident fig number based approaches provide support small shown fig similarly fig suggests accuracy show lot approaches two simulation units accuracy particularly important interactions simulation units general observed classification lack research approaches based leverage extra features simulation units concluding remarks work shows many interesting challenges explored play key role enabling virtual development complex heterogeneous systems decades come early success attributed large number reported applications however application references covered see fig large majority represent couplings two simulators two different domains network simulator power grid one hvac simulator building envelop one however excludes potentially many unreported applications systems become complex demand cosimulation scenarios large hierarchical heterogeneous accurate protected increase survey presents attempt covering main challenges tackle broad topic covered two main domains time discrete event based separately surveyed challenges arise two domains combined taxonomy proposed classification works related last five years carried using taxonomy challenges highlight semantic adaptation modular coupling stability accuracy finding standard hybrid particular important early system analysis adaptations required combine simulators different formalisms even conforming standard difficult generalize scenario possible work around allow system integrator describe proposed one main conclusions classification lack research modular stable accurate coupling simulators dynamic structure scenarios approaches play key role use ports solution inspired work already direction finally document attempt summarize bridge enhance future research wherever may lead acknowledgment authors wish thank yentl van tendeloo depth review based kenneth guldbrandt lausdahl providing valuable input discussions throughout making survey twt gmbh valuable input everyday challenges faced cosimulation master research partially supported flanders make vzw strategic research centre manufacturing industry phd fellowship grant agency innovation science technology flanders iwt addition work presented partially supported project funded european commission horizon programme grant agreement number project financially supported swedish foundation strategic research simulator requirements causality causal rollback support none availability local none scaled static model dynamic solver outputs deadreckoning model detailed model nominal values discontinuity indicator preferred outputs state order accuracy state frequency multiple remote information exposed outputs single time constraints next values wcet signal kind serialization input extrapolation causality state derivative time outputs jacobian state outputs propagation feedthrough delay dependency kind state linear abstract feature feature fmi mandatory exclusive optional figure simulation unit requirements features provided fmi standard cosimulation version framework requirements dynamic structure simulators domain comm approach rate jacobi feature single multiple comm step size fixed mandatory exclusive optional standard hla fdmu fmi results visualization adaptive coupling assignment live algebraic constraints postmortem interactive strong coupling support none fully implicit category figure framework requirements platform independence performance protection hierarchy fault tolerance parallelism distribution accuracy extensibility config reusability year total reports figure classification respect requirements causal category locally available remotely available time constraints fixed constraints dynamic constraints rollback single rollback reports multi rollback year category figure classification respect simulation unit requirements execution capabilities state nominal values output nominal values input extrapolation model kind signal frequency outputs preferred step size next step size feedthrough worst case exec time state values serialized state outputs state jacobian output jacobian state derivatives output derivatives reports year figure classification respect simulation unit requirements information exposed category dynamic domain domain single rate fixed comm step adaptive comm step coupling alg constraints coupling strong coupling partial strong coupling full strong coupling post mortem visualization live visualization interactive visualization fmi based hla based fdmu based jacobi communication communication two simulators three simulators reports year domain reports domain reports simulators accuracy simulators yes accuracy protection figure classification respect framework requirements reports domain figure formalisms figure formalisms figure accuracy forprotection ulation units malisms simulation units references modelica unified language physical systems modeling url https ieee standard modeling simulation high level architecture hla federate interface specification url https gian abad lady mari faeda guerrero jasper kendall ignacio dianne magtibay mark angelo purio evelyn raguindin simulation power surge monitoring suppression system using labview multisim tool international conference humanoid nanotechnology information technology communication control environment management hnicem pages ieee dec isbn doi andreas abel torsten blochwitz alexander eichberger peter hamann udo rein functional interface mechatronic gearshift simulation commercial vehicles international modelica conference munich ahmad comprehensive platform systems computer communications dec doi alur courcoubetis halbwachs henzinger nicollin olivero sifakis yovine algorithmic analysis hybrid systems theoretical computer science feb issn doi alvarez cabrera krijn woestenenk tetsuo tomiyama architecture model support cooperative design mechatronic products control design case mechatronics apr issn doi christian andersson methods tools dynamic systems functional interface phd thesis lund university christian andersson claus johan efficient predictor multistep solvers technical report mathematical sciences issn url http martin arnold stability sequential modular time integration methods coupled multibody system models journal computational nonlinear dynamics may issn doi martin arnold michael preconditioned dynamic iteration coupled systems bit numerical mathematics doi martin arnold antonio carrarini andreas heckmann gerhard hippmann simulation techniques multidisciplinary problems vehicle system dynamics 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krammer markus schratter peter wimmer daniel watzenig christianmichael gruber comprehensive approach modeling simulation virtual validation integrated safety systems jan gereon meyer editors advanced microsystems automotive applications lecture notes mobility pages springer international publishing isbn doi abir ben khaled mongi ben gaid daniel simon gregory font multicore simulation powertrains using weakly synchronized model partitioning ifac proceedings volumes volume pages france oct doi abir ben khaled laurent duval mohamed mongi ben daniel simon contextbased polynomial extrapolation slackened synchronization fast simulation using fmi international modelica conference pages link ping university electronic press james ellis kleckner advanced simulation techniques phd thesis ernesto kofman approximation devs simulation continuous systems simulation feb issn doi ernesto kofman sergio junco systems devs approach continuous system simulation transactions society modeling simulation 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vehicle esp system based adams matlab journal software monti luo dougal vpnet framework analyzing communication channel effects power systems ieee electric ship technologies symposium pages ieee apr isbn doi weilin min luo lin zhu antonello monti ferdinanda ponci method enabler jointly analysis design electrical power protection communication simulation mar weilin xiaobin zhang huimin platforms networked control systems overview control engineering practice feb issn doi liu cai zhu tsai wong calculation wing flutter coupled method journal aircraft mar issn doi john lygeros lecture notes hybrid systems oded maler zohar manna amir pnueli timed hybrid systems theory practice doi manbachi sadu farhangi monti palizban ponci arzanpour impact penetration optimization distribution networks using monitoring platform applied energy may issn doi toni mancini federico mari annalisa massini igor melatti fabio merli enrico tronci system level formal verification via model checking driven simulation natasha sharygina helmut veith editors computer aided verification volume lecture notes computer science pages springer berlin heidelberg isbn doi zohar manna amir pnueli verifying hybrid systems robertl grossman anil nerode andersp ravn hans rischel editors hybrid systems volume lecture notes computer science pages springer berlin heidelberg isbn doi url http martinez kurzweg levitan marchand chiarulli mixedtechnology simulation analog integrated circuits signal processing issn doi william mccalla fundamentals circuit simulation volume springer science business media isbn mews svacina weissleder autosar models miltesting automotive domain software testing verification validation icst ieee fifth international conference pages isbn doi bart meyers joachim denil boulanger hardebolle christophe jacquet hans vangheluwe dsl explicit semantic adaptation edward jones christophe jacquet daniel balasubramanian editor mpm number ceur workshop proceedings pages miami united states sep miekkala 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dynamic behavior integrated together dynamic iteration large circuits decomposed coupled constituent systems dynamic iteration techniques used software hardware developed simulated concurrently multiple levels abstraction orchestration methods explored carothers fey frey tseng hulbert late early standard interfaces recognized key protection scale important enhance industrial applicability cosimulation historical perspective section provides historical perspective relates major concepts time recognized studied state art summarized table one formalism dynamic iteration traditionally equations describing dynamical behavior large circuits integrated together systems sparsely coupled reflecting connections corresponding circuits many techniques developed take advantage structure crucial idea improved simulation speed two orders magnitude decompose large system set coupled constituent systems integrate independently decomposition circuit implies definition inputs outputs resulting constituent systems coupling assignment outputs inputs subsystem call subsystems whose outputs assigned inputs neighbor subsystems essence dynamic iteration approach integrate subsystem independently period time using extrapolated outputs neighbor subsystems inputs naturally fact outputs extrapolated introduces inaccuracy solution subsystem integration repeated period time corrected outputs form convergence criteria met extrapolated outputs subsystem corrected collecting outputs integration easy see approach requires communication constituent systems times integration subsystem done independently parallel using numerical method step size control policy signals exchanged functions interval advantages independent step size control policy become evident one observes many circuits components change different rates whole system simulated simulation unit would use smallest time step ensures sufficient accuracy fastest changing component would huge waste computational effort slow components similarity numerical methods best knowledge dynamic iteration techniques numerical first resemble coordination software implements techniques expect number subsystems assumes subsystems specific formalism differential equations two formalisms digital analog modern definition applied enable virtual development coupled software hardware systems application domain decreases need build prototype board circuits validate composition software hardware part enables software hardware developed validated concurrently best knowledge one first uses modern sense frameworks developed application domain typically assumed two simulation units two formalisms systems quickly became complex new idea introduced use multiple models different levels abstraction subsystem simulations could made arbitrarily faster intervals solving abstract models arbitrarily accurate intervals solving detailed ones particular case level abstraction solved different tool continuous time tool discrete event tool separation continuous time discrete event made abstract synchronization problem synchronization methods simulation units two domains developed could call first master algorithms heterogeneity aspect comes play time multiple formalisms used describe subsystem multiple levels abstraction state machines describe rough approximation modes differential equations describe detailed dynamics electronic circuit depending purpose subsystem neighbors solved detail whereas subsystems farther away simulated higher levels abstraction domain rtl tlm classify multiple abstraction levels models switching multiple levels abstraction studied number heterogeneity simulation tools coupled increases need provide common interface couple number tools recognized hickey kuhl petrellis zwolinski later blochwitz parallel previous advancements also use heterogeneous physical systems automotive railway hvac name common motivation fact enables specialized simulation units cooperatively simulated system huge savings time cost compared monolithic modeling approach later distributed concurrent development processes enabled studied protection identified desired characteristic enable suppliers integrators exchange units without disclose sensitive information avoiding vendor contracts furthermore used every stage development process early system validation bringing hard constraints set challenges many simulation units large scale recently acknowledgment need able simulate even larger systems systems scale distribution become inherent challenges state art frameworks section provides detailed classification reference ref distributed engine control system network model using fmi scnsl summary work describes master context maritime industry causality causal rel time analytic rollback none availability local results visualization post mortem alg loop explicit sim step size fixed sim rate single domain coupling model assignments standard fmi communication model jacobi num sim three ref power system communication network smart grid applications summary work describes power system network simulator info predict step sizes causality causal rel time analytic rollback none availability local results visualization post mortem alg loop explicit sim rate single coupling model assignments num sim two domain sim step size variable communication model gauss seidel ref towards comprehensive framework simulation dynamic models emphasis time stepping summary work describes approach finds appropriate step size nfr performance nfr accuracy nfr protection availability local info derivatives info derivatives state info statevars causality causal rollback none rel time analytic num sim three domain sim rate single sim step size variable communication model jacobi communication model gauss seidel alg loop explicit results visualization post mortem ref methods simulation systems application automotive domain summary work addresses simulators nfr performance nfr parallelism availability local rel time fixed real scaled time simulation info wcet rollback none results visualization post mortem num sim three coupling model assignments domain communication model jacobi alg loop explicit sim rate single sim step size fixed ref simulation multibody systems use coupling techniques case study summary work discusses method couplings algebraic constraints one results kind coupling done many derivatives coupling variables nfr accuracy rollback single availability local rel time analytic causality causal info derivatives info jacobian results visualization post mortem communication model jacobi alg loop implicit coupling model algebraic constraints domain num sim three sim rate single sim step size fixed ref combining advantages specialized simulation tools modelica models using functional interface fmi summary work describes application power production domain nfr protection nfr performance causality causal rel time analytic rollback none availability local results visualization post mortem sim rate single coupling model assignments standard fmi domain communication model gauss seidel num sim two alg loop explicit sim step size variable ref master using fmi summary work describes approach nfr protection nfr platform independence nfr parallelism causality causal rollback none info stateserial info jacobian availability local results visualization post mortem coupling model assignments sim step size fixed sim rate single alg loop implicit domain num sim three standard fmi communication model jacobi communication model gauss seidel ref parallel mechatronic systems summary work describes framework based jacobi iteration scheme nfr parallelism nfr performance nfr distribution nfr protection availability remote causality causal rel time analytic rollback none results visualization post mortem alg loop explicit domain coupling model assignments coupling model algebraic constraints num sim three communication model jacobi sim rate single sim step size fixed ref effect multirate techniques efficiency accuracy multibody system dynamics summary work deals essentially one simulators fast one drives simulation slow one provides extrapolated inputs avoid excessive computation nfr accuracy nfr performance causality causal rollback none availability local rel time analytic results visualization post mortem alg loop explicit communication model gauss seidel coupling model assignments num sim two domain sim rate multi sim step size fixed ref designing power system simulators smart grid combining controls communications dynamics summary work describes tool formed coupling devs simulator modules wrap devs simulators nfr distribution nfr accuracy causality causal rollback single rel time analytic availability local alg loop explicit domain communication model gauss seidel num sim two sim rate single sim step size variable coupling model assignments results visualization post mortem ref asynchronous method coupled simulation mechatronic systems summary work describes approaches two simulation tools main contribution method applies correction based jacobian subsystem coupling variables nfr accuracy nfr distribution causality causal rollback single availability remote info jacobian rel time analytic sim rate single sim step size fixed results visualization post mortem communication model gauss seidel alg loop semi implicit alg loop explicit alg loop implicit coupling model assignments domain num sim two ref generating functional mockup units software specifications summary work describes application robotics ref convergence study explicit approaches respect subsystem solver settings summary paper describes global error analysis takes account solvers instead analytical solvers commonly done nfr accuracy rollback none info full model causality causal rel time analytic availability local alg loop explicit domain num sim two coupling model assignments communication model jacobi sim rate single sim step size fixed results visualization post mortem ref hybrid systems modelling simulation destecs approach summary work present coupling tools crescendo causality causal rel time analytic rollback none availability local communication model gauss seidel results visualization post mortem coupling model assignments alg loop explicit domain sim rate single sim step size fixed num sim two ref new environment jpeg algorithm summary work introduces guidelines implementation matlab systemc case study jpeg algorithm info full model causality causal rel time analytic availability local domain coupling model assignments sim step size fixed sim rate single alg loop explicit communication model gauss seidel results visualization live num sim two ref stabilized overlapping modular time integration coupled differentialalgebraic equations summary work discusses techniques simulators coupled via algebraic constraints nfr accuracy availability local rel time analytic causality causal rollback none info full model info jacobian results visualization post mortem sim rate single num sim three alg loop explicit sim step size fixed domain coupling model algebraic constraints communication model gauss seidel communication model jacobi ref modular technique automotive system simulation summary work describes mdpcosim framework nfr performance decomposition system done performance reasons nfr parallelism nfr accuracy availability local ipc communication used causality causal info derivatives rollback none info predict step sizes results visualization post mortem alg loop explicit coupling model assignments domain communication model jacobi sim step size variable step size control approach based looking derivatives sim rate single num sim three ref tool networked control systems summary work describing another tool coupling nfr distribution causality causal rel time analytic rollback none availability remote results visualization post mortem alg loop explicit coupling model assignments domain num sim two sim rate single sim step size fixed communication model gauss seidel ref comprehensive platform systems summary work describes integration two tools modelica causality causal rel time analytic availability local communication done named pipes rollback none results visualization post mortem num sim two coupling model assignments alg loop explicit domain communication model gauss seidel domain sim rate single sim step size variable ref ncswt integrated modeling simulation tool networked control systems summary work describes coupling two tools matlab coupling done hla standard preliminary version tool described nfr platform independence nfr performance nfr distribution info predict step sizes causality causal rel time analytic rollback none availability remote alg loop explicit num sim two coupling model assignments domain standard hla communication model gauss seidel sim rate single sim step size variable results visualization live ref networked control system wind tunnel ncswt evaluation tool networked systems summary work describes coupling two tools matlab coupling done hla standard nfr platform independence nfr performance nfr distribution info predict step sizes causality causal rel time analytic rollback none availability remote alg loop explicit num sim two coupling model assignments domain standard hla communication model gauss seidel sim rate single sim step size variable results visualization live ref framework tools power systems analysis software summary work describes two tools power grid domain matlab running nfr distribution nfr open source causality causal rel time analytic rollback none availability remote coupling model assignments domain communication model gauss seidel alg loop explicit results visualization post mortem num sim two sim rate single sim step size fixed ref collaborative modelling development dependable embedded systems summary work describes coupling two tools overture nfr accuracy nfr distribution nfr platform independence availability remote causality causal rel time analytic sim step size variable sim rate single domain domain num sim two coupling model assignments alg loop explicit results visualization live communication model gauss seidel ref formal approach collaborative modelling embedded systems summary work describes coupling two tools overture already described nfr accuracy nfr distribution nfr platform independence availability remote causality causal rel time analytic sim step size variable sim rate single domain domain num sim two coupling model assignments alg loop explicit results visualization live communication model gauss seidel ref robonetsim integrated framework network simulation summary work describes integration three simulators argos used scenarios two simulators nfr distribution nfr platform independence rollback none causality causal rel time analytic availability remote results visualization post mortem alg loop explicit communication model jacobi domain num sim two sim rate single sim step size fixed coupling model assignments ref determinate composition fmus summary work describes master algorithm ensures determinate execution nfr protection availability local rollback none rel time analytic causality causal info causality feedthrough info predict step sizes info stateserial coupling model assignments standard fmi sim step size variable domain communication model jacobi num sim three sim rate single alg loop explicit results visualization post mortem ref guidelines application coupling method summary work describes approach energy information signals used errors compensated corrector step nfr accuracy nfr performance rollback none rel time analytic causality causal info record outputs availability local results visualization post mortem alg loop explicit num sim two coupling model assignments communication model gauss seidel domain sim rate single sim step size fixed ref selection monitoring coupoling error weak coupled subsystems summary work describes method finding appropriate communication step sizes cosimulations lti systems essentially provides rules thumb chose communication step size based maximum instantaneous frequency components nfr accuracy availability local rollback none rel time analytic causality causal info frequency outputs results visualization post mortem sim rate single coupling model assignments alg loop explicit communication model gauss seidel domain sim step size fixed num sim two ref communication simulations power system applications summary work describes two tools gridlabd smart grid development nfr scalability nfr faulttolerance nfr protection nfr distribution tools keeps track messages transit causality causal rel time analytic availability remote rollback none coupling model assignments alg loop explicit domain num sim two sim rate single sim step size variable results visualization post mortem communication model gauss seidel ref environment heterogeneous systems summary work describes coupling two simulation tools nfr distribution rollback none causality causal rollback single rel time real scaled time simulation availability remote alg loop explicit coupling model assignments communication model gauss seidel domain num sim two results visualization interactive live sim rate single sim step size fixed ref hybridsim modeling toolchain systems summary approach described reference allows arrange process units modelica models tinyos applications sysml used configure master coordination simulators done fmi standard nfr protection nfr config reusability causality causal rel time analytic rollback none availability local coupling model assignments alg loop explicit communication model gauss seidel num sim two domain standard fmi sim rate single sim step size variable results visualization post mortem ref investigation loose coupling bcvtb summary work discusses consistency stability jacobi cosimulation methods later presents case study hvac systems nfr accuracy causality causal rel time analytic rollback none availability local coupling model assignments results visualization post mortem communication model jacobi communication model gauss seidel alg loop explicit domain num sim three sim rate single sim step size fixed ref research application active distribution network based ptolemy simulink summary work describes ptolemy simulink nfr distribution causality causal rel time analytic rollback none availability remote coupling model assignments domain communication model gauss seidel num sim two sim rate single sim step size fixed alg loop explicit results visualization post mortem ref vpnet framework analyzing communication channel effects power systems summary work describes coupling two simulation tools vtb opnet achieve availability local rollback none causality causal rel time analytic results visualization post mortem alg loop explicit coupling model assignments communication model gauss seidel domain coordination sample discrete time system num sim two sim rate single sim step size fixed ref distributed hybrid simulation using hla functional interface summary main difference work proposes variable step size wrapper around components approach taken quantization nfr distribution nfr parallelism rollback none causality causal rel time analytic availability remote availability local alg loop explicit communication model gauss seidel communication model jacobi standard fmi standard hla num sim three sim rate multi sim step size variable coupling model assignments domain results visualization post mortem ref using hla distributed continuous simulations summary work addresses need adapt simulators simulators order used hybrid scenario fundamentally oriented nfr distribution nfr parallelism rollback none causality causal rel time analytic availability remote availability local alg loop explicit communication model gauss seidel communication model jacobi standard fmi standard hla num sim three sim rate multi sim step size fixed coupling model assignments domain results visualization post mortem ref feral framework simulator coupling requirements architecture level summary describe framework borrows many concepts ptolemy fundamentally event based allows specialization basic directors semantic adaptation simulation units nfr protection nfr extensibility info signal causality causal rollback none availability local rel time analytic sim rate multi communication model gauss seidel standard fmi domain domain num sim three sim step size variable ref implementing stabilized strongly coupled systems using functional interface summary work describes implementation method described context fmi standard nfr accuracy nfr protection info jacobian info input extrapolation info record outputs info stateserial causality causal rel time analytic availability local rollback none domain num sim three sim rate single sim step size fixed alg loop implicit results visualization post mortem coupling model algebraic constraints communication model gauss seidel standard fmi ref interface summary describes method makes use jacobian information fixed point computations nfr performance nfr accuracy availability local rel time analytic rollback single info jacobian causality causal results visualization post mortem communication model gauss seidel communication model jacobi coupling model algebraic constraints domain alg loop implicit num sim three sim rate single sim step size fixed ref framework design automotive cyber physical systems summary work describes integrates systemc carsim availability local rollback none causality causal info full model rel time analytic coupling model assignments domain domain alg loop explicit results visualization post mortem sim step size fixed sim rate single num sim two ref microgrid framework summary describes coupling two simulators written matlab nfr performance availability local rel time analytic rollback none dev orchestration conservative causality causal info predict step sizes coupling model assignments results visualization post mortem communication model gauss seidel domain num sim two sim rate single sim step size variable alg loop explicit ref hybrid systems spaceex uppaal summary orchestration algorithm one described work exploits standard allowing zero step transitions info causality feedthrough rollback none info stateserial causality causal rel time analytic availability local coupling model assignments standard fmi num sim two communication model jacobi alg loop explicit sim rate single sim step size variable due rejection steps due accuracy domain domain abuse fmi standard able support state transitions results visualization post mortem ref platform using opnet analyzing smart grid performance summary many details provided orchestration however due fact infer certain features nfr performance rollback none causality causal rel time fixed real scaled time simulation availability remote communication done udp sim step size fixed sim rate single two simulators give domain coupling model assignments num sim two alg loop explicit results visualization post mortem ref coupling multizone airflow contaminant transport software contam energyplus using summary work described coupling contam energyplus tools achieve hvac simulation coupling done fmi coupling done compiled binaries case study highlights problems explicit method even instabilities occur nfr protection rollback none rel time analytic causality causal availability remote results visualization post mortem domain num sim two standard fmi alg loop explicit coupling model assignments communication model gauss seidel sim step size fixed uses synchronization step sim rate single ref multicore simulation powertrains using weakly synchronized model partitioning summary according work explores variable step solvers nfr parallelism nfr performance info causality feedthrough info full model rel time fixed real scaled time simulation rollback none availability local standard fmi coupling model assignments num sim three domain sim rate single sim step size fixed alg loop explicit results visualization post mortem communication model jacobi communication model gauss seidel ref fast systems application internal combustion engines summary paper focus parallelization approach start single model partition multiple models executed separate fmus parallel partitioning important accuracy reasons break algebraic loops less sensitive variables nfr parallelism nfr accuracy nfr performance nfr scalability info full model info wcet info causality feedthrough causality causal rel time analytic rollback none availability local communication model gauss seidel sim step size fixed sim rate single standard fmi domain num sim three alg loop explicit breaks loops establishing order delaying one variables loop ref acceleration fmu architectures summary paper addresses problem performance fmi solution proposed parallel parallelization approach one presented since fmi enforce thread safety across multiple instances fmu work presented ensures execute concurrently using mutexes changing scheduling policy nfr parallelism nfr performance nfr protection nfr scalability info wcet info causality feedthrough causality causal rel time analytic rollback none availability local communication model jacobi standard fmi domain num sim three alg loop explicit ref adas virtual prototyping using modelica unity via openmeta summary framework includes tools communication tools realized using openmeta work uses unity modelling simulation environment allowing live interaction communication udp report extra caution due network delays failures nfr parallelism info full model causality causal rel time analytic rollback none availability remote num sim three domain results visualization live results visualization interactive live coupling model assignments sim rate single alg loop explicit ref combining devs concepts design simulate multimodels complex systems wip summary work preliminary description nfr parallelism nfr distribution nfr protection nfr accuracy rollback none info stateserial causality causal rel time analytic availability local domain alg loop explicit num sim three sim rate multi sim step size variable standard fmi coupling model assignments communication model gauss seidel results visualization post mortem ref hybrid fmus using dev dess mecsyco summary proposes use fmu wrapper around dev dess models meaning proceeds using approach handles black box fmus algorithm used drive conservative parallel devs simulator requires fmu able perform rollback use state set get nfr parallelism nfr distribution nfr protection nfr accuracy rollback none info stateserial causality causal rel time analytic availability local domain alg loop explicit num sim three sim rate multi sim step size variable standard fmi coupling model assignments communication model gauss seidel results visualization post mortem ref fmi embedded control software summary paper describes adaptation embedded system comply fmi thus interface fmus validate implementation run nfr distribution nfr parallelism rel time fixed real scaled time simulation domain num sim two sim rate single sim step size fixed communication model gauss seidel standard fmi coupling model assignments alg loop explicit results visualization live ref framework power system analysis summary paper proposes framework takes account network delays compensates proposes use cubic spline extrapolation compensate delay recognizes faults line resulting voltage drops derivatives used extrapolation assume gigantic proportions thus wreaking havoc simulation address framework employes algorithm detect discontinuities detection simple check derivative signal see whether exceeds empirically threshold basically looks dirac delta figure shows effect handling discontinuity nfr distribution nfr accuracy nfr parallelism num sim two sim rate single sim step size fixed communication model gauss seidel due parallel interface protocol use coupling model assignments domain alg loop explicit availability remote causality causal rel time analytic rollback none ref mechanical electrical ssr application practical shaft failure event summary two simulators explained paper prior common approach would run two simulations one complete one another second using first inputs open loop approach whose results misleading due ignoring feedback loops simulator advances parallel communication made barrier nfr parallelism communication model jacobi num sim two domain sim step size fixed results visualization post mortem sim rate single availability local causality causal rel time analytic rollback none alg loop explicit coupling model assignments ref impact penetration optimization distribution networks using monitoring platform summary describes application distribution energy smart grids supported framework simulators involved rtds simulates distribution network model vvo engine coded matlab rel time fixed real scaled time simulation num sim two causality causal rollback none availability local coupling model assignments sim rate single sim step size fixed alg loop explicit communication model jacobi ref communication step size control summary describes master algorithm allow interpolation inputs needs rollback touches upon accuracy suggests adaptive step size control mechanism address algebraic loops assumes feedthrough information nfr performance nfr accuracy info derivatives info stateserial causality causal rel time analytic rollback single availability local standard fmi coupling model assignments num sim three domain sim rate single sim step size variable alg loop explicit results visualization post mortem communication model jacobi ref based platform wireless protocols design explorations summary application wireless network development one simulators actual linux operating system represents wireless network protocol simulator causality causal rel time analytic availability local coupling model assignments num sim two domain communication model jacobi ref calculation wing flutter coupled method summary fully implicit method dealing parallelism nfr parallelism nfr performance causality causal rel time analytic rollback single availability remote coupling model assignments num sim two domain alg loop implicit results visualization post mortem ref coupled simulation interaction turbomachinery summary relates application algorithm simulation deformation blades transonic compressor rotor airflow one simulators calculates deformation blades calculates flow dynamics around blades communication orchestration algorithm use shifted half step nfr performance highlight need computation rotor expensive info derivatives causality causal rel time analytic rollback none availability remote seems perform computation separate computers coupling model assignments num sim two domain sim rate single sim step size fixed alg loop explicit results visualization post mortem communication model gauss seidel although gauss seidel shifted time ref coupling approach summary proposes address challenges using model based coupling approach master keep track two values packet data receiving time delay time takes packet reach master simulator sending time delay time takes packet leave master reach simulator sample delayed master acts replacement basically dead reckoning model nfr performance nfr parallelism nfr accuracy causality causal domain num sim two availability local coupling model assignments rollback none rel time fixed real scaled time simulation sim rate single sim step size fixed alg loop explicit results visualization post mortem communication model jacobi communication model gauss seidel ref automated configuration summary describes master configure parameters throughout execution idea behind adaptive master algorithms nfr protection nfr accuracy info causality feedthrough causality causal availability local rollback none rel time analytic domain num sim three coupling model assignments sim rate single sim step size variable alg loop explicit results visualization post mortem ref explicit approach controlling size methods summary presents approach estimate local truncation error caused extrapolations inputs assumed make error require rollback categories nfr accuracy study global error control local error nfr protection nfr performance control step size increases performance study get optimal step size info causality feedthrough rollback none causality causal domain num sim three rel time analytic sim rate multi sim step size variable alg loop explicit availability local results visualization post mortem coupling model assignments communication model jacobi theory seem support communication model paper studied assuming jacobi ref devs coupling spatial ordinary differential equations vle framework summary proposes way wrap continuous time ode simulator devs model requires state variables derivatives available categories nfr hierarchy nfr open source info derivatives info statevars info predict step sizes causality causal domain num sim three rel time analytic sim rate multi discrete event framework definition sim step size variable discrete event framework definition category alg loop explicit availability local results visualization post mortem coupling model assignments communication model gauss seidel discrete event framework category extrapolation inputs also gauss seidel violate causality inputs outputs sorts according dependencies events processed retain causality ref error analysis coupling summary describes fmi based master called snimowrapper categories nfr accuracy study global error control local error nfr protection info causality feedthrough rollback none causality causal rel time analytic availability local domain num sim three sim rate multi sim step size fixed alg loop explicit results visualization post mortem coupling model assignments communication model jacobi standard fmi ref error analysis error estimates fmi model exchange summary studies error control method known richard extrapolation categories nfr accuracy study global error control local error nfr protection info causality feedthrough info statevars rollback single causality causal domain num sim three rel time analytic sim rate multi sim step size variable alg loop explicit availability local results visualization post mortem coupling model assignments communication model jacobi communication model gauss seidel standard fmi ref preconditioned dynamic iteration coupled systems summary studies convergence dynamic iteration method proposes way ensure way though requires information model categories nfr accuracy study global error info jacobian info record outputs info full model rollback single causality causal rel time analytic availability local domain num sim three sim rate multi sim step size fixed alg loop implicit results visualization post mortem coupling model algebraic constraints communication model gauss seidel ref approach solver coupling summary proposes predictor corrector master evaluates macro step twice uses perturbation inputs get estimate required partial derivatives approach generalized multiple kinds joints mechanical domain double pendulum double slider crank mechanism used numerical examples categories rollback single causality causal rel time analytic availability local domain num sim two sim rate multi sim step size fixed alg loop semi implicit results visualization post mortem coupling model algebraic constraints communication model jacobi ref stabilized implicit methods solver coupling based constitutive laws summary presents implicit methods scenarios coupled via applied forces difference paper previous ones author seems fact coupling constraints integrated differentiated enrich information used ensure original coupling constraints met categories rollback single causality causal rel time analytic availability local domain num sim three sim rate multi sim step size fixed alg loop semi implicit results visualization post mortem coupling model algebraic constraints communication model jacobi ref energy conservation power bonds adaptive step size control error estimation summary proposes master requires identification power bonds assumes scenario energy conserving thus calculate energy residual error minimized step size adapted via step size reduced next step method explicit nfr protection nfr accuracy due step size control rollback none causality causal rel time analytic availability local domain num sim three sim rate multi sim step size variable alg loop explicit results visualization post mortem coupling model assignments communication model jacobi ref continuous approximation techniques methods analysis numerical stability local error summary analyses stability local error multiple approaches multiple extrapolation approaches inputs considers jacobi also talks method called extrapolated interpolation method ensure discontinuities inputs subsystems rollback none method explicit causality causal domain num sim three rel time analytic sim rate single sim step size fixed alg loop explicit availability local results visualization post mortem coupling model assignments communication model jacobi communication model gauss seidel ref stability sequential modular time integration methods coupled multibody system models summary studies stability gauss seidel method proposed previous work based analysis proposes implicit stabilization technique uses iteration resulting method implicit equations solved linear categories nfr accuracy study global error info jacobian info record outputs info full model rollback single causality causal rel time analytic availability local domain num sim three sim rate single sim step size fixed alg loop implicit results visualization post mortem coupling model algebraic constraints communication model gauss seidel ref algebraically coupled dynamic subsystems summary describes technique deal algebraically coupled using control theoretic approach highlights method supports scenarios arbitrary index boundary condition coordinator seen unit elegant approach method explicit beauty making bcc unit like run different rate paper show running higher rate stability increases rollback none causality causal domain num sim two rel time analytic sim rate multi sim step size fixed alg loop explicit availability local results visualization post mortem coupling model algebraic constraints communication model jacobi ref algebraically coupled dynamic subsystems without disclosure proprietary subsystem models summary describes technique solve causal conflicts using boundary condition coordinator bcc causal conflicts arise naturally coupling different relevant challenge needs overcome order perform correct bcc requires knowledge state variables simulations modifications made ensure information required nfr protection nfr distribution rollback none causality causal domain num sim three rel time analytic sim rate multi sim step size fixed alg loop explicit availability local results visualization post mortem coupling model algebraic constraints communication model jacobi ref method solver coupling algebraic constraints incorporating relaxation techniques summary master algorithm capable dealing algebraic constraints described requires derivatives coupled variables available executes communication step twice method uses predict step corrector step final corrected coupling variables obtained polynomial extrapolation relaxation avoid instabilities categories rollback single causality causal domain num sim three rel time analytic sim rate multi info jacobian sim step size fixed alg loop semi implicit availability local results visualization post mortem coupling model algebraic constraints communication model jacobi ref approaches solver coupling algebraic constraints summary proposes predictor corrector master evaluates macro step twice uses perturbation inputs get estimate required partial derivatives categories rollback single causality causal domain num sim three rel time analytic sim rate multi sim step size fixed alg loop semi implicit info jacobian availability local results visualization post mortem coupling model algebraic constraints communication model jacobi ref stabilized approach solver coupling algebraic constraints summary master algorithm capable dealing algebraic constraints described requires derivatives coupled variables available executes communication step twice method uses predict step corrector step predictor step allows method estimate sensitivity state variables respect applied categories rollback single causality causal domain num sim three rel time analytic sim rate multi info jacobian sim step size fixed alg loop semi implicit availability local results visualization post mortem coupling model algebraic constraints communication model jacobi ref methods tools dynamic systems functional interface summary phd thesis linear extrapolation based master proposed convergent require fixed point iterations modification methods proposed increase performance executing environment modification avoids need restart dealing discontinuities categories nfr protection nfr platform independence nfr open source rollback none causality causal domain num sim three rel time analytic sim step size fixed availability local results visualization post mortem coupling model assignments communication model jacobi standard fmi ref configuration automotive scenarios summary language developed addition three novel diagrams represent different aspects configuration architectural coupling executable units tools assignment tools models connections connections clear diagram addition define couple well formedness properties checked easily approach give brief summary tool icos categories nfr config reusability nfr parallelism nfr hierarchy nfr extensibility domain num sim three rel time analytic availability local coupling model assignments results visualization post mortem info full model ref distributed daccosim summary daccosim able perform distributed simulations simulations term computation node used collection fmu wrappers include fmu local master global master fmu wrappers thereby masters responsible passing outputs connected inputs avoid bottlenecks component node contains master fmus wrapped masters take responsibility coordinated step sizes case fmu needs roll back nfr performance possibility splitting simulation cluster focus performance article additionally use variable step size nfr config reusability possible create multiple configuration files simulation configuration gui stored therefore reused nfr protection level protection fmi nfr parallelism possibility splitting simulation cluster nfr distribution executed cluster computers availability remote availability local nfr hierarchy daccosim weak hierarchical notion local global masters nfr scalability framework considered scalable distributed architecture nfr platform independence two versions daccosim library version relying java windows version using qtronic sdk nfr accuracy article provides example result using daccosim compared simulation using dymola results close accuracy ensured fmu examining outputs estimating far exact value nfr open source framework distributed open source license january info stateserial framework perform single rollback using state variable serialization causality causal framework based fmi considered causal domain framework supports multiple formalisms based fmi num sim three frameworks capable supporting many fmus thereby many simulation units daccosim offers algorithm depending masters coupling model assignments rel time analytic mentioning time models article sim rate single simulation rate fmus sim step size variable framework uses coordinated variable step alg loop explicit framework uses euler method richardson method whether default parameterizable fully customizable unknown based article communication model jacobi see bullet article standard fmi based fmi standard results visualization post mortem ref parallel synchronization continuous time discrete event simulators summary presents two synchronization approaches detailed three different synchronization protocols coordinate simulation scenarios include one discrete event simulator one continuous time simulator discrete event simulator implement parallel simulation approach know means even internally simulator forced rollback due straggler messages focus parallel approaches categories nfr performance nfr protection nfr parallelism nfr accuracy domain domain num sim two sim rate multi sim step size fixed results visualization post mortem rel time analytic availability local rollback single coupling model assignments info full model ref generation optimised master algorithm fmi summary essentially paper shows compiled approach increases performance also shows many decisions made designing master compiled approach allows elegant specifically tailored master generated nfr performance nfr protection nfr config reusability nfr open source rel time analytic availability local info causality feedthrough info statevars rollback none causality causal info preferred step sizes domain num sim three sim rate multi sim step size fixed results visualization post mortem standard fmi communication model gauss seidel alg loop implicit coupling model assignments ref functional digital functional interface two complementary approaches comprehensive investigation heterogeneous systems summary paper describes compares two approaches performing heterogeneous systems namely functional digital fdmu functional interface fmi besides describing approaches also introduces fdmu framework framework implements functional digital approach furthermore proposals presented combining fdmu fmi approaches fdmu approach web framework build web service standards capable coupling different simulation tools provide visualization based cad models fdmu consists three main concepts functional building blocks fbb wrappers fdmu master fdmu console functional building block wrap geometric information cad models behavioral models simulator tool responsibility wrappers establish connection different simulation tools fdmu master simulator finally fdmu master ensures correct communication simulators fdmu console user nfr performance communication overhead web approach nfr protection web approach protection possible nfr parallelism web approach parallel nature uses queues transmission data nfr distribution web approach easy distributable nfr scalability distributed systems paradigm ensures scalability nfr platform independence web approach nfr extensibility new wrapper implemented causality causal every input fbb must appropriate output belonging another fbb domain coupling model assignments num sim three sim rate multi availability remote results visualization live framework provides interactive visualization based cad standard fdmu ref heterogeneous platform efficient analysis flexraybased automotive distributed embedded systems summary motivation flexray wired network automotive control applications solutions exist simulates parts network platform called teodacs simulation approach used platform covers mechanics parts network physical layer application layer done solutions framework cisc syad used perform microelectronics inmotion mechanics bridged teodacs platform uses interesting approach faster cosimulations namely use model switching less detailed model replaces detailed model parts simulation paper provides overview existing approaches transaction based modeling hdls systemc verilog cable harness topology modeling along contain shortcomings domain furthermore paper provides details implementation models used showcases platform analyse system specific examples nfr accuracy model switching nfr performance model switching dynamic structure structure changed via model switching domain syad supprots multiple formalisms carmaker avl inmotion domain coupling model assignments num sim three multiple flexray nodes added rel time analytic model switching similar clear analytic simulation used results visualization post mortem ref moka framework fmi summary paper describes moka framework performing creating fmus using fmi framework turns creation fmus process using fmu created inheriting one classes implementing virtual functions thereby avoiding writing boilerplate code implementation fmus realised concepts fmublock inherited fmuport fmustatevariables fmublock extended concrete fmu slave implements common computation phase functions slaves contains fmuport data exchange fmustatevariables state tracking simulation fmuport classes provides data exchange interface slave abstracts value references automatically assigning value reference variable basestatevariable class also functions base extended provides virtual functions state variable services statevariable inherits basestatevariable represents state variables slave framework also provides template fmu master master code changes minimally different scenarios article exemplifies application moka framework two fmus used bouncing ball integer counter example qtronic sdk bouncing ball moka future work stated development dsl current study different scenarios executed without altering master code nfr protection nfr config reusability causality causal rel time analytic rollback none availability local coupling model assignments num sim three standard fmi domain alg loop explicit sim rate single communication model gauss seidel results visualization post mortem ref building energy control systems building controls virtual test bed summary describes framework called building controls virtual test bed bcvtb used simulation modular extensible platform interface different simulation programs intention give users option use best tools suited model various aspects building energy control systems use programs expertise middleware couple number simulation programs also provides libraries extended furthermore paper describes gathered capabilities framework support framework based ptolemy extended java packages simulator package adds functionality allows actor perform system calls start executable windows osx linux simply starts simulation program sends input tokens simulation program receives new values sends output port algorithms also provided simulators coupled also exemplified specific simulators also described connect client programs article describes interfaces created simulink matlab modelica system cals furthermore specific example presented nfr config reusability nfr distribution nfr platform independence nfr extensibility nfr open source causality causal rel time analytic rel time fixed real scaled time simulation domain paper explanation focused domain num sim three sim rate single sim step size fixed results visualization post mortem coupling model assignments communication model jacobi ref integration platform fmi heterogeneous simulations systems summary article concerns integrating fmi hla federate extending command control wind tunnel metamodel include enables tool use fmus part simulation tool describes integration platform allows users model synthesize complex heterogeneous command control simulations tool therefore support multiple simulation engines introduction tool given paper furthermore case study vehicle thermal management using fmus presented focompared simulation different environment work sponsored dod nfr platform independence nfr distribution nfr config reusability domain num sim three sim rate multi sim step size fixed sim step size variable rel time fixed real scaled time simulation rel time analytic results visualization live standard hla standard fmi communication model jacobi ref integrated tool chain design systems summary article presents overview project thereby tool projects concerns production tool chain design cpss therefore consists semantic foundation several baseline tools modelio overture openmodelica furthermore application called application entry point configuring uses orchestration engine coe perform actual simulations coe based fmi standard entire tool chain semantic foundation presented paper related nfr config reusability nfr protection info nominal values output info nominal values state info derivatives info derivatives state info jacobian info jacobian state info preferred step sizes info causality feedthrough causality causal availability local info statevars info record outputs rel time analytic info stateserial standard fmi info signal num sim three domain domain sim rate single results visualization post mortem results visualization live ref design platform summary deliverable related project contains technical documentation platform end first year project part project orchestration engine coe related info predict step sizes supports fmi suggested extension nfr performance coe supports variable step size increase performance rollback single supports rollback last successful state nfr performance coe supports variable step size increase performance nfr accuracy contains various constraints bounded difference sampling rate furthermore support allowed min max values scenario categorization section describes category lists references belong category classified previous section requirements fault tolerance platform fault tolerant example one simulation unit fails take place particularly important long running simulations fault tolerant certain features need available periodically store state simulation units record inputs simulation unit simulation unit fails state periodically stored simulation paused state restored new instance simulation unit history input values passed simulation unit used bring simulation unit current state references category configuration reusability category refers fact frameworks provide way configure scenarios reused means configuration considered external execution external context means configuration reused without altering binaries application provide way reuse configurations process set references category performance performance relative measure platform performant able simulate great deal short amount time needing little resources achieved using variable step integration methods signal extrapolation techniques parallelism also plays role mostly time aspect performance references category protection protection deals requiring models participating provide detailed structure variables etc multiple levels protection ranging fully protected protected good protection enables component suppliers provide system integrators detailed simulations components avoiding expensive contracts multiple techniques employed ensure degree protection instance making models corresponding simulation units available web service possible solution another example framework implements fmi standard allows models simulation units exported single functional unit binary format imported references category parallelism framework parallel makes use multiple perform typically computer local network techniques signal extrapolation help improve gained parallelism furthermore waveform relaxation techniques jacobi iterations promote parallelism references category distribution framework parallel distributed allows simulation unit remote across wide area network important since suppliers instead transferring simulation units executable form across computers make available web offers much control simulation units used techniques used parallelism used promote distribution fault tolerance also important references category hierarchy hierarchical framework able abstract scenario black box simulation unit intuitive promotes abstraction complex systems references category scalability framework scalable supports large number simulation units intimately related performance paralelism references category platform independence framework platform independent works multiple computing platforms achieved platform independent language java used coordinate simulation references category extensibility framework extensible easily extended support new kinds simulation units new kinds capabilities higher level domain specific language used specify behaviour platform agnostic way code generated description hypothesis high level description easily extended describe new behaviour code generation process adapted accordingly references category accuracy accurate error trace produced correct trace minimal achieved error control mechanisms references category open source consider open source frameworks make available source code certain licenses paid way references category simulator requirements sub section covers taxonomy focuses individual simulators capabilities information exposed frequency state instantaneous frequency state used adjust communication step size frequency outputs instantaneous frequency output used adjust communication step size done references category detailed model simulators make equations dynamic system available fall category references category nominal values outputs information indicates order magnitude output signals nominal values state information indicates order magnitude state signals signal kind kind output signal helps master algorithm understand assumptions signal references category time derivative output references category state references category jacobian output references category state discontinuity indicator discontinuity indicator signal indicates presence discontinuity output simulation unit deadreckoning model deadreckoning model function used simulation units extrapolate behavior simulation unit preferred step size references category next step size next step size indicates next communication time appropriate current simulator references category order accuracy order accuracy used determine appropriate input extrapolation functions used scenario causality feedthrough references category propagation delay propagation delay indicates many performed change input affects output simulink number delay blocks chain input output input extrapolation information denotes kind input extrapolation performed simulation unit references category state variables references category serialized state references category outputs information denotes output simulation unit evaluated references category wcet denotes worst case excution time references category causality causal references category time constraints analytic simulation references category scaled real time simulation fixed simulator fixed scaled real time simulated time progresses according fixed linear relationship real time references category dynamic simulator dynamic scaled real time relation simulated time real time changed throughout simulation references category rollback support none references category single single rollback means simulation unit capable saving certain state past simulated time revert state reverted simulation unit revert past references category multiple multiple rollback means simulation unit capable saving certain state past simulated time revert state reverted simulation unit revert past many times necessary availability remote references category local references category framework requirements standard high level architecture references category functional interface references category functional digital references category coupling assignments references category algebraic constraints references category number simulation units two references category three references category domain references category references category dynamic structure references category rate single references category multi denotes framework distinguishes slow fast dimensions communication step size accordingly providing slow systems references category communication step size fixed references category variable references category strong coupling support none explicit method references category partial method references category full implicit method references category results visualization postmortem results available simulation references category live references category interactive references category communication approach jacobi references category references category list acronyms cps devs dtss fmi gvt ivp nfr ode system continuous time discrete event discrete event system specification discrete time system specification functional interface framework requirement global virtual time intellectual property initial value problem requirement ordinary differential equation simulator requirement
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arxiv mar generalized strong preservation abstract interpretation rancesco anzato rancesco tapparo dipartimento matematica pura applicata padova via belzoni padova italy franz tapparo abstract standard abstract model checking relies abstract kripke structures approximate concrete models gluing together indistinguishable states namely partition concrete state space strong preservation specification language encodes equivalence concrete abstract model checking formulas show abstract interpretation used design abstract models general abstract kripke structures accordingly strong preservation generalized abstract models precisely related concept completeness abstract interpretation problem minimally refining abstract model order make strongly preserving language formulated minimal domain refinement abstract interpretation order get completeness operators turns refined strongly preserving abstract model always exists characterized greatest fixed point consequence behavioural equivalences like bisimulation simulation stuttering corresponding partition refinement algorithms elegantly characterized abstract interpretation completeness properties refinements keywords abstract interpretation abstract model checking strong preservation completeness refinement behavioural equivalence introduction design abstract model checking framework always includes preservation result roughly stating formula specified temporal language holds abstract model also holds concrete model hand strong preservation means formula holds abstract model holds concrete model strong preservation highly desirable since allows draw consequences negative answers abstract side generalized strong preservation relationship abstract interpretation abstract model checking subject number works see paper follows standard abstract interpretation approach abstract domains specified galois connections namely pairs abstraction concretization maps deal generic temporal languages state formulae inductively generated given sets atomic propositions operators interpretation atomic propositions subsets states operators mappings states determined suitable semantic structure kripke structure concrete semantics states formula set states making true abstract semantics systematically defined standard abstract interpretation powerset states plays role concrete semantic domain abstract domains range absdom states abstract domain absdom states induces abstract semantic structure atoms abstracted operators interpreted best correct approximations thus determines abstract semantics evaluates formulae abstract domain turns approach generalizes standard abstract model checking given kripke structure states simplicity omit labeling function atomic propositions standard abstract model specified abstract kripke structure astates set astates abstract states defined surjective map states astates thus astates determines partition states vice versa turns state partitions particular abstract domains fact lattice partitions states abstract interpretation lattice abstract domains absdom states abstract state space astates corresponds particular abstract domain astates absdom states abstract domains derived state partition called partitioning interpretation language abstract kripke structure determines abstract semantic function astates abstract kripke structure strongly preserves states turns strong preservation generalized standard abstract models abstract interpretationbased models given generalized abstract model absdom states induced abstract semana tics strongly preserving states turns abstract domain property abstract semantics evaluates formulae abstract domain strongly preserving standard strong preservation becomes particular instance namely abstract kripke structure strongly preserves corresponding partitioning abstract model strongly preserves hand generalized strong preservation may work standard strong preservation may fail namely may happen although strongly preserving abstract semantics partitioning abstract model astates exists derived strongly preserving abstract kripke structure astates generalized strong preservation complete abstract interpretations given language kripke structure states key problem compute smallest abstract state space astates exists one define abstract kripke structure astates strongly preserves problem admits solution number temporal languages like ctl equivalently actl ctl without operator number algorithms solving problem exist like paige tarjan ctl henzinger bustan grumberg tan cleaveland actl groote vaandrager coarsest partition refinement algorithms given language partition states determined state labeling algorithms viewed computing coarsest partition refines strongly preserves worth remarking algorithms designed computing behavioural equivalences used process algebra like bisimulation ctl simulation actl stuttering equivalence abstract framework allows give generalized view partition refinement algorithms show abstract domain adl absdom states strongly preserves given language always exists turns adl partitioning abstract domain includes full propositional logic closed logical conjunction negation otherwise proper loss information occurs abstracting adl corresponding partition moreover languages may happen one define abstract kripke structure abstract state space strongly preserves whereas abstract strongly preserving semantics absdom states instead exists concept complete abstract interpretation well known encodes ideal situation abstract semantics coincides abstraction concrete semantics establish precise correspondence generalized strong preservation abstract models completeness abstract interpretation results based notion forward complete abstract domain abstract domain forward complete concrete semantic function namely loss precision occurs approximating computation notion forward completeness dual orthogonal standard definition completeness abstract interpretation giacobazzi showed complete abstract domains systematically constructively derived noncomplete abstract domains minimal refinements done forward completeness well given domain abstract domain refines forward complete exist characterized greatest fixpoint domain called forward complete shell turns strong preservation related forward completeness follows described abstract domain adl strongly preserves always exists turns adl coincides forward complete shell operators basic abstract domain determined state labeling characterization provides elegant generalization partition refinement algorithms used standard abstract model checking consequence results derive novel characterization corresponding behavioural equivalences terms forward completeness abstract domains example turns partition bisimulation kripke structure corresponding partitioning abstract domain forward complete standard predecessor transformer basic notions notation preliminaries let set fun denotes set functions called arity following standard convention meant specific object arity also denoted denotes identity map fun def namely set images function image also denoted img denotes composition complement operator universe set writing set subsets given set like partition often write compact form like stand ord denotes proper class ordinals ord denotes first infinite ordinal let poset posets often denoted also use symbol denote pointwise ordering functions set mapping posets continuous preserves least upper bounds lub countable chains dually preserves greatest lower bounds glb countable chains complete lattice also denoted denote respectively lub glb greatest element least element mapping complete lattices additive denote lfp gfp respectively least greatest fixpoint exist operator poset theorem states monotone operator complete lattice admits least fixpoint following characterization holds lfp upper iteration sequence defined transfinite induction usual successor ordinal limit ordinal well known continuous lfp dually also admits greatest fixpoint following characterization holds gfp lower iteration sequence defined upper iteration sequence case limit ordinals let set preord denotes set preorder relations preorder reflexive transitive part denotes set partitions sets partition called blocks equivalence relation denote part corresponding partition vice versa part denotes corresponding equivalence relation part endowed following standard partial order coarser refines iff well known hpart complete lattice transition system consists possibly infinite set states transition relation usual assume relation total exists maximal path necessarily infinite finitely branching finite set transformers defined usual def def pre def def post let observe additive operators pre post relation relations defined follows iff iff abstract interpretation completeness abstract domains standard cousot cousot abstract interpretation abstract domains equivalently specified either galois connections adjunctions upper closure operators uco let recall standard notions galois connections insertions posets monotone functions quadruple called galois connection short addition galois insertion short onto let also recall notion equivalent adjunction iff map called adjoint turns one adjoint map uniquely determines adjoint map follows one hand map admits necessarily unique map iff preserves def arbitrary lub case hand map admits necessarily unique map iff preserves arbitrary glb def case particular complete lattices turns additive also complete lattice complete lattice well himg assume standard abstract interpretation framework concrete abstract domains complete lattices related abstraction concretization maps forming called abstraction concretization ordering relations concrete abstract domains describe relative precision domain values means approximation equivalently precise galois connections allow relate concrete abstract notions relative precision abstract value approximates concrete value equivalently adjunction key consequence requiring galois connection turns best possible approximation holds value abstract domain useful representing values represent distinct members lifted identifying equivalence class values abstract domain concretization abs denotes set abstract domains write abs mean abstract domain related abstract domain disjunctive corresponding concretization map additive closure operators upper closure operator simply closure poset operator monotone idempotent extensive dually lower closure operators monotone idempotent restrictive uco denotes set closure operators let complete lattice closure uco uniquely determined image img coincides set fixpoints follows img also image closure operator iff def terms case corresponding closure operator called least set inclusion subset contains moreover turns uco img thus closure operators bijection allows consider closure operator uco function img particularly useful give rise ambiguity since one distinguish use closure function set according context turns complete meet subsemilattice glb general complete sublattice since lub defined might different fact turns complete sublattice namely img also iff additive complete lattice uco endowed pointwise ordering complete lattice denoted huco every uco uco iff iff img img greatest element whereas least element thus glb uco defined pointwise lub set closures uco closure whose image given lattice abstract domains well known since abstract domains equivalently specified either galois insertions closures two approaches completely equivalent one hand uco complete lattice isomorphic img img img provide isomorphism hand def uco closure associated himg complete lattice isomorphic furthermore two constructions inverse let also remark abstract domain disjunctive iff additive given abstract domain specified associated closure thought logical meaning since shared abstract representation objects thus closure operator approach particularly convenient reasoning properties abstract domains independently representation objects abstract domains specified gis precision follows abs precise concrete abstraction denoted pointwise ordering uco corresponds therefore standard ordering used compare abstract domains respect precision also equivalent denoted associated closures coincide hence quotient abs gives rise poset slight abuse notation simply denoted habs thus write abs mean representative equivalence class abs specified galois insertition turns habs complete lattice called lattice abstract interpretations isomorphic complete lattice huco lub glb abs therefore following reading operators domains let abs concrete among domains abstractions abstract among domains concrete every latter domain also known reduced product completeness let concrete domain concrete semantic let corresponding abstract function abstract domain abs specified sound abstract interpretation holds abstract function called correct approximation means concrete computation correctly approximated namely abstract function precise since holds iff holds abstract def function called best correct approximation completeness abstract interpretation corresponds requiring addition soundness loss precision occurs approximated thus completeness encoded equation also called backward completeness dual form forward completeness may considered simple example let consider abstract domain sign representing sign integer variable namely sign abs let def consider binary concrete operation integer addition sets integers def square operator sets integers turns best correct approximation integer addition sign sound complete sign easy check best correct approximation square operation sign instead complete dual form completeness may considered soundness condition equivalently formulated forward completeness corresponds requiring equation holds therefore means loss precision occurs concrete computation abstract value approximated let notice backward forward completeness orthogonal concepts fact observed backward complete forward complete sign best correct approximation square operator sign forward complete observed instead backward complete giacobazzi observed completeness uniquely depends upon abstraction map upon abstract domain means backward complete best correct approximation backward complete well case indeed coincides hence abstract domain one define backward complete abstract operation backward complete thus abstract domain abs defined backward complete iff equation holds simple observation makes backward completeness abstract domain property namely intrinsic characteristic abstract domain let observe holds iff holds backward complete thus closure uco defines abstract domain backward complete holds analogous observations apply forward completeness also abstract domain property abs forward complete forward closure uco forward complete holds let also recall result see theorem fact lemma backward complete abstract domains fixpoint complete well means abs backward complete concrete monotone function lfp lfp moreover also holds greatest fixpoints namely simplicity notation consider unary functions since extension generic functions straightforward gfp gfp far forward completeness concerned following result holds lemma abs forward complete monotone gfp gfp moreover continuous lfp lfp proof let show gfp gfp one hand since gfp gfp gfp gfp gfp therefore using forward completeness gfp gfp thus gfp gfp follows gfp gfp hand using forward completeness gfp gfp gfp gfp gfp therefore applying obtain gfp gfp gfp assume continuous let show induction hypothesis forward completeness inductive hypothesis thus applying obtain since continuous always additive continuous composition continuous functions hence lfp theorem additive theorem lfp concludes proof worth noting concretization maps abstract domains satisfies ascending chain conditions every ascending chain eventually stationary always trivially continuous shells refinements abstract domains studied beginning abstract interpretation led notion shell abstract domain given generic poset semantic objects intuitively means refinement property objects generic notion shell goes follows object defined object satisties property refinement iii greatest among objects satisfying note exists unique moreover exists object turns operator mapping lower closure operator monotone idempotent reductive operator called refinement particularly interested shells abstract domains partitions namely shells complete lattices abstract domains partitions given state space partition property part part coarsest refinement satisfying exists also given concrete domain domain property abs abs exists abstract domain satisfies refines giacobazzi gave constructive characterization backward complete abstract domains assumption dealing continuous concrete functions consequence showed backward complete shells always exist concrete functions continuous section follow idea forward completeness provide link strongly preserving abstract models complete abstract interpretations abstract model checking strong preservation standard temporal languages like ctl actl ltl interpreted models specified kripke structures given set atomic propositions language kripke structure consists transition system together state labeling funcdef tion use following notation def part denotes state partition induced notation means state satisfies state formula language specific definition satisfaction relation depends language interpretations standard operators found standard abstract model checking relies abstract kripke structures defined partitions concrete state space set abstract states related surjective abstraction maps concrete states abstract states thus gives rise state partition def part thus standard abstract model checking formulae interpreted abstract kripke structure whose states abstract representation block partition given specification language state formulae weak preservation result guarantees formula holds abstract ktipke structure also holds corresponding concrete structure moreover strong preservation short encodes equivalence abstract concrete validity formulae definition preserving abstract kripke structures depends language let recall examples let concrete kripke structure surjection consider language abstract kripke structure weakly preserves defined let psim part partition induced simulation equivalence psim also holds psim abstract kripke structure strongly preserves iii let pbis part partition induced bisimulation equivalence pbis also holds pbis abstract kripke structure strongly preserves ctl following dams section henzinger section notion strong preservation also given mere state partition rather abstract kripke structure let semantic function state formulae kripke structure def semantic interpretation induces following logical gfed abc stop stop onml hijk gfed abc gfed abc figure traffic light equivalence iff let part partition induced index denoting kripke structure omitted partition part strongly interpreted thus coarsest partition strongly preserving number well known temporal languages like see respectively points iii fragments described henzinger turns strongly preserving abstract kripke structure strongly preserving particular strongly preserving additionally smallest possible abstract state space namely abstract kripke structure strongly preserves however given language kripke structure formulae interpreted following example shows always possible define abstract kripke structure partition strongly preserves example consider following simple language stop kripke structure depicted figure superscripts determine labeling function models traffic light controller like germany red redyellow green yellow according standard semantics axx iff path starting happens turns jaxxstopkk jaxxgokk thus however let show exists abstract transition relation abstract kripke structure strongly preserves assume contradiction abstract kripke structure exists let since axxgo axxstop strong preservation must axxgo axxstop hence necessarily leads contradiction axxgo fact would axxgo hand instead case analogous would still axxgo even along lines hard show proper abstract kripke structure strongly preserves defined even either split still define abstract transition relation strongly preserving partitions abstract domains let possibly infinite set states following section partition part viewed abstraction follows approximated unique minimal cover namely union blocks graphical example depicted side figure abstraction formalized def def hence partition part induces abstract domain adp abs abstract domain abs called partitioning equivalent adp partition observe closure adp associated partitioning abstract domain defined dams uses term fine instead strongly preserving figure partitions abstract domains left right adp accordingly closure uco coincides partition called partitioning denote abspar ucopar sets respectively partitioning abstract domains closures noted surjective abstraction used standard abstract model checking maps concrete states abstract states section gives rise partitioning galois insertion def def partitions also viewed dual abstractions set approximated union blocks graphical example approximation depicted side figure dual abstraction formalized ordering concrete domain given subset relation def def following interested viewing partitions approximations partitions abstract domains thus partitions viewed representations abstract domains hand turns abstract domains abstracted partitions abstract domain abs induces state equivalence identifying states distinguished iff def block state partition par induced def par thus par abs part mapping abstract domains partitions example let let specify abstract domains uco uco induce partition par part example par observe partitioning abstract domain adp abstract domains carry additional information underlying state partition additional information allows distinguish turns precisely stated abstract interpretation since mappings par adp allows show whole lattice partitions viewed abstraction lattice abstract domains theorem par abs part adp galois insertion proof let abs part let uco closure associated abstract domain let prove par adp prove adp consider adp hence exists let since par exists block par thus particular consequently since therefore consider block show namely since adp adp therefore likewise turn finally observe adp adjunction indeed galois insertion let observe recalled section adjoint maps par adp give rise order isomorphism lattices hpart habspar corollary let abs following statements equivalent partitioning additive partition case par forward complete complement operator proof let abs let uco corresponding uco theorem abspar iff adp par thus adp par obviously additive moreover iff iff therefore par since part fact hence therefore thus par moreover since additive hence since adp adp par assume abspar enough prove fact additive therefore additive composition additive maps therefore let observe following fact consequence fact partition additivity fact assume forward complete closed complements enough prove additive part additive observe additive iff additive iff closed arbitrary unions closed complements arbitrary intersections part clearly consider let show order show let prove notice closed complements would would imply namely thus would obtain contradiction hence therefore swapping roles also obtain def let remark adp par lower closure operator habs abs partitioning iff hence exactly refinement namely abstract refinement partitioning abstract semantics languages concrete semantics consider temporal specification languages whose state formulae inductively defined ranges typically finite set atomic propositions ranges finite set operators also denoted respectively operator formulae interpreted semantic structure possibly infinite set states interpretation function fun maps also denoted respectively def def moreover note def interpretation induces state labeling concrete state semantic function evaluates formula set states making true semantic structure jpks semantic structures generalize role kripke structures fact standard model checking semantic structure usually defined kripke structure interpretation operators defined terms standard logical operators paths following freely use standard logical temporal operators together corresponding usual interpretations example prer pre etc example consider standard semantics ctl ctl respect kripke structure hence determines corresponding interpretation atoms operators ctl namely pre prer defines concrete semantic function ctl operator arity whose interpretation given semantic structure say language closed exists instance includes negation standard interpretations prer closed standard interpretation pre pre prer notion extended straightforward way infinitary operators instance closed infinite logical conjunction iff exists particular let remark closed infinite logical conjunction must exist namely able express tautology true let remark state space finite closed logical conjunction always mean exists finally note closed negation infinite logical conjunction includes propositional logic would possible consider generic operators whose arity possibly infinite ordinal thus allowing example infinite conjunctions disjunctions ooo oooo figure kripke structre left abstract domain right abstract semantics following apply standard abstract interpretation approach defining abstract semantics let language semantic structure abstract semantic structure given abstract domain abs abstract interpretation function fun abstract semantic structure therefore induces abstract semantic function evaluates formulae abstract values abstract interpretation correct respectively respectively respectively correct respectively underapproximation semantic operations monotone abstract semantics respectively concrete semantics namely respectively particular abstract domain always induces abstract semantic structure best correct approximation interprets atoms operators best correct approximations respectively def def thus abstract domain systematically induces abstract semantic function also denoted therefore defined jpka usual abstract interpretation observe concrete semantics particular abstract semantics namely abstract semantics induced identical abstraction example let let consider kripke structure lattice depicted figure let semantic structure induced kripke structure let consider formulae exr whose concrete semantics follows jexrks jex abstract domain galois insertion determined following concretization map note corollary partitioning additive turns jexrka jrka jex jpka jqks observe abstract semantics jexrka proper jexrks jexrks hand concrete semantics jex precisely represented jexrka jex jex generalized strong preservation showed section state partition viewed partitioning abstract domain adp specified thus given language corresponding semantic structure turns partition part systematically induces corresponddef adp adp evaluates formula possibly ing abstract semantics empty union blocks strong preservation partition characterized terms corresponding abstract domain adp follows lemma part iff proof let first observe fact block containing since turn let prove structural induction using observation jpks jpks jpkp observation definition inductive hypothesis definition consider conversely consider block hypothesis iff iff iff thus states partition part check whether set states satisfies formula equivalent check whether abstract state precise abstract semantics key observation abstract framework partitions particular abstract domains allows generalize notion strong preservation partitions generic abstract semantic functions follows definition let language semantic structure corresponding abstract semantic structure abstract semantics strongly preserving definition generalizes standard strong preservation partitions characterized lemma arbitrary abstract domain abs corresponding abstract interpretation function likewise standard weak preservation generalized follows let concrete kripke structure induces concrete semantics let surjective abstraction let corresponding partitioning abstract domain let abstract kripke structure gives rise abstract semantics weakly preserves hence weak preservation generalized generic abstract domains abstract semantics accordingly definition onml hijk hijk onml figure kripke structure left abstract kripke structure right strong preservation abstract domain property definition direct natural generalization standard notion strong preservation abstract model checking equivalently stated follows lemma iff proof one hand iff iff trivially true hand iff iff trivially true iff iff particular worth noting holds lemma let abs let abstract semantic structures let abstract semantic structure proof lemma applying let first observe fact consequence fact structural induction analogously proof lemma easy prove thus lemma thus turns strong preservation abstract domain property means given abstract domain abs possible define abstract semantic structure corresponding abstract semantics induced abstract semantics particular also holds standard approach abstract kripke structure corresponding surjection standard abstract semantics strongly preserves abstract semantics induced partitioning abstract domain strongly preserves case abstract semantics coincides standard abstract semantics strong preservation abstract domain property therefore defined without loss generality follows definition abstract domain abs strongly preserving semantic structure denote spl abs set abstract domains example let consider example turns abstract domain lemma jexrks jexrka example let consider simple language kripke structure depicted figure kripke structure induces semantic structure hence jpks jexpks jexk pks let consider partitioning abstract domain induced partition related let consider two different abstract semantic structures abstract semantic structure induced best correct approximation abstract semantic structure instead induced abstract kripke structure figure hence different fact let show abstract semantics jpka jexpks jex pks thus jpks jexpks jexk pks thus consequently lemma abstract semantics abstract strongly preserving domain recalled section language semantic structure induce corresponding logical partition part lemma turns coarsest strongly preserving partitioning abstract domain generalized arbitrary abstract domains follows let define adl def adl hence adl closure arbitrary intersections set concrete semantics formulae observe adl abs theorem abs spl iff adl proof let uco let uco uco associated adl recall adl iff lemma hypothesis let show structural induction using hypothesis jpks jpks jpka hypothesis definition inductive hypothesis definition thus lemma spl thus adl abstract domain consequence turns represents loss precision concrete semantics formula lemma states abstract semantics given abstract domain exists unique nevertheless example shows unique abstract semantics may induced different abstract semantic structures different abstract interpretation functions however closed conjunction turns abstract domain adl abstract interpretation function unique given best correct approximation adl theorem let closed infinite logical conjunction let adl abstract semantic structure adl adl proof since closed arbritrary logical conjunctions adl thus adl exists fact adl lemmata let lemma jpks jpkad adl let kad adl kad kad adl observation definition lemma definition observation thus adl hence abstract domain adl unique choice interpreting atoms operations generalized framework strong preservation partitions becomes particular instance galois insertion moreover closed infinite conjunction turns abstract domain adl partitioning also closed negation proposition par adl adp adl strongly preserving iff par adl iff adp adl let closed conjunction adl partitioning iff closed logical negation proof let uco uco associated adl par adl adl adl also iff iff par adl moreover adp adp par adl adl iff iff point par iff theorem adp adl since closed infinite logical conjunction adl thus closed logical negation iff adl closed complementation exactly means adl forward complete complement corollary latter fact happens iff adl partitioning particular closed conjunction negation turns adp adl proper loss information occurs domain adl abstracted partition par adl hand closed conjunction negation adp adl therefore theorem abstract interpretation function partitioning abstract domain adp uniquely determined example let consider traffic light controller example already observed formulae following semantics jstopkk jgokk jaxxstopkk jaxxgokk adl par adl denote uco associated adl shown example turns abstract kripke structure properly abstracts strongly gfed abc gfed abc gfed abc abc gfed figure concrete left abstract right kripke structures preserves defined approach abstract domain adl induces corresponding strongly preserving abstract semantics adl best correct approximation operator axx adl axx example consider language ctl kripke structure depicted figure interpretation temporal operators ctl standard well known coarsest partition pctl obtained refining initial partition induced labeling algorithm since pctl coincides bisimulation equivalence easy check pctl partition determines see point section abstract kripke structure depicted figure since ctl closed conjunction negation proposition turns abstract domain actl partitioning coincides following partitioning closure adp pctl let consider following language time bounded reachability operator useful quantitative temporal analysis discrete systems chapter standard interpretation follows iff exists path starting let characterize semantics formulae jpkk jqkk jef pkk jef qkk jef qkk jef thus adl hand proposition par adl case turns adp adl moreover analogously example let show exists abstract transition relation determines abstract kripke structure strongly preserves let blocks assume contradiction abstract kripke structure exists concrete model thus strong preservation must hand therefore weak preservation would contradiction thus necessarily let observe hence strong preservation point would still contradiction hence necessarily iii would obtain observed point contradiction thus shows possible define abstract kripke structure abstract state space strongly preserves abstract domain adl induces corresponding abstract semantics instead strongly preserves case best correct approximation operator adl strong preservation completeness section establish precise correspondence generalized strong preservation abstract models completeness abstract interpretations problem minimally refining abstract model order get strong preservation formulated complete domain refinement abstract interpretation forward complete shells let consider forward completeness abstract domains abs generic concrete operations hence forward complete simply equivalently set operations fun observe abstract domain means associated closure closed image functions namely also note constant iff precisely represented let also note abstract domain abs always forward uco let first note forward shells always exist let abs abs defined def abs lemma shell proof let uco uco let trivially consider uco since therefore thus forward complete shell concrete abstraction characterize useful view abstract domains closure operators concrete domain subsets hence viewed subset img concrete domain characterized least subset contains img forward complete need characterize least amount concrete information must added order get forward completeness turns forward complete shells admit constructive fixpoint def characterization let uco uco uco defined follows uco namely uco abstract domain contains image observe operator uco uco uco monotone lemma gfp uco proof observe uco iff iff uco iff uco thus uco uco uco uco uco gfp uco thus turns lower iteration sequence uco uco converges complete shell example let consider square operator sets integers abstract domain sign observed section sign forward complete square operator let apply lemma order compute forward complete shell ssq sign observe sign sign sign thus first step iteration refines sign sign notice abstract domain sign second step iteration obtain sign general step iteration provides sign complete shell ssq sign coincides least fixpoint sign finally following easy observation useful later lemma let fun abs proof therefore well follows abs abs strong preservation complete shells let language atoms operators let semantic structure opl denote respectively corresponding sets semantic interpretations atoms operators turns forward completeness opl implies strong preservation lemma abs forward complete opl proof theorem show adl let show induction since forward complete jpks jpks definition inductive hypothesis since forward complete inductive hypothesis definition hand converse true strong preservation imply forward completeness shown following example example let consider example showed partitioning abstract domain however forward complete opl fact instead turns abstract domains characterized forward complete shells complete shells strongly preserving abstract domains partition refinement algorithms computing behavioural equivalences like bisimulation simulation equivalence divergence blind stuttering equivalence used standard abstract model checking compute coarsest strongly preserving partition temporal languages like case bisimulation equivalence simulation equivalence stuttering equivalence given language concrete state space partition refinement algorithms work iteratively refining initial partition within lattice partitions part fixpoint reached input partition determines set atoms interpretation foldef def lows general determines set atoms interpretation particular done abstract domain abs considering concretization namely viewed set atoms interpretation thus abstract domain abs together set functions fun determine language atoms operations endowed semantic structure therefore abstract domain adla generalizes framework output partition refinement algorithm language accordingly aim characterizing adla output refinement process initial domain within lattice abs abstract domains following result shows forward completeness operations right notion refinement used case abstract domains theorem let abs fun assume closed infinite logical conjunction adla proof since closed conjunction adla let first prove structural induction jaksa ksa ksa ksa inductive hypothesis ksa therefore since forward ksa ksa let prove opposite inclusion let first observe adla forward simplicity notation consider adla ksa adla lemma know sufficient prove transfinite induction ord adla adla inductive hypothesis adla moreover adla forward hence closed thus adla namely adla limit ordinal follows inductive hypothesis adla strongly preserving abstract domains complete shells let consider language atoms operators semantic structure immediate consequence theorem abstract domain adl characterized forward opl shell abstract domain corollary let closed infinite logical conjunction adl sap let also observe adl equivalently characterized forward opl shell initial abstract domain induced atoms adl sopl strongly preserving partitions theorem corollary provide elegant generalization partition refinement algorithms strong preservation abstract interpretation perspective given language operators corresponding semantic structure recalled section input partition part partition refinement algorithm determines set atoms interpretation thus turns coarsest partition characterized abstract approach follows corollary let closed infinite logical conjunction par sopl let closed logical negation adp sopl proof corollary adl sopl proposition par adl par sopl proposition corollary point adp adp par adl adl sopl worth remarking closed negation proposition corollary turns adp sopl means closed negation output partition partition refinement algorithm achieving strong preservation optimal within lattice abstract domains example let consider language concrete kripke structure example labeling determines initial partition part abs abstract domains sopl sef let compute sef fixpoint already observed example adp possible define strongly preserving abstract kripke structure abstract space application behavioural equivalences well known temporal languages like ctl actl induce state logical equivalences coincide standard behavioural equivalences like bisimulation equivalence ctl divergence blind stuttering equivalence simulation equivalence actl derive novel characterization behavioural equivalences terms forward completeness abstract interpretations bisimulation equivalence let kripke structure set atomic propositions relation bisimulation exists symmetric since empty relation bisimulation bisimulations closed union turns largest set bisimulation relation exists largest bisimulation equivalence relation called bisimulation equivalence denoted pbis part denotes corresponding partition thus partition part bisimulation pbis well known finitely branching bisimulation equivalence coincides state equivalence induced ctl pbis pctl holds see lemma moreover known see section enough consider finitary logic language including propositional logic existential next operator order pbis usual interpretation number algorithms computing bisimulation equivalence exists algorithm runs log time algorithm computes bisimulation equivalence recalled pctl framework obtained consequence fact abstract domains ctl coincide lemma let finitely branching adctl adp pbis proof let opctl set standard interpretations operators ctl pre show uco forward complete opctl iff forward complete assume forward complete let first prove forward complete pre pre definition pre complete complete complete complete complete pre definition pre following fixpoint characterizations well known lfp pre lfp gfp pre gfp let show forward complete proofs remaining operators opctl analogous need show lfp pre lfp pre let show forward complete function pre pre pre pre pre pre pre complete pre complete complete complete complete pre observe since additive therefore continuous moreover let show hypothesis finitely branching follows pre continuous first notice pre continuous iff hence let check let decreasing chain subsets let since finitely branching finite exists hence exists therefore since pre continuous also pre continuous therefore apply lemma lfp pre lfp pre thus lemma sopctl corollary adctl finally since finitely branching closed conjunction negation adp adp pbis adp consequence results section particular corollary partition refinement algorithm algbis computing bisimulation equivalence finitely branching kripke structure like characterized complete shell refinement follows algbis par thus algbis viewed algorithm computing particular abstraction par particular complete shell particular holds algorithm leads design generalized procedure computing abstract strongly preserving domains finally abstract approach allows give following nice characterization bisimulation partition terms forward completeness corresponding partitioning abstract domain adp theorem let part bisimulation iff adp forward complete proof view adp uco adp let first observe iff adp forward complete one hand since hence union blocks therefore adp hand adp contains union blocks thus either consequently let note adp forward complete iff block possibly empty union blocks holds additive therefore fact blocks implies namely condition bisimulation holds hand condition bisimulation holds therefore union blocks closes proof smallest abstract transition relation recalled section abstract kripke structure pbis strongly preserves ctl iff exist simple elegant consequence approach easy show unique therefore smallest abstract transition relation pbis induces strong preservation ctl let finitely branching lemma adp pbis pbis recall concrete interpretation induced theorem unique interpretation atoms operations abstract domain pbis gives rise abstract semantics best correct approximation pbis hence pbis strongly preserving ctl interpretation induced must coincide pbis consequently pbis iff therefore conclude observing iff believe similar reasoning could also useful languages order prove smallest abstract transition relation induces strong preservation exists example proved case actl bustan grumberg stuttering equivalence lamport criticism operator well known motivated study temporal logics obtained removing operator led study notions behavioural equivalences interested divergence blind stuttering dbs short equivalence let kripke structure set atoms relation divergence blind stuttering relation exist srti iii trtk symmetric observe condition allows case simply boils requiring since empty relation dbs relation dbs relations closed union turns largest dbs relation relation exists turns largest dbs relation equivalence relation called dbs equivalence denoted pdbs part denotes corresponding partition particular partition part dbs relation pdbs nicola vaandrager theorem showed finite kripke structures interpretation path quantifiers possibly finite prefixes dbs equivalence coincides state equivalence induced language also holds pdbs true standard interpretation path quantifiers infinite paths since requires divergence sensitive notion stuttering see details groote vaandrager presented partition refinement algorithm computes partition pdbs time provide characterization divergence blind stuttering equivalence state equivalence induced following language includes propositional logic existential operator interpretation existential path quantifier standard infinite paths since transition relation assumed total let recall standard semantics existential operator follows iii following result characterizes dbs partition terms forward completeness corresponding partitioning abstract domain adp theorem let part part dbs partition iff adp forward complete proof already shown proof theorem turns iff adp forward complete thus remains show part satisfies condition definition dbs relation iff adp forward complete let first observe part satisfies condition iff otherwise assume exists thus condition implies let condition satisfied otherwise thus therefore means condition satisfied complete proof sufficient show adp forward complete function additive second argument thus need show namely exist let prove induction case hence therefore hypothesis moreover monotone first component therefore suppose exist let since finite path length inductive hypothesis hence exist moreover since hypothesis therefore thus exist thus found following finite path states sequence last one belong means consequence obtain characterization dbs equivalence state equivalence induced standard interpretation language corollary let finite pdbs proof definition pdbs gpart part dbs relation theorem pdbs gpart part adp complete theorem adp part adp preserves lub part hence adp pdbs adp abs part adp complete theorem abspar adp part adp pdbs abspar complete corollary abspar iff forward complete adp pdbs abs complete note forward complete iff hence adp pdbs abs complete finally since finite therefore closure infinite conjunction boils closure finite conjunction corollary thus proposition adp pdbs pdbs par adp pdbs par consequence corollary algorithm computing dsb equivalence finite kripke structure characterized complete shell refinement follows par simulation preorder equivalence simulations possibly nonsymmetric bisimulations simulation kripke structure exists empty relation simulation simulation relations closed union largest simulation relation exists turns largest simulation preorder relation called similarity preorder denoted rsim preord therefore preorder relation preord simulation rsim simulation equivalence symmetric closure rsim iff exist two simulation relations psimeq part denotes partition corresponding number algorithms computing simulation equivalence proposed like first compute similarity preorder obtain simulation equivalence problem computing simulation equivalence important model checking recalled section simulation equivalence strongly preserves actl psimeq pactl see section recall actl obtained restricting ctl defined section universal quantifiers allowing negation atomic propositions actl turns abstract domain actl obtained abstract domain following sublanguage lemma let finitely branching adactl proof let opactl set standard interpretations operators actl pre analogously proof lemma consequence fixpoint characterizations turns abs forward complete opactl iff forward complete pre thus lemma corollary pre actl actl thus proposition pactl par adactl par psimeq consequence corollary algorithm algsimeq computes simulation equivalence viewed partitioning abstraction pre shell refinement algsimeq par instantiation generalized procedure complete shell allows design new efficient abstract algorithm computing simulation equivalence whose space time complexity comparable algorithms like preorders abstract domains simulations give rise preorders rather equivalences like case bisimulations dbs relations thus order characterize simulation preorders forward completeness abstract domains need view preorders abstract domains obtained generalizing abstraction section partitions preorders def let preord let define rpre prer preorder gives rise abstract domain rpre related following abstraction concretization maps def prer def easy check hypothesis preorder follows rpre indeed hence preord induces abstract domain denoted add abs also note prer namely prer closure associated add notation add comes fact abstract domain equivalent add disjunctive lemma add abs preord abs disjunctive proof observe trivially additive add disjunctive hand let abs disjunctive consider relation trivially preorder thus add disjunctive order conclude add equivalent enough observe prera true prera let observe add indeed generalizes adp partitions preorders part adp add simple consequence fact partition viewed equivalence relation exactly block prep hand abstract domain abs induces preorder relation preord preord follows preord iff turns maps add preord allows view lattice preorder relations abstraction lattice abstract domains theorem preord abs preord add proof let abs preord let prove preord add let let show add prer prer xry preord thus applying let let show note prer add namely preord def let remark add preord lower closure operator habs lemma abs disjunctive iff hence coincides refinement also known disjunctive completion namely abstract disjunctive refinement provide characterization simulation preorders terms forward completeness theorem let preord simulation iff add forward complete pre proof recall prer closure associated add first observe iff prer forward complete one hand prer obtain therefore prer hand prer prer prer prer prer therefore prer obtain thus remains show satisfies condition definition simulation iff prer forward complete prove prer pre prer pre prer let prer pre prer exists pre prer xry simulation exists hence prer together transitive gives prer therefore pre prer observe order show simulation enough show xry pre prer following implications hold prer holds prer uco prer pre pre prer pre prer prer prer pre prer prer pre prer pre prer pre monotone pre prer monotone prer forward complete pre prer closes proof related work loiseaux generalized standard approach abstract model checking general abstract models abstraction relation states used instead surjective function states however results strong preservation given theorems require hypothesis relation difunctional case abstraction relation indeed derived function class strongly preserving abstract models loiseaux framework really larger class standard abstract models see detailed discussion dams section giacobazzi quintarelli first noted strong preservation related completeness abstract interpretation studying relationship complete abstract interpretations clarke spurious counterexamples given formula actl model checker running standard abstract kripke structure defined state partition may provide spurious counterexample namely path abstract states namely blocks correspond real concrete counterexample case exploiting spurious counterexample partition refined splitting single block result refined partition admit spurious counterexample given new refined abstract model model checker giacobazzi quintarelli cast spurious counterexamples partition lack standard completeness abstract interpretation sense corresponding partitioning abstract domain adp applying results put forward method systematically refining abstract domains order eliminate spurious counterexamples relationship completeness spurious counterexamples studied also shown block splitting operation paige tarjan partition refinement algorithm characterized terms complete abstract interpretations general idea systematically enhancing precision abstract interpretations refining underlying abstract domains dates back early works cousot cousot evolved systematic design abstract interpretations abstract domain refinements conclusion work shows abstract interpretation technique allows generalize notion strong preservation standard abstract models specified abstract kripke structures generic domains abstract interpretation inductively defined language turns strong preservation standard abstract model checking framework based partitions space state becomes particular instance property forward completeness abstract domains semantic operators language particular generalized abstract model always refined fixpoint iteration abstract domain strongly preserves generalizes framework idea partition refinement algorithms reduce state space order obtain minimal abstract kripke structure strongly preserving temporal language work deals generic temporal languages consisting state formulae future work would interesting study whether ideas abstract approach applied linear languages like ltl consisting formulae interpreted sets paths kripke structure idea investigate whether standard strong preservation ltl generalized abstract interpretations powerset traces corresponding completeness properties fairness also interesting topic investigation namely study whether abstract framework allows handle fair semantics fairness constraints finally let mention results presented paper led design generalized refinement algorithm based abstract interpretation computing abstract strongly preserving domains shown section abstract strongly preserving domain characterized greatest fixpoint computation abs shown algorithm viewed exactly corresponding abstract greatest fixpoint computation part leads abstract refinement algorithm parameteric abstract interpretation lattice abs abstract domains generic inductive language acknowledgements wish thank mila dalla preda roberto giacobazzi contributed early stage work paper extended revised version work partially supported firb project abstract interpretation model checking verification embedded systems project aida abstract interpretation design applications references apt plotkin countable nondeterminism random assignment acm bloom paige transformational design implementation new efficient solution ready simulation problem sci comp bouajjani fernandez halbwachs minimal model generation proc internat conf computer aided verification cav lncs springer browne clarke grumberg characterizing finite kripke structures propositional temporal logic theoret comp bustan grumberg minimization acm trans comput clarke grumberg jha veith abstraction refinement proc internat conf computer aided verification cav lncs springer clarke grumberg jha veith abstraction refinement symbolic model checking acm clarke jha veith counterexamples model checking proc ieee symp logic computer science lics ieee press clarke grumberg long model checking abstraction acm trans program lang clarke grumberg peled model checking mit press cleaveland iyer yankelevich optimality abstractions model checking proc intern static analysis symposium sas lncs springer cleaveland parrow steffen concurrency workbench semantics based tool verification concurrent systems acm trans program lang cousot cousot abstract interpretation unified lattice model static analysis programs construction approximation fixpoints proc acm popl cousot cousot systematic design program analysis frameworks proc acm popl cousot cousot abstract interpretation application comportment analysis generalizing strictness termination projection per analysis functional languages proc ieee int conf computer languages iccl cousot cousot refining model checking abstract interpretation automated software engineering journal cousot cousot temporal abstract interpretation proc acm popl dalla preda completeness stability abstract model checking laurea thesis italian univ verona italy dams abstract interpretation partition refinement model checking thesis eindhoven university technology netherlands dams grumberg gerth abstract interpretation reactive systems acm trans program lang bakker meyer zucker infinite computations denotational semantics theoret comp nicola vaandrager three logics branching bisimulation acm dovier piazza policriti efficient algorithm computing bisimulation equivalence theoret comp emerson mok sistla srinivasen quantitative temporal reasoning proc internat conf computer aided verification cav lncs springer emerson clarke characterizing correctness properties parallel programs using fixpoints proc icalp lncs springer giacobazzi ranzato unifying view abstract domain design acm comput gentilini piazza policriti bisimulation simulation coarsest partition problems automated reasoning giacobazzi quintarelli incompleteness counterexamples refinements abstract model checking proc intern static analysis symposium sas lncs springer giacobazzi ranzato refining compressing abstract domains proc icalp lncs springer giacobazzi ranzato optimal domains disjunctive abstract interpretation sci comp giacobazzi ranzato scozzari making abstract interpretations complete acm groote vaandrager efficient algorithm branching bisimulation stuttering equivalence proc icalp lncs springer grumberg long model checking modular verification acm trans program lang hennessy milner algebraic laws nondeterminism concurrency acm henzinger henzinger kopke computing simulations finite infinite graphs proc focs ieee press henzinger maujumdar raskin classification symbolic transition systems acm trans comput lamport good temporal logic information processing ifip lee yannakakis online minimization transition systems proc acm stoc loiseaux graf sifakis bouajjani bensalem property preserving abstractions verification concurrent systems formal methods system design semantics abstract static analyzes temporal properties proc intern static analysis symposium sas lncs springer abstract domains property checking driven analysis temporal properties proc intern conf algebraic methodology software technology amast lncs springer paige tarjan three partition refinement algorithms siam ranzato tapparo making abstract model checking strongly preserving proc intern static analysis symposium sas lncs springer ranzato tapparo strong preservation completeness abstract interpretation proc european symposium programming esop lncs springer ranzato tapparo abstract refinement algorithm strong preservation proc intern conf tools algorithms construction analysis systems tacas lncs springer ranzato tapparo efficient algorithm computing simulation equivalence based abstract interpretation preparation schmidt closed logical relations powersets proc intern static analysis symposium sas lncs springer tan cleaveland simulation revisited proc intern conf tools algorithms construction analysis systems tacas 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arxiv apr consideration publication theory practice logic programming chain programs writing deterministic metainterpreters david rosenblueth instituto investigaciones aplicadas sistemas universidad nacional apdo abstract many metainterpreters found logic programming literature nondeterministic sense selection program clauses determined examples familiar demo vanilla metainterpreters applications nondeterminism convenient cases however deterministic metainterpreter explicit selection clauses needed cases include conversion parallelism parallelism processors implementation search strategies simulation reasoning deterministic metainterpreters write programmer must concerned set children node derivation tree argue possible advantageous write metainterpreters reasoning terms object programs converted syntactically restricted form call chain form forget except unit clauses give two transformations converting logic programs chain form one moded programs implicit two existing methods execution one arbitrary programs illustrations approach show examples three applications mentioned introduction perhaps common use metalogic definition implementation metainterpreters safra shapiro abramson rogers kowalski apt turini many applications metainterpreters based concise definitions like vanilla metainterpreter easily elaborated required applications however neglected possibly employing convoluted definitions examples deterministic metainterpreters exhaustively traversing search spaces purpose present technique simplifying design deterministic metainterpreters technique converts object program form severely restricted syntax thereby facilitating reasoning search space early works exploiting metainterpreters great advantage example bowen kowalski amalgamation language metalanguage bowen kowalski sergot facility sergot shapiro algorithmic debugger shapiro metainterpreters familiar vanilla david rosenblueth demo metainterpreters nondeterministic definitions consider example following demo predicate kowalski kowalski demo axiom demo demo demo demo demo true definition nondeterministic determined first clause definition axiom theory conclusion might needed demonstrate kowalski applications nondeterminism convenient others however deterministic metainterpreter explicit selection axioms desired problem amenable deterministic metainterpretation exhaustively traversing using processor search space generated logic program goal ueda tamaki another problem describing search strategies logic programs equivalent flowcharts like programs clark van emden yet another application kowalski approach reconciling reactive rational agents metainterpreters kowalski course possible write deterministic metainterpreters logic programs clark gregory perhaps among first publish clark gregory one metainterpreter show slightly modified version metainterpreter fig intended meanings predicates metainterpreter follows assume set answer substitutions goal subgoal predicate single call set intended hold list form predicate set generalisation single call set case goal may one subgoal form true form set answer substitutions terminates true simplify code predicate set abs turn viewed generalisation set abs list clauses bsm form set answer substitutions bsi refer reader clark gregory thorough discussion metainterpreter exhibited fig programmer must concerned set unifiers children node derivation tree term instances predicate reason concern see variables logic program appear anywhere clause additional difficulty absent usual nondeterministic metainterpreters study deterministic metainterpreters viewed attempt narrow gap form logic programming needed applications form useful controlling execution logic programs kowalski observation kowalski fgcs project experienced gap suggests looking work done connection project chain programs writing deterministic metainterpreters example representation definition true true example goal single call set metainterpreter single call set definition abs matches abs set set set set set abs append set true set single call set term instances set matches matches copy unify matches abs matches abs matches abs abs term instances term instances copy unify term instances figure deterministic metainterpreter arbitrary definite programs readability use string name variable taking value instance value similarly name variable taking value list instances value first ueda ueda tamaki tamaki published methods converting nondeterministic logic program deterministic version motivation developing methods allowing execution programs processors parallel understanding methods work might lead key ideas obtaining deterministicevaluation methods however considerable intricacy methods obstacle giving clear concise explanation central mechanisms david rosenblueth trying elucidate principles methods based found important common characteristic tend hide certain occurrences variables list behaving like stack particular occurrences variables receive substitution subgoal selected occur subgoal selected succeeded variables occurrences called variables tamaki hence consider programs lacking variables fortiori transformations get simplified kind program lacking variables chain programs clauses form well clauses formally define chain programs sect apparently concentrating chain programs simplify task devising deterministictraversal methods general defining deterministic metainterpreters particular confirm possibility deriving sect deterministic metainterpreter reasoning terms relational union composition metainterpreter useful provide ways transforming logic programs chain form form methods ueda tamaki handle moded programs necessarily chain form programs argument place predicate used either input instantiated output uninstantiated comparing original methods versions simplified chain programs uncovered transformation converting moded programs chain form sect give transformation fact implicit ueda tamaki methods ueda tamaki hide variables list behaving like stack previously used transformation adapting parsers languages obtaining inference systems moded logic programs rosenblueth rosenblueth peralta next sect give another transformation converting arbitrary definite programs chain form inspired previous one general transformation easily extend sect existing methods deterministic exhaustive traversal ueda tamaki handle arbitrary definite programs section shows write deterministic metainterpreters applications first exhibit metainterpreter behaviour prolog systems next give metainterpreter assume familiarity logic program transformation rules pettorossi proietti code appearing throughout sequel well examples programs converted chain form may found http chain programs writing deterministic metainterpreters deterministic metainterpreter chain programs section write deterministic metainterpreter chain programs reasoning terms relational union composition chain program viewed defining system equations relational expressions systems relational equations studied example engelfriet bakker meertens blikle translation chain programs relational equations enables regard logical inference programs evaluation relational expressions built union composition chain programs similar different certain programs occurring literature name difference allow function symbols constants clarity discuss metainterpreter assuming ground terms constructed however sect observe addition variable renaming full unification opposed matching unification metainterpreter also valid constructing terms variables chain programs systems relational equations define chain program logic program consisting clauses form distinct variables term first argument place predicate called input second argument place output useful single kind chain program answers program goal leftmost ground input ground assuming leftmost computation rule var var every clause form program called program throughout sequel use var denote set variables expression clearly clause denotes inclusion name relations denoted respectively denotes relational let define clause defining predicate symbol form hand clause defining predicate symbol form ground instance composition relations defined david rosenblueth hence chain program denotes system relational expressions inclusion form largest set clauses defining predicate symbol sometimes meaning logic program defined least herbrand model interested least solution system shown solution equal unique solution system obtained replacing inclusions equalities notational convenience extend relational composition case first argument set let denotes image also sometimes omit curly braces singletons denote identity relation empty relation well empty set given object chain program use metavariable taking value ground term set ground terms relation denoted objectlevel predicate compositions relations denoted predicates relation denoted single clause pjs unions relations denoted single clause computing answers chain program goal ground term translates evaluating expression names relation denoted evaluation evaluate relational expressions form represent branches sld tree metainterpreter evaluator let establish representation simplicity use ambivalent syntax jiang clause defining predicate symbol form following clause metalevel nonunit interpreted infix list constructor constant clause defining predicate symbol form unit unit unit predicate clearly unnecessary could defined using unit predicate use readability chain programs writing deterministic metainterpreters also largest set clauses defining predicate symbol defn interpreted infix list constructor denoting set union constant general could two main evaluation strategies termwise decomposing relationwise decomposing concentrate termwise definition see rosenblueth application metainterpreter using relationwise definition consider predicate intended hold next two clauses constitute termwise definition predicate follow distributivity composition union yszs union yszs meant hold union yszs meant hold yszs decompose definition follows definition composition represents composition single term single relation following clause translates composition composition union pjs relations defn pjs pjs pjs intended hold pjs inductively define decomposing pjs definition follows distributivity composition union pjs yszs pjs union yszs predicate assumed hold predicate union yszs assumed hold yszs next write definition definition uses auxiliary predicate case represents unit clause hand represents nonunit clause predicate uses definition david rosenblueth translate previously defined predicate used unit nonunit remains define intended hold represents unit clause unit unit clause covers case matches input unit clause named whereas clause covers case match input unit clause named finally give complete metainterpreter standard prolog notation addition approximated set union list concatenation efficiently executable metainterpreter would use difference lists clarity prefer ordinary lists call abcde metainterpreter yszs append yszs defn pjs pjs yszs pjs append yszs unit nonunit unit unit conversion moded programs chain form written deterministic metainterpreter chain programs aim develop transformation converting moded programs apt chain chain programs writing deterministic metainterpreters form derive transformation rules pettorossi proietti transformation fact implicit two existing methods deterministic exhaustive traversal ueda tamaki methods previously used transformation adapting parsers grammars obtaining inference systems moded logic programs rosenblueth rosenblueth peralta clause called moded var var var var var program called moded consists moded clauses condition causes every input ground input initial goal also ground use leftmost computation rule subgoal succeeds condition causes constructed term effect input subgoals thus avoiding speculative bindings rosenblueth rosenblueth peralta third condition variable occurring occurs meant simplifying transformation however observe possible eliminate condition without excessively elaborating transformation use standard equality theory given program theory consists following axioms xnf ynf xnf ynf function symbol occurring xnp xnp ynp ynp predicate symbol occurring called reflexivity symmetry transitivity function substitutivity predicate substitutivity respectively one way obtaining chain clause moded clause would resolve first moded clause predicate substitutivity replace argument subgoal variable fold resulting clause using new predicates remove introduced equations see however folding operation may always sound gardner shepherdson tamaki sato consider following append program used splitting lists hli david rosenblueth employ angled brackets instead ordinary brackets grouping input output arguments clarity predicate substitutivity symmetry possible derive next could fold using following definitions naive naive case naive folding operation would replace subgoals enclosed rectangles definiendum naive however observed tamaki sato general incorrect fold clause using definition naive variable like appears atoms replaced definiendum appears atoms replaced definiendum unify variable appearing definiendum see incorrectness fold subsequently unfold naive generalisation obtained note unlike cause incorrectness tamaki sato occurs suggests minimal strengthening syntactic conditions defining moded clauses avoid situation would requiring variables occur hence define clause prechain form var var var var clause prechain form possible arrive chain form first applying predicate substitutivity folding completed definitions predicates form foldings satisfy also gardner shepherdson stronger condition gardner shepherdson folding face problem converting moded clause prechain form recall first variables deterministic methods ueda tamaki variables receive substitution subgoal selected occur subgoal selected succeeded observe next chain programs writing deterministic metainterpreters prechain form clauses lack variables finally note fact methods use stack hide variables suggests also use stack achieve prechain form since chain form prechain form lack variables could chosen prechain form object programs deterministic metainterpreter resulting metainterpreter however would concise converting moded clause prechain form often apply following sequence unfolding steps group lemma lemma equation introduction let definite clause occurrence variable clause obtained replacing occurrence adding equation logically implied standard equality theory proof first resolve predicate substitutivity follows occurrence head select subgoal predicate substitutivity equation next apply symmetry equations resolvent resulting clause form hand occurrence subgoal select subgoal resulting clause form either case occurrence equation next apply function substitutivity many times necessary make occurrence appear top level equation finally apply reflexivity equations except thus disposing unwanted equations resulting clause claimed form hence lemma holds often use equation introduction followed symmetry also refer equation introduction example let convert first prechain form chain form achieve prechain form use auxiliary stack introduce following predicate hst hst first apply equation introduction definition hst next apply equation introduction david rosenblueth subsequently resolve unifying two underlined atoms hst next fold using function substitutivity hst finally fold using systematically rename variables prechain form apply predicate substitutivity symmetry fold completed definitions arriving end example rosenblueth rosenblueth peralta required variable occurring occurs note one occurrence variable like thus violating condition would extra equation applying equation introduction twice note wish eliminate reflexivity since would obtain clause variable occurring input output head preventing folding however equations one eliminated factoring equations side hence withdrawn condition derivation suggests proof theorem relating moded clause chain form theorem let moded clause let var var var var chain programs writing deterministic metainterpreters clause logically implied standard equality theory iff version function substitutivity axiom function symbol completed definitions list form xdj xdj predicate symbols occurring proof first apply equation introduction definitions hst let renaming substitutions lloyd var var rhs rhs since variables rhs occur application variable renames uniquely hence think application addition subscript next apply equation introduction rename variables obtaining var var subsequently resolve rename hstn stn let occurs one define var max var david rosenblueth apply transitivity factoring replacing equations variable renaming stn note var var iff equivalently var var iff hence exactly equations constructing next repetitively fold using function substitutivity arrive hstn prechain form obtained folding hstn finally apply predicate substitutivity symmetry fold completed definitions predicates resulting clause desired form hence conclude theorem holds example let apply theorem clause example var var var var var var resulting clause chain form definitions variable renaming end example chain programs writing deterministic metainterpreters let moded program define program resulting applying theorem every clause way predicate symbols clause occur clause theorem associates chain program moded program way logical consequence conservative extension implication direction also holds logically implied conservative extension seen first resolving clause head predicate symbol part definition unfolding definitions finally note chain program moded program program program every unit clause var var answers subgoal ground input ground using leftmost computation rule sect example example explicitly linking transformation abcde metainterpreter give representation chain form clauses defn defn defn nonunit unit hst hst unit hst unit hst unit unit unit end example conversion definite programs chain form section first give transformation inspired previous one converting arbitrary definite program chain form next explain couple abcde metainterpreter unmoded transformation transformation roughly moded transformation takes clause predicates one input argument place one output argument place disposes variables adding stack replaces arguments subgoal variables hence first apparent obstacle find trying convert arbitrary unmoded clause chain form arguments predicates clause david rosenblueth predetermined roles fact argument predicate may play role either input output suggests treating arguments uniformly one way yet binary predicates could replicate arguments predicate two copies set arguments one copy behaving single input possibly variables subgoal selected copy behaving single output naturally give groundness property runtime terms translates use full unification instead matching thus associate predicate another predicate defined denotes subset identity relation herbrand universe also add stack hst hst another decision make variables push onto stack fact could push variables clause economical pushing variables occurring atoms clause example consider usual append program first write definition hst hst apply equation introduction getting hst need later folding application next obtain instance apply equation introduction hst resolve unifying two underlined atoms get hst subsequently fold using function substitutivity hst chain programs writing deterministic metainterpreters finally fold using systematically rename variables apply predicate substitutivity symmetry fold completed definitions predicates manipulating stack moded transformation thus arriving chain form end example arbitrary definite program converted chain form following theorem theorem let definite clause let var var var var clause logically implied standard equality theory iff version function substitutivity axiom function symbol completed definitions hst hst hst xri hst xri xri list form proof first use equation introduction definitions hst xri xri david rosenblueth next apply equation introduction way two variables common except obtain theorem var var var var var apply definition mgu use equation introduction resulting instance getting variable renaming hstn stn var subsequently resolve hstn stn folding add following equations recalling subgoal addition preserves soundness var equations applying transitivity enable subgoal resulting clause prechain form add equations var equations enable fold apply transitivity factoring way equations replaced variable renaming stn var next repetitively fold using function substitutivity arrive hstn chain programs writing deterministic metainterpreters term term prechain form obtained folding hstn finally apply predicate substitutivity symmetry fold completed definitions predicates resulting clause desired form hence conclude theorem holds moded transformation logically implied conservative extension seen unfolding definitions deterministic metainterpreter arbitrary chain programs sect wrote deterministic metainterpreter assuming leftmost input every goal tree apt sld tree leftmost computation rule ground modify metainterpreter also correct trees necessarily groundness property possible handle terms variables generalising matching full unification metainterpreter explicitly using unification bowen kowalski demo metainterpreter bowen kowalski case unification clauses form reduces argument passing need incorporate clauses form consider instance previous definition included clause unit following demo metainterpreter would replace clause unit rename match sub apply sub rename holds result renaming variables distinct variables match sub holds sub mgu apply sub holds result applying sub prolog provides way approximate effect extralogical copy predicate copy unit david rosenblueth note addition unification abcde metainterpreter occurs single point unlike unifiers metainterpreter fig pervasive variables name extending existing traversal methods arbitrary definite programs section deals first reconstruction extension existing tamaki ueda deterministic methods objective methods executing programs processors parallel existing versions methods restricted moded programs hence reconstruction uses transformation moded programs form extension modifies methods make applicable arbitrary definite programs essentially replacing moded transformation sect definite transformation sect fall short proposing practical methods eliminate layer interpretation one way eliminating layer would feed metainterpreter object program partial evaluator mixtus sahlin however resulting residual program may enormous another possibility would compile away layer interpretation hand done present work reconstruction derivations methods start abcde metainterpreter brevity omit detailed derivations indicate derivations could obtained also instead using difference lists original methods employ ordinary lists clarity reconstruction method observed tamaki programs originally degree parallelism may lose parallelism capture parallelism thus treats clauses parallelism special way simplicity concerned special treatment concentrate main component method converts parallelism parallelism obtain programs produced method unfold using yszs append yszs let consider first chain program transformed method perhaps interesting clause program produced chain programs writing deterministic metainterpreters method clause associated subgoal form yszs append yszs intended hold set answers observe first intended meaning next easy obtain identifying predicate predicate rest clauses resulting transforming chain program readily obtainable abcde metainterpreter consider arbitrary moded program clause corresponding case yszs append yszs term var defined theorem recall object clause transformed moded transformation subgoal predicate symbol partially evaluated abcde metainterpreter results subgoal form easily unfolded away producing following clause similar yszs append yszs difference method uses parameter predicates recording values variables whereas keeps variables part term even difference compiled program abcde metainterpreter follow search strategy using separate parameter however method economical exploiting fact predicates may modify stack predicates hold relations output stack equal input stack see observe stack definitions theorem hence compute answers goal first computing answers affixing front answer predicate list obtained concatenating lists affixed front every list let first rewrite clause astream yszs astream astream append yszs david rosenblueth predicate astream yszs intended hold iff yszs hence asserts throughout section binds stronger denotes list concatenation let consider abcde metainterpreter clarity rename variables styszs stys stys stxs stzs append stzs styszs stxs styszs holds stxs styszs stack occurs front every element stxs styszs hence asserts definition obtain specifically recursive equation definition obtain side obtain side reconstruction method let turn attention method starting also abcde metainterpreter use completed definitions pjs pjs acts like list continuations underlined subgoals indicate forthcoming unfolding application first unfold using pjs append using identity follows right distributivity composition union rewrite pjs append chain programs writing deterministic metainterpreters rectangle indicates forthcoming folding application fold using definitions predicates pjs append arriving clause metainterpreter another interesting clause obtained unfolding using nonunit rewrite nonunit append step justified using associativity composition figure shows resulting metainterpreter applied unfolding step using definition defn pjs pjs yszs pjs append yszs unit unit unit unit nonunit append figure deterministic metainterpreter unmoded versions methods reconstructed methods chain programs replace moded transformation definite transformation however sect must also rename variables using unit predicate either david rosenblueth applications prolog metainterpreter together definite transformation far designed metainterpreters performing traversals committedchoice processors observe processors deterministic bindings standard deterministic imperative languages suggests possibility using metainterpreters describing search strategies one imperative language particular see obtain imperative implementation prolog search strategy slightly modifying continuationbased metainterpreter sect fig difference previous metainterpreters standard implementations prolog whereas perform exhaustive traversals prolog systems may may however easily modify metainterpreter ask user whether answers requested also note prolog systems usually remember answers query allowing eliminate answer list halt write write read cont haltcont defn pjs pjs haltcont cont haltcont halt cont pjs haltcont haltcont unit unit haltcont cont unit unit haltcont nonunit append haltcont halt cont pjs halt halt halt cont pjs cont haltcont pjs haltcont metainterpreter meant constructing ground terms obtain true pure prolog system handling terms variables would include variable renaming unification manner similar either deterministic metainterpreters arbitrary definite programs written directly without using transformations chain form example metainterpreter fig programmer however aware set unifiers children node derivation tree contrast approach programmer write metainterpreter without considering chain programs writing deterministic metainterpreters unifiers except using unit predicate case object clause body simplifying treatment unifiers metainterpreter final application exhibit metainterpreter usual give metainterpreter chain programs transformations moded definite programs chain form make metainterpreter applicable programs necessarily chain form next variant abcde metainterpreter constructs one proof extra argument indicate amount resources needed construct proof ans defn pjs pjs ans halt cont pjs unit unit ans unit unit nonunit append halt cont pjs ans ans halt cont pjs ans pjs style writing metainterpreters may viewed alternative appearing kowalski kowalski sadri concluding remarks contributions applications metainterpreters neglected perhaps based convoluted definitions comparison demo predicate deterministic metainterpreters example result especially elaborate since programmer must consider set unifiers children node derivation tree thus metainterpreters converting parallelism parallelism ueda tamaki describing david rosenblueth languages clark van emden simulating boundedresource reasoning kowalski kowalski sadri received due attention compilation methods converting parallelism parallelism developed first ueda ueda tamaki tamaki studying methods identified chain programs important exhibiting essence techniques use chain programs stepping stone methods ueda tamaki viewed comprising two parts conversion moded program chain form application partial deduction deterministic metainterpreter chain programs contribution part consisted first extracted ueda tamaki implicit transformation converts moded program equivalent chain form next using generalisation transformation given another unmoded transformation converts arbitrary definite programs chain form part contributed showing write deterministic metainterpreters chain programs one metainterpreter served first reconstruct extend arbitrary unmoded definite programs existing methods ueda tamaki finally observed deterministic metainterpreters applications exhaustive traversals gave metainterpreter follows prolog search strategy another one counts number steps search refutation opposed number steps refutation methodology designing methods follows write deterministic metainterpreter chain programs ignoring unification incorporate metainterpreter one transformations converting either moded unmoded programs chain form case unmoded transformation selected add renaming unification metaclauses dealing object unit clauses observed even adding unification need concerned substitutions single point metainterpreter whereas metainterpreter written directly unifiers pervasive fig performance study made study illustrating performance programs degraded result transforming programs chain form study used sicstus prolog version ran redhat linux version table next columns labeled show data source program columns labeled show data corresponding transformed program first exhibit number clauses program size measured bytes compiled code chain programs writing deterministic metainterpreters program num clauses split moded append append splitting quicksort ord lists quicksort diff lists program size give relative execution times according sicstus memory requirements local global stacks according sicstus splitting list prefixes suffixes sorting reverse sorted list quicksort programs program split moded append append splitting quicksort ord lists quicksort diff lists execution time stacks related work work stemmed exhaustivetraversal methods however various publications studying deterministic traversals search spaces within logic programming hirakawa chikayama furukawa bansal sterling codish shapiro lichtenstein codish shapiro shapiro sato tamaki demoen contributions demoen perhaps closest motivation authors sketch reconstruction continuationbased method give metainterpreter method based recomputation similarly number transformations converting logic programs syntactically restricted form sato tamaki tarau boyer tarau one sato tamaki common work connection method transformations differ however producing programs every clause binary one atom body future work argued writing deterministic metainterpreters advantageous approach task amounts describing evaluation strategy david rosenblueth relational expressions form defined evaluation strategies expressions close connection implementation functional programming languages peyton jones rewrite systems dershowitz jouannaud investigating different evaluation strategies relational expressions lead different search strategies spaces determined chain programs would one way extending contributions presentation results came across need eliminating layer interpretation work would greater practical impact combined algorithmic elimination layer without producing excessively large residual program chain programs also proved useful devising rosenblueth rosenblueth peralta inference systems derived parsers need consider unification treatment unit clauses hence need modify treatment terminals studying applications chain programs might helpful relegate role played unifiers would another avenue research acknowledgments work owes much carlos velarde contributed motivating discussions carefully read previous versions paper spotted errors theorem proofs also thank referees whose comments substantially improved presentation results gratefully acknowledge facilities provided iimas unam references abramson rogers eds logic programming mit press apt turini eds logic programming mit press apt logic programming prolog prentice hall bansal sterling compiling programs committedchoice technical report case western reserve university cleveland ohio blikle comparative review program methods gruska mathematical foundations computer science lecture notes computer science bowen kowalski amalgamating language metalanguage logic programming clark eds logic programming academic press chain programs writing deterministic metainterpreters clark gregory notes implementation parlog journal logic programming clark van emden consequence ieee transactions software engineering codish shapiro compiling shapiro concurrent prolog collected papers vol mit press bakker meertens completeness inductive assertion method journal computer system sciences dershowitz jouannaud rewrite systems van leeuwen handbook theoretical computer science vol elsevier science publishers engelfriet simple program schemes formal languages lecture notes computer science gardner shepherdson transformations logic programs lassez plotkin eds computational logic essays honor alan robinson mit press hirakawa chikayama furukawa eager lazy enumerations concurrent prolog proc second international logic programming conference uppsala sweden jiang ambivalent logic semantic basis metalogic programming hentenryck proc eleventh international conference logic programming kowalski problems promises computational logic lloyd computational logic symposium proceedings kowalski springboard information processing century icot journal pannel discussion fifth generation computer systems conference kowalski using reconcile reactive rational agents apt turini eds logic programming mit press kowalski sadri towards agent architecture combines rationality reactivity pedreschi zaniolo eds proc international workshop logic databases san miniato italy lecture notes computer science lichtenstein codish shapiro representation enumeration flat concurrent prolog computations shapiro concurrent prolog collected papers vol mit press lloyd foundations logic programming edn demoen findall without warren proc tenth international conference logic programming mit press pettorossi proietti transformation logic programs foundations techniques journal logic programming peyton jones implementation functional languages prentice hall rosenblueth chart parsers inference systems logic programs new generation computing rosenblueth method using layered streams obtained chain programs extended abstracts lopstr eighth international workshop program synthesis david rosenblueth tion june manchester technical report series department computer science university manchester issn report number http summary appeared lecture notes computer science rosenblueth peralta slr inference inference system logic programs based slr parsing journal logic programming safra shapiro meta interpreters real kugler information processing ifip elsevier science publishers sahlin mixtus automatic partial evaluator full prolog new generation computing sato tamaki existential continuation new generation computing sato tamaki first order compiler deterministic logic program synthesis algorithm symbolic computation sergot facility logic programming degano sandewall eds proc european conference integrated interactive computing systems north holland shapiro prolog flat concurrent prolog shapiro concurrent prolog collected papers vol mit press shapiro algorithmic program debugging mit press tamaki compilation ground prolog languages proc fourth international conference logic programming melbourne australia tamaki sato transformation logic programs proc second international logic programming conference tarau program transformations compilation metaprograms voronkov proc russian conference logic programming lecture notes intelligence tarau boyer elementary logic programs deransart eds proc programming language implementation logic programming sweden lecture notes computer science ueda making exhaustive search programs deterministic new generation computing
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map moving horizon state estimation binary measurements apr giorgio battistelli luigi chisci nicola forti stefano gherardini paper addresses state estimation discretetime systems binary threshold measurements following maximum posteriori probability map approach exploiting moving horizon approximation map shown linear system noise distributions probability density function proposed state estimator involves solution sampling interval convex optimization problem application estimator dynamic estimation diffusion field given binary measurements field also illustrated finally simulation results relative application shown demonstrate effectiveness proposed approach ntroduction binary sensors whose output indicates whether noisy measurement sensed variable analog measurement exceeds given threshold frequently employed monitoring control applications idea multitude sensing devices possible achieve estimation accuracy possibly single one expensive highresolution sensors could provide significant practical benefits terms ease sensor deployment minimization communication requirements fact binary threshold measurement conveys minimal amount single bit information implying communication bandwidth savings consequently greater energy efficiency makes paramount importance fully exploit little available information means smart estimation algorithms respect work recently addressed system identification parameter state estimation binary measurements following either deterministic probabilistic approach deterministic context available information essentially concentrated sampling instants binary measurement signal switched value shown additional information exploited non switching sampling instants penalizing values estimated quantity corresponding predicted measurement opposite side respect binary sensor reading far away threshold nevertheless clear little information available estimation purposes whenever binary sensor switchings occur hence authors dipartimento ingegneria dell informazione dinfo degli studi firenze via santa marta gherardini also dipartimento fisica astronomia degli studi firenze via sansone istituto nazionale fisica nucleare infn sezione firenze possible way achieve high estimation accuracy many binary sensors measuring variable different thresholds would clearly increase number switchings actually emulating number sensors tends infinity availability single analog measurement conversely following probabilistic approach binary sensor readings could exploited infer information probability distribution variable interest clarify point let assume large number binary sensors type measuring variable threshold available distribution measurement noise gaussian zero mean given standard deviation known thanks numerosity measurements relative frequency values occurring sensor readings could considered reasonable estimate probability sensed variable threshold turn exploiting knowledge measurement noise distribution allows extract information location value sensed variable respect threshold example found binary measurement equal sensors gaussian measurement noise hypothesized turns expected measurement sensed variable threshold amount equal times standard deviation measurement noise notice sensors noiseless provide either output paradoxically case minimal information sensed variable takes values interval either threshold extracted set binary measurements arguments suggest adopting probabilistic approach estimation using binary measurements presence measurement noise helpful source information words said procedures devised estimation binary measurements exploiting fact measurement noise randomly shifts analog measurement thus making possible infer statistical information sensed variable relying stated paradigm paper presents novel approach recursive estimation state dynamical system given binary measurements proposed approach based movinghorizon approximation maximum probability map estimation extends previous work concerning parameter estimation recursive state estimation contribution paper show linear system optimization problem arising formulation turns convex hence practically feasible implementation rest paper organized follows section introduces map problem formulation state estimation binary measurements section iii presents approximation map estimation referred mhmap algorithm analyzes properties resulting optimization problem section discusses possible application proposed approach dynamic estimation diffusion field binary field measurements section presents simulation results relative dynamic field estimation finally section concludes paper perspectives future work aximum posteriori state estimation binary sensors following notation used throughout paper col denotes matrix obtained stacking arguments one top diag denotes diagonal matrix diagonal entries indicate vectors respectively zero unit entries let consider problem recursively estimating state nonlinear dynamical system zti vti set measurements provided binary sensors zti yti zti zti state estimated known input threshold binary sensor sake simplicity define col zti col yti vector additive disturbance affecting system dynamics accounts uncertainties mathematical model col vti measurement noise vector let denote usual normal distribution mean variance statistical behaviour system characterized diag notice sensor produces binary measurements yti depending whether noisy system output zti threshold according available probabilistic description problem estimating state system binary measurement model formulated hereafter bayesian framework resorting maximum posteriori probability map criterion remainder section preliminary step map state estimation problem formulated end notice binary measurement yti provides intrinsically relevant information state taken account means posteriori probabilities yti particular binary measurement yti bernoulli random variable binary sensor time instant posteriori probability yti given yti yti yti yti yti yti function complementary cumulative distribution function cdf random variable since vti conditional probability yti written terms qfunction follows let denote col vector binary measurements collected time col vector state trajectory let denote col estimates made stage time instant given posteriori probability estimate state trajectory obtained solving following map estimation problem arg max arg min bayes rule notice latter equation considered markov property dynamical system state normally distributed vectors moreover likelihood function binary measurement vector written yik latter equality exploited statistical independence binary sensors accordingly loglikelihood yik cost function minimized map estimation problem turns additive constant terms yik yik unfortunately expression global minimum exist hence optimal map estimate determined resorting numerical optimization routine respect main drawback number optimization variables grows linearly time since vector size consequence grows solution full information map state estimation problem becomes eventually unfeasible approximation introduced iii oving horizon approximation section approximate solution map state estimation problem proposed resorting mhe approach accordingly defining sliding window goal find estimate partial state trajectory col using information available place full information cost time instant minimization following cost addressed jtmh present work consists assigning arrival cost fixed structure penalizing distance state beginning sliding window prediction computed previous time instant thus making estimation scheme recursive natural choice quadratic arrival cost form yik yik initial penalty function known mhe literature arrival cost introduced summarize past data explicitly accounted objective function matter fact form arrival cost plays important role behavior performance overall estimation scheme principle could chosen minimization yields estimate would obtained minimizing algebraic expression true arrival cost seldom exists even sensors provide continuous measurements hence approximation must used respect common choice also followed bayesian point view corresponds approximating pdf state conditioned measurements collected time gaussian mean covariance choice weight matrix case continuous measurements shown stability estimation error dynamics ensured provided large avoid overconfidence available estimates recently similar results proven hold also case binary sensors deterministic context practice seen design parameter tuned pursuing suitable tradeoff stability considerations necessity neglecting already available information since limit going zero approach becomes finite memory one summing stage following problem solved problem given prediction sequence measurement yti find optimal minimize cost function arrival cost input sequences estimates concerning propagation estimation procedure problem problem prediction set equal value estimate made time instant clearly recursion initialized priori expected value initial state vector general solving problem entails solution optimization problem however equations linear resulting optimization problem turns convex standard optimization routines used order find global minimum see let consider following assumption functions linear axt constant matrices suitable dimensions proposition assumption holds cdf complementary function logconcave hence cost function arrival cost convex remark assumption convexity cost function guaranteed also general case statistical behaviour random variables described logarithmically concave distribution functions indeed pdf also cumulative distribution function hence contribution related binary measurements turns convex next section focus case discretetime linear system particular considering diffusion process governed partial differential equation pde spatially discretized means finite element method fem dynamic field estimation section consider problem reconstructing diffusion field sampled network binary sensors arbitrarily deployed spatial domain interest diffusion process governed following parabolic pde models various physical phenomena spread pollutant fluid case represents dependent substance concentration denotes constant diffusivity medium laplace operator spatial variables furthermore let assume mixed boundary conditions dirichlet condition accordingly unknown function approximated outward pointing unit normal vector objective estimate values dynamic field interest given binary measurements pde system simulated mesh finite elements via finite element approximation described specifically domain subdivided suitable set non overlapping regions elements suitable set basis functions defined elements choices basis functions elements key points method specific case investigation elements triangle define mesh vertices basis function affine function vanishes outside fes around denoting kronecker delta order account mixed boundary conditions basis functions supposed ordered first correspond vertices mesh lie either interior last correspond vertices lying unknown expansion coefficient function relative time basis function known expansion coefficient function relative basis function notice second summation needed impose dirichlet condition boundary pde recast following integral form generic weight function applying green identity one obtains choosing test function equal selected basis functions exploiting approximation galerkin weighted residual method applied following equation obtained specifies value concentration boundary homogeneous neumann condition assumed impermeable contaminant notice latter equation boundary integral equation omitted since equal thanks homogeneous neumann condition fact construction basis functions vanish interested reader referred details fem theory particular convert case inhomogeneous boundary conditions homogeneous one defining state vector col vector boundary conditions col equation written compact form stiffness matrix representing diffusion mass matrix captures physical interconnections among vertices affected boundary condition remaining nodes mesh applying example implicit euler method latter equation discretized time thus obtaining linear model time integration interval process disturbance taking account also discretization errors notice linear system dimension equal number vertices mesh lying linear system assumed monitored network threshold sensors sensor binary quantization applied directly measure concentration contaminant point spatial domain exploiting concentration written linear combination concentrations grid points resulting output function takes form zti vti assumption fulfilled fig mesh used estimator elements nodes umerical results rmse concentration section present simulation results proposed approach applied problem state estimation spatially distributed processes discussed previous section consider simulated system triangular elements vertices fixed integration step length initial condition field vector field interest defined bounded spatial domain covers area see fig boundary condition bottom edge condition remaining portions compared time fig rmse concentration state estimator function time random network threshold sensors fig concentration field time monitored random network binary sensors red ground truth simulator proposed estimator implements coarser mesh see fig nodes runs slower sample rate model uncertainty taken account initial condition estimated dynamic field set moving window size weight matrices chosen true concentrations first corrupted gaussian noise variance binary observations obtained applying different threshold sensor network note order receive informative binary measurements generated uniformly distributed random numbers interval range nominal concentration values throughout experiment duration simulation experiment fixed samples fig shows performance novel state estimator implemented matlab terms root mean square error rmse estimated concentration field ket rmse ket norm estimation error time simulation run averaged sampling points evenly spread within independent monte carlo realizations estimation error computed time basis estimate observed proposed estimator successfully estimates dynamic field even observed network randomly deployed binary sensors effect measurement noise mean value rmse seen fig becomes apparent ppendix rmse concentration fig rmse concentration function measurement noise variance fixed constellation binary sensors shown operating noisy environment turns beneficial certain values state estimation problem arbitrary functions first second derivatives function respect respectively equal rmse concentration proof proposition assumption cost function convex functions function domain concave namely let consider cdf complementary function positive functions dti theor rem calculus namely number sensors fig rmse concentration estimates function number sensors deployed monitoring area certain values including observation noise higher variance actually improve quality overall estimates results fig numerically demonstrates validity stated paradigm recursive state estimation binary measurements thus represents interesting contribution work finally fig shows evolution rmse function number binary observations available fusion center state estimation binary sensors formulated moving horizon maximum posteriori probability map optimization problem shown problem turns convex linear system case simulation results relative dynamic field estimation exhibited conjectured feature proposed estimator estimation accuracy improves starting null measurement noise variance latter achieves optimal value beyond estimation performance deteriorates future work topic concern stability properties state estimator application target tracking binary proximity sensors hence follows concave conversely depends sign term convexity properties function easily verified variable hence holds onclusions future work since simple change variable stated proving consequence qfunction using complement rule cumulative distribution function written remaining case noting observed sign term negative thus proving cdf convexity whole cost function eferences wang zhang yin system identification using binary sensors ieee transactions automatic control vol wang yin zhang joint identification plant rational models noise distribution functions using observations automatica vol ristic gunatilaka gailis achievable accuracy gaussian plume parameter estimation using network binary sensors information fusion vol vijayakumaran levinbook wong maximum likelihood localization diffusive point source using binary observations ieee transactions signal processing vol ribeiro giannakis distributed estimation wireless sensor networks part gaussian case ieee transactions signal processing vol ribeiro giannakis distributed estimation wireless sensor networks part unknown probability density function ieee transactions signal processing vol wang yin guo state observability observers systems irregular sampling sensor limitations ieee transactions automatic control vol battistelli chisci gherardini moving horizon state estimation linear systems binary sensors accepted ieee conference decision control osaka japan bai mudumbai dasgupta robust tracking piecewise linear trajectories binary sensor networks automatica vol aslam butler constantin crespi cybenko rus tracking moving object binary sensor network proc acm conf embedded networked sensor systems los angeles usa djuric vemula bugallo signal processing particle filtering binary sensor networks proc digital signal processing workshop djuric vemula bugallo target tracking particle filtering binary sensor networks ieee transactions signal processing vol teng snoussi richard decentralized variational filtering simultaneous sensor localization target tracking binary sensor networks ieee transactions mobile computing vol capponi fatkullin ling shi stochastic filtering diffusion processes level crossings ieee transactions automatic control vol mignone morari moving horizon estimation hybrid systems ieee transactions automatic control vol rao rawlings mayne constrained state estimation nonlinear systems stability moving horizon approximations ieee transactions automatic control vol alessandri baglietto battistelli receding horizon estimation linear systems ieee transactions automatic control vol alessandri baglietto battistelli moving horizon state estimation nonlinear systems new stability results approximation schemes automatica vol alessandri baglietto battistelli zavala advances moving horizon estimation nonlinear systems proc ieee conference decision control cdc liu zhang chen moving horizon estimation networked systems quantized measurements packet dropouts ieee transactions circuits systems regular papers vol delgado goodwin combined map bayesian scheme finite data moving horizon estimation automatica vol battistelli chisci forti pelosi selleri point source estimation via finite element kalman filtering accepted ieee conference decision control osaka japan battistelli chisci forti pelosi selleri distributed finite element kalman filter ieee european control conference linz austria brenner scott mathematical theory finite element methods new york boyd vandenderghe convex optimization cambridge university press cambridge
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toward robust sparse data representation wireless sensor networks mohammad abu shaowei dusit school computer engineering nanyang technological university singapore programme institute infocomm research singapore school information systems singapore management university singapore aug sense sensing successfully used optimized operations wireless sensor networks however raw data collected sensors may neither originally sparse easily transformed sparse data representation paper addresses problem transforming source data collected sensor nodes sparse representation nonzero elements contributions address three major issues include effective method extracts population sparsity data sparsity ratio guarantee scheme customized learning algorithm sparsifying dictionary introduce unsupervised neural network extract intrinsic sparse coding data sparse codes generated activation hidden layer using sparsity nomination constraint shrinking mechanism analysis using real data samples shows proposed method outperforms conventional methods coding compressive sensing sparse autoencoders wireless sensor networks ntroduction data nonzero elements sample vector may naturally exist compressive sensing applications wireless sensor networks wsns path reconstruction problem indoor localization sparse event detection hand sparse data representation easily induced many contexts meteorological applications environmental data gathering particular noise patterns usually presented collected data wsns greatly affect performance conventional transformation algorithms haar wavelet discrete cosine transforms motivates quest effective methods wsns one breakthroughs recent deep learning paradigms finding high level data abstractions achieved introducing sparsity constraints data representations divergence rectifier function topographic coding methods introduced extracting intrinsic features data similar way human brain encoding sensory organ data low percentage spikes visual cortex particular sparse deep learning methods generate sparse representations across training data single unit lifetime sparsity neither guarantee sparsity input signal assert number nonzero values sparse codes however practical implementation wsns requires sparse representation input signal population sparsity sparsity ratio guarantee specifically solution underdetermined system number unknowns number equations dependent sparsity ratio signal mechanism must assert upper limit sparsity ratio sparsity bounding necessary wsns enables using one flat acquisition matrix data encoding node therefore reduces overhead terms memory storing many measurement matrices transmitting node data control exchange need send rate control messages main contributions paper summarized three folds follows paper introduces effective population sparsityinducing algorithm sparsity ratio guarantee algorithm based customized unsupervised neural network model three layers also called autoencoder network generates required sparse coding second hidden layer proposed shrinking sparse autoencoder ssae sparsity achieved introducing regularization term cost function basic autoencoder customize learning algorithm meet wsn characteristics example activations hidden layer parameter learning stage rounded three places consider limited computational precision node rounding considers low precision computations sensor nodes reduces compressed data size data transmission load present customized learning method optimizes ssae cost function basically back propagated error used update nonzero active neurons dominant output values input pattern moreover shrinking mechanism guarantees sparsity bound also used learning ssae parameters therefore ssae asserts number nonzero elements generated time instant literature rich methods designed optimized wsn operations nonetheless much less attention given sparsityinducing stage using straightforward methods extract short distance connection region interest roi backhaul connection base station sensor nodes motes gateway energy constrained fig compressive sensing based data aggregation model roi assumed relatively far therefore gateway designed transmit compressed data costly long distance wireless connection sparsity basis common previous studies using principal component analysis pca discrete cosine transform dct discrete fourier transform dft discrete wavelet transforms difference matrices however sparse coding discipline evolved considerable advances significantly enhance hence overall wsn operations therefore paper intended introduce robust effective method proposed method consists three steps data collection offline training modeling iii online sparse code generation example online sparse code generation application shown figure described details later rest paper organized follows section problem formulation presented section iii describes proposed algorithm ssae structure section discusses important practical issues training fitting proposed model section numerical results using realworld data set presented finally section summarizes paper roblem ormulation consider dense wireless sensor network consisting nodes figure collects data region interest roi sensor collects sample temperature measurements predefined sampling period transmits packets configured transmission power sufficient reach base station due long distance propagation therefore gateway used collect data vector sensor nodes relay analysis processing thereafter historical data matrix formulated containing collected data vectors rows number collected vectors sensors oscillators assumed synchronized clock collecting sufficient historical samples details data collection elucidated section energy bandwidth constrained employs spatially compress data smaller data size radio transceiver energy consuming unit ordinary sensor node thereby energy consumption becomes critical unit transmits huge data backhaul connection sensor nodes assumed transmit short distances important note algorithm also temporally applied individual sensor node however data delivery latency provoked temporal samples must collected node transmitted one compressed chunk next give overview framework implementation device data reconstruction unit compressive sensing signal processing method effective data recovery data samples nyquist rate assuming sparse signal nonzero elements therefore called signal sparsity ratio equal moreover suppose measurement sensing matrix obeys restricted isometry property rip assumed much smaller therefore flat matrix columns rows sensing system consideration executed compress data expressed resulted measurement vector sampled different distributions meet rip gaussian distribution moreover high probability recovery must also meet following constraint constant unit reconstruction achieved minimizing following relaxed problem arg min small constant optimization problem solved using regularized least square method called least absolute shrinking selection operator lasso clearly whole framework based sparsity assumption natural signal sound images transformed sparse form projecting suitable basis however case dealing wsn data precisely sensor nodes produce noisy readings form noiseless data vector physical phenomenon added noise vector noise values assumed independent gaussian variables zero mean variance therefore even neighbor sensors spatially correlated hence compressible noise existence hampers accurate approximation source signal using linear projection methods particular smooth signal representable using linear combinations fourier bases source signal reconstructed signal hidden layer input layer method sparse signal non zero values compressive sensing decoding compressive sensing encoding transmitted signal neuron computational unit base station shrinking mechanism method output layer fig illustration ssae structure gateway fig example data compression transmission recovery operations using models smooth piecewise signals linearly representable wavelet bases nonetheless smoothness condition guaranteed sensor data data samples usually affected noise patterns commercial sensors sense phenomenon finite precision noise robust example noise readings destroy sparsity pattern dct transformed data main aim robust mechanism transform source signal sparse signal upper bound guarantee sparsity ratio generated signal feature sparsitybased applications particular guarantee enables designing low memory low communication overhead applications wsns single sensing matrix used unit compress data require information recover reconstruction signal measurement vector reconstruction noiseless data vector example system online operational procedure shown figure includes components next section presents proposed sparsityinducing mechanism iii hrinking parse autoencoder ssae section introduce autoencoder variant call shrinking sparse autoencoder ssae shown figure ssae network specially designed transform sensory data original domain sparse domain ssae structure consists three neural computational unit layers firstly input layer connected input signal number sensor nodes network briefly sphered version raw sensor data described section secondly hidden layer used generate intrinsic code activation thirdly output layer includes number neurons input layer generates recovery input data layers connected using following formulations weight matrix connecting input hidden layers weight matrix connecting hidden output layers biases input hidden layers respectively additionally sparse data representation obtained applying shrinking operation described section simplicity contain ssae parameters moreover hyperbolic tangent function ssae cost function includes two terms follows log training matrix historical data input vector stored row matrix hidden layer activation moreover training set size configured offline training algorithm details given section autoencoder first term average sum difference input vectors reconstructions output layer term used encourage neural network reconstruct input data output layer second term used encourage sparsity generated coding hidden layer sparsity penalty parameter manage weights term optimization problem words using big value results highly sparse representation poor reconstruction capability delta rule used update ssae weights biases follows wij wij learning rate layer number within ssae network update rules executed iteration gradient descent method partial derivative given algorithm shrinking operation hidden layer neurons input hidden layer activation shrinking input maximum nonzero activations copy operation end end smallest value end output cost function defined single sample means overall partial derivative found averaging partial derivatives input samples second term affects partial derivative hidden layer computed follows log loge derivative thereby ssae designed generate many zeros hidden layer one think neuron active output equal zero inactive neuron participate forwarding input data output generates signals end two important issues second term must noted follows second term minimizes hidden layer activation still ensure exactly zero activations guarantee sparsity ratio generated codes accordingly shrinking mechanism must applied hidden layer activation propagating output layer reconstruct input particular one think second term used mechanism nominating promising neurons deactivated shrinking mechanism described next section shrinking pruning scheme even though cost function ssae designed generate sparse data coding hidden layer still neither guarantee coding population sparsity sparsity input vector assert maximum nonzeros input equally important likely generate values close absolutely zero therefore propose simple shrinking mechanism complete design cycle input vector proposed shrinking mechanism zero least dominant neurons hidden layer therefore switching deactivation mode least dominant neurons ones least effect data reconstruction output layer hence minimum activation values result sparsity restrictions therefore active neurons hidden layer forward propagate input ssae network remaining neurons switched optimized implementation shrinking scheme given algorithm absolute value function offline training ssae parameter adjustment done offline training stage resource demanding process training must performed unit ssae parameters disseminated online data compression learning stage ssae parameters tuned using resourceful relatively high precision operations however usually constrained terms computational resources computational precision machine epsilon value therefore rounding activation hidden layer needed learning stage match low precision moreover rounding less data transmitted learn ssae parameters minimize using method called lbfgs method however firstly collected historical data must randomly shuffled sensors readings highly correlated time nonshuffled data causes ssae dominantly learn training data patterns training data therefore shuffling step ensures training testing data sets contain possible data patterns moreover cross validation technique effective method testing model algorithm offline training algorithm input historical sensor data input maximum nonzero activations input sparsity input number folds cross validation randomly shuffle divide folds held testing sphere input get using append end repeat forwardly propagate compute using shrink get algorithm compute using end compute cost value using compute gradient vector update using method learning converges compute accuracy using end compute average accuracy folds output input layer words ensures neuron hidden layer active input patterns hence neuron exists increases model performance generalizing non local data hence performs well extremely non linear data neurons participating increases possible code formulations number distinct combinations increased active neurons figure shows hidden layer activations time two main desirable properties observed population sparsity achieved maximum number active neurons time instant guaranteed ssae network upper bound nonzeros generated sparse code considers tradeoff recovery error compression ratio data aggregation model therefore single sensing matrix needed using create measurement vector node neurons participating sparse code generation without neuron moreover activation values active neurons concentrated around small values near zero feature achieved conventional average activity ratio sparse autoencoders divergence designed lifetime sparsity iscussion ractical onsiderations generalization capabilities benefiting available samples training cross validation divides training data groups groups time one group held testing using remaining model fitting model performance found averaging errors cross validation groups offline learning described algorithm learning algorithm computationally intensive sensor nodes must performed moreover statistical parameters underlying phenomenon change offline training must updated disseminated nodes computational complexity online encoding decoding sparse codes lightweight particular sensor node generate sparse codes using algorithm overall time complexity data recovery performed using similar time complexity sparse codes verification analysis following sections meteorological data set sensorscope project used data set contains surface temperature samples sensors learning curve ssae shown figure important indication successful ssae training ensuring hidden neurons connected zero weights section practical issues ssae training fitting discussed data collection crucial aspect machine approaches ssae network training data requirement system designer may access large historical data set collected past historical data used train ssae model however case new wsn deployments lack sensor data hinders accurate fitting ssae parameters clearly ssae model needs globally generalize unseen data samples machine learning method training data improve generalization performance data solution wsns following issues must considered using ssae sparsity inducing method assumed sensor nodes densely deployed hence spatially correlated figure sensorscope project data ssae learns spatial correlation redundant patterns nodes collected data therefore underlying phenomenon becomes different way changes nodes spatial correlation new data collection offline model fitting must performed amount data required fit ssae model depends underlying sensed phenomenon fig data sphering effects data showing histograms basic statistical values raw data range sphered data scaled new range distribution stage data sphering simply achieved applying following operation sensors raw input vector max min standard deviation historical training matrix arithmetic mean input vector ssae input vector resulting data vector sphering unlike standard standardization subtracts arithmetic mean input vector whole training matrix mean value effect data sphering training data shown figure clearly data transformed smoother curve zero mean statistical parameters also shown equally important resulting scale sphered data interval makes suitable operation hyperbolic tangent function particular hyperbolic tangent function generates output range without data range ssae produce outputs similar input data reverse operation data sphering required reconstruct original raw sensors vector ssae output values reverse operation given desphere sphere fig ssae designed produce maximum nonzero values time instant sensors learning curve shows convergence offline learning algorithm activation values hidden layer neurons fig surface temperature readings neighbor sensors sensorscope deployment day sample every minutes shows spatial correlation among sensors measurements hence data compressibility constant recovered vectors therefore stored however must sent along compressed data therefore transmitted data size using umerical esults section evaluate performance ssaebased sparsity inducing method ssae tunning complex correlation patterns among sensors data samples needed data sphering using historical sensor data train ssae operation required namely data sphering one main difficulties applying neural networkbased methods numerical tuning network hyperparameters setting autoencoder variants facilitated searching scale values logdomain values value minimizes cross validation error selected rmse sparsity penalty fig sparsity parameter setting bars represent root mean square error rmse values runs maximum performance achieved rmse sparsity ratio fig root mean square error rmse versus sparsity ratio comparison widely used literature two important observations made sparsity inducing algorithms achieve relatively similar recovery error high values however high sparsity ratio values typical practical applications reduction data size noticeable therefore values used applications measurement vector size similar source signal size hand ssae significantly outperforms conventional methods practical low sparsity ratios nonzero values generated sparse codes required minimized conventional methods use minimization model raw data linear combinations sparse bases paper used scikitlearn library testing dictionary learning method coordinate descent method used find lasso problem solution similar algorithm implementation enables setting required sparsity ratio defining number nonzero coefficients sparse code set remaining parameters default values normalize data zero mean unit variance learning dictionary model addition slightly better performance also noticed learning time ssae method also shorter method significant wsns noisy data accordingly figure shows setting sparsity sparsity ratio sparsity term interpreted sparsity nomination term fed shrinking mechanism generate sparse codes therefore trying different values useful achieve maximum signal reconstruction performance ssae next experiments following function used sparsity penalty settings found manually fitting two values described finding line connecting two manually fitted points comparing benchmarks using difference matrix captures difference adjacent correlated values sparse basis used similar noted difference matrix poor performance sparsifying data hence included comparison analysis figure shows comparison ssae recovery performance conventional methods including principal component analysis pca discrete fourier transform dft discrete cosine transform dct dictionary learning conventional methods chosen sensors may report imprecise measurements due external noise sources inaccurate sensor calibration unstable power supply imperfect node design section assume noise values independent gaussian variables zero mean variance added noise vector noticed ssae method allows compression sensors data also helps estimating noiseless data vector physical phenomenon overcomplete sparse representation achieved number hidden layer neurons sparse code size greater input layer neurons however measurement vector size proportional sparse code size therefore number nonzero items must minimized less nonzero coefficients defined overcomplete sparse code hand using neurons hidden layers result overfitting problem overfitting degrades neural network reconstruction performance increases learning time parameters table summarizes experiments using overcomplete sparse representation results also include case adding external noise sensors measurements shows overcomplete case useful unreliable network reduce noise effects producing sparse codes however networks using overcomplete codes degrade table system performance different numbers hidden neurons rmse external noise best overfitting overfitting rmse noise unreliable unreliable best algorithm performance due overfitting problem ummary paper introduced algorithm data aggregation signal wireless sensor networks proposed method consists three steps data collection offline training modeling online sparse code generation modeling scheme based neural network three layers sparse codes exposed hidden layer neurons cost function introduced sparsity nomination scheme shrinking mechanism used switch least dominant neurons hidden layer asserting number generated nonzero values sparse code resulting scheme used many applications compressive sensingbased data aggregation schemes future research analytically study energy consumption computational burdens proposed scheme eferences liu xing path reconstruction dynamic wireless sensor networks using compressive sensing proc acm int symp mobile hoc networking computing acm kuai zheng yang hou compressive sensing approach based mapping localization mobile robot indoor wireless sensor network proc int conf networking sensing control ieee meng han sparse event detection wireless sensor networks using compressive sensing proc annu conf inform sciences syst ieee quer masiero munaretto rossi widmer zorzi interplay routing signal representation compressive sensing wireless sensor networks proc inform theory applicat workshop ieee lee ekanadham sparse deep belief net model visual area proc conf advances neural inform process glorot bordes bengio deep sparse rectifier networks proc int conf artificial intell vol lecun learning invariant feature hierarchies proc european conf comput vision springer lennie cost cortical computation current biology vol fazel fazel stojanovic random access compressed sensing fading noisy communication channels ieee trans wireless vol masiero quer munaretto rossi widmer zorzi data acquisition joint compressive sensing principal component analysis proc global telecommun conf ieee bajwa haupt sayeed nowak compressive wireless sensing proc int conf inform process sensor networks acm luo sun chen compressive data gathering wireless sensor networks proc annu int conf mobile computing networking acm griffin tsakalides compressed sensing audio signals using multiple sensors proc european signal process misra yang jha efficient via sparse representation sensor networks proc int conf inf processing sensor networks acm liu soil moisture sensing measurement scheduling estimation using compressive sensing proc int conf inform process sensor networks acm pottie kaiser wireless integrated network sensors commun acm vol qaisar bilal iqbal naureen lee compressive sensing theory applications survey commun networks vol candes restricted isometry property implications compressed sensing comptes rendus mathematique vol chen dongarra condition numbers gaussian random matrices siam matrix analysis vol romberg tao robust uncertainty principles exact signal reconstruction highly incomplete frequency information ieee trans inf theory vol candes romberg tao stable signal recovery incomplete inaccurate measurements commun pure applied mathematics vol tibshirani regression shrinkage selection via lasso royal statistical rumelhart hinton williams learning representations errors cognitive modeling byrd nocedal zhu limited memory algorithm bound constrained optimization siam scientific computing vol kohavi study bootstrap accuracy estimation model selection proc int joint conf artificial vol sensorscope sensor networks environmental monitoring online available http domingos useful things know machine learning commun acm vol bengio practical recommendations training deep architectures neural networks tricks trade springer lee battle raina efficient sparse coding algorithms proc advances neural inform process mairal bach ponce sapiro online dictionary learning sparse coding proc annu int conf machine learning acm pedregosa varoquaux gramfort michel thirion grisel blondel prettenhofer weiss dubourg machine learning python machine learning research vol ramanathan chehade balzano nair zahedi kohler pottie hansen srivastava sensor network data fault types acm trans sensor networks vol liu starzyk zhu optimized approximation algorithm neural networks without overfitting ieee trans neural vol
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deep residual learning pdes manifold jan zhen zuoqiang abstract paper formulate deep residual network resnet control problem transport equation resnet transport equation solved along characteristics based observation deep neural network closely related control problem pdes manifold propose several models based transport equation equation equation discretization pdes point cloud also discussed keywords deep residual network control problem manifold learning point cloud transport equation equation deep residual network deep convolution neural networks achieved great successes image classification recently approach deep residual learning proposed tackle degradation classical deep neural network deep residual network realized adding shortcut connections classical cnn building block shown fig formally building block defined input output vectors layers function represents residual mapping learned fig relu denotes composition relu department mathematics hong kong university science technology hong kong email mazli yau mathematical sciences center tsinghua university beijing china email zqshi relu conv relu conv figure building block residual learning transport equation resnet consider terminal value problem linear transport equation given velocity field composition output function fully connected layer use softmax activation function softmax weight fully connected layer softmax function given exp softmax exp models posterior probability instance belonging class solution approximately solved along characteristics know along characteristics constant let partition characteristic transport equation solved using simple forward euler discretization time step choose velocity field corresponds weight layers residual block relu one step forward euler discretization actually recovers one layer deep resnets fig numerical solution transport equation given exactly output get resnets point training set already labeled value want match output value given train parameters velocity filed terminal value based analysis see training process resnet formulated control problem transport equation denotes training set labeled value sample function may scalar vector value different applications far formulate resnet control problem transport equation model inspire get new models replacing different component control problem modified pde model pde point view resnet consists five components pde transport equation numerical method euler velocity filed model terminal value softmax initial value label training set five components listed last one initial value given data choice four components consider replace options recently many works replacing forward euler ode solver forward euler simplest ode solver replacing solver may get different network sense densenet forward euler replaced scheme many numerical schemes solve ode numerical scheme may complicated used dnn constructing good ode solver dnn interesting problem worth exploit future work terminal value control problem point view softmax output function good choice terminal vaule since may far real value semisupevised learning ssl seems provides good way get terminal value instead softmax function shown fig dnn dnn dnn softmax ssl wnll output output output figure standard dnn dnn learning dnn last layer replaced wnll recently try use weighted nonlocal laplacian replace softmax results pretty encouraging velocity field model transport equation need model high dimensional velocity filed general difficult compute high dimensional vector field application associated images successes cnn resnet proved velocity filed model based convolutional operators effective powerful however way model velocity field moreover applications convolutional operator makes sense propose alternative velocity model propose model based equation reduce degree freedom velocity field notice even though velocity high dimensional vector field tranport equation component along useful based observation one idea model component along introducing transport equation becomes equation get control problem equation velocity field vector field meanwhile model need model scalar function number parameters reduced tremendously model scalar function much easier model high dimensional vector field already many ways approximate simplest way model linear function respect although linear function whole model linear model since equation nonlinear equation neural network consider effective way approximate scalar function high dimensioanl space one option simple mlp one hidden layer shown fig weight weight relu figure mlp model one hidden layer many neural networks literature approximate scalar function high dimensioanl space radial basis function another way approximate radial functions centered sample point used basis function approximate coefficients basis function one often used basis function gaussian function exp another difficulty solving model efficient numerical solver equation high dimensional space equation nonlinear equation difficult solve linear transport equation recently fast solver equation high dimensional space attracts lots attentions many efficient methods developed pdes point cloud another thing consider change characteristic method solving transport equation many numerical method transport equation based eulerian grid however methods need discretize whole space eulerian grid impossible high high dimensional space characteristic method seems practical numerical method solve transport equation hand need solve transport equation dataset instead whole space usually assume data set sample low dimensional manifold pde confined manifold manifold sampled point cloud consists data set including training set test set numerical methods point cloud used solve pde transport equation manifold transport equation model rewritten follows denotes gradient manifold let local parametrization differentiable function let define defined gij first fundamental form gij defined manifold sampled data set collection unstructured high dimensional points unlike classical numerical methods solve pde regular grids meshes case need discretize pde unstructured high dimensional point cloud handle kind problems recently point integral method pim developed solve pde point cloud pim gradient point cloud approximate integral formula kernel function assumed function compact support corresponding discretization volume weight depends distribution point cloud manifold equation also confine equation manifold point cloud one possible choice discretize modeled way discussed previous section pdes dissipation also consider add dissipation stabilize pdes model dissipation need add constraints subset training set otherwise solution smooth fit data due viscosity choice issue simplest way choose random sense viscous term maintains regularity solution convection term used fit data point cloud operator along constraints discretized weighted nonlocal laplacian discussion paper establish connection deep residual network resnet transport equation resnet formulated solving control problem transport equation along characteristics based observation propose several pde models manifold sampled data set consider control problem transport equation equation viscous equation preliminary discussion relation deep learning pdes many important issues remaining unresolved including model velocity field numerical solver control problem first step exploring relation deep learning control problems pdes references chow darbon osher yin algorithm overcoming curse dimensionality certain equations projections differential games ucla chow darbon osher yin algorithm overcoming curse dimensionality equations arising optimal control differential games problems ucla chow darbon osher yin algorithm overcoming curse dimensionality equations ucla darbon osher algorithms overcoming curse dimensionality certain equations arising control theory elsewhere ucla camreport darbon osher splitting enables overcoming curse dimensionality ucla han jentzen overcoming curse dimensionality solving highdimensional partial differential equations using deep learning july zhang ren sun deep residual learning image recognition ieee conference computer vision pattern recognition cvpr pages zhang ren sun identity mappings deep residual networks huang liu maaten weinberger densely connected convolutional networks ieee conference computer vision pattern recognition cvpr pages shi sun point integral method elliptic equations variable coefficients point cloud shi osher zhu weighted nonlocal laplacian interpolation sparse data journal scientific computing wang luo zhu shi osher deep neural networks data dependent implicit activation function preparation
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face synthesis video face recognition single sample per person jan fania mokhayeri student member ieee eric granger member ieee bilodeau member ieee video surveillance face recognition systems employed detect individuals interest appearing distributed network cameras performance systems decline significantly faces captured unconstrained operational domain multiple video cameras different underlying data distribution compared faces captured controlled conditions enrollment domain still camera particularly true individuals enrolled system using single reference still improve robustness systems possible augment reference set generating synthetic faces based original still however without knowledge many synthetic images must generated account possible capture conditions systems may therefore require complex implementations yield lower accuracy training many less relevant images paper introduces algorithm face synthesis dsfs exploits representative variation information available prior operation camera calibration compact set faces unknown persons appearing selected affinity propagation clustering captured condition space defined pose illumination estimation variations face images projected onto reference still individual integrating face relighting technique inside morphable model framework compact set synthetic faces generated resemble individuals interest capture conditions relevant particular implementation based sparse representation classification synthetic faces generated dsfs employed form dictionary accounts structured sparsity dictionary blocks combine original synthetic faces individual experimental results obtained videos chokepoint datasets reveal augmenting reference gallery set systems using proposed dsfs approach provide significantly higher level accuracy compared approaches moderate increase computational complexity index recognition single sample per person face synthesis face reconstruction illumination transferring sparse classification video surveillance ntroduction face recognition important function several video surveillance applications particularly screening given one reference still images target individual interest systems fania mokhayeri eric granger laboratory imagery vision artificial intelligence livia technologie canada fmokhayeri bilodeau image video processing laboratory litiv canada gabilodeau seeks accurately detect presence videos captured multiple distributed surveillance cameras despite recent progress computer vision machine learning designing robust system remains challenging problem surveillance applications one key issue visual domain shift faces enrollment domain reference still images typically captured controlled conditions operational domain video frames captured uncontrolled conditions variations pose illumination blurriness etc appearance faces captured videos corresponds multiple data distributions differ considerably faces captured enrollment another key issue limited number reference stills available per target individual design facial models although still faces cohort persons trajectories video frames unknown individuals typically available many surveillance applications screening single reference still per person available design corresponds single sample per person sspp problem performance systems decline significantly due limited information available represent variations seen video frames many discriminant subspaces manifold learning algorithms directly employed sspp problem also difficult apply methods sparse classification src different techniques sspp problems proposed improve robustness systems using multiple face representations face frontalization generating synthetic faces original reference stills incorporating generic auxiliary paper focus methods augmenting reference gallery set either synthetic set generated based original reference still variation information transferred generic set challenge strategies augmenting reference gallery set selecting sufficient number synthetic generic faces cover intraclass variations many synthetic faces generic auxiliary faces may generated collected respectively account possible capture conditions case systems would therefore require complex implementations generic set defined auxiliary set contains multiple video frames per person unknown individuals provides abundance information variations capture conditions characterizing capture condition unlabeled rois video trajectories operation domain pose illumination representative reference faces labeled reference still rois enrollment domain generating synthetic faces based knowledge extended reference set synthetic faces faces synthetic figure overview proposed dsfs algorithm augment reference gallery set assume gallery set initially contains one reference still roi per individual interest may yield lower accuracy training many facial images provide less relevant information paper new approach proposed exploits discriminant information generic set face synthesis process new algorithm called face synthesis dsfs maps representative variation information generic set original facial regions interest rois isolated reference stills way compact set synthetic faces generated represent still reference rois probe video rois common capture condition depicted fig dsfs technique involves two main steps characterizing capture condition information generating synthetic face rois based information obtained first step prior operation camera calibration process generic set facial rois collected video captured compact representative subset rois selected clustering generic set capture condition space defined pose illumination blur contrast estimation measures model reference still roi reconstructed via morphable model rendered based pose representatives finally layers lighting representatives extracted projected rendered reference rois pose manner variations effectively transferred onto reference still rois particular implementation original synthetic rois employed design structural dictionary powerful variation representation ability src dictionary blocks represent variations computed either reference faces synthetic faces cooperation src proposed dsfs improves robustness src sspp scenario domain variations order validate performance proposed dsfs algorithm sspp src implementation evaluated compared two public face databases main advantage proposed approach ability provide compact set accurately represent original reference face relevant variations pose illumination motion blur corresponding capture condition instance context src implementations set prevent refines informative classes sparse coding process accordingly require traditional dictionary learning process rest paper organized follows section provides overview related works sspp section iii describes proposed face synthesizing algorithm section presents particular implementation dsfs system section experimental methodology dataset protocol performance metrics validation systems described experimental results presented finally section vii concludes paper discusses future research directions elated ork face ecognition ingle till several techniques proposed literature improve robustness systems designed using sspp categorized techniques multiple face representation generic learning generation synthetic faces overview techniques presented bellow multiple face representation one effective approach address sspp problem extract discriminant features face images bashbaghi proposed robust system based diverse face representations applied multiple appearanceinvariant feature extraction techniques patches isolated reference still images order produce multiple face representations generate pool diverse exemplarsvms pool provides robustness common nuisance factors encountered surveillance applications jiwen proposed discriminative analysis method learning discriminative features image patches technique patches individual considered form manifold sample per person projection matrix learned maximizing manifold margin different persons face image processed several posespecific deep convolutional neural network cnn models generate multiple features multiple face representation techniques however able compensate small variations consequently effective tackle variations practical applications extreme illumination pose expression variations generic learning early finding compensate visual domain shift systems employ generic set enrich diversity reference gallery set generic learning concept generic learning widely discussed many researchers proposed adaptive generic learning method utilized external data estimate scatter matrix individual applies information reference set recent years integration sparse representationbased classification src generic learning attracted significant attention deng added generic leaning src framework proposed extended src esrc provide additional information face datasets construct variation dictionary represent changes training probe images research deng proposed generic learning method projects generic samples data null space order reduce negative influence variation information yang introduced sparse variation dictionary learning svdl technique taking relationship reference set external generic set account obtained projection learning generic reference set variation information integrated reference set domain adaptation enhance facial models authors proposed robust auxiliary dictionary learning radl technique extracts representative information generic dataset via dictionary learning without assuming prior knowledge occlusion probe images zhu proposed local generic framework lgr sspp builds gallery dictionary extracting neighboring patches gallery dataset variation dictionary constructed using external generic training dataset predict variations authors proposed robust using system classifier trained reference face still versus many faces individuals captured videos system auxiliary set collected videos unknown people employed select discriminant feature sets ensemble fusion functions supervised autoencoder network proposed system generate canonical face representations unknown video rois robust appearance variations commonly found operational video scene despite significant improvements reported generic learning several critical issues remain addressed generic variation may similar gallery individuals extraction discriminative information generic set may guaranteed moreover large number images collected external data may contain redundant information could lead complex implementations degrade capability covering variations synthetic face generation augmenting reference gallery set synthetically another strategy compensate appearance variations sspp shao presented algorithm extends dictionary using set synthetic faces generated calculating image difference pair faces authors augmented reference gallery set generating set synthetic face images cameraspecific lighting conditions design robust system surveillance conditions blanz vetter proposed morphable model reconstruct face single face image accordingly synthesize new face images authors employed cnn regress shape texture parameters directly input image without optimization process renders face compares image zhang proposed spherical harmonic basis morphable model shbmm integration spherical harmonics framework richardson presented face reconstruction technique single image introducing cnn framework derives shape fashion deep convolutional autoencoder face reconstructing single proposed convolutional encoder network combined expertdesigned generative model serves decoder apart reconstruction techniques techniques generate synthetic images various illumination conditions transferring illumination target images reference face images authors proposed imagebased framework relight face adaptive layer decomposition although synthetic images improve robustness systems designed sspp may covering range variations practical scenarios redundancy learned discriminative subspace since highly correlated original face images many synthetic images generated account possible capture conditions ods without selection representative face images reference gallery external data generating synthetic faces may require complex implementations yield lower accuracy training many less relevant images overcome challenges discussed paper presents framework exploits face synthesis generic learning technique proposed section iii generates compact set synthetic facial images per individual interest corresponds relevant capture conditions mapping variations representative set video rois selected original reference still rois iii omain pecific face ynthesis paper focuses augmenting reference face set cover variations individual appearing ods compact set synthetic rois new face synthesis dsfs technique proposed employs knowledge generate compact set synthetic rois design systems prior operation camera calibration process dsfs selects video rois representative pose angles illumination conditions facial trajectories unknown persons captured video rois selected via clustering facial trajectories captured condition space defined pose illumination conditions next dsfs exploits shape reconstruction method illumination transferring technique generate synthetic rois representative pose angles illumination conditions reference still rois models reference still rois reconstructed rendered representative pose angles layers representative illumination conditions extracted projected onto rendered images view applying morphing layers words layers video rois replaced still reference roi fig shows pipeline dsfs technique characterizing capture conditions important concern reference set augmentation selection representative pose angles illumination conditions represent relevant capture conditions mentioned adding large number potentially redundant images reference set significantly increase time memory complexity may degrade recognition performance due dsfs technique representative pose angles illumination conditions cover relevant variations approximated characterizing capture conditions large generic set video rois set formed multiple rois isolated several facial trajectories unknown persons captured let define set rois still reference individuals video rois generic set denotes number individuals reference gallery set total number rois generic set respectively proposed technique see fig estimation luminance contrast pose measured video rois generic set next clustering process applied video rois measurement space defined pose luminance contrast first step applied rois metric space defined pose tilt yaw roll second step applied rois pose cluster space defined luminance contrast metrics prototype cluster considered exemplar generic variational information obtained step transferred reference still rois face synthesizing step see section iii although many algorithms also suitable implement dsfs following subsections describe dsfs specific algorithms estimation head pose estimate head pose ith video roi generic set defined euler angles used represent pitch yaw roll rotation around axis axis axis global coordinate system respectively order estimate head pose discriminative response map fitting method employed current method terms fitting accuracy efficiency suitable handling occlusions changing illumination conditions distortion luminance contrast distortion measures estimate distortion video roi corresponding reference still roi components structural similarity index measure presented employed measure proximity average luminance contrast locally utilizing sliding window global luminance distortion image quality glq factor calculated sliding window pixels corner bottomright corner image total sliding steps sliding step denotes mean values image window constant value defined dynamic range pixel values similarly contrast distortion estimated using global contrast distortion image quality gcq factor defined denotes standard deviation image window constant value defined dynamic range pixel values representative selection affinity propagation applied cluster video rois generic set defined normalized space defined measures clustering algorithm aims maximize net similarity average distortion rois pose angles produce set exemplars two types messages responsibility availability exchanged data points highquality set exemplars corresponding clusters emerges suitable clustering technique dsfs automatically determine number clusters based data distribution produced exemplars correspond actual rois indeed cluster centroids produced many clustering methods necessarily actual rois interpretation given clustering samples simultaneously terms may favor certain common pose angles clustering algorithm proposed preserves diversity pose angles illumination effects first step clustering reference roi generic set decomposition characterization capture conditions iuij decomposition rendering lij texture mapping onto face shape morphing synthetic face images representative pose angles layers lighting representative samples shape recovery statistical deformable model trained shape vectors synthetic face images representative pose illumination synthetic set figure block diagram dsfs technique applied reference still roi estimation pose angles roll yaw clustering pitch generic set estimation luminance distortion pose clusters clustering inside pose clusters estimation contrast distortion pose lighting exemplars figure pipeline characterizing capture conditions video rois operational domain performed pose angle vector population pose cluster clustered according glq gcq metrics find representative luminance contrast samples representative luminance contrast samples called lighting exemplar found along representative pose angles called pose exemplar fig pitch yaw illumination roll ujnj contrast figure illustration clustering process clustering algorithm inputs set pose similarities indicating well sample index similar sample generic set pose responsibility defined accumulated evidence sample serve exemplar sample taking account potential exemplars sample evidence whether pose candidate exemplar would good exemplar obtained application pose availability availability reflects accumulated evidence appropriate would sample choose sample exemplar taking account support samples sample exemplar availabilities initialized zero pose responsibilities computed iteratively using rule availabilities updated iteration using max min max value maximizes either identifies sample exemplar identifies sample exemplar sample procedure terminated fixed number iterations local cost functions remain constant number iterations end pose clustering pose clusters determined second clustering applied pose cluster measure space find lighting exemplars first step computes similarities corresponding responsibility availability obtained according max min max estimated rcl acl combined itor exemplar decisions algorithm terminated decisions change several iterations end clustering pose cluster number lighting clusters ujnj obtained central representative samples clusters pose cluster considered pose lighting exemplars pose uij lji cij illumination contrast center ith cluster uji pose cluster larger clusters represent greater number generic samples influence classification therefore weight assigned exemplar uij indicate importance approximated based cluster size wij nij nij number samples cluster uij total number generic samples selection strengthens classes representative reconstructing probe sample shape vector defined principal components analysis performed estimate statistics shape faces xns yns zns base layer shading material texture components pixel respectively face reconstruction face model reference rois reconstructed using morphable model technique study customized version employed texture fitting original replaced image mapping replacing texture fitting original image mapping efficient method implemented face reconstruction one frontal face image basically shape model defined convex combination shape vectors set examples reference roi reconstruct shape face synthesis intrinsic image decomposition still reference image decomposed materialdependent layer albedo extracted based image model defined image decomposition method explicitly models separate texture layer addition shading layer material layer order avoid ambiguity caused textures employ simple constraints shading materials layers effectively model presented follows shape represented probability distribution faces around averages shape calculated basis vectors number basis vectors vector stores reconstructed shape terms vertices highresolution mesh generating synthetic rois based representative pose lighting conditions models reference rois reconstructed layers extracted rendering process extracted material layers employed texture model model rendered pose exemplars following layers lighting exemplars extracted finally lighting layers projected rendered images view applying morphing layers following subsections describe steps proposed face synthesizing dsfs shape parameters reconstructed shape ith reference still roi optimization algorithm presented employed find optimal reference still roi next step extracted material layers projected geometry reference gallery set given facial shape texture novel poses rendered various forms pose adjusting parameters camera model rendering procedure face projected onto image plane weak perspective projection linear approximation full perspective projection vij rij positions model vertexes reconstructed pose scale factor orthographic projection matrix rij rotation matrix constructed rotation angles pitch yaw roll translation vector since image directly mapped model corresponding color information available vertices occluded frontal face image consequently possible still blank areas generated texture map order correct blank space areas bilinear interpolation algorithm utilized fill areas unknown texture using known colors vicinity illumination transferring pose exemplar set samples ijk corresponding ujk selected lighting exemplars layer ijk extracted using process described section guided filter guidance input illumination layer applied preserve structure input face pose exemplar illumination layers ljk projected rendered reference vij performed morphing filtered version according following steps detect landmark points ljk vij using active shape model locate corresponding feature points landmark points ljk vij denoted ljk vij respectively define triangular mesh ljk vij via delauij nay triangulation technique obtain djk iii coordinate transformations affine projections points warp triangle separately source destination using mesh warping technique moves triangular patches newly established location align two rois triangulated layers considering warped pixel locations omain invariant till ideo face ecognition dsfs several studies revealed limit robustness system number training images per class limited underlying distribution still reference probe video rois differ section particular implementation considered see fig assess impact using dsfs generate synthetic rois address limitations phase compact set synthetic rois dsfs reference still rois synthetic set reference set generic set reference still rois video rois synthetic rois face detection design dictionary video frames operational phase way number synthetic rois generated reference still roi therefore total number synthetic rois qtotal synthetic set rois ith reference still roi presented aiq ird aij concatenated synthetic roi ith reference still roi overall process dsfs face generation technique formalized algorithm probe video frame face detection probe video roi sparse coefficients compute reconstruction error output identity stream frames captured video cameras dictionary probe roi sparse coefficients associated rois figure block diagram proposed srcalgorithm dsfs algorithm generic set input reference set estimate pose angles calculate luminance contrast distortion measures clustering pose space obtain clustering illumination contrast space obtain uji end extract layer recover face model using map texture render pose obtain vij uji extract layers ljk apply guided filter ljk morphing filtered ljk vij obtain end end end output sets synthetic face rois representative pose illumination conditions aiq ird based system augmented dictionary constructed employing synthetic rois generated via dsfs technique classification performed via structured src approach since synthetic rois individual including synthetic poses illuminations etc form block dictionary src considered structured sparse recovery problem main steps proposed dictionary augmentation summarized follows step generation synthetic facial rois first step synthetic rois aiq generated reference gallery set using dsfs technique number synthetic rois class step augmentation dictionary synthetic rois generated dsfs technique added reference dictionary design dictionary let reference gallery dictionary concatenated result dictionary integrates original synthetic rois linear model set synthetic rois added class since synthetic rois added class total class label probe rois determined based block sparse reconstruction error follows number synthetic rois dictionary presented dictionary design work enables src perform recognition one reference still roi makes robust visual domain shift label arg min order solve src problem equation classical alternating direction method adm considered adm efficient algorithm global convergence step classification given probe video roi general src represents sparse linear combination codebook derives sparse coefficients solving minimization problem follows min order apply adm problem first define auxiliary variable transform equivalent problem since generated synthetic rois individual form block dictionary better classification arise representation probe roi produced minimum number blocks dictionary instead looking representation probe roi dictionary training data using structured src goal find representation probe roi uses minimum number blocks dictionary dictionary blocks block sparsity formulated terms mixed norm min min zgi finally multipliers updated standard way sequence generated adm algorithm initial point converges solution max minimizing respect obtained shrinkage formula follows dtc dtc finally weighted matrix obtained shows cluster weights multiplied term min kwi multipliers penalty parameters apply adm approach minimize augmented lagrangian problem respect alternately namely minimizing respect given following linear system zgi minx indicator function ith block sparse coefficient vector corresponding dictionary block since dictionary block corresponds specific class represents class index ranging well optimization problem seeks minimum number coefficient blocks reconstruct probe roi note optimization program since requires searching possible blocks checking whether span given relaxation problem obtained replacing norm solving min note problem two blocks variables objective function separable form since involves thus adm applicable augmented lagrangian problem form step validation practical systems important detect reject outlier invalid probe rois use sparsity concentration index sci criteria defined maxi sci total number classes probe set coefficient vector probe roi accepted valid sci otherwise rejected invalid threshold face recognition process dictionary augmentation formalized algorithm algorithm system input reference face models classes enlisted gallery generic set threshold probe roi generate synthetic rois class using dsfs method build cross domain dictionary adding synthetic rois reference gallery set solve problem using adm technique sci find class label else reject invalid output class label xperimental ethodology databases order validate proposed dsfs surveillance conditions extensive experiments conducted two publicly available datasets chokepoint datasets selected representative screening applications contain reference image per subject captured controlled condition still camera surveillance videos subject captured uncontrolled conditions surveillance cameras videos captured distributed network cameras covers range variations pose illumination scale dataset contains individuals still image video sequences per individual simulating video surveillance scenario video individual walk designed route changes illumination expression scale pose chokepoint dataset consists individuals walking trough portal individuals male female walking trough portal recording portal portal one month apart camera rig cameras placed door used simultaneously recording entry person four sessions total dataset consists video sequences face images appearance face captured variations terms illumination conditions pose misalignment experimental protocol chokepoint database individuals randomly chosen individuals individual includes frontal captured image prior experiment video data split parts rois extracted video sequences individuals selected random generic set represent capture conditions rois video sequences remaining individuals along video sequences already selected individuals employed testing order obtain representative results process repeated times different random selection generic set individuals average accuracy reported mean standard deviation runs individuals randomly considered individuals including captured image per individual corresponding lowquality video sequences along rois video sequences individuals employed testing rois extracted video sequences individuals selected random generic set represent capture conditions process replicated times different stills videos individuals average accuracy reported mean standard deviation runs enrollment generic set faces captured video trajectories across ods global modeling rois extracted using face detection algorithm face detection also applied still images prior face synthesis clustering applied generic set representative video rois selected various pose illumination contrast conditions weight assigned exemplar according cluster size synthetic face images generated individual based information obtained selected exemplars recall clustering seeks exemplars samples representative clusters automatically determines number clusters independent replication dictionary designed using reference still synthetic rois operational phase recognition performed coding probe image dictionary regarding weights obtained enrollment domain throughout experiments sparsity parameter fixed reference system based svms also evaluated enrollment svm classifier rbf kernel trained individual using target rois reference still individual plus related synthetically face images versus rois reference still cohort persons plus synthetic face images performance measures assess ability face synthesizing techniques address shifts domain domain shift quantification dsq measure employed measure similarity dictionary designed using synthetic rois compared dictionary formed images collected measuring mean pixel error dictionaries given two dictionaries number images dsq measure defined esults iscussion section first presents examples synthetic faces generated using dsfs technique compares synthetic faces generated using face synthesizing methods shbmm performance systems based svms src presented using synthetic facial rois system design performance assessed increasing number synthetic rois per individual according pose angles lighting effects characterize impact performance systems tested growing number synthetic rois generic training set compared several relevant systems esrc radl svdl lgr face frontalization final experiment compares performance system designed synthetic rois obtained dsfs system designed growing number synthetic rois face synthesis subsection presents examples pose lighting exemplars obtained clustering facial trajectories captured condition space figs show example pose clusters obtained chokepoint video trajectories individuals video trajectories individuals experiment pose clusters exemplars typically determined chokepoint videos respectively second level clustering applied illumination contrast measure space pose clusters figs show exemplars selected based pose lighting proposed representative selection dsfs see section overall exemplars typically selected based pose lighting clusters determined chokepoint videos respectively figs show examples synthetic rois generated different pose illumination contrast conditions pose cluster pose cluster illumination illumination qdsq kdtr higher value indicates less domain shift accuracy system assessed per individual interest transaction level using receiver operating characteristic roc space true positive rates tprs plotted function false positive rates fprs threshold values tpr proportion target rois correctly classified individuals interest total number target rois sequence fpr proportion rois incorrectly classified individuals interest total number rois area roc curve global scalar measure accuracy interpreted probability correct classification range tpr fpr accordingly accuracy systems estimated using partial area roc curve pauc using auc since number target data imbalanced area curves aupr also used estimate performance systems contrast contrast figure examples representative selection results using clustering technique terms pose angles luminance contrast chokepoint dataset video sequences individuals dataset video sequences individuals respectively center clusters show exemplars using dsfs face synthesizing technique chokepoint datasets respectively basel face model bfm used generative shape model appendix compares quality synthetic face generated via dsfs techniques synthetic rois generated using dsfs also evaluated quantitatively table shows dsq values dsfs face synthesizing methods including shbmm chokepoint datasets higher dsq values indicate smaller domain shift potentially higher recognition rate corresponding two domains results provided two following scenarios frontal view first experiment individuals randomly selected set synthetic face generated frontal view various lighting effects still roi individual design corresponding video rois frontal view collected form finally dsq measured dictionaries profile view second experiment individuals randomly selected synthetic rois generated profile view different illumination conditions form corresponding video rois profile view collected construct finally dsq estimated dictionaries shown table dsq values dsfs method higher followed closely shbmm scenarios accordingly dictionary designed synthetic rois generated via dsfs method suitable reduce visual domain shifts potentially achieve higher level accuracy results line recognition performance results figure examples synthetic rois generated different capture conditions using dsfs technique chokepoint datasets table average dsq value frontal profile views chokepoint datasets svm src dsfq database frontal viewprofile view frontal viewprofile view proposed dsfs shbmm aupr chokepoint database pauc technique baseline baseline number synthetic pose rois per individual number synthetic pose rois per individual face recognition aupr pauc subsection performance achieved using system based src dsfs see section assessed experimentally reference system based svms also evaluated pose variations system evaluated versus number synthetic rois incorporate growing facial pose figs show average auc aupr obtained increasing number synthetic rois generated using dsfs representative pose angles fixed lighting condition results indicate adding extra synthetic rois generated representative pose angles allows outperform baseline systems designed original reference still roi alone auc aupr accuracy increases typically synthetic pose rois chokepoint datasets respectively mixed pose illumination variations performance systems assessed versus number synthetic rois generated pose lighting effects figs show average auc aupr obtained increasing number synthetic rois used design src svm classifiers chokepoint databases set synthetic rois generated using dsfs technique various pose illumination conditions adding synthetic rois generated various pose illumination contrast conditions allows significantly outperform baseline system designed original reference still roi alone auc aupr accuracy increases typically synthetic rois chokepoint datasets baseline number pose samples per individual baseline number pose samples per individual figure average auc aupr versus number synthetic rois generated dsfs according various pose fixed illumination system employs either svm src classifiers chokepoint databases respectively shown figs accuracy trends stabilize maximum value size generic set greater dsfs view performance stabilizing synthetic rois additional samples selected randomly among clusters note chokepoint dataset contains faces captured range illumination conditions various densities hence exemplars may represent many video rois method assigns higher weights distributions may yield higher level performance results obtained dsfs also compared systems exploit face synthesis techniques including randomly selected images shbmm spherical harmonic basis images shown fig dsfs always outperforms techniques impact representative selection without prior knowledge synthetic faces generated according uniform distribution adding large number synthetic rois dictionary needed cover possible cases baseline baseline number synthetic pose illumination rois per individual number synthetic pose illumination rois per individual baseline baseline aupr pauc pauc possible cond dsfs capture cond dsfs capture cond aupr pauc conditions third dictionary employs synthetic pose rois generated different pose illumination results fig suggest augmenting dictionary using representative synthetic rois dsfs technique yields higher level accuracy particularly pose illumination conditions aupr synthetic pose rois synthetic pose rois synthetic pose illuination rois synthetic pose illuination rois baseline baseline number synthetic pose illumination rois per individual number synthetic pose illumination rois per individual rois generated dsfs shbmm according pose lighting effects system employs either svm src classifiers chokepoint databases aupr pauc figure average auc aupr versus number synthetic baseline synthetic pose rois synthetic pose illuination rois significantly increase time memory complexity systems importantly may cause proposed dsfs technique extracts representative information produce compact set synthetic rois robust variations order evaluate impact generating synthetic rois based representative information pose lighting cluster instances three dictionaries designed src dictionary designed representative synthetic rois dsfs technique dictionary designed synthetic rois capture conditions dsfs without clustering dictionary designed possible conditions figs compares auc aupr system according different number synthetic rois terms pose illumination conditions first scenario evaluates impact representative selection terms pose three dictionaries designed src first dictionary typically employs representative synthetic pose rois generated dsfs technique chokepoint datasets respectively second dictionary employs synthetic pose rois generated dsfs technique capture conditions third dictionary employs synthetic pose rois generated set rotation angles ranging second scenario evaluates impact representative selection terms pose illumination conditions three dictionaries designed src first dictionary typically employs representative synthetic rois generated different pose illumination dsfs technique chokepoint datasets respectively second dictionary employs synthetic pose rois generated different pose illumination dsfs technique capture baseline synthetic pose rois synthetic pose illuination rois figure average auc aupr system designed representative synthetic rois system designed randomly generated synthetic rois chokepoint datasets comparison reference techniques experimental setting compare recognition rate dsfs technique recent face synthesizing methods shbmm framework also present impact using face synthesizing along ksvd dictionary learning following recognition rate dsfs technique existing generic learning techniques esrc radl svdl lgr compared regularization parameter set note performance face synthesizing techniques evaluated dictionary learning also compared dsfs results results obtained face frontalization method quality frontalization via face frontalization technique shown fig appendix table lists compares recognition performance results recognition rate illustrated mean standard deviation runs next compare computational complexity terms average running time individual well number inner products needed per iteration figs shows computational complexity terms number inner products growing number synthetic rois table iii compares complexity proposed algorithm radl lgr flowbased frontalization techniques chokepoint datasets per iteration experiments conducted matlab linux version workstation intel cpu ram table comparative transaction level analysis proposed approach related art methods chokepoint databases category pauc aupr pauc aupr baseline src esrc src radl src lgr src svdl src src src shbmm src face frontalization src face synthesizing generic learning proposed dsfs src face synthesizing lgr radl cost methods main reason dictionary designed dsfs technique able represent capture conditions require traditional dictionary learning process number inner products number inner products database classifier generic learning chokepoint database technique vii onclusions number synthetic rois gallery dictionary number synthetic rois gallery dictionary figure time complexity versus number synthetic rois chokepoint data table iii average computational complexity time dsfs techniques chokepoint datasets technique chokepoint database database productsrun time productsrun time radl lgr face proposed results show proposed method joint use generic learning face synthesizing achieves superior recognition results compared methods configuration verifies face synthesizing technique better preserves identity information although radl lgr face frontalization techniques achieve comparable accuracy approach computationally expensive concluded augmenting src synthetic rois generated dsfs technique good recognition rate less computational paper proposes face synthesizing dsfs technique improve performance systems surveillance videos captured various uncontrolled controlled conditions individuals recognized based single facial image proposed approach takes advantage operational domain information generic set effectively represent probe rois compact set synthetic faces generated resemble individuals interest capture conditions relevant operational domain validation augmented dictionary block structure designed based dsfs face classification performed within src framework experiments chokepoint datasets show augmenting reference discretionary systems using proposed dsfs approach provide significantly higher level accuracy compared approaches moderate increase computational complexity result indicated face synthesis alone effectively resolve challenges sspp visual domain shift problems dsfs generic learning face synthesis operate complementarity proposed dsfs technique could improved generate synthetic faces expression variations robust addition improve performance representative synthetic rois generated using dsfs could applied generate local rois dsfs general synthetic rois could applied train multitude face recognition systems like deep cnns information robust models specific operational domains eferences dewan granger marcialis sabourin roli adaptive appearance model tracking face recognition pattern recognition bashbaghi granger sabourin bilodeau dynamic ensembles face recognition pattern recognition wright yang ganesh sastry robust face recognition via sparse representation ieee transactions pattern analysis machine intelligence wanger wright ganesh zhou towards practical face recognition system robust registration illumination sparse representation iieee transactions pattern analysis machine intelligence hassner harel paz enbar effective face frontalization unconstrained images cvpr liu wassell new face recognition algorithm based dictionary learning single training sample per person bmvc masi tran hassner leksut medioni really need collect millions faces effective face recognition eccv deng guo extended src undersampled face recognition via intraclass variant dictionary ieee transactions pattern analysis machine intelligence deng zhou guo equidistant prototypes embedding single sample based face recognition generic learning incremental learning pattern recognition elhamifar vidal robust classification using structured sparse representation cvpr bashbaghi granger sabourin bilodeau robust watchlist screening using dynamic ensembles svms based multiple face representations machine vision applications tan wang discriminative multimanifold analysis face recognition single training sample per person ieee transactions pattern analysis machine intelligence abdalmageed rawls harel hassner masi choi lekust kim natarajan face recognition using deep representations wacv chen gao adaptive generic learning face recognition single sample per person cvpr gao yuille sparse representation based classification face recognition insufficient labeled samples ieee transactions image processing huang shang customized sparse representation model mixed norm undersampled face recognition ieee transactions information forensics security yang van gool zhang sparse variation dictionary learning face recognition single training sample per person iccv nourbakhsh granger fumera extended sparse classification framework domain adaptation video surveillance accv wei wang undersampled face recognition via robust auxiliary dictionary learning ieee transactions image processing zhu yang zhang lee local generic representation face recognition single sample per person accv bashbaghi granger sabourin bilodeau ensembles video face recognition single sample per person avss parchami bashbaghi granger sayed using deep autoencoders learn robust representations face recognition avss shao song feng zheng dynamic dictionary optimization face classification using local difference images information sciences mokhayeri granger bilodeau synthetic face generation various operational conditions video surveillance icip blanz vetter face recognition based fitting morphable model ieee transactions pattern analysis machine intelligence tran hassner masi medioni regressing robust discriminative morphable models deep neural network cvpr zhang samaras face recognition single training image arbitrary unknown lighting using spherical harmonics ieee transactions pattern analysis machine intelligence richardson sela kimmel learning detailed face reconstruction single image cvpr tewari kim garrido bernard theobalt mofa deep convolutional face autoencoder unsupervised monocular reconstruction arxiv preprint chen jin zhao face illumination manipulation using single reference image adaptive layer decomposition ieee transactions image processing asthana zafeiriou cheng pantic robust discriminative response map fitting constrained local models cvpr wang bovik sheikh simoncelli image quality assessment error visibility structural similarity ieee transactions image processing frey dueck clustering passing messages data points science jeon cho tong lee intrinsic image decomposition using separation surface normals eccv paysan knothe amberg romdhani vetter face model pose illumination invariant face recognition avss zhu lei liu shi face alignment across large poses solution cvpr jiang yan zhang zhang gao efficient reconstruction face recognition pattern recognition deng yin zhang group sparse optimization alternating direction method spie vol huang wang zhang lao kuerban chen benchmark comparative study face recognition cox face database ieee transactions image processing wong chen mau sanderson lovell probabilistic image quality assessment face selection improved face recognition cvprw viola jones robust face detection international journal computer vision qiu chellappa subspace interpolation via dictionary learning unsupervised domain adaptation cvpr aharon elad bruckstein algorithm designing overcomplete dictionaries sparse representation ieee transactions signal processing ppendix face synthesizing results examples synthetic face images generated via dsfs techniques individual chokepoint dataset fig shown face frontalization results fig shows example face frontalized facial probe roi individual chokepoint dataset via face frontalization technique iii iii iii figure synthetic face images generated via dsfs chockpoint data figure face frontalization via technique probe face image profile view frontalized face image
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differentially private ordinary least squares sheffet aug abstract linear regression one prevalent techniques machine learning however also common use linear regression explanatory capabilities rather label prediction ordinary least squares ols often used statistics establish correlation attribute gender label income presence potentially correlated features ols assumes particular model randomly generates data derives tvalues representing likelihood real value true correlation using ols release confidence interval interval reals likely contain true correlation interval intersect origin reject null hypothesis likely true correlation work aims achieving similar guarantees data differentially private estimators first show wellspread data gaussian transform jlt gives good approximation secondly jlt approximates ridge regression linear regression derive certain conditions confidence intervals using projected data lastly derive different conditions confidence intervals analyze gauss algorithm dwork introduction since early days differential privacy main goal design privacy preserving versions existing techniques data analysis therefore surprise several first differentially private algorithms machine learning algorithms special emphasis ubiquitous problem linear regression kasiviswanathan chaudhuri kifer computing science university alberta edmonton canada work done author harvard university supported nsf grant correspondence sheffet osheffet ily however existing body work differentially private linear regression measures utility bounding distance linear regressor found standard algorithm regressor found algorithm motivated perspective since bounds difference estimators translate error bounds prediction loss function bounds highly interesting yet little use situations one uses linear regression establish correlations rather predict labels statistics literature ordinary least squares ols technique uses linear regression order infer correlation variable outcome especially presence factors paper draw distinction linear regression refer machine learning technique finding specific estimator specific loss function ordinary least squares refer statistical inference done assuming specific model generating data uses linear regression many argue ols prevalent technique social sciences agresti finlay works make claim labels new unlabeled batch samples rather aim establish existence strong correlation label feature needless say works privacy individuals data concern order determine certain variable positively resp negatively correlated outcome ols assumes model outcome noisy version linear mapping variables denoting random gaussian noise predetermined ols unknown given many samples establishes two things fitting linear function best predict sample via computing yix coefficient positive resp negative inferring based true likely reside resp fact crux ols describing using probability distribution reals indicating likely fall derived computing values take account variance data well variance noise based probability distribution one example imagine run linear regression certain coordinates results vector differentially private ordinary least squares define interval interval centered whose likelihood contain particular importance notion rejecting interval contain origin one able say high confidence positive resp negative details regarding ols appear section work give first analysis statistical inference ols using differentially private estimators emphasize novelty work lie algorithms discuss next based transform jlt additive gaussian noise already known differentially private blocki dwork instead novelty work lies analyses algorithms proving output algorithms useful statistical inference algorithms first algorithm algorithm adaptation gaussian jlt proving adaptation remains private straightforward proof appears appendix described algorithm takes input parameter addition parameters problem indicates number rows later analyze one set value second algorithm taken algorithm outputting private projection matrix input matrix bound row privacy parameters parameter indicating number rows resulting matrix set sample lap let denote smallest singular value sample whose entries samples normal gaussian return matrix unaltered else let denote result appending sample whose entries samples normal gaussian matrix altered end verbatim work dwork yet column contains many column mostly populated zeros setting ols gives likely whereas guarantees given liberately focus algorithms approximate moment matrix data run output two reasons first enable sharing running unboundedly many since deal ols based private erm algorithms chaudhuri bassily inference requires use matrix loss function algorithms minimize private rather prove outputting minimizer perturbed private means ols based erm algorithms requires devise new versions algorithms making second step line work first understanding using existing algorithms leave approach well performing private hypothesis testing using algorithm dwork lei output merely reject decision without justification releasing relevant tests judging dwork future work contribution organization analyze performances algorithms matrix form coordinate generated according homoscedastic model gaussian noise classical model statistics assume existence vector every sampled study result running algorithm data two cases altered algorithm appended algorithm former case algorithm boils projecting data gaussian jlt sarlos already shown jlt useful linear regression yet work bounds difference estimated regression projection following sarlos work works statistics analyzed compressed researcher collects data uses approximation matrix test ols hypothesis approximation published researcher use test completely different hypothesis model may seem objectionable assumptions like noise independence sampled gaussian distribution called question past yet due prevalence model see fit initiate line work differentially private least squares ordinary model algorithm analyze gauss algorithm dwork input matrix bound row privacy parameters symmetric upper triangle entries sampled return differentially private ordinary least squares linear regression zhou pilanci wainwright however none works give confidence intervals based projected data presumably three reasons firstly works motivated computational speedups use fast jlt opposed analysis leverages fact composed gaussians secondly focus works ols rather newer versions linear regression lasso lies convex set lastly evident smallest confidence interval derived data since works consider privacy applications actually zhou pilanci wainwright consider privacy applications jlt quite different differential privacy assume analyst access data need give confidence intervals projected data analysis therefore first best knowledge derive therefore achieve rich expressivity one infers tvalues confidence bounds rejection ols estimations without access also show certain conditions sample complexity correctly rejecting increases certain bound without privacy bound privacy denotes condition number matrix appears section section analyze case algorithm append data jlt applied case solving linear regression problem projected approximates solution ridge regression tikhonov hoerl kennard aim regression solve minz means penalize vectors whose large general known derive ridge regression literature deriving confidence intervals solely ridge regression virtually indeed prior work need calculations access data general freely given deriving confidence intervals could done appealing back ols unable derive approximated general case additional assumptions data admittedly depend part verified solely data show solving linear regression problem allows give confidence intervals thus correctly determining correlation sign section discuss analyze gauss algorithm dwork outputs noisy version covariance given matrix using additive noise rather multiplicative noise empirical work shows analyze gauss output might input small singular values results truly bad regressors nonetheless additional conditions imply output psd derive confidence bounds dwork analyze gauss algorithm finally section experiment heuristic computing directly outputs algorithms show algorithm conservative algorithm sense tends reject number examples large enough give strong indication rejection contrast algorithm may wrongly rejects even true discussion works already looked intersection differentially privacy statistics dwork lei smith chaudhuri hsu duchi dwork especially focusing robust statistics rate convergence handful works studied significance power hypotheses testing differential privacy without arguing noise introduced differential privacy vanishes asymptotically slavkovic uhler wang rogers works experimentally promising yet focus different statistical tests mostly independence testing able prove results case simple single hypothesis efficient procedure repeated simulations cumbersome time consuming approach contrast deal composite hypothesis simultaneously reject sign sign altering confidence interval critical region one potential reason avoiding analysis differentially private hypotheses testing involve existing results typically statistical inference sole source randomness lies underlying model data generation whereas estimators deterministic function dataset contrast differentially private estimators inherently random computation statistical inference considers randomness data randomness computation highly uncommon work best knowledge first deal randomness ols hypothesis testing therefore strive analysis separate two sources randomness classic hypothesis testing use denote bound bad event depends solely homoscedastic model use bound bad event depends randomized thus result originally form converted result null hypothesis randomness generating feature matrix standard ols theory assumes fixed see theorems differentially private ordinary least squares preliminaries ols background taking quantity notation throughout paper use letters denote scalars bold characters denote vectors letters denote matrices zero vector denoted zeros denoted use model denote specific vector though reader may find bit confusing hopefully clear context also use denote elements natural basis unit length vector direction coordinate use denote privacy parameters algorithms use denote confidence parameters referring bad events hold resp based homoscedastic model randomized algorithm resp also stick notation algorithm use denote tive scalar throughout paper use standard notation svd composition matrix singular values inverse gaussian distribution univariate gaussian denotes gaussian distribution whose mean variance standard concentration bounds gaussians give multivariate gaussian positive denotes multivariate gaussian distribution mean coordinate covariance coordinates pdf gaussian defined subspace colspan matrix gaussian distribution denoted mean independence among rows variance columns also require following property gaussian random variables let two random gaussians see proposition additional distributions denote lap laplace distribution whose mean variance referred degrees freedom distribution distribution squared sum independent normal gaussians given def holds existing tail bounds distribution laurent massart give distribution referred degrees freedom distribution denotes distribution reals created independently sampling known fact thus common practice apply gaussian tail bounds sufficiently large differential privacy work deal input form row bounded two inputs called neighbors differ single row definition dwork algorithm alg maps range differential privacy holds alg alg neighboring inputs subsets background ols unfamiliar reader give brief overview main points ols details explanations proofs appear section given observations assume existence label derived independently also known homoscedastic gaussian model use matrix notation denotes feature matrix denotes labels assume full rank parameters model therefore set discover end minimize minz kyy xzz coordinate def quantity distributed according satisfying measurable thus describes likelihood give estimation likely known value particular given denote number interval contains probability mass derive corresponding confidence interval centered confidence level def particular importance quantity since correlation likelihood seeing depends ratio magnitude standard deviation mentioned earlier since rather viewing differentially private ordinary least squares sampled common think sample normal gaussian allows associate estimating event different specifically given null hypothesis let number means null hypothesis lower bound number sample points needed order null hypothesis bound basis comparison standard ols differentially private theorem fix positive definite matrix fix parameters coordinate let matrix whose rows samples sampled vector fix ols interval length provided sufficiently large constant furthermore exists constant ols correctly rejects null hypothesis provided max number ols projected data section deal output algorithm special case algorithm outputs matrix unaltered work clarify setting follows denote concatenation vector clearly denote column subset remaining columns matrix therefore denote output ryy simplicity denote denote svd decomposition orthonormal basis columnspan orthonormal basis finally work examine linear regression problem derived projected data denote ryy ree give main theorem estimating based theorem also illustrates separate two sources privacy case bounds probability bad events depend sampling rows bounds probability bad event depends sampling coordinates theorem let parameters generate vector coordinate sampled indepeny dently assume algorithm projects matrix without altering fix fix coordinate equations deriving pivot quantity distribution satisfying pdfd denote implications theorem immediate estimations one based true data based modulo approximation factor exp particular theorem enables deduce corresponding confidence interval based corollary setting theorem following fix let denote number interval contains probability mass compare confidence interval corollary confidence intervalqof standard ols model jlwhose length matrix known results regarding transform give therefore values get confidence interval theorem factor standard ols confidence interval observe common case dominating factor bound intuitively makes sense contracted observations observations hence model based confidence intervals derived rather supplementary material give discussion compare work bounds one gets plugging sarlos work also compare bounds derived alternative works differentially private linear regression moreover interval essentially optimal note interval contains probability mass qof differentially private ordinary least squares rejecting null hypothesis due theorem mimic ols technique rejecting null pothesis denote ject indeed associated ing slightly truncated much like theorem establish lower bound end correctly rejecting theorem fix positive definite matrix fix parameters coordinate let matrix whose rows sampled let vector sampled fix exist constants run algorithm parameter correctly reject null hypothesis using algorithm returns matrix unaltered estimate verify thatnindeed provided max max min defined min discussion culln minates following corollary thus conclude result theorem min holds setting min corollary denoting interesting note know also bound recall variance since every sample independent gaussian draw upper bound log lower bound using denote number condition max observe overall result similar nature many results differentially private learning bassily form without privacy order achieve total loss sample complexity bound differential privacy sample complexity increases however subtlety worth noting proportional additional min dependence follows fact differential privacy adds noise proportional upper bound norm row setting value deriving bound comparing lower bound given theorem bound bound hold rather yet theorem also introduces additional dency require since otherwise algorithm might alter projecting definition proportional precisely focus discussion subsection would like set value high possible larger observations better confidence bounds depend satisfying min recall sample point drawn sampled sample defined proof theorem rem gives lower bound following lower bounds means projected ridge regression turn deal case matrix pass algorithm case matrix appended dea noting algorithm output similarly going denote decompose standard assumption sampled need introduce additional notation denote appended matrix vectors using output algorithm solve linear regression problem derived ryy set ryy ryy sarlos results regarding johnson lindenstrauss transform give sufficiently many rows solving latter optimization problem gives good approximation solution optimization problem kyy arg minz kyy xzz kzz latter problem differentially private ordinary least squares known ridge regression problem invented tikhonov hoerl kennard ridge regression often motivated perspective penalizing linear vectors whose coefficients large also often applied case full rank close one show minimizer unique solution ridge regression problem rhs always solution ridge regression problem might smaller risk ols solution known derive reject null hypothesis ridge regression except using manipulate relying ols back fact prior work need analysis confidence intervals one could use standard ols access given therefore much reason unable derive projected ridge clearly situations confidence bounds simply additional assumptions data work give confidence intervals case interval intersect origin assure sign sign detailed supplementary material give overview analysis first discuss fixed data fixed model algorithm sole source randomness prove model approximation summary section give conditions length interval dominated factor derived theorem confidence intervals analyze gauss section analyze analyze gauss algorithm dwork algorithm works adding random gaussian noise noise symmetric coordinate diagonal sampled log using tation denote output algorithm denotes number contains mass however goal remains argue serves good approximation end combine standard ols confidence interval says randomness picking homoscedastic model approximate thus resp argue possible use get confidence interval certain conditions theorem fix assume exists homoscedastic model given assume also holds theorem fix define let ryy denote result applying algorithm matrix algorithm appends data matrix fix coordinate computing holds upper bound details appear supplementary material note assumptions fairly benign assume row bounded key assumption yet model row sampled assumption merely means large enough namely experiment output confidence interval theorem ing note approach using ryy interpolate ryy apply theorem using estimations ryy ignores noise added appending matrix therefore leads inaccurate estimations goal set experiment outputs algorithms theorem guarantees computing output algorithm matrix unaltered case give good approximation wondering computing directly output get good approximation true get conclusion rejecting nullhypothesis answers ever mixed two main differentially private ordinary least squares observations notice algorithms improve number examples increases algorithm conservative algorithm setting tested algorithms two settings first synthetic data much like setting theorems generated using independent normal gaussian features generated using homoscedastic model chose first coordinate twice big second opposite sign moreover independent feature variance label also set variance homosedastic noise equals number observations ranges second setting data ran two algorithms diabetes dataset collected ten years taken uci repository strack truncated data attributes sex binary age buckets years number medications numeric diagnosis numeric naturally added column intercept omitting entry missing values nine attributes left entries shuffled fed algorithm varying sizes running ols entire observation yields get fairly high magnitude result falsely reject based analyze gauss quite often even large values shown figure additional figures including plotting distribution approximations appear supplementary material results show approximations take account inherent randomness dpalgorithms lead erroneous conclusions one approach would follow conservative approach advocate paper algorithm may allow get true approximation otherwise reject based confidence interval algorithm intersecting origin another approach leave future work replace new distribution one takes account randomness estimator well however open challenge since first works statistics see slavkovic dwork lei requires move hypothesis testing algorithms ran version algorithm uses finds largest use without altering input yet alter input approximates ridge regression ran algorithm verbatim set repeated algorithm times synthetic data coordinate results plot get algorithms decide reject based tvalue larger corresponds fairly conservative surprisingly increases become closer expected value tvalue analyze gauss close algorithm factor smaller detailed see corollary result false analyze gauss tends produce larger thus reject values algorithm still reject shown figure exacerbated real data setting actual least singular value fairly small comparison size however fairly surprising case rejected since synthetic case close case analyze gauss tvalue approximation fairly large variance still synthetic data coordinate figure correctly wrongly rejecting differentially private ordinary least squares acknowledgements bulk work done author postdoctoral fellow harvard university supported nsf grant also unpaid collaborator nsf grant author wishes wholeheartedly thank salil vadhan tremendous help shaping paper author would also like thank jelani nelson members privacy tools sharing research data project harvard university especially james honaker vito orazio vishesh karwa kobbi nissim gary king many helpful discussions suggestions well abhradeep thakurta clarifying similarity result lastly author thanks anonymous referees many helpful suggestions general reference ullman particular references agresti finlay statistical methods social sciences pearson hall bassily smith thakurta private empirical risk minimization efficient algorithms tight error bounds focs blocki blum datta sheffet transform preserves differential privacy focs chaudhuri kamalika hsu daniel convergence rates differentially private statistical estimation icml dwork cynthia weijie zhang private false discovery rate control corr hoerl kennard ridge regression biased estimation nonorthogonal problems technometrics kasiviswanathan lee nissim raskhodnikova smith learn privately focs kifer daniel smith adam thakurta abhradeep private convex optimization empirical risk minimization applications regression colt laurent massart adaptive estimation quadratic functional model selection annals statistics zarowski christopher lower bounds smallest eigenvalue hermitian positivedefinite matrix ieee transactions information theory muller keith stewart paul linear model theory univariate multivariate mixed models john wiley sons pilanci wainwright randomized sketches convex programs sharp guarantees isit pilanci mert wainwright martin iterative hessian sketch fast accurate solution approximation constrained corr chaudhuri kamalika monteleoni claire sarwate anand differentially private empirical risk minimization journal machine learning research rao radhakrishna linear statistical inference applications wiley duchi john jordan michael wainwright martin local privacy statistical minimax rates focs rogers ryan vadhan salil lim gaboardi marco differentially private hypothesis testing goodness fit independence testing icml dwork lei differential privacy robust statistics stoc dwork cynthia kenthapadi krishnaram mcsherry frank mironov ilya naor moni data privacy via distributed noise generation eurocrypt dwork cynthia mcsherry frank nissim kobbi smith adam calibrating noise sensitivity private data analysis tcc dwork cynthia talwar kunal thakurta abhradeep zhang analyze gauss optimal bounds privacy preserving principal component analysis stoc rudelson mark vershynin roman smallest singular value random rectangular matrix comm pure appl math improved approx algs large matrices via random projections focs sheffet private approximations matrix using existing techniques linear regression corr url http smith adam statistical estimation optimal convergence rates stoc differentially private ordinary least squares strack deshazo gennings olmo ventura cios clore impact measurement hospital readmission rates analysis clinical database patient records biomed research international pages tao topics random matrix theory american mathematical thakurta abhradeep smith adam differentially private feature selection via stability arguments robustness lasso colt tikhonov solution incorrectly formulated problems regularization method soviet math uhler caroline slavkovic aleksandra fienberg stephen data sharing association studies journal privacy confidentiality available http ullman private multiplicative weights beyond linear queries pods slavkovic differential privacy clinical trial data preliminary evaluations icdm wang yue lee jaewoo kifer daniel entially private hypothesis testing revisited differcorr kantarcioglu inan mixture gaussian models bayes error differential privacy codaspy acm zhou lafferty wasserman compressed regression nips differentially private ordinary least squares extended introductory discussion omitted preliminary details due space constraint details introductory parts sections omitted bring appendix especially recommend uninformed reader extended ols background provide appendix linear algebra given matrix denote svd orthonormal matrices diagonal matrix whose entries singular values use denote largest smallest singular value resp despite risk confusion stick standard notation using denote variance gaussian use denote singular value use denote inverse defined matrix proof privacy algorithm theorem algorithm private proof proof theorem based fact algorithm result composing differentially private algorithm dwork lei differentially private analysis transform sheffet specifically use theorem sheffet states given matrix whose singular values greater publishing differentially private matrix whose entries sampled normal gaussians since singular values greater specified algorithm outputting private rest proof boils showing private matrix whose smallest singular value smaller passes step facts hold knowing whether pass private output algorithm private hence basic composition gives overall bound differential privacy prove pair neighboring matrices differ row denoted applying weyl inequality hence adding lap private prove note standard laplace distribution therefore holds matrix passes algorithm must also note similar argument shows matrix passes algorithm gaussian distribution univariate gaussian denotes gaussian distribution whose mean variance pdf exp bounds standard concentration gaussians give multivariate gaussian positive denotes multivariate gaussian distribution mean coordinate coordinates pdf gaussian defined subspace colspan every colspan pdf rank det exp det multiplication singular values matrix gaussian distribution denoted mean variance rows variance columns full rank holds pdfn det det exp trace case use matrix gaussian distributions row matrix sample multivariate gaussian repeatedly use rules regarding linear operations gaussians holds holds holds particular viewed holds mcc mcc also require following proposition proposition given constant let two random gaussians follows pdfy pdfx cpdfcy corollary notation proposition set holds differentially private ordinary least squares haustive account ols refer interested reader rao muller stewart proof proof mere calculation given observations assume existence pdimensional vector label derived independently also known homoscedastic gaussian model use matrix notation denotes whose rows use denote vectors whose entry resp simplify discussion assume full rank pdfx pdfcy exp exp exp exp pdfx exp pdfy exp exp exp parameters model therefore set discover end minimize minz kyy xzz solve referred degrees freedom distribution denotes distribution reals created independently sampling taking quantity pdf given pdftk known fact increases becomes closer closer normal gaussian often used determine suitable bounds rate converges illustrate section existing tail boundsp form constant pdftk often cumbersome work therefore many cases practice common assume commonly use existing normal gaussians differential privacy facts known dwork alg outputs vector holds kalg alg adding laplace noise lap coordinate output alg satisfies privacy similarly showed neighboring holds kalg alg adding gaussian noise coordinate output alg satisfies privacy another standard result dwork gives composition output private algorithm output private algorithm results private algorithm detailed background ordinary least squares unfamiliar reader give short description model ols operates well confidence bounds one derives using ols means holds alternatively every coorn dinate holds hence get addition note vector since symmetric projection matrix therefore equivalent summing squares samples words quantity sampled degrees freedom sidetrack ols discussion give following next bounds claim shows claim following holds randomness model randomness log log proof since denoting svd decomposition denoting diagonal trix whose entries coordinate distributed like gaussian differentially private ordinary least squares non gaussians factor greater standard deviation holds since log trace trace bound proven immediate corollary using bound triangle bound follows tail bounds distribution detailed section returning ols important note depends solely independent one another note whereas depends spherically symmetric two projections independent one another independent result two calculations quantity def distributed like degrees freedom therefore compute exact probability estimation quantity measurable satisfying importance lies fact fully estimated observed data value makes pivotal quantity therefore given use describe likelihood give estimation likely enable perform multitude statistical inferences example say two hypotheses likely much likely hypothesis true hypothesis true compare two coordinates report confident even compare among get across multiple datasets datasets get subsampling rows single dataset particular use unlikely values given denote number interval contains probability mass derive observe though spherically symmetric likely necessarily hold spherically symmetric therefore result using triangle bound bounding corresponding confidence interval centered confidence level comment actual meaning confidence interval analysis thus far applied vector derived according model result quantity distributed like distribution given model parameters random variable therefore guarantee derived according model def event happens however analysis done given dataset drawn views quantity given unknown therefore event either holds hold alternative terms likelihood confidence used instead probability confidence indeed event happen fraction datasets generated according model rejecting null hypothesis one important implication quantity refer specifically hypothesis called null hypothesis def quantity represents large relatively empirical estimation standard deviation since known number degrees freedom tends infinity distribution becomes normal gaussian common think sample normal gaussian allows associate estimating event different formally define common reject null hypothesis sufficiently small typically specifically given say null hypothesis let number standard bounds give means null hypothesis meaning lower bound number sample points needed order null hypothesis bound basis comparison standard ols differentially private indeed accurate associate value check value however uses take constant often asymptotically threshold get rejecting null hypothesis theorem far new except maybe differentially private ordinary least squares theorem theorem fix positive definite matrix fix parameters coordinate let matrix whose rows samples sampled vector fix interval length provided sufficiently large constant furthermore exists constant correctly reject null hypothesis provided max denotes number pdf content approximating normal gaussian one set proof discussion shows orderqto null hypothesis must therefore sufficient condition ols large enough therefore argue inequality indeed holds assume row vector recall according model straightforward concentration bounds gaussians give holds part standard ols analysis holds rudelson vershynin therefore due lower bound none hold case events implies confidence interval level length order suffices plugging lower bound see inequality holds comment sufficiently large constants cusing setting every row sampled multivariate gaussians stated way discussing solely power ols discussions sample size calculations see muller stewart holds constants hidden proof close within interval small given projecting data using gaussian transform main theorem restated discussion theorem theorem let matrix parameters coordiwe generate vector nate sampled independently assume sufficiently large singular values matrix greater large constant algorithm projects matrix without altering publishes ryy fix fix coordinate deriving follows ryy ree ree pivot quantity distribution satisfying pdfd denote comparison existing bounds sarlos work utilizes fact numbers rows large enough matrix specifically given denote arg minz krxzz ryy let denote setting sarlos work theorem guarantees log log using bounding section confidence interval approximate tail bound distribution tail bound gaussian use approximation differentially private ordinary least squares gives confidence interval level cenp log tered length log log implies confidence interval decreased degrees freedom roughly furthermore longer depends rather due fact rely gaussians mimicking carefully original proof deduce roughly degrees freedom depends solely worst case proportional uncommon matrices former much larger latter mentioned introduction alternative techniques chaudhuri bassily ullman finding estimator linear regression give bound bounds harder compare interval length given corollary indeed discuss section rejecting enough samples multivariate gaussian whose well conditioned give bound well worstupper bound yet possible techniques also much better data works sarlos alternative works regrading differentially private linear regression take account questions generating likelihood discuss rejecting null hypothesis proof theorem goal turn analysis show distribution specified theorem distribution assume existence fixed later vectors return homoscedastic model coordinate sampled words first examine case sole source randomness estimation based assumption fixed argue following claim model given output kpu kpu denotes projection operator onto subspace orthogonal colspan words independent proof matrix sampled given learn projection row onto subspace spanned columns denoting row row recall initially spherically symmetric gaussian result denote two projections independent samples resp however know learn exactly whereas get information still sampled gaussian know row therefore rpu rpu rpu rpu rely existing results linearity gaussians ree kpu ree kpu ree implies kpu claim based assumption fixed however given many different ways however assign vectors distributions get claim unique see recall equations therefore discuss section able better analyze explicit distributions estimators section able argue distributions far considered case fixed whereas goal argue case coordinate sampled end switch intermediate model sampled multivariate gaussian fixed arbitrary vector length formally let denote distribution fixed specific vector whose length denoted kpu claim assumptions claim given differentially private ordinary least squares ree proof recall rpu assumption sum two independent gaussians kpu rpu kpu summing two independent gaussians means furthermore variances gives distribution claim already established fixed kpu hence still easy verify chain derivations applicable corollary given coordinate proof corollary follows immediately fact definition distribution spherically symmetric gaussian defined subspace colspan dimension continue need following claim claim given given independent proof recall given specific vector depends projection distribution colspan projection row onto colspan distribution ree rpu rpu depends projection onto time fix specific vector length projection row onto since independent since row independent since chosen independently independent formally consider pair coordinates rpu rpu recall given therefore know cov eet rpu eet rpu rpu rpu rpu eet rpu kpu gaussians covariance implies independence independent gaussians established specified distributions continue proof theorem assume exists small due corollary denoting distributions holds specifically denote function observe sample independently distributed like fact possible use standard techniques differential privacy argue similar result probabilities event depends function close differential privacy sense differentially private ordinary least squares degrees freedom way sample deduce pdf function sample real value let denote suitable set values independently lies range using corollary claim distributions precisely stated pdf def distribution pdf point sandwiched next aim argue characterization pdf still holds would convenient think sample use equation since fixed precisely deduce sampled spherical gaussians reason still holds lies fact depend details pre pdf dvv pdfpu pdf kvv pdfpu dvv pdf words shown interval pdfpu dvv last transition possible precisely independent kvv precisely makes pivot quantity proof lower bound symmetric equality follows fact since interval analogously also show pdfpu dvv lower bounded upper bounded repeat argument using analogous definition conclude shown equation holds every interval pre lower bounded upper bounded conclude proof theorem need show equation exists claim homoscedastic model gaussian noise satisfy log theorem follows using plugging discussion differentially private ordinary least squares proof lower bound immediate therej fore prove upper bound parameter correctly null hypothesis using algorithm returns matrix unaltered estimate verify first observe kpu sampled dimension therefore holds indeed provided max assuming therefore secondly argue holds see first observe picking distribution product identical picking taking product therefore distribution identical denoting kvv claim sheffet gives max min denote done comment analysis proof claim implicitly assumes think projection dimensionality reduction ratio small however similar analysis holds comparable would argue small pdfn numbers resp proof first need use lower bound show indeed algorithm alter various quantities far expected values formally claim following proposition lower bounds theorem theorem holds also kpu proof proposition first need argue enough samples gap sufficiently large since also concatenation sampled gaussian clearly txi similarly therefore row sample proof theorem theorem theorem fix positive definite matrix fix parameters coordinate let matrix whose rows sampled let vector sampled fix exist constants run algorithm denote combining two inequalities get implies required denote argue large use lower bound zarowski theorem combining simple arithmetic manipulations deduce min differentially private ordinary least squares established lower bound follows draws min conditioned large enough randomness algorithm matrix pass output algorithm conditioned algorithm outputting due lower bound result theorem hold deduce result theorem holds since argue theorem holds following two bounds used also hold projected ridge regression section deal case matrix pass algorithm case matrix appended denoting algorithm output similarly going denote decompose sampled standard assumption need introduce additional notation denote appended matrix vectors meaning kpu lastly proof theorem argue given length distributed like kpu appealing fact holds given lower bound indeed holds plugging value kpu concludes proof proposition based proposition show indeed reject theorem holds reject iff def holds iff implies respectively denote vector denoted matrix hence reject note bound based determines null hypothesis holds due lower bound ryy based bounds stated sufficient condition rejecting nullhypothesis arg min kryy ryy using output algorithm solve linear regression problem derived set accurately bounds shown claim sarlos results regarding johnson lindenstrauss transform give sufficiently many rows solving latter optimization problem gives good approximation solution optimization problem arg minz kyy arg minz kyy xzz kzz possible denote single column subset remaining columns differentially private ordinary least squares latter problem known ridge regression problem invented tikhonov hoerl kennard ridge regression often motivated perspective penalizing linear vectors whose coefficients large also often applied case full rank close ridge regression problem always solvable one show minimizer unique solution ridge regression problem rhs always defined even singular original focus ridge regression penalizing large coefficients therefore ridge regression actually poses family linear regression problems minz xzz one may set scalar much literature ridge regression devoted art penalty term either empirically based var yields best risk propose fundamentally different approach choice normalization factor set solution regression problem would satisfy privacy projecting problem onto lower dimension solution ridge regression problem might smaller risk ols solution known derive reject null hypothesis ridge regression except using manipulate relying ols back fact prior work need analysis confidence intervals one could use standard ols access given therefore much reason unable derive projected ridge clearly situations confidence bounds simply derived consider example case draws obviously gives information nonetheless additional assumptions data work give confidence intervals case interval intersect origin assure sign sign clearly sarlos work gives upper bound however distance bound distance come coordinate coordinate confidence guarantee would like fact even clear though obvious sarlos work see ryy often show equal comment notation throughout section assume full rank one simply replace occurrence makes formulas general case running ols projected data section analyze projected ridge regression assumption fixed assume source randomness comes picking matrix analyze distribution see equation value function optimize denoting formally express estimators ryy ryy fixed given claim given kpu furthermore independent one another proof first write explicitly based projection matrices ryy ryy denoting colspan denoting projection onto subspace ridge regression opposed ols yield biased estimator note approach using ryy interpolate ryy apply theorem using estimations ryy ignores noise added appending matrix therefore bound produce inaccurate estimations break orthogonal composition colspan hence colspan therefore differentially private ordinary least squares whereas essentially kpu finally observe independent former depends projection spherical gaussian latter depends projection multivariate gaussian equality holds aim describe distribution given know since independent independent therefore given induced distribution remains similarly given rows remain independent one another row distributed like spherical gaussian colspan turn implies observe claim assumes given may seem somewhat strange since without assuming anything many combinations however always similarly always case recall ols definitions equation therefore distribution unique set given dataset serves approximation multiplying random matrix vector get immediate corollary claim fixed holds quantity multiplying random vector matrix get denotes vector direction similar rpu rpu kpu therefore distribution sum independent gaussians required also sum independent gaussians implies distribution tributed like therefore following theorem follows immediately theorem fix define let ryy denote result applying algorithm matrix algorithm appends data matrix fix coordinate computing equations holds denotes number contains mass note theorem much like rest discussion section builds fixed means serves approximation yet goal argue similarity proximity end combine standard ols confidence interval says randomness picking homoscedastic model confidence terval theorem deduce differentially private ordinary least squares randomness next section goal give conditions interval equation much larger comparison interval length get theorem importantly conditions make interval theorem useful large note tion large interval example situations likely inform sign motivating example good motivating example discussion following section strict submatrix dataset data contains many variables entry dimensionality entry large yet regression made modest subset variables case least singular value might small causing algorithm alter however could sufficiently large run algorithm would alter input indeed differentially private way finding subset variables induce submatrix high interesting open question partially answered single regression work thakurta smith thakurta smith indeed conditions specify following section depend data minimal variance data direction motivating example indeed variance necessarily small conditions deriving confidence interval ridge regression proof former bound follows known results transform shown proof claim latter bound follows standard concentration bounds plugging result proposition equation get difference also use following proposition proposition proof looking interval specified equation give upper bound random quantities interval first give bound dependent randomness continue view fixed proposition probability latter holds diagonalizable matrix matrix since observe randomness randomness technically give confidence interval around contains need use instead resp avoid overburdening reader already see many parameters switch asymptotic notation min clear deduce based proposition get equation differentially private ordinary least squares happens case exists small satisfy moreover case sign sign precisely claims claim exists proof based discussion enough argue conditions claim constraint equation holds since require evident show conditions claim show required left algebraic manipulations suffices holds assume hold claim fix iii homoscedastic model probability sign sign assume closer normal gaussian proof based discussion aim show homoscedastic model coordinate independently holds magnitude greater show invoke claim argue since since whereas also use fact deduce due requirement observe conditions specified claim condition merely guarantees sample large enough argue estimations close expect value condition merely guarantee condition iii hold especially together conditions claim pose constraints regards various parameters play interesting compare requirements lower bound get theorem especially latter bound two bounds strikingly similar exception also require greater part unfortunate effect altering matrix give confidence bounds coordinates small relative summary require compathat contains enough sample points set convenient rable think small constant say implies differentially private ordinary least squares interval around intersect origin comment conditions sufficient necessary furthermore even conditions holding make claims optimality confidence bound proposition onwards discussion uses upper bounds corresponding lower bounds best knowledge confidence intervals analyze gauss algorithm complete picture analyze analyze gauss algorithm dwork dwork algorithm works adding random gaussian noise noise symmetric nate diagonal sampled log using notation denote output algorithm symmetric vector scalar whose coordinates sampled using output algorithm simple derive equations logues upper bound defined def large constant comment practice instead using might better use mle namely def instead upper bound derived replacing unknown variable mle estimator common approach applied statistics note fairly benign assume tion simrow bounded assumption ply assumes reasonable estimation likely hold assume assumption magnitude least singular value therefore major one nonetheless case considered row sampled assumption merely means large enough order prove theorem require following proposition proposition fix fix matrix let vector coordinate sampled independently gaussian kmvv get argue possible use confidence interval certain conditions theorem fix assume exists homoscedastic model given assume also holds proof given mvv denoting singular values svp rotate mvv without affecting infer kmvv distributed like sum independent gaussians sampled standard union bound gives non gaussians exceeds standard deviation factor hence holds kmvv trace easy see sensitivity mapping fix differ one row vector kvv zero vector kvvv trace vvv proof also requires use following equality holds invertible matrix invertible differentially private ordinary least squares case bound variance appealing def proof theorem fix first apply standard results gaussian matrices tao used also dwork analysis see remainder proof fix subject bounded operator norm note fixing fix recall homoscedastic model coordinate sampled therefore gaussian tion bounds give plugging bounds terms appear equation denoting row deduce samto bound note pled independently therefore gaussian concentration bounds absolute value gaussian note due symmetry bound size term note sen independently since since bound trix thus terms appearing equation known except parameter given model next derive upper bound plug equation complete proof theorem derive confidence interval differentially private ordinary least squares kzz upper bound kzz term magnitude upper bounded recall equation according given assumption least singular value bound min psd therefore symmetric matrix strictly term positive applying proposition holds recall ntx plugging bounds equation get holds course chosen independently bound term equation first give bound kzk recall recall assumption given statement theop rem plies hence kzk moreover implies armed bounds operator norms bound magnitude different terms equation term exact term standard ols know distributed like distribution therefore greater scalar sampled bounded upper since assume term bounded kzk thus denote kzz magnitude indeed rhs definition statement theorem experiment additional figures complete discussion experiments conducted attach additional figures plotting approximations get algorithms decision whether correctly reject null hypothesis sign first show distribution approximation coordinates rejected figure decision whether reject based whether right conservative reject needed wrong rejected wrong sign rejected rejected figure one see algorithm far lower expected therefore much conservative fact tends coordinate even largest value figure however algorithm also much smaller variance also reject ought notreject whereas algorithm erroneiously rejects nullhypotheses seen figures differentially private ordinary least squares synthetic data coordinate synthetic data coordinate synthetic data coordinate synthetic data coordinate data coordinate figure distribution approximations selected experiments synthetic data null hypothesis rejected data coordinate figure correctness decision reject nullhypothesis based approximated null hypothesis rejected differentially private ordinary least squares synthetic data coordinate synthetic data coordinate data coordinate data coordinate figure distribution approximations selected experiments synthetic data null hypothesis essentially true figure correctness decision reject nullhypothesis based approximated null hypothesis essentially true
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nov semidefinite program structured blockmodels david choi november abstract semidefinite programs recently developed problem community detection may viewed special case stochastic blockmodel develop semidefinite program tailored instances blockmodel networks overlapping communities establish label recovery sparse settings conditions analogous recent results community detection settings data generated blockmodel give oracle inequality bounds excess risk relative best blockmodel approximation simulations presented community detection overlapping communities latent space models introduction stochastic blockmodel popular class models network data node assumed belong latent class various blockmodel exist community structure hierarchical community structure overlapping blockmodels well relatives latent space models mixed membership degreecorrected blockmodels blockmodels models estimation latent nodal classes active area research blockmodels spectral methods known yield asymptotically consistent estimates provided network sufficiently large dense special case community structure additionally known specialized methods achieve weakly consistent estimates even spectral methods fail completely due sparsity examples methods include semidefinite programming message passing variants blockmodel relatives estimation methods also exist less understood particular theory analogous community detection yet seem exist cases address gap show paper semidefinite programming applied community detection also blockmodel well specifically propose semidefinite program tailored specific instance blockmodel program prove estimation bounds analogous already known community detection including weak consistency bounded degree setting data generated blockmodel semidefinite program used construct version data matrix provide oracle inequality bounding error relative best blockmodel approximation organization paper follows section presents semidefinite program section presents convergence analysis sparse data section discusses numerical optimization section gives simulations results proofs contained appendix problem formulation section present generative models consider paper derive semidefinite relaxation combinatorial problem estimating latent nodal classes present estimators blockmodel settings preliminary notation given matrix let denote submatrix size similarly rnk let denote ith subvector length use mab denote entry submatrix likewise use denote ath entry subvector implies mab generative models stochastic blockmodel let denote symmetric adjacency matrix undirected network nodes stochastic blockmodel classes node random latent class upper triangular entries independent bernoulli conditioned discrete aij bernoulli aji aij probability distribution giving expected class frequencies symmetric matrix gives connection probabilities class type general model general model random matrix generated aij bernoulli pij aji aij symmetric satisfies pii seen stochastic blockmodel special case pij semidefinite program assume observed estimation task find maximizing generic combinatorial problem maxn fij choice objective functions fij paper let fij equal likelihood function fij aij log aij log case finds maximum likelihood assignment specified parameter matrix note may differ actual generative model optimizing computationally tractable relax semidefinite program let denote matrix submatrices given fab fij expressed maxn etzi ezj denote canonical basis rewritten max xxt subject ezn suggests following semidefinite program relaxation xxt approximated positive semidefinite matrix max xab denotes positive semidefinite denotes elementwise feasible submatrix nonnegative sums one viewed relaxed version indicator matrix ezi etzj encoding class pair matrix denoising blockmodel estimation let denote solution semidefinite program let generated general model generative matrix let denote map estimate constructed treating submatrix probability distribution arg max alternatively let denote randomized estimate dyad independent random variable distribution probability let denote cluster labels found spectral clustering applying first eigencoordinates generated blockmodel generative block structured blocks induced case use estimate permutation estimate permutation let denote matrix densities induced aij est discussion related work semidefinite programs used community detection well allowed outliers allowed degree heterogeneity works network required exhibit assortatitve block structure general model without restrictions estimation considered dense settings best knowledge networks sparse bounded degree presented sections considered previous work additionally semidefinite program presented bears resemblance one appearing lower bounding optimal objective value quadratic assignment problem without finding feasible solution also recent work estimating pairwise alignments beween objects convergence analysis section analyze solution semidefinite program matrix denoising label recovery analogous existing results community detection results imply weak consistency performance better random guessing regime average degree asymptotically bounded constant consistency well vanishing excess risk average degree organization section following section defines basic notation section states required conditions section presents convergence results proven appendix preliminaries following notation used given generated let denote expected density pij given let denote objective function semidefinite program submatrices given fab aij log aij log let denote idealized version replaced expectation submatrices given pij log pij log let denote feasible region semidefinite program xab used intermediate step algorithm let denote solution semidefinite program written maximize matrix let function given bcd bab identifies subsets equal values assumptions assumption apply used estimate general model assumption requires density exceeding assumption bounds entries differ roughly constant factor assumption let generated evolves satisfying assumption matrix may evolve satisfies assumptions apply generated stochastic blockmodel sufficient show converges true label switching assumption describes parametrization commonly used sparse blockmodels assumption places bounds misspecification sparse blockmodel setting assumption let generated stochastic blockmodel parameters let constant let satisfies constant rank satisfies bab aij assumption let fixed matrix satisfy bab log assumption states need identical values structure given additionally entry closest element bab terms bregman divergence associated poisson likelihood results theorem holds generated general model including gives oracle inequality quality randomized estimate given relative best blockmodel approximation generative theorem let assumptions hold let denote randomized estimate given min constant terms depending appears assumption theorem assumes sparse blockmodel setting assumptions shows randomized estimate asymptotically recover vanishing fraction incorrect values theorem let assumptions hold let denote estimate given result replaced given corollary follows theorem states eigencoordinates used compute converge unitary transformation block structured suggests converge permutation labels proven appendix direct application theorem corollary let assumptions hold let denote eigenvectors let denote eigenvectors largest eigenvalues absolute value let denote set orthogonal matrices holds min remark weak consistency bounded degree regime implied results estimation error bounded away performance random guessing provided theorem theorem constant numerical optimization semidefinite program solved alternating direction method multipliers admm semidefinite program much larger previously introduced community detection speedups often achieved exploiting problem structure resulting competitive computation times solve using admm introduce decision variable auxiliary variables initialized zero evolve according following rule max given positive stepsize parameter operator denotes projection onto set positive semidefinite matrices operator denotes projection onto affine subspace matrices satisfying linear constraints xab slowest step admm iterations computation requires eigendecomposition matrix comparison semidefinite programs community detection require decomposition matrix admm iteration however many settings interest highly structured fast eigendecomposition methods particular let submatrices symmetric share common set orthonormal eigenvectors holds partition vjt eigenvalue corresponding eigenvectors case computed following manner find eigendecomposition submatrix yielding eigenvectors sets partitioning eigenvalues satisfying let given vjt return holds matrices orthogonal positive semidefinite since steps require eigendecompositions matrices computation time eigendecomposition scales cubically matrix size resulting speedup quite significant practice example speedup factor roughly going hours minute solve semidefinite program decomposition hold sufficient condition span association scheme evidently fundamental concept combinatorics defined follows definition sec set symmetric matrices form association scheme following properties hold holds span key property association schemes following result theorem follows result states share common set eigenvectors lemma let form association scheme eigenvectors partition scalars holds vkt additionally vector one eigenvectors theorem proven appendix states parameter matrix span association scheme decomposition hold equal number matrices association scheme precomputable theorem let defined symmetric span form association scheme let initialized zero evolve admm equations holds satisfying section give three examples semidefinite programs shown belong association schemes respectively remark association schemes originally invented statisticians study experiment design used semidefinite programming lower bound optimal objective value traveling salesperson problem without finding feasible solution simulation results illustrate usage performance semidefinite program show simulated results three examples community structure overlapping communities latent space models community structure blockmodel setting generated parameters equals model nodes connect probability class probability different classes parameter matrix written manual verification seen satisfy requirements given definition association scheme fast methods used evaluate admm iterations figure shows adjacency matrix generated estimated class labels found spectral clustering discussed section estimates found direct spectral clustering instance semidefinite program yields nearly perfect class estimates spectral clustering fails due sparsity figure shows average simulated performance range values network size average degree see adj matrix spectral clusters missclassification rate figure community structured blockmodel nodes classes average degree unbalanced class sizes adjacency matrix similarity matrix resulted nodes shows similarity matrix class estimates found spectral clustering split largest community failed find smallest two communities resulted nodes sdp spectral number nodes figure misclassification rates found semidefinite programming blue line solid spectral clustering red line dotted community structured blockmodel expected average degree simulations per trial standard errors shown semidefinite program roughly constant error average degree fixed consistent theorem known results community detection contrast misclassification rate spectral clustering increases sparsity graph exemplified subplot spectral clustering performs well small networks degrades severely increases two overlapping groups communities let parameters satisfy let integer representation let equal otherwise model exist two types community structure type comprised communities node belongs one community type two nodes higher probability connection share least one community common observe given manual verification seen satisfy requirements given definition association scheme model may interest compute estimate label switching also estimate overlapping communities denote seen given zim representation equivalently may also defined follows let denote graph vertices edges induced thresholding gab otherwise seen maximal cliques communities satisfy permutation thus estimate construct given estimate est maximal cliques using maximal cliques estimate overlapping communities remark even settings model valid subsets still interpretable overlapping subsets densely connected nodes adjacency spectral matrix ping model clustering figure overlapping community model overlapping communities adjacency matrix similarity matrix similarity matrix similarity matrix assuming model similarity matrix using spectral clustering errors using model errors spectral clustering errors figure shows adjacency matrix generated pattern clearly visible figures show estimated communities using comparison figure shows semidefinite program assumes instead figure shows estimate spectral clustering instance accurately estimate true communities misclassification rate consistent theorem predicts nearly implying subsequent steps estimating succeed well contrast two alternative methods give poor estimates misclassification rates respectively figure shows average misclassification rate simulations range values network size average degree comparison misclassification rate spectral clustering estimates instead shown well misclassification rate semidefinite program quite constant fixed suggesting asymptotic results theorem may require larger model compared results shown figure however increases semidefinite program estimates overlapping communities hence well much better accuracy spectral clustering shows little improvement increasing latent space models consider latent space model reminiscent node assigned latent coordinate vector conditional dyad independent bernoulli log odds given log odds aij bandwidth parameter general equivalent blockmodel missclassification rate sdp spectral number nodes figure misclassification rates found semidefinite programming blue line solid estimates found spectral clustering red line dotted overlapping community model expected average degree unbalanced class sizes simulations per trial standard errors shown let denote matrix squared distances latent coordinates given dij kyi known first eigencoordinates recover unitary transformation estimate approximate blockmodel classes class represents coordinate equals logit distance metric order belong association scheme choose extending beyond form grid choose toric distance given let denote representation let given let equal distance torus circumference min jth element vector since depends differences follows written weighted sum matrices circulant matrix given min cab otherwise manual inspection seen satisfies requirements association scheme hence also association scheme well see chapters complete treatment let denote solution semidefinite program given estimate let equal estimate arg max let denote randomized estimate probability xab given take first eigencoordinates estimate similarly using corollary bounds error randomized estimate true distances relative rounding coordinates closest points grid corollary let denote loss function let form grid denotes representation min dij proof holds dij minn dij min dij dij min first inequality holds theorem definition second inequality holds minimization strictly smaller set previous line last inequality holds figure shows latent coordinates arranged circle adjacency matrix generated figure also shows estimated coordinates derived randomized given given applying usvt method adjacency matrix spectral estimate constructed inverted form estimate instance yield estimates similar substantially accurate usvt approach failed due sparsity figure shows different configuration similar results figure shows average simulated estimation accuracy using range values network size average degree comparison performance spectral usvt method shown well see estimation error average degree fixed contrast estimation error spectral usvt method worsens sparsity exemplified subplot usvt method performs well small networks degrades severely increases proof theorems intermediate results first present intermediate results used proof theorems let denote solution idealized problem maximize lemma main technical result states general matrix model nearly optimizes desired objective function even though noisy proxy available proof closely follows approach usvt figure latent space model latent coordinates arranged circle latent coordinates estimated using randomized estimated using estimated using usvt spectral method directly rms errors respectively usvt estimation error figure latent space model latent coordinates arranged cross latent coordinates estimated using randomized estimated using estimated using usvt spectral method directly rms errors respectively sdp usvt number nodes figure estimation error found semidefinite programing blue line solid usvt method red line dotted latent space model expected average degree latent coordinates arranged circle simulations per trial standard errors shown lemma let assumptions hold holds depend lemma gives condition approximately block structured roughly states exists arg maxab asymptotically block structure corresponding lemma let assumptions hold exists satisfy holds lemma states error randomized estimate converges expectation lemma let assumptions hold pij pij bernstein inequality states independent satisfying variance expectation exi holds exp grothendieck inequality states exists universal constant matrix max max rows unit ball lemmas proven section proof theorems proof theorem given let vector given etzn theorem holds following steps pij pij pij pij pij maxn pij minn holds lemma follow identity pij pij additionally using definition holds lemma holds implying proof theorem define bernstein inequality seen satisfies result assumptions seen hold probability let defined min bab log min seen assumption bound apply lemma observe pij log log log log pij log log log terms bounded uniformly implies holds implying conditions lemma hold probability lemma thus implies let eij given eij show apply follows eij last equality follows show holds place observe following bounds eij var var eij applying chebychev states var yields proof lemmas proof lemma let version submatrices given aij pij log alegebraic manipulation yields max max holds implying holds maximizes implying follows grothendieck inequality remains bound right hand side definition seen fab aij pij log aij pij log given define xij xij xij aij pij log xij using holds assumption letting log seen var xij xji pij pij bernstein inequality follows xij exp used fact pij pij letting value implies xij exp applying union bound implies max exp implies satisfying log max xij log since combining proves lemma xij proof lemma holds rearranging using implied yields dividing sides using lemma yields probability least proving lemma proof lemma let xij given xij pij pij log pij seen xij independent random variables distributions xij pij log log probability xji xij xii definition xij holds pij pij exij xij bound right hand side observe assumption pij log log pij log hence holds max exij var xij xji applying bernstein inequality thus yields xij exij exp letting implies xij combining bound proves lemma exij proof theorem lemma found theorem proof theorem let span let denote set observe following properties submatrices must symmetric eigenvectors since span association scheme specifically holds linear space well initial values let weights binary disjoint support holds max max log log log log well property implies matrix given since submatrix linear combination log log max also property eigenvectors submatrix orthogonal include vector lemma result effect projection change eigenvalue associated vector submatrix implying hence hold implies using properties show induction base case holds property suppose follows properties properties properties property completes induction argument since follows property property implies holds proving theorem proof corollary prove corollary states eigencoordinates approximate unitary transform theorem holds suggests computed spectral clustering converge label permutation intermediate results lemma bounds eigenvalues assumptions lemma let assumptions hold let denote matrix diag let respectively denote sorted eigenvalues holds use following version theorem taken theorem let symmetric singular values respectively fix assume min let let orthonormal columns satisfying exists orthogonal kop min proof corollary lemma proof corollary let rank theorem holds kop follows first inequality follows lemma also kop kop kop max pij max max bab max theorem holds pij pij max pij substituting yields completing proof proof lemma let given let diag let denote eigenvector eigenvalue let let given seen showing eigenvector eigenvalue since follows eigenvalues converge completing proof references edoardo airoldi david blei stephen fienberg eric xing mixed membership stochastic blockmodels journal machine learning research sep arash amini aiyou chen peter bickel elizaveta levina methods community detection large sparse networks annals statistics arash amini elizaveta levina semidefinite relaxations block model arxiv preprint bailey designs association schemes lecture series pages rosemary bailey association schemes designed experiments algebra combinatorics volume cambridge university press afonso bandeira yutong chen amit singer games compact groups orientation estimation arxiv preprint peter bickel aiyou chen nonparametric view network models modularities proceedings national academy sciences stephen boyd neal parikh eric chu borja peleato jonathan eckstein distributed optimization statistical learning via alternating direction method multipliers foundations trends machine learning tony cai xiaodong robust computationally feasible community detection presence arbitrary outlier nodes annals statistics sourav chatterjee matrix estimation universal singular value thresholding annals statistics yudong chen xiaodong jiaming convexified modularity maximization stochastic block models arxiv preprint etienne klerk dmitrii pasechnik renata sotirov semidefinite programming relaxations traveling salesman problem siam journal optimization aurelien decelle florent krzakala cristopher moore lenka inference phase transitions detection modules sparse networks physical review letters chao gao zongming anderson zhang harrison zhou achieving optimal misclassification proportion stochastic block model arxiv preprint michel goemans franz rendl semidefinite programs association schemes computing ronald graham handbook combinatorics volume olivier roman vershynin community detection sparse networks via grothendiecks inequality probability theory related fields pages peter hoff adrian raftery mark handcock latent space approaches social network analysis journal american statistical association paul holland kathryn blackmond laskey samuel leinhardt stochastic blockmodels first steps social networks brian karrer mark newman stochastic blockmodels community structure networks physical review florent krzakala cristopher moore elchanan mossel joe neeman allan sly lenka pan zhang spectral redemption clustering sparse networks proceedings national academy sciences vince lyzinski daniel sussman minh tang avanti athreya carey priebe perfect clustering stochastic blockmodel graphs via adjacency spectral embedding electronic journal statistics vince lyzinski minh tang avanti athreya youngser park carey priebe community detection classification hierarchical stochastic blockmodels arxiv preprint catherine matias vincent miele statistical clustering temporal networks dynamic stochastic block model arxiv preprint andrea montanari subhabrata semidefinite programs sparse random graphs application community detection arxiv preprint elchanan mossel joe neeman allan sly stochastic block models reconstruction arxiv preprint tiago peixoto hierarchical block structures model selection large networks physical review karl rohe sourav chatterjee bin spectral clustering stochastic blockmodel annals statistics pages jaewon yang jure leskovec overlapping community detection scale nonnegative matrix factorization approach proceedings sixth acm international conference web search data mining pages acm tengyao wang richard samworth useful variant theorem statisticians biometrika anderson zhang harrison zhou minimax rates community detection stochastic block models arxiv preprint pan zhang cristopher moore scalable detection statistically significant communities hierarchies using message passing modularity proceedings national academy sciences yuan zhang elizaveta levina zhu detecting overlapping communities networks using spectral methods arxiv preprint qing zhao stefan karisch franz rendl henry wolkowicz semidefinite programming relaxations quadratic assignment problem journal combinatorial optimization
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preprint version manuscript submitted ieee signal processing exact solution decomposition panos markopoulos dimitris chachlakis evangelos papalexakis november oct abstract study decomposition tensors treated collection matrices jointly decomposed contributions follows prove problem equivalent combinatorial optimization variables derive first two algorithms literature exact solution first algorithm cost exponential second one cost polynomial mild assumption algorithms accompanied formal complexity analysis iii conduct numerical studies compare performance exact proposed standard hosvd hooi glram pca studies show outperforms tensor approximation counterparts processed data outlier corrupted index terms data analysis outliers robust tucker decomposition tensors ntroduction roblem tatement introduced tucker tucker decomposition fundamental method tensor analysis applications wide range fields including machine learning computer vision wireless communications biomedical signal processing data analysis name considering tensor processing formed concatenation say across mode loss generality number coherent class distribution coherent tensor measurements tucker decomposition simplifies decomposition strives jointly decompose collected tensors unveil structure class distribution svd hosvd orthogonal iteration hooi algorithms wellknown solvers tucker decompositions detailed presentation tucker markopoulos chachlakis department electrical microelectronic engineering rochester institute technology rochester usa panos dimitris papalexakis department computer science engineering university california riverside riverside usa epapalex corresponding author preprint version manuscript submitted ieee signal processing respective solvers offered note types solvers generally guarantee locally optimal solution take familiar form analysis pca thus similar pca sensitive outliers within processed tensor hand analysis substituting pca illustrated remarkable extending formulation tensor processing one similarly endow robustness tucker decompositions substituting formulations indeed approximate algorithm proposed however remains date unsolved work offer first time exact solution special case approximation provide two optimal algorithms formal problem statement follows consider collection matrices equal size rank min decomposition strives jointly analyze maximizing vkf approximated squared frobenius norm returns summation squared entries matrix argument among methods literature coincides multilinear pca matrices generalized approximation matrices glram clearly simplifies approximation matrix solved means familiar decomposition svd optimal arguments built singular vectors respectively counteract impact outliers work consider reformulation maximize returns summation absolute values matrix argument problem studied title analysis authors presented approximate algorithm solution employed image reconstruction date solved exactly literature even special case approximation work deliver first time exact solution means two novel algorithms addition provide numerical studies demonstrate exact work refer problem highlight connection formulation instead general tucker formulation preprint version manuscript submitted ieee signal processing superiority decomposition reconstruction standard glram pca xact olution reformulation combinatorial optimization rank takes form maximize first focus absolute value notice sgn sgn sgn returns vector argument view lemma follows lemma given holds max maximum attained sgn sgn sgn addition following lemma derives optimality svd lemma given holds max returns highest singular value matrix argument maximum attained dominant singular vectors respectively compact notation concatenate holds denotes kronecker matrix product view lemma lemma rewrite max max max preprint version manuscript submitted ieee signal processing clear combinatorial problem feasibility set following proposition derives straightforwardly lemma lemma concludes transformation combinatorial problem proposition let bopt solution combinatorial maximize denote uopt vopt singular vectors bopt respectively uopt vopt optimal solution also bopt sgn opt vopt sgn uopt vopt vopt uopt bopt vopt bopt special case uopt vopt bopt set effect metric given bopt uopt vopt obtained svd bopt thus proposition solution obtained solution combinatorial problem svd connection hardness sequel show simplifies specifically matrix vector satisfying vec rewritten max clear every optimal value trivially equivalently thus becomes max max exact formulation problem notice also combinatorial optimization proposition becomes max max since maximum vector coincides euclidean norm accordance analysis based equivalence proven formally jointly asymptotic rank thus equivalence also jointly asymptotic rank preprint version manuscript submitted ieee signal processing fig draw holds plot nullspaces columns colored planes observe planes partition coherent cells visible cells cyan hyperplane cells exact algorithm exhaustive search proposition shows solution obtained solution combinatorial problem first exact algorithm solves straightforwardly exhaustive search feasibility set fact noticing invariant negations matrix argument obtain solution bopt exhaustive search set bex every value takes bex conduct svd calculate cost min since entails svd calculations cost algorithm min thus exponential number jointly processed matrices quadratic matrix sizes exact algorithm search cost polynomial sequel focus case constant present algorithm solves polynomial cost emerges case interest signal processing applications measurements sensing system images proposition optimal solutions bopt uopt vopt respectively holds sgn bopt sgn vopt opt opt opt sgn opt vopt uopt vopt addition every find vopt opt opt opt vopt uopt preprint version manuscript submitted ieee signal processing therefore defining rdm rewritten bopt sgn vopt uopt svd consider rank min admits svd qsw diagonal matrix carries defining popt vopt uopt rewritten bopt sgn popt view since sgn invariant positive scalings vector argument optimal solution bopt found binary set sgn certainly definition subset thus finite size upper bounded turn implies exist instances yield value sgn delve observation build tight superset polynomial size following mild general position assumption assumption every holds rank collection columns linearly independent define denote nullspace every angle equal accordingly kwi cos clearly hyperplane partitions two halfspaces sgn every sgn every accordance proposition consider closed set includes boundary whereas open overlap view definitions proceed following illustrative example consider two column indices hyperplanes divide halfspace pairs respectively assumption one two halfspaces defined intersect halfspaces defined forming four cells clear sgn every example every holds sgn sgn next one step consider arrangement hyperplanes similar discussion linearly independent intersect coincide preprint version manuscript submitted ieee signal processing hyperplanes partition cells depends formally every set defined complementary index sets satisfy definition accordance example every lies intersection halfpsaces thus yields exact value sgn specifically every holds sgn sgn view every define signature cell sgn moreover observe every observations definitions rewritten sgn importantly shown exact number coherent cells formed nullspaces points general position assumption exactly equality accordingly per cardinality equal clarity fig plot nullspaces colored planes columns arbitrary satisfies assumption interesting exactly coherent cells emerge intersection formed halfspaces sequel rely develop conceptually simple method computing tight superset cell signatures assumption hyperplane intersection line subspace definition line verge cells jointly bounded hyperplanes consider vector crosses verge point crossing value sgn change sgn adjusts signature new cell entered time crossing simultaneously hyperplanes assumption hyperplanes intersect therefore clear sgn remain preprint version manuscript submitted ieee signal processing algorithm polynomial input vec vec vec svd every build every svd max diag output bopt uopt vopt fig algorithm exact solution cost invariant crossing fact equal sgn view set sgn contains signatures sets bounded verge moreover shown see every cell exists least one verge bounds therefore derives set bpol includes cell signatures thus superset notice every size since take distinct values find bpol upper bounded thus polynomial order practically every calculated orthogonalization cost keeping dominant terms construction bpol costs parallelized processes testing every entry bpol optimality costs additional thus overall cost second algorithm taking also account constant svd cost formation presented algorithm summarized fig iii umerical tudies kuk kvk consider entry additive white gaussian noise awgn consider useful data want reconstruct joint analysis irregular corruption entries matrices entries total entries preprint version manuscript submitted ieee signal processing fig reconstruction mse versus corruption variance corrupted additively noise reconstruct follow one two approaches first approach vectorize matrix samples perform standard matrix analysis obtain first principal component vec vec vec every approximate mat mat reshapes vector argument matrix accordance vec second approach process samples natural form matrices analyzing solution pair approximate first approach obtain pca svd second approach conduct hosvd hooi glram proposed exact reconstruction method measure mean squared error kai independent realizations corruption variance fig plot reconstruction mean squared error mse every method versus observe pca exhibit highest mse due vectorization operation outperforms pca clearly across values methods perform similarly well low outlier variance increases performance hosvd hooi glram deteriorates severely hand exhibits robustness proposed exact maintains sturdiest resistance corruption outperforming counterparts across board preprint version manuscript submitted ieee signal processing eferences tucker mathematical notes factor analysis psychometrika vol vasilescu terzopoulos multilinear analysis image ensembles tensorfaces proc european conf comput vision eccv copenhagen denmark may lathauwer moor vandewalle multilinear singular value decomposition siam matrix anal vol gomes almeida costal tensor method blind spatial signature estimation proc ieee int conf speech signal florence italy may hansen lars arnfred algorithms sparse nonnegative tucker decompositions neural computation mit press vol sun papadimitriou lin cao liu qian multivis social network exploration visual analysis proc siam int conf data mining sdm sparks may kolda sun scalable tensor decompositions data mining proc ieee int conf data mining pisa italy lathauwer moor vandewalle best approximation tensors siam matrix anal vol may kolda bader tensor decompositions applications siam review vol sun tao papadimitriou faloutsos incremental tensor analysis theory applications acm trans knowl disc data vol sidiropoulos lathauwer huang papalexakis faloutsos tensor decomposition signal processing machine learning ieee trans signal vol jul pang yuan robust tensor analysis ieee trans circuits syst video vol huang sidiropoulos bro joint tensor factorization outlying slab suppression applications ieee trans signal vol cao wei han lin robust face clustering via tensor decomposition ieee trans vol markopoulos karystinos pados options signal processing proc intern sym wireless commun sys ieee iswcs ilmenau germany markopoulos karystinos pados optimal algorithms signal processing ieee trans signal vol markopoulos kundu chamadia pados efficient analysis via bit flipping ieee trans signal vol plataniotis venetsanopoulos mpca multilinear principal component analysis tensor objects ieee trans neural vol generalized low rank approximations matrices mach vol golub van loan matrix computations baltimore johns hopkins university press van loan ubiquitous kronecker product comput appl yale geometry symmetry san francisco orlik terao arrangements hyperplanes new york preprint version manuscript submitted ieee signal processing winder partitions hyperplanes siam appl vol brown diaconis random walks hyperplane arrangements ann probability karystinos liavas efficient computation binary vector maximizes quadratic form ieee trans inf theory vol jul
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round subspaces new spectral clustering algorithm ali kemal oct october abstract basic problem spectral clustering following solution obtained spectral relaxation close integral solution possible find integral solution even though might completely different basis paper propose new spectral clustering algorithm recover subspace corresponding span indicator vectors opt close original subspace spectral norm opt minimum possible opt always moreover algorithm impose restriction cluster sizes previously algorithm known could find closer opt present two applications algorithm first one finds disjoint union bounded degree expanders approximate given graph spectral norm second one approximating sparsest graph cluster expansion provided eigenvalue laplacian matrix significantly improves upon previous algorithms required introduction paper study following problem solution spectral relaxation partitioning problem close integral solution still find integral solution main difficulty due rotational invariance spectral relaxation basis integral solution might completely different basis solutions spectral relaxation arguably important problem spectral clustering widely used approach many data clustering graph partitioning problems arising practice spectral clustering one uses top bottom matrix derived input usually laplacian adjacency matrix graph derived distances nearest neighbors find clusters separated nice way close arbitrary rotation hence crucial part spectral clustering methods round formally study problem approximating linear subspace another subspace constant every vector subspace coordinates email asinop work done author visiting simons institute theory computing university california berkeley material based upon work supported national science foundation grant comprised distinct values equivalently given orthonormal matrix form think embedding points problem find minimize total variance direction min max mean points cluster use denote matrix cluster means row one cluster means objective stated concisely minc spectral norm geometrically speaking corresponds finding constant subspace makes minimum angle problem clustering spectral norm optimal solution corresponds one threshold cuts perspective problem seen generalization thresholding higher dimensions main contribution new spectral clustering algorithm recover whose center matrix satisfies opt opt minimum possible observe opt furthermore recovered opt jaccard index optimum partition cluster found close unique cluster among optimum previously algorithm known find closer opt also study two closely related problems first one goal approximate matrix spectral norm block diagonal matrix every block normalized adjacency matrix clique second application turn problem expansion given undirected weighted graph find nodes minimize maximum expansion def min max min denotes total weight edges crossing second application approximating optimum expansion graphs whose spectrum grows faster make precise later choice spectral norm measure closeness associated subspaces quite natural perspective second application given subspace show construct graphs polynomial time approximating expansion graphs implies solution spectral clustering problem perspective see subspace rounding problem prerequisite toward obtaining approximation algorithm problem expansion best known due related work spectral methods successfully used clustering tasks arising many different areas vlsi machine learning data analysis computer vision usually obtained formulating clustering task combinatorial optimization problem cuts solving corresponding basic sdp relaxation whose solution often given extremal eigenvectors associated matrix one first spectral clustering algorithms worst case guarantees given graph partitioning problem assuming certain conditions internal versus external conductance problem finding minimize spectral norm first introduced context learning mixtures gaussians best known approximation factor due problem closely related spectral clustering expansion defined cluster sizes constrained nearly equal problem admits log log approximation hand approximation sought one find clusters expansion log log times optimum look basic sdp relaxation expansion optimal fractional solution given smallest eigenvectors corresponding graph laplacian matrix fact main motivation behind usage clustering tasks practice natural question whether one round eigenvectors called cheeger inequalities shown simple thresholding yields expansion optimal value expansion later better bound given assuming gap eigenvalues versions cheeger inequality known guarantees hides dependencies logarithmic factors expansion form algorithms find parts problem becomes significantly harder exactly clusters desired case shown method similar one proposed yield partition maximum expansion best known approximation algorithm expansion problem yet algorithm known achieves polylogarithmic approximation perhaps simplest case expansion gap smallest one might think eigenvalue laplacian matrix form stability criteria implies maximum expansion put another way approximating optimum least easy finding minimum possible expansion among clusters case trivial show thresholding second smallest eigenvector laplacian yields partition optimal one hand best prior result due find optimal one words algorithm known find approximation optimum organization first introduce useful notation background section state main contributions section propose new spectral clustering algorithm section section prove algorithm always finds given subspace optimum section discuss applications algorithm main applications section approximating graph using disjoint union expanders section expansion finally section present simple reduction expansion problem means algorithm expansion solve subspace rounding problem well notation background def let associate set nodes vector def def kqk given subset use denote indicator vector else matrices use denote set real matrices likewise use denote set symmetric positive semidefinite matrices respectively finally let set orthonormal matrices stiefel manifold def given matrix use min refer ith largest singular value define minimum singular value likewise kakf denotes frobenius norm kakf given matrix use refer minor corresponding rows columns finally use denote projection matrices onto column space respectively observe aat aat one way measuring closeness two subspaces look much degrees need rotate vector one subspace closest vector subspace well known quantity related spectral norm completeness provide formal version statement along proof proposition given two linear subspaces orthonormal basis respectively cosine largest angle two subspaces given following def cos min max sin proof definition easy see measures maximum degrees necessary rotate point point vice versa prove second statement point span written moreover orthonormal thus kxk kapk kpk allows rewrite cos follows min max min max best given moreover thus min max consequently sin maxp definition let setv family sets subsets use disjv setv denote set disjoint subsets disjv setv order compare subspaces need identify canonical representation subspaces associated natural representation use basis vector normalized indicator one clusters notation basis matrices given let def corresponding normalized incidence matrix eak use denote associated projection matrices multiplication either projection matrices natural correspondence means differences means proposition disjv orthonormal matrix laplacian matrix ith column mean points cluster provided cluster otherwise difference associated center defined example measures sum squared distances point center cluster origin cluster measure distance sets way similar cosine distance def notation given define note convenience use particular measure set similarity closely related jaccard index proposition pair subsets proof since immediately see direction suppose implies therefore particular generalize set similarity measure setv define notation given def min max whenever resp say resp resp observe notion proximity strong bound example subset size preserved exactly next theorem says similarity measure use notation tightly related spectral norm distance corresponding basis disjv moreover appropriately theorem given ordering columns proof theorem given section proposition given proof aat aat since kat consider two subspaces basis angle two subspaces small one might intuitively expect aat close also next lemma make intuition formal also include proof completeness lemma given kaat proof prove upper bounding spectral norm aat since aat aat matrix zero claim trivially true suppose consider largest eigenvalue corresponding eigenvector means either without loss generality may assume def consequently eigenvector eigenvalue particular kaat aat prove multiply sides see aat aat implies aat graph partitioning given undirected graph nodes edge weights use denote adjacency laplacian matrices consider following graph partitioning problem goal minimize maximum ratio total weight edges cut number nodes inside among clusters definition expansion given undirected graph nodes edge weights define expansion following def min max denotes total weight unordered edges fixed use disjv refer achieves first glance notion expansion might seem different usual definition given however indeed proposition disjv max max min particular proof min therefore min prove direction let maxt consequently recall min capture objective function using spectral norm within factor lemma given disjv maxt def proof let maxt need prove lower bound trivial give proof upper bound note matrix whose diagonals columns indicator vectors every define matrix equal along diagonals everywhere else since laplacian matrix therefore diagonal whose entries let consequently given lemma simple relaxation basic sdp relaxation following min kqt note disjv feasible indeed relaxation moreover principle implies optimum value corresponding optimal solution smallest therefore contributions main problem given matrix form think embedding points find disjv minimize total variance direction min max mean points cluster one exists otherwise problem clustering spectral norm remark covering points simplicity allow points left uncovered set however guarantees still hold even require cover points arbitrarily assign uncovered points clusters making sure relative cluster sizes change procedure changes approximation ratio factor express succinctly following min proposition two closely related problems whose optimum within square root lemma finding rotation matrix rrt disjv minimize following min kry approximate gram matrix using block diagonal matrices block constant equivalent min main contribution new spectral clustering algorithm whose given algorithms prove following guarantee outputs theorem restatement theorem let disjv pectral lustering disjv close cluster size remark small clusters main guarantee stated theorem works example consider case optimal cluster size cluster exactly equal words algorithm recover exactly easy consequence show approximate graph disjoint union expanders provided one exists polynomial time corollary restatement corollary given graph exists disjv laplacian spectral norm laplacian corresponding disjoint union normalized cliques polynomial time find disjv next significantly improve known bounds recovering clusters small expansion definition previous spectral clustering algorithms guarantee recovering smallest eigenvalue associated laplacian matrix satisfies new algorithm significantly relaxes requirement theorem restatement theorem given graph laplacian matrix let obtained running algorithm smallest eigenvectors finally show approximation algorithm expansion implies approximation bound spectral clustering problem restricted orthonormal matrices words spectral clustering problem prerequisite approximating expansion even graphs whose normalized laplacian matrix eigenvalue larger constant def theorem restatement theorem given let graph whose normalized laplacian exists weighted undirected matrix smallest eigenvalue least small expansion disjv moreover constructed polynomial time algorithm clustering algorithm listed algorithms pectral lustering algorithm main procedure invoked matrix smallest laplacian matrix output close use denote closest refer true clusters intuition first start discussion main challenges involved spectral clustering intuition behind major components algorithm finding cluster since directions clusters think direction associated one clusters moreover one true clusters total correlation center remaining directions small utilizing intuition easily find subset say algorithm however property need true even though every direction total correlation might thus overwhelming correlation associated direction fact reason type procedures fail find every cluster remedy time find try peel natural approach project columns onto orthogonal complement center similar ideas used context learning mixtures anisotropic gaussians column based matrix reconstruction projection obtain new orthonormal matrix corresponding remaining clusters boosting unfortunately iterate approach matter accurate error best guarantee iterations error much worse algorithm keep error accumulating via boosting step algorithm unraveling even boosting remains one issue clusters found may overlap one another unlike distance based clustering problems assignment problem cluster node belong quite spectral clustering even given simple local procedure figure assignment node looking cluster centers deal issue reducing ownership problem finding matching bipartite graph algorithm approach similar one used special case santa claus problem unfortunately operation cascading effect adding new cluster might considerably change previous clusters centers dealing challenge causes final algorithm rather involved final algorithm final difficulty face boosted cluster might cannibalize much smaller clusters overcome issue maintaining estimates every true cluster coarse estimate core originally found finer estimate obtained boosting due cascading effect unraveling whenever add new cluster centers project onto orthogonal complement every round overview algorithm proceeds iteratively rth iteration finds core one true clusters say main invariant need core set following lemma noticeably close say remaining true clusters small overlap sense boosting work however found algorithm needs boost theorem need invariant true nravel since close still mostly overlapping theorem hence overlap disjoint invariant consequently use instead boosting obtain much closer require using centers boosted sets instead core sets make sure error lemma accumulate projection step one last time output iterations apply nravel boosted sets result remark algorithms last step involves computing top singular vector cases good initial guess indicator vector large separation thus simply use power method log many iterations compute sufficiently accurate approximation top right singular vector obtain approximation top left singular vector easily previously shown power method sufficient context spectral clustering analysis algorithm section prove correctness algorithm main result following proof given end section theorem restatement theorem let disjv pectral lustering disjv close order keep analysis simple make effort toward optimizing constants algorithm oost top left singular vector return ound algorithm ound squ sqv return pectral lustering algorithm algorithm ind luster rank choose minimum following returns nravel ind luster nravel oost return nravel top right singular vector iii return ound see figure sample graph construction algorithm choose minimum following returns matching construct bipartite graph left side block identical nodes edge nodes find matching covers nodes matched block return input bipartite graph figure graph constructed nravel input ordering min preliminaries following proposition show pair symmetric matrices close spectral norm gap largest second largest eigenvalues largest eigenvectors matrices close also obtained using wedin theorem chose give simple proof proposition given maximum eigenvectors def def proof suppose kpk kqk let hence main tool use identify clusters eigenvalues principal minors gram matrix basically eigenvalues measure much true clusters overlap given principal minor make connection formal following claim claim given disjv subset let ith largest element proof kyst observe eigenvectors corresponding eigenvalue thus eigenvalues consider principal minor corresponding whose largest eigenvalue large second largest eigenvalue small previous claim implies unique optimal cluster almost contained however still sufficient might much larger next lemma show one take top right singular vector round threshold obtain another subset close lemma initial guess given disjv subset def def let top right singular vector define subset obtained ound satisfies following argmaxt def def proof use since top right singular vector kys kqk claim implies via proposition see provided esbi sign therefore het esbi ihq esbi kkq esbk consequently using fact see found new clusters iterate projecting onto orthogonal complement center following lemma prove long clusters close optimal ones projection preserves remaining clusters lemma given disjv cluster centers linearly independent suppose exists disjv form disjv disjv def good spectral clustering sense addition proof note singular values either moreover rank rank rank since spectral clustering invariant change basis assume singular values means orthonormal obtained linear transformation therefore good spectral clustering also words upper bound simple cauchyschwarz consequence start proof correctness unraveling procedure correctness nravel algorithm prove input nravel algorithm list possibly overlapping sets close ground truth output list disjoint sets also close ground truth algorithm based formulating simple maximum bipartite matching problem lemma given setv exists disjv nravel alb disjv gorithm output exists proof easy see blocks matched resulting assignment collection subsets first property second property consider corresponding subsets hence implies prove disjv exists always matching blocks suppose matching minimizes maxs hall theorem show set right nodes neighbors left observe contains nodes block adding whole block increase nodes set neighbors hence subsets need consider form correctness ind luster algorithm prove given orthonormal basis close rotation ind luster algorithm output set close one clusters lemma given whose singular values exists disjv small enough constant ind luster algorithm output exists proof implies therefore words kyt max kyt consequence exists kyt kyu kyu kyu kyu def kyu kyu let define sort nodes ascending order provided simple markov inequality sum kyup atp least consequently smallest integer kyu satisfies kyu assume subset kys kyu recall variance lower bounded sum squares largest singular values kyu kys hence provided implies lemma tells new subset obtained rounding top right def singular vector satisfies argmaxt correctness oost algorithm mentioned earlier keep finding clusters removing iteratively error quickly accumulate degrade quality remaining clusters prevent apply boosting procedure described oost algorithm every time find new cluster main idea close cluster ground truth say far others top left singular vector say vectors associated close ones unfortunately use simple perturbation bounds wedin theorem make full use instead projection vectors tend stay together therefore vectors close vectors hence indeed close addition theorem boosting given disjv consider subset suppose exists oost algorithm satisfies constant remark note theorem allows convert subset overlap subset close unfortunately new subset obtain longer guaranteed small intersection def theorem lemma def use claim def since left top singular vector kyst define kqs kqs since proposition implies het kqs kqs def let use denote mean subset hea assume without loss generality hence hand het kqs using lower bound quantity since kqk het using proposition see ound satisfies correctness ound algorithm possibly simplest case problem vector argued introduction algorithm boils simple thresholding case proposition given ound algorithm satisfies def proof let without loss generality may assume kqk max correctness pectral lustering algorithm finally put everything together prove correctness pectral lustering algorithm following lemma show algorithm iteratively find sets think coarse approximations think fine approximations iteration sbi correspond unique moreover small overlap remaining coming even though might still large overlap previous easily use nravel algorithm rectify issue lemma let disjv conb found stant consider sequences pectral lustering start rth iteration exists ordering def following properties def every every sbi proof induction trivially true given suppose true described means beginning iteration nravel lemma disjv using invoke lemma implies theorem see provided small enough constant use lemma see subset ind luster satisfies reorder holds true since remain true consider nravel lemma know particular subset means whenever prove case using fact disjoint consequently executing oost noting see via theorem combined fact sbi remain also remains true induction see true pectral lustering theorem let disjv disjv close lemma implies nravel proof lemma outputs disjoint collection second bound second last inequality used theorem applications section show applications spectral clustering algorithm expansion first application approximating graphs one may also interpret applying subspace rounding algorithm basic sdp relaxation expansion problem theorem given graph laplacian matrix let obtained running algorithm smallest eigenvectors remark faster algorithm slightly modifying algorithm take advantage underlying graph structure one obtain faster randomized algorithm guarantees theorem expected running time proof lemma know consider matrix whose columns smallest eigenvectors means implies kik thus kik claim follows theorem matrix graph approximations next application approximating matrix terms diagonal matrices corresponding adjacency matrices normalized cliques spectral norm def theorem given matrix let polynomial time find disjv proof let matrix whose consider pectral lustering rows top eigenvectors means theorem final application approximating graph laplacian via another laplacian corresponding graph formed disjoint union normalized cliques expanders spectral norm since working laplacian matrices means new graph approximates cuts original graph also corollary given graph exists disjv laplacian spectral norm laplacian corresponding disjoint union normalized cliques find disjv polynomial time proof since apply theorem matrix rest follows easily expansion implies spectral clustering section show approximation algorithms various graph partitioning problems imply similar approximation guarantees clustering problem def theorem given let exists weighted undirected regular graph whose normalized laplacian matrix smallest eigenvalue least small expansion disjv moreover constructed polynomial time proof consider following sdp chose minimum value sdp remains feasible iii doubly stochastic diagonally dominant psd trace easy see feasible solution lemma moreover feasible solution corresponds adjacency matrix graph undirected degrees equal show properties maxt recall maxs particular using theorem omitted proofs proof theorem disjv theorem restatement theorem given moreover appropriately ordering columns upper bound recall laplacian matrix lemma see let matching set within maximum diagonal element given corresponding diagonal element proving lower bound show bounds therefore following lower bound define def argmax def argmax consider claims indeed perfect matching consider matched pair without loss generality say claim since finish proof claims claim similarly thus proof consider matrix min particular diagonals least consider diagonal corresponding ets max max construction equal proves first part claim second part follows immediately applying argument claim bijections proof suppose since disjoint contradiction similar argument shows bijection well claim bijections claim exists cycle form proof suppose since construction means since disjoint construction therefore implies since similar argument also show consequently contradiction cycles length implies acknowledgments thank moses charikar ravishankar krishnaswamy stimulating discussions problem also thank anup rao pointing lemma holds equality references sanjeev arora boaz barak david steurer subexponential algorithms unique games related problems annual ieee symposium foundations computer science focs october las vegas nevada usa pages charles alpert andrew kahng yao spectral partitioning multiple eigenvectors discrete applied mathematics noga alon milman isoperimetric inequalities graphs superconcentrators comb theory ser pranjal awasthi sheffet improved bounds clustering approximation randomization combinatorial optimization algorithms techniques international workshop approx international workshop random cambridge usa august proceedings pages nikhil bansal uriel feige robert krauthgamer konstantin makarychev viswanath nagarajan joseph naor roy schwartz graph partitioning small set expansion ieee annual symposium foundations computer science focs palm springs usa october pages marianna bolla spectral clustering biclustering learning large graphs contingency tables wiley nikhil bansal maxim sviridenko santa claus problem proceedings annual acm symposium theory computing seattle usa may pages charles brubaker santosh vempala isotropic pca clustering annual ieee symposium foundations computer science focs october philadelphia usa pages amit deshpande luis rademacher efficient volume sampling subset selection annual ieee symposium foundations computer science focs october las vegas nevada usa pages alex gittens prabhanjan kambadur christos boutsidis spectral clustering via power method provably corr venkatesan guruswami ali kemal sinop optimal matrix reconstruction proceedings annual symposium discrete algorithms soda kyoto japan january pages amit kumar ravindran kannan clustering spectral norm algorithm annual ieee symposium foundations computer science focs october las vegas nevada usa pages tsz chiu kwok lap chi lau yin tat lee shayan oveis gharan luca trevisan improved cheeger inequality analysis spectral partitioning algorithms higher order spectral gap symposium theory computing conference stoc palo alto usa june pages ravi kannan santosh vempala adrian vetta clusterings good bad spectral journal acm james lee shayan oveis gharan luca trevisan multiway spectral partitioning cheeger inequalities journal acm anand louis konstantin makarychev approximation algorithm sparsest kpartitioning proceedings annual symposium discrete algorithms soda portland oregon usa january pages anand louis prasad raghavendra prasad tetali santosh vempala many sparse cuts via higher eigenvalues proceedings symposium theory computing conference stoc new york usa may pages andrew michael jordan yair weiss spectral clustering analysis algorithm advances neural information processing systems neural information processing systems natural synthetic nips december vancouver british columbia canada pages jianbo shi jitendra malik normalized cuts image segmentation ieee trans pattern anal mach stewart sun matrix perturbation theory academic press stella jianbo shi multiclass spectral clustering ieee international conference computer vision iccv october nice france pages
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oct extensions rings gabriel picavet martine hermitte abstract introduce ring extensions order relativize domains take also account contexts recent papers extensions appear hidden form extension inc pair class extensions nice stability properties also define extensions converse true extensions closely linked finiteness properties fibers applications given fmc extensions introduction notation consider category commutative unital rings epimorphism epimorphism category let ring extension set denoted extension said fip finitely many intermediate algebras property finite chain set elements pairwise comparable respect inclusion say extension fcp finite chain property chain finite dobbs authors characterized fcp fip extensions clearly extension satisfies fip must also satisfy fcp extension called fmc finite maximal chain extensions begin explaining motivations aims reader familiar notions used find scholia sequel well necessary definitions exist literature knebusch zang introduced extensions book actually extensions nothing normal pairs intensively studied literature intend give extensive list recent papers written ayache ben nasr dobbs jaballah jarboui others indebted authors papers rich source suggestions observed mathematics subject classification secondary key words phrases flat epimorphism fip fcp extension minimal extension integral extension morita hull support module picavet picavet dealing fcp fip fmc extensions followed extension perhaps hidden form extensions reminded domains see therefore introduced extensions extensions factored first extension integral second note fmc extensions give systematic study extensions section section class extensions nice behavior respect classical operations commutative algebra important result extensions coincide another one class stable forming subextensions composition striking result stability class extensions absolutely flat base change like localizations henselizations ring extension admits closure contained examples provided laskerian pairs open pairs pairs section deals extensions special kind extensions form first extension second integral ring extension admits closure contained class extensions seems less properties class extensions advantage commutation closures localizations prime ideals examine transfer quasi almost properties subextensions section study transfer quasi almost properties nagata extensions section complete generalize results respect finiteness fibers authors evidently considered particular cases extensions main result finite fibers particular recover result fmc extensions section gives calculations respect closure closure case fcp recalls results definitions reader warned mostly use definition extensions flat epimorphic subextensions investigated results needed may found scholium flat epimorphic extensions results summarized scholium powers give quick proofs results generalizations results literature extensions long fcp fmc extensions concerned use minimal ring extensions concept introduced extension called minimal known minimal extension either flat epimorphism conditions mutually exclusive three types integral minimal extensions ramified decomposed inert theorem minimal extension admits crucial ideal maximal spec moreover integral minimal extension key connection ideas fcp fmc maximal necessarily finite chain length results juxtaposing minimal extensions following define length supremum lengths chains particular integer exists maximal chain length usual spec max min tot respectively set prime ideals maximal ideals minimal prime ideals units total ring fractions ring residual field spec extension conductor spec localization denote integral closure local ring called elsewhere ring support suppr spec msuppr suppr max finally denotes proper inclusion cardinality set scholium give recalls flat epimorphisms see chapitre except proposition flat epimorphism spec either isomorphism flat epimorphism spec flat epimorphism domain surjective ring morphisms injective flat epimorphism injective let tower extensions flat epimorphism flat epimorphism need extension remedies defect picavet picavet faithfully flat epimorphism isomorphism hence integral flat epimorphism flat epimorphism ideal epimorphism spectrally injective residual extensions isomorphisms flat epimorphisms remain flat epimorphisms base change particular localization respect multiplicatively closed subset flat epimorphisms descended faithfully flat morphisms recalls results extensions recall definitions properties ring extensions rings lot characterizations extensions keep useful paper give two definitions dual emphasize characterizations local case scholium called flat epimorphism called normal pair integrally closed normal pair theorem called finitely generated regular ideals invertible equivalently tot theorem hence extensions relativization rings clearly minimal extension flat epimorphism use extensions terminology minimal extensions reader may find properties minimal extensions proposition lemma proposition asserted dechene dissertation addition must supposed local reason word surprisingly disappeared printing process need two next results explicitely appear deserve emphasized refer definition definition manis extensions proposition let ring extension spec respectively supp extensions manis max proof class extensions stable localization proposition get converse use scholium follows proposition definition proposition let ring extension local manis case integrally closed manis exists spec valuation domain conditions quotient field proof theorem scholium theorem next result shows fcp extensions described special manner proposition let ring extension fcp integrally closed composite minimal extensions integrally closed fcp supp finite proof assume fcp integrally closed composed minimal extensions lemma conversely composed minimal extensions integrally closed since minimal extension extension obviously integrally closed fcp integrally closed extension theorem logical equivalence theorem definition ring extension called greatest flat epimorphic subextension morita hull called hull greatest subextension confusion occur set called denoted weakly note surjective hull terminology justified picavet picavet morita work earlier corollary morita hull computed using transfinite induction let set ideal subextension set fcp extension integer stage interesting point result showing integral closedness extensions closely related proposition olivier corollary extension integrally closed pullback square ring total quotient ring case extension since ring whose localizations prime ideals valuation domains absolutely flat ring exist integrally closed extensions see passing pullback construction may descend extensions result companion minimal extensions proposition proposition let extension therefore greatest extenproof obvious since hull sion contained fcp show later cases extensions introduced following definition definition extension rings called one following equivalent statements holds extension factored integral case see observe holds integral flat injective epimorphism scholium extensions observe extensions akin extensions refer zariski main theorem explored section see example theorem hence integral extensions extension clearly integrally closed extensions allow avoid fcp hypotheses give definitions involved ring extensions fiber spec fibr spec subspace fibr spec homeomorphic spectrum fiber ring homeomorphism given spectral map fiber morphism definition ring extension called incomparable pair prime ideals equivalently ring spec incomparable residually algebraic algebraic spec residually algebraic pair extension residually algebraic following characterization announced unaware result also proved corollary present arxiv however proof largely shorter use powerful results theorem extension residually algebraic pair proof suppose let set flat epimorphism definition extension hence incomparable follows incomparable since integral follows incomparable conversely since integrally closed theorem second equivalence proposition theorem corollary extension picavet picavet follows properties described integrally closed domains valid arbitrary ring extensions moreover result dobbs easily gotten integral extension spectrally surjective theorem follows scholium property example domains quotient fields characterized reader may consult theorem view theorem domain spec spec bijective give another example extension extension called pair subextensions property pair incomparability prime ideal lemma let maximal ideal whose fiber void pair corollary dimension therefore pair dim max also pairs proposition extension spec spec injective equivalently spectrally injective proposition extensions appear frequently integral domains context another examples given extensions spec spec sets see later properties extensions develop machinery extensions proposition extension spec msupp proof proof easy use property definition extension see also proposition proposition let extension integral ring morphism integral closure proof enough apply theorem extension use definition result applies integral morphism therefore integrality ascends property extensions know composite extensions theorem following corollary contains theorem corollary let tower extensions follows proof consider tower extensions composite two extensions using proposition see obtained writing left integral extension right extension therefore prove converse tower extensions whenever converse consequence theorem last statement corollary using corollary exhibit new examples extensions recall ring called laskerian ideals finite intersection primary ideals ring extension laskerian pair laskerian ring proposition shows integral domain quotient field field extension laskerian pair algebraic laskerian domain follows easily next result generalizes proposition corollary fmc extension proof composite finitely many minimal extensions corollary enough observe minimal extension either integral corollary let extension tower integrally closed proof observe next result deals extensions integral domains called pairs suppose local call extension exists spec spec corollary arbitrary extension called max view corollary enough characterize extensions type spec spec picavet picavet corollary let extension spec spec local spec spec algebraic field extension case proof follows max part proof gotten observing inc extension spec spec spectrally another one proved corollary surjective flat epimorphism scholium let extension ideal shared easy show using proposition case able give general statement lemma let extension ideal extension maximal ideal valuation domain quotient field proof assume first scholium flat epimorpism therefore form write flat epimorphism follows get easily since flat epimorphism case easy consequence lemma generalize complete proposition proposition let extension rings following statements equivalent spec max fiber morphisms integral proof entailed lemma assume holds let max contains minimal prime ideal lain minimal prime ideal follows proposition applied holds argue paragraph proposition get pextension whence proposition integral extensions incomparability see corollary shows reverse implication holds extensions extension integral fiber morphisms spec extension lemma ring flat epimorphism therefore surjective scholium follows fiber morphism integral remark logical equivalence still valid replace integral proposition enough show extension integral integral spec suppose indeterminate ideal varies min extension element min lain element min set unitary polynomials assumptions show element spec containing meets multiplicatively closed subset whence integral similar result hold replace except suppose integrally closed see apply proposition get extension actually situation already occurs rings factor domains lucas paper shows precisely proposition third paragraph shows ring tot absolutely flat ring domain spec example example shows necessarily observe domain max case integral field prove observe factored see class extensions stable flat base change example let valuation domain quotient field example thus consider ideal may imply except happens instance prime ideal lain prime ideal proposition let quasi extension flat epimorphism quasi addition subrings ring extension quasi picavet picavet proof first part enough consider case well known following diagram pushout spec lying isomorphism since flat epimorphism scholium follows identifies result follows extensions localize globalize case flat epimorphic extension surjective maps isomorphisms resp injective resp flat epimorphism enough use scholium reader may find corollary extensions proposition let two extensions subrings ring proof let integral closures integral corollary applies using corollary find proposition entails flat epimorphism finally since composite extensions known arbitrary product extensions components proposition following result easy consequence proposition let finite family extensions particular finite family extensions way following result deduced remark proposition let extension rings upward directed family elements extensions proof enough use proposition integral closure ring morphism preserves integral closure ring mort phisms every ring morphism absolutely flat morphism flat preserves integral closure theorem flat epimorphisms henselizations morphisms absolutely flat another examples morphisms essentially finite type absolutely reduced proposition morphisms flat reduced proposition prove ascent result absolutely flat ring morphisms proved using base changes need introduce concepts ring called aic ring monic polynomial zero recalled ring faithfully flat integral extension aic ring moreover aic ring localization prime ideal strict henselian ring lemma theorem let extension absolutely flat ring morphism extension proof suppose aic ring see enough use base change set first observe following reason composite extension last extension integral moreover absolutely flat case faithfully flat deduced faithfully flat base change enough apply proposition thus assume aic ring let spec lying absolutely flat proposition observe therefore suppose local local injective deduce theorem isomorphism therefore proof complete case case need observe absolutely flat morphisms preserve integral closure extension integrally closed picavet picavet proposition let extension rings base change preserves integral closure fcp proof result holds fcp extension integrally closed needs isomorphism since observe property may fail even localization prime ideal proposition let extension rings faithfully flat ring morphism set respectively fcp proof case clear faithfully flat morphisms descend flat epimorphisms scholium case use characterization fact faithfully flat spec lying corollaire fcp case proved theorem proposition let ring extension spectrally surjective ring morphism example either faithfully flat injective integral injective example faithfully flat proof let spec set spec lying spectrally surjective faithfully flat morphism corollaire theorem result follows faithful flatness theorem let ring extension greatest subextension proof see use proposition tells set subextensions upward directed use tion prove existence let tower integral deduce factored tower integral extensions extensions clearly integral closure closure respective intersections last result means long integral closures closures subsets concerned suppose extensions next give definition dual definition extension arbitrary extensions definition ring extension called extension factored integral proposition extension integral follows subring definition proof factorization integral flat epimorphism scholium scholium therefore corollary let extension let moreover proof corollary integrally closed corollary moreover integral extension note integral extensions extensions hence minimal extensions extensions enough consider example let two minimal extensions local integral example shows composite extensions may reverse implication holds picavet picavet theorem let extension spec moreover case flat epimorphic subextension extension proof let morita integral result follows hull hull coincide proposition way flat epimorphism integral could define rings rings tot theotot case rem ring converse evidently holds therefore concept define something new observed remark fmc extension minimal extension minimal integral fcp extension proposition let extension verifying hypotheses factored flat epimorphism following commutative diagram pushout moreover spec addition integrally closed pullback proof consider injective composite map flat epimorphism deduced base change get surjective map isomorphism scholium fibers transitivity corollaire isomorphism scholium get lemma first statement conductors proof lemma second holds flat epimorphism see scholium extensions theorem let extension diagram pushout pullback factored first extension second greatest subextension supp supp supp supp replaced msupp proof show view theorem enough apply almostproposition whence keeping mind extension integrally closed whereas integral extension trivial moree integral integrally closed obvious consider subextension integral applying see view proposition obviously supp supp supp conversely let spec absurd entails corollary let extension following conditions equivalent supp supp supp supp supp supp proof since get spec moreover supp supp supp supp supp supp supp supp assume exists supp neither integral picavet picavet supp supp supp supp supp contradiction supp supp integral contradiction assume exists supp supp neither ine supp tegral supp supp supp supp integral contradiction supp supp contradiction assume exists supp supp neither integral supp supp supp supp supp contradiction supp supp integral contradiction proposition following similar statement proved ayache dobbs reduces theorem case fcp proposition proposition let extension integral minimal extension integrally closed diagram pullback proof lemma proposition let two towers extensions proof denote hulls deduce corollary moreover clearly integral hull greatest subextension deduce proposition let two towers extensions flat epimorphism proof mimic proof proposition use theorem proposition let extension flat epimorphism extensions proof enough use proposition definition proposition extension spec suppose proof arbitrary extension theorem conversely locally whence holds locally spec theorem corollary fcp extension spec proof enough show spec using proposition integral definition assume minimal extension exists minimal extension crucial maximal ideal particular max integral may assume exists using minimal extension supp lemma exists minimal extension crucial maximal ideal contradic integral whence tion intend demonstrate methods allow prove easily results instance next statement generalizes corollary fruitful algebraic number theory proposition let local ring extension suppose tower integrally closed proof integrally closed prime ideal lying faithfully flat epimorphism whence isomorphism scholium absurd let prime ideal zerodimensional isomorphic therefore follows since deduce scholium proof complete picavet picavet also generalize proposition follows proposition let extension local maximal ideal local addition either integral minimal prime ideal valuation domain quotient field proof obviously local let proposition follows exists integer giving equivalently rsn obtain assume integral complete proof use proposition fcp extensions case consider fcp extensions obtain results proposition let fcp extension following statements equivalent either integral spec supp supp supp supp proof equivalence proposition shows holds local ring fcp extension either integral proposition moreover either integral easy show next show equivalent proposition supp let supp almoste since giving follows view dichotomy principle proposition since local ring supp conversely assume seen hence integrally closed follows theorem moreover supp supp extensions implies supp conclude supp supp fcp extension corollary supe pose theorem letting get deduce proposition supp supp suppose last condition holds proposition factored integrally closed whence proposition integral therefore lemma let two integral minimal extensions fcp integral proof set integral proposition assume max proposition let integral closure also ideal prime maximal fcp extension flat epimorphism since field follows integral extension proposition let fcp extension least integral proof may assume integral integral set minimal element integral follows least element going show let set length exist maximal chain maximal chain length let intend show enough choose minimal element consider induction hypothesis minimal let first show minimal assume integral contradicts lemma shows beginning proof contradiction proved assume holds let minimal picavet picavet get integral also hull moreover since holds get set get minimal integral using lemma get integral also hull since holds contradiction therefore proved need relative version support let ring morphism relative support suppt msr particular ring extension suppr proposition let fcp extension following statements hold supp supp supp supp supp supp msupp msupp msupp proof consequence proposition prove first part supp supposed max set mrm fcp fcp integral absurdity proposition show second part assume supp supp supp first part contradiction giving obviously msupp propositions msupp supp supp conclude msupp msupp msupp proposition let fcp extension msupp msupp msupp proof fact going show msupp msupp extensions msupp let msupp since proposition giving therefore proposition since local conversely seen integrally closed follows msupp hence msupp proposition moreover conclude msupp msupp msupp ring extension dim max indeed scholium flat epimorphism bijective well conclusion still valid another context corollary let fcp extension assume one following conditions satisfied msupp msupp equivalently max proof proposition proposition proposition let fcp extension integral closure since proof set also hull integral msuppr either integral proposition corollary also hull let integral closure integral closure assume integral giving assume giving conclude get msuppr since max get whence integral closure picavet picavet build example fcp extension max particular example let integral domain quotient field spec two maximal ideals prime ideal satisfying assume minimal integral minimal minimal max last condition satisfied either ramified inert indeed cases moreover ramified case inert case theorem apply proposition lemma several times set integral minimal minimal max moreover spec prime ideal lying proposition set minimal max minimal max follows spec prime ideal lying end assume minimal hence integral closure particular fcp theorems since integrally closed assume exists minimal contradiction proposition since integral follows msupp minimal supp give msupp proposition giving intend refine theorem following scheme used proposition extensions integral domains proposition let integral following cases supp example fcp preserves integral closure extensions proof msupp since let msupp msupp msupp identifies let max set max since integral supp maximal ideal lying follows lemma identifies supp gives localizing precedent equality still therefore locally whence proposition usual reasoning shows integrally closed since contained get observe whence follows next propositions generalize ayache results proposition proposition let extension following statements hold let supp supp assumption holds fcp supp supp proof observe corollary since integral closure get follows thus prove reverse inclusion set integral since tower therefore moreover since integral entails follows since extension integrally closed observe integrally closed whence follows picavet picavet set view thus suppose follows integral deduce proposition tet supp supp therefore assume supp gives view proposition let extension subextension set following statements hold integral assume integral minimal integral minimal type assume minimal minimal assume minimal set integral integral minimal max minimal exactly one prime ideal lying proof integral extensions extensions moreover also integral closure gotten considering extension gotten considering extension assume integral minimal integral set suppu max giving theorem get implies suppu follows therefore identifies minimal type proposition integral minimal type assume minimal set extensions max since max max max lying follows integral suppt get minimal view proposition assume minimal extension set assume integral use proposition getting max spec follows suppu integral minimal suppu max minimal properties crucial maximal ideal assume max let max minimal extension gotten assume moreover gives let max lying gives suppv moreover let max gives follows suppt suppv max proposition minimal exactly one prime ideal lying lemma let fcp extension fcp proof obviously fcp integral closure proposition entails suppr suppr claim suppu suppu deny let suppu suppu get giving contradiction another use proposition shows proposition let fcp extension subextension set let hull also hull fcp extension proof lemma get fcp extension let hull since gives subextension picavet picavet moreover view proposition least integral since integral get integral subextension follows also hull fcp extension case nagata extensions section transfer properties nagata extensions proposition let fcp extension fcp extension proof suppose local order use proposition enough know following facts valuation domain spec addition fcp enough use theorem fcp fcp proposition proof enough use proposition third assertion results proposition proposition proposition follows extension proof integral integral whence lemma let fcp ring extension proof minimal set msupp msupp lemma msupp msupp proposition giving msupp proposition entails proposition minimal fcp extension extensions choose first minimal extension verifying preceding property therefore supp lemma get exists minimal contradiction proposition fcp extension proof assume set lary giving lemma extension contradicting definition hence proposition let fcp extension proof tower first third equalities come theorem second proposition end section special result proposition let extension fip proof map defined bijective theorem whence flat epimorphism moreover result follows since faithfully flat fibers extensions intend complete results begin recalling features ring morphisms ring morphism called finite type finite space spec proposition proposition ring morphism finite type incomparable fibers finite proof use corollary definition picavet picavet theorem extension respectively finite fibers finite type integral fiber morphisms proof clear implies condition proposition prove converse let write union finite type let prime ideals lying prime ideal set incomparable last statement proposition corollary integrally closed extension subextensions finite type finite fibers proof enough observe fibers flat epimorphism cardinal epimorphism spectrally injective ring extension called strongly affine subextensions finite type considerations show case subextensions finite fibers example fcp extension strongly affine also interested extensions necessarily strongly affine subextensions finite fibers next lemma useful proof obvious lemma let extension spectrally injective finite fibers finite fibers spectrally injective finite fibers finite fibers remark let extension finite fibers let spec integral extension study finiteness fibr reduced follows epimorphism spectrally injective see scholium hypotheses proposition hold examine three cases case well known spectrally injective suppose deduce lain spec fibr fibt conclusion follows thus remaining case extensions assume fibr fibt spec scholium proposition let extension maximal ideal finite fiber morphisms finite fibers suppr proof closure commutes localization prime let prime ideal ideals proposition set canonical morphism clearly fibr fibrp therefore localize data assume local case get factorization since follows extension integrally closed proposition applied get spec valuation ring quotient field follows hence therefore pushout diagram theorem valuation domain quotient field proposition injective flat epimorphism bijective map min min fiber therefore finite therefore min finite set maximal ideals lying minimal prime ideals lying infer lemma whence integrally closed therefore integral domain integral closure maximal ideal contains conclude enough use result gilmer corollary number maximal ideals less separable degree extension fields maximal ideal remark suppose hypotheses clearly proposition hold picavet picavet tower finitely many integral minimal case extensions suppre ideals dif max integral minimal ferent localization result may apply generalizes result supposed integral minimal lemma proposition let ring extension finite fibers finite fibers finite fibers extension finite fibers proof let spec morphisms first second morphism integral flat epimorphism deduced base change integral morphism flat epimorphism therefore ring zero dimensional second morphism surjective scholium set thus module finite flat ring morphism hence free spec proposition contains therefore injective follows suppose finite fibers let flat epimorphism proposition since spec spec injective finite fibers finite fibers integral therefore spectrally surjective remark actually statement valid suppose flat epimorphism next result contains lemma gotten long proof corollary let extension finite fibers finite fibers finite fibers proof proposition first equivalence clear second consequence lemma following result clear theorem let extension finite fibers finite fibers corollary module finite finite fibers extensions proof zariski main theorem factorization module finite flat epimorphism corollaire conclude use scholium rest injective proof map flat epimorphism surjective since integral flat flat epimorphism epimorphism corollary fmc extension finite fibers proof extension use corollary example exhibits fmc extension fcp actually infinite maximal chain proposition let extension finite fibers proof obviously tower deduce follows lemma integral get following proposition gives kind converse proposition let extension proof let set integral normal pair follows lemma residually algebraic pair theorem generalized arbitrary extensions max whence theorem remark proposition proposition numerical properties fcp extensions lemma let fcp extension map suppt defined bijective particular fip picavet picavet proof let also integral closure let set assume giving suppt suppt assume view proposition get suppt hence well defined let assume another use proposition gives follows injective reference gives bijective proposition let fcp extension define two follows defined defined proof follows lemma proposition recall supp corollary fcp supp supp supp supp supp supp proof set supp supp supp supp proposition assume let contradiction follows intersecting two members equality get way intersecting equality get corollary let fcp extension define two orderisomorphisms proof use notation proposition begin remark play symmetric roles since let proposition get injective similar argument shows injective extensions exists let let supp supp supp corollary supp supp proposition giving supp reasoning gives follows giving surjective hence bijective similar argument shows surjective hence bijective corollary fcp defined particular fip proof using notation proposition corollary may remark since fip case obvious gathering previous results get following theorem theorem fcp next statements equivalent supp supp map defined supp supp defined map defined map defined map supp one conditions holds supp fip former conditions equivalent following conditions proof lemma picavet picavet statement holds exists proposition gives supp supp proposition use corollary get corollary get corollary get moreover supp corollary give supp proposition case assume fip obviously corollary gives using map lemma get suppt suppt suppr suppr corollary example give example results theorem hold fcp set minimal max integral minimal set max local ring follows supp supp neither integral similarly supp supp indeed fcp end paper length computations fcp case proposition let fcp extension following statements hold proof prove use maps corollary follows theorem proposition references anderson dobbs pairs rings prime ideals math xxxii anderson dobbs fontana treed nagata rings pure appl algebra extensions ayache ben nasr echi jarboui universally catenarian pairs rings math ayache constructive study set intermediate rings comm algebra ayache dobbs finite maximal chains commutative rings jaa ayache jaballah residually algebraic pairs rings math ben nasr jarboui new results normal pairs rings ricerche mat chatham pairs commutative rings rendiconti del circolo matematico palermo serie tomo chang fontana uppers polynomial rings domains comm algebra dobbs picavet hermitte characterizing ring extensions satisfy fip fcp algebra dobbs picavet hermitte transfer results fip fcp properties ring extensions comm algebra dobbs picavet hermitte extension nagata rings finitely many intermediate rings nagata ring int math math article dobbs picavet hermitte shapiro intersections composites minimal ring extensions algebra number theory dobbs characterizations integrality involving incomparability properties comm algebra dobbs shapiro pairs integral domains houston endo rings math soc japan ferrand olivier homomorphismes minimaux anneaux algebra fontana huckaba papick domains dekker new york gilmer multiplicative ideal theory dekker new york griffin rings zero divisors journal fur die reine angewande mathematik grandet une des morphismes minimaux non entiers acad paris grothendieck springer verlag berlin houston uppers zero polynomial rings multiplicative ideal theory commutative algebra new york jaballah finiteness set intermediary rings normal pairs saitama math picavet picavet jarboui massaoud finite saturated chains overrings comm algebra knebusch zhang manis valuations extensions springer berlin lazard autour platitude bull soc math france lazarus fermeture changement base ann fac sci toulouse lucas results rings pacific morita flat modules injective modules quotient rings math olivier anneaux absolument plats universels buts samuel commutative tome exp olivier des par morphismes absolument plats alg pure des sciences technique languedoc montpellier france olivier going along absolutely flat morphisms pure appl algebra picavet applications notion contenu comm picavet universally rings rings absolute integral closure comm algebra picavet hermitte minimal morphisms multiplicative ideal theory commutative algebra springer new york picavet seminormal schemes rees rings algebra repr theory picavet hermitte combinatorics results nagata extensions palestine picavet hermitte morita hulls fcp extensions comm algebra raynaud anneaux locaux lect notes springer vol uda incomparability ring extensions hiroshima math visweswaran laskerian pairs pure appl algebra blaise pascal laboratoire cnrs avenue des landais cedex france address address
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stateful behavioral types abs eduard kamburjan chen feb department computer science technische darmstadt germany kamburjan abstract notoriously hard correctly implement multiparty protocol involves interactions constraints states multiple participants assist developers implementing protocols propose novel specification language specify interactions within multiple actors sideeffects heap memory actors analysis presented type checking specification language formalizes protocol global type describes procedure asynchronous method calls usage futures heap firstorder logic characterize runs instances types give modeltheoretic semantics types translate logical constraints traces prove protocol adherence program protocol every trace program adheres protocol every trace model formula type introduction combination actors languages scala abs sometimes called active objects active research area system models frequently used practice general active objects communicate internally within object via object heap memory communicate externally asynchronous method calls futures constructs synchronizing executions invoked calls encapsulated heap memory asynchronous calls explicit synchronization points specified futures make hard specify verify protocols active objects main obstacle din owe pointed bridge gap local perspectives single objects global perspectives whole system precisely specify communication within object heap memory global perspective multiparty session types short mpst one important member behavioral types established theories typing globally stateless concurrent interactions method calls among multiple participants objects ensure communication safety current works mpst attempted specify states communication using global values assuming channels communication concept however global values sufficient specify interplay processes heap memory inside object channels able fully represent usage futures translation execution integrate stateful analysis specification din mpst since mpst able formally specify scenario global interactions specifying communication global perspective functional properties local states communication aim ensure correctness heap accesses methods actors deadlock freedom processes interact verification local specifications compositionally guarantees global specification specify passed data modifications heap memory logic fol formulas transform behavioral types logical constraints traces moreover perspective define protocol adherence every generated trace program model translation type following scenario illustrates approach challenges specifying verifying protocols active objects assume gui computation server program protocol interface server without knowing wants compute data sending via method call executing call gets ready next action setting field intern value expect trace constraint terminating process stay responsive delegates task remains responsive requests without waiting computation invoking new method call future carry computation result bigger back code figure implement scenario object tstate intern init int resume futhinti return int return unit start int futhuniti cmp expect object unit cmp int dat futhinti cmp dat futhinti resume object int cmp int main start code denotes call cmp calls method cmp start calls initial method cmp resume calls method resume continuation cmp cmp starts actual computation challenge resume formal specifications express transparent get must pass data received read return value changes heap expect reads correct future contributions propose specification language actors behaviors integrates fol specify heap memory type system integrating validity calculus semantics protocol adherence roadmap section gives overview approach illustrates workflow first present core ideas without loops branching section introduces core language active objects sync dynamic logic section gives types operations section gives type system section gives language type constructs loops section gives language type constructs branching section gives conclusion related work challenges overview workflow consider actors use method calls futures heap memory communication every method call asynchronous starts new process callee object call active process caller gets fresh future identity one may synchronize termination started process object delete may switches active process another process currently active process terminates using futures gives programmers control make synchronization however leads following complications unexposed state active object concurrency model process exactly one future thus reading future synchronizing unknown process depends state process object futures analyzed isolation reading future must take unexposed state object account mixed communication paradigms communication heap memory hard describe data types requires specification computation explicit caller callee thus difficult isolate parts program realizing communication protocol additionally method calls asynchronous future reads synchronous endpoints active object model callee endpoints methods calls objects caller endpoints endpoints future synchronization processes interplay multiple objects contain multiple processes must captured analysis notion endpoints objects processes endpoints example specifying global types specification language fol specifying local memory instead global values since global values natural active object setting logic memory locations variables fields allows use theory first order dynamic logic capture semantics methods formalize scenario section following global type specification language expecti comp comphi dat result resumehx end denotes true following syntax honda comp denotes message comp call method comp formula expect postcondition process started call two formulas provided first precondition describing state caller second postcondition describing state callee prp step generating local types objects step propagating guarantees objects step generating local types methods fig workflow phase global type denotes projection object resp method function function propagating guarantees return value denoted keyword result annotation denotes memory location future denoted call stored formula dat states dat parameter carries value ceived parameter formula requires parameter call method resume carries future previous call comp finally describes read future stored location note specify locations formulas avoid situation endpoint must guarantee obligation containing values access example bocchi allow situation thus require additional analyses analysis adopt approach similar mpst project global type endpoints defined inside automatically derive local specifications objects methods additionally formulas used specify conditions heap memory projected logical substructure callee callee access caller fields analysis analysis requires protocol encoded global type defines order method calls future read operations objects annotated specifications heap memory passed data analysis mechanism two phases phase global type used generate local types endpoints phase endpoints type checked local types via generating causality graph workflow phase based mpst approach adjusted active object concurrency model phase workflow phase shown fig step global type projected onto participating objects generates object types type specifies obligation object running methods certain order guaranteeing fol specifications object state projection projected onto substructure object object prove specifications memory access step propagated within object type order method executions specified specification postcondition method assumed precondition next method step object type projected methods producing method types global type encodes following obligations short obl implementation obl whole system deadlock obl object observable events calls reads ordered specified global type obl method observable events ordered specified local type derived global type adhere following demonstrate phase workflow global type example formally introduce syntax point step object types continue example projecting object get starth comph state result type starth denotes starting point runtime execution type comph denotes invocation method comp type put specifies termination currently active process state holds position put derived automatically specify global type process terminates precondition resume weakened location visible callee use information caller check information able access syntactically checked locations visible type read specifies synchronization future stored step propagation next step propagate postcondition last process precondition next process process specified active start resume heap modified thus postcondition start still holds resume starts adding state expect precondition resume strengthens assumption type checking resume propagation conditions results starth comph put state expect state expecti read put result step method types generate method type specify method isolation projecting object type step method resume generates resume state result method types share syntax object types projection object types splitting positions one method ends another starts phase generating method types phase analysis type checks implementation methods method types formulas checked validity type checking method types guarantees correct local order events obl state specifications checked integrating validity calculus type system guarantee obl require following analyses causality graph generate causality graph ensure deadlock freedom obl deadlock free causality graph active objects causality graph also used ensure methods one object executed order specified global type object obeys obl nodes local types projected object types edge connecting two nodes models statement first type causes statement second type example edges call corresponding receiving type graph partially generated partially generated code edge connecting gray nodes added analysis maps location future methods resolving future dotted line considered termination method causes next method order check semantics one contribution defining verifying protocol adherence view property program follows specified scenario sense every generated trace model translation global type thus define protocol adherence logical characterization global types translate types constraints traces sequences configurations generated program declarative approach defining protocol adherence allows connect properties embedded type execution methods using dynamic logic capture body method dynamic logic formula expresses formula holds executing holds beginning dynamic logic formula valid every generated trace every position terminates holds corresponding position starts holds core language using futures section introduces sync simple active object language based abs due space limitation present basic constructs sync projection future locations type system validity calculus guarantees guarantees guarantees guarantees deadlock freedom local action order protocol adherence state specification composition guarantees fig workflow phase analysis leave constructs branching repetition later sections consists main statement set actors objects fields method share states inside object processes interleave active process must terminate another scheduled therefore methods considered sequential assume standard operations literals types booleans integers lists object definition syntax sync let denote expressions denote data types denote variable field names denote object names fut denote future types represents possibly empty lists represents optional elements prgm main object return fut skip objects communicate asynchronous method calls using futures upon method call fresh future generated callee side passed caller callee writes return value future upon termination corresponding process anyone access future read write sync standard imperative language two additional statements calls method parameters object generated future stored caller continues execution callee computing call scheduling later another process currently active reads value future stored process computing future terminated reading process blocks define reduction relation events semantics sync first define event process action communication definition events let range futures event denoted defined following grammar irev fev frev noev invocation iev models calls method using passing parameters invocation reaction irev models starts execution resolve resolving fev models resolves contains contains moment finishing execution fetch frev models reads value fig defines reduction relation denoted semantics denotes evaluation expression stores rule call executes method call object stores generating fresh future invocation event new process set active rule start new process becomes active upon creation object must inactive invocation reaction event generated rule sync synchronizes future stored checking whether configuration contains prc val resolved reads return value rule end terminates process rules parameter noev configuration composed processes objects process unique future store maps fields literals object object unique name active future store maps variables literals contain call iev prc prc prc start irev prc prc prc val sync end frev prc prc fev prc return prc val fig semantics rules definition runtime syntax processes objects following grammar defines runtime processes objects configurations prc prc val process either executing method request carried object represented prc returned represented prc val object name future active process heap write indicate inactive composition configurations commutative associative denote initial configuration program prgm prgm processes configuration terminated configuration also terminates method body method denoted write initial local store task executing parameters use traces sequences pairs events configurations capture behavior program consider terminating runs define semantics prgm finite traces definition run semantics run sequence configurations events trace run sequence evm every run evj noev holds let prgm sync program prgm generates denoted prgm run initial configuration terminated configuration trace run dynamic logic dynamic logic combines heap symbolic executions statements symbolic execution uses symbolic values describe possible set actual values reason one execution statement describes set executions example formula describes number bigger smaller value stored executing variable contains positive value based absdl present sync dynamic logic short sdl extends logic program variables heap memory modalities model effect statements logic method parameters special variables modality formula holds configuration say holds every configuration reached executing configurations models evaluate sdl formulas definition formulas define set formulas terms following grammar ranges predicate symbols ranges function symbols ranges logical variables ranges logical program variables set formulas denoted sdl local program variables modeled special function symbols model heap accesses following schmitt use two function symbols store select connecting axiom select store heap value value heap special local program variable modeling heap explicitly special function symbol result interpreted return value method logical variable free bound quantifier definition formula valid evaluates true every configuration differentiate global formulas refer heap multiple objects refer containing function symbols elements special function symbol self models reference reasons proven holds given state checking locally code validity calculus superset sdl presented definition let formula weakened obtained replacing function symbols exclusive refer fields objects free variables existentially quantifying example let field object parameter method class consider formula weakening object reason still information stateful specifications futures definition defines specification language writing global types specify behavior system due space limitation represent key constructs leave constructs later sections definition syntax global types let range sdl formulas range object names denotes optional elements main end calling type specifies method call written arrow future call must stored location specifies call parameters passed callee memory moment call formula postcondition callee process specifies state return value initial method call main specifies postcondition process running type specifies synchronization future stored parameter expression possible specify reading action future stored list every synchronization must specified end specifies end communication denotes complete protocol initializing method call denotes partial types even without fields formula implementation referenced specifications endpoints object names object method types use share syntax together call local types grammar local types defined follows definition syntax local types let range sdl formulas range object names denotes optional elements put read end receiving type denotes start process computing state formula holds precondition describes local state method parameters type put denotes end process state holds postcondition describes return value local store contrary global types postcondition process annotated call point termination point termination explicit type corresponds caller side type read models read skip denotes communication use denote complete local type denote partial types projection three steps projection global types objects condition propagation projection object types methods projection objects projection objects needs ensure every object access locations occurring specification add put correct position requires additional parameter projection keep track process specified active postcondition track postcondition last active method object use partial function sdl map objects formulas method active yet undefined written projection object denoted selected projection rules methods calls termination given fig write function defined updates write otherwise main main skip put skip read skip put end end fig selected rules projection objects projecting caller sending local type generated active process precondition proven caller callee active process process termination type added receiving type callee specified inactive receiving type added projecting object skip added case updated maps callee new postcondition rest paper omit write propagation concurrency model heap change process active guarantees last active process still hold next process propagation formulas added postcondition one method precondition next propagation moves formulas must hold points still assumed hold propagation replaces partial local type partial type matches given pattern definition propagation propagation function prp defined via term rewriting denoted follows denotes fixpoint rewriting put put target object projection methods projection method say results set method types method may multiple methods types long method types distinguishable call method types result set distinguishable preconditions two preconditions formula valid rules projection method straightforward refer section example full rules given figure definition global type projections methods defined types method distinguishable put skip otherwise read read skip otherwise skip otherwise skip skip put skip otherwise end skip fig projection methods semantics types constraints traces formalize behavioral types last section transform constraints traces define function transforming global types constraints traces recall defined configurations events primitive references ith configuration references ith event trace use events formulas colors thus include futures method names literals object names domain constraints refer sdl formulas meaning ith configuration holds translation models position holds subtrace holds subtrace construct constraint use relativization syntactic restriction constraint substructure described another constraint definition let constraint free variable data type constraint denote relativization quantifiers type constraint relativization adds restrictions main rule relativization following example relativizes constraint natural numbers iseven iseven rules translating constraint defined follows definition semantics global types predicate res holds resolving event holds active main irev fev res iev irev fev res iev irev fev res frev res end true constraint describing call type three events modeling call start process termination process exist configurations events projected formulas hold every event fev exact position termination fev events specified global types constrain reading location defined analogously sequential composition described expressing global type constraint traces also enables connect types sdl semantics sdl defined terms traces consider example statement modality body method precondition postcondition formula expresses invocation reaction event position corresponding resolving event holds also give translation local types irev iev iev put fev read frev end true analysis aim verify protocol adherence also verify deadlock freedom additionally requires analysis deadlock freedom equivalent causality graphs active objects causality graph global type node singular partial local type edge models must happen definition causality graph let global type nodes causality graph local types derived projecting endpoints edge added either partial type projection object sending type receiving type derived projection calling type note global types contain sufficient information deduce causality causality get statements deduced global type synchronizations futures specified locations use analysis futures instead generating causality graph first derive partial causality graph global type apply analysis type checking graph completion deducing missing edges analysis defined determines methods responsible resolve futures given expression definition given expression program pointsto analysis determines set methods reachable configuration prc whenever checked type read edges node termination type methods point node current type read added graph page admissible definition admissibility causality graph admissible every path every object pair receiving types exists connecting path without edge form put type system analysis auxiliary post models value every future resolved satisfies formula post represents conjunction postconditions specified figure gives selected typing rules invokes validity calculus analysis introducing typing rules define roles set set edges added term set nodes method node set nodes referring types typed define three kinds type judgments type judgment programs prgm checks prgm global type ensured type checking rule tmain checks every endpoint implemented prgm main block makes correct initializing call checks object object type edges collected typing rules objects added partial causality graph resulting graph checked admissibility type judgment objects checks whether given set causality edges set sdl formulas rule projects methods checks method collects resulting edges type judgment statements checks whether welltyped given environment statements far whenever sdl formula checked validity check performed added modality consider heap memory far recorded causality edges record object iroles admissible main post main skip object return return put term node read fig selected typing rules method get statement synchronizes rule checks executing statements return statement results state holds rule also checks formula describes state call executed rule additionally executes analysis adds edges described previous section theorem deadlock freedom protocol adherence let prgm program global type prgm prgm deadlock every generated trace prgm satisfies prgm prgm decidability types validation judgment prgm undecidable validity logic used specifying undecidable developer choose fol fragment decidable validity trade expressiveness analyzability developer analysis chooses restricted fragment may limit expressiveness specification validity logic used specifying may become decidable using undecidable fol fragment approach used validation tool check whether implemented system behaving expected approach able integrated development process similarly approaches applies techniques proposed mpst connecting global local views concurrent programs notoriously difficult problem using contracts invariants protocol adherence current work mpst defines protocol adherence fidelity theorem defines every sequence interactions session follows scenario declared mpst follows operational semantics types defined shown semantics language refinement semantics types similarly behavioral contracts define protocol adherence compliance compares interaction contracts operational approaches specification define protocol adherence declarative perspective define property require hold traces program declarative specification analyzed tools logical specification enable easier integration static analysis tools consider state since required logical characterization specified traces loops section extend syntax loop use types resp repeats type resp finitely often formula loop invariant satisfied whenever loop iteration starts ends definition syntax repetition syntactic restrictions local type object form allowed start loop reason restriction every loop invariant object must guarantee executing next iteration object active loop guarantee invariant beginning following give projection rules loop definition projection rules loops auxiliary predicate rcv holds specified called put skip rcv skip skip skip skip skip skip first rule projects global types object types first case applied object participates repetition inner type repeatedly called last active process must terminate first repeatedly called method must terminate within repetition termination inside loop ensured projecting inner type appended end second case applied object participates one active process whole repeated communication skip finally last case removes repetition object participate note case object repeatedly called active following syntactic restriction discussed second rule projects object types methods rule distinguishes cases checking whole process inside repetition process completely inside repetition removed visible method definition rules propagation loops put put put put fixpoint operator propagation used presence loops loop invariants hold first repetition first rule ensures last process repetition satisfies invariant terminating second rule adds invariant next process reason first rule third rule another case first one case two repetitions succeeding finally last rule adds invariant processes inside repetition rule enables type system use invariant first method repetition ensures last method reestablishes invariant loops constraints however expressive enough constraint kleene star resemble regular languages thus use monadic second order logic mso capture repetition mso extends firstorder logic quantifier quantities subsets primitive express membership sets definition translation mso global local types repetition uses set boundary indices inner translation must hold plus constraints invariant give following typing rule loops resembles loop invariant rules hoare sequent calculi definition typing rule loops post skip post post skip first premise continues type checking program environment information invariant global information post defined section available second premise checks invariant holds initially third premise checks invariant preserved loop body forth premise checks loop body last premise combines derived causality edges extension causality graph described example consider big data analysis webtool like amazon cloud clientside gui computational server model following scenario sends data computational server stay responsive ends initial process called repeatedly server update progress whenever updated server also gets information whether gui hidden users screen sequence diagram figure illustrates protocol comp update run fig sequence diagram example example important specify behavior state one needs ensure updating stays state expecting receive updates server field expect saves data nil specification main runh comp update end formula nil invariant field expect nonempty list type describes server reads return value variable previous action stores future identifies call update every read futures must annotated critical deadlock process running start would attempt read future comp system would deadlock active process waiting active process waiting call update processed already active process local type method update updatehself self nil repetition repetition visible object single process thus postcondition derived invariant similarly end end type models termination protocol visible object incoming messages arrive single process example code figure implements behavior gui server example checked types example object list int expect nil unit run list int comp expect bool update int val int res return updatescreen expect val res called wrong state object unit comp list int int length int res compute fut bool update res fig code example loop branching active object multiple ways communicate choice continue protocol object reacts choice communicated via method selection branch corresponds different method call choice communicated via futures objects must react choice object reading future choice communicated via heap memory processes must behave according condition memory aim stick standard imperative statements must regard restriction statement choose two branches protocol may describe two analysis branching choice communicated method calls condition passed data new process running objects condition future already running processes running possibly objects via processes running later object definition syntax branching else xij global type xij describes chooses branch formulas additional postconditions choosing process process read choice reading future xij describe currently active process xij additional postcondition local type active choice passive choice branch must taken reading future evaluating definition projection rules branching given ith branch xij denote updated function aci xij xij xij auxiliary predicate allact states mentioned objects occur branches active dist states set formulas overlap allact xij xij dist unsatisfiable figure shows projection rules branching projection rule global types four cases first two straightforward choosing process currently active reacting processes third case handles objects behave branches forth handles objects active one projection passive choice moves read type position choice front global type explicit point process terminates thus read must choice adds postcondition choosing process however get statement must statement relies read value guard xij allact xij allact skip read dist read fig projection rules branching definition translation mso branching translation mso constraints use auxiliary predicate firstterm states ith position trace refers first resolving event auxiliary predicate lastterm states ith position trace refers last resolving event firstterm fev lastterm fev additionally translation branches encodes choosing process terminates process relies communication choice via return value rules follows xij firstterm firstterm xij lastterm following present rules branching typing rules split branches two disjoint sets shows guard statement together added branch selects correct continuation type sets branches singletons choice operators removed definition typing rules post post post post else else extension causality graph described use following example illustrate handle branching example consider scenario client wants access data server sends login data calling method acc decides login data invalid logs denied access calling logging server returns access succeeds returns data value reacts return value returns boolean indicating whether access successful formalized following type main start hresult hresult hresulti local type following note read type branching put put result conclusion related work paper propose analysis gives developers means specify verify protocols information transmitted via asynchronous method calls also heap memory active objects semantics give declarative definition protocol adherence following works related actors objects previous type system active objects specify verify state future passing furthermore need check scheduler work crafa padovani investigate behavioral types join calculus typestate concurrency model similar actors encoding futures gay model channels objects integrating mpst classes use mpst language moose types describe communication shared channels ensure deadlock freedom approach similar giachino ensure deadlock freedom inferring behavioral contracts applying cycle detection algorithm however consider protocol adherence state contracts setting bocchi develop mpst discipline assertions endpoint state specifications use global values global types require complex checks ensure endpoint proves obligations explicitly track values several endpoints implicitly equations locations stateless setting laneve padovani combine contracts behavioral types toninho yoshida use dependent mpst reason passed data logics session types formulas examined caires carbone intuitionistic linear logics work examine dynamic logic combines called sdl heap symbolic executions statements examine declarative formulation protocol adherence references anand harrold heap cloning enabling dynamic symbolic execution java programs ase ieee computer society ancona bono bravetti behavioral types programming languages publishers hanover usa baker hewitt incremental garbage collection processes sigart newsletter bocchi demangeon yoshida multiparty logic tgc vol lncs springer bocchi honda tuosto yoshida theory distributed multiparty interactions concur vol lncs springer bocchi lange tuosto three algorithms methodology amending contracts choreographies sci ann comp sci boer serbanescu henrio rochas din johnsen sirjani khamespanah yang survey active object languages acm comput surv cadar sen symbolic execution vol lncs springer caires pfenning session types intuitionistic linear propositions concur vol lncs springer carbone lindley montesi wadler coherence generalises duality logical explanation multiparty session types concur vol lipics schloss dagstuhl fuer informatik castagna gesbert padovani theory contracts web services acm trans program lang syst crafa padovani chemical approach programming acm trans program lang syst drossopoulou mostrous yoshida objects session types information computation din bubel deductive verification tool concurrent modelling language abs cade vol lncs springer din owe sound complete reasoning system asynchronous communication shared futures log algebr meth program din tarifa johnsen specification verification scalable concurrent distributed systems icfem vol lncs springer albert genaim based deadlock analysis concurrent objects vol lncs springer gay gesbert ravara vasconcelos modular session types objects logical methods computer science gay vasconcelos wadler yoshida theory applications behavioural types dagstuhl seminar dagstuhl reports giachino henrio laneve mastandrea actors may synchronize safely ppdp acm giachino laneve lienhardt framework deadlock detection coreabs software systems modeling oct halstead multilisp language concurrent symbolic computation acm toplas harel dynamic logic new york secaucus usa henkin relativization respect formulas use proofs independence compositio mathematica henrio laneve mastandrea analysis synchronisations stateful active objects ifm vol lncs springer hewitt bishop steiger universal modular actor formalism artificial intelligence proceedings international joint conference artificial intelligence ijcai morgan kaufmann publishers honda yoshida carbone multiparty asynchronous session types acm mar johnsen schlatte steffen abs core language abstract behavioral specification fmco kamburjan session types abs tech darmstadt http kamburjan din chen compositional analysis languages using futures icfem vol lncs springer laneve padovani pairing contracts session types concurrency graphs models vol lncs springer lavender schmidt pattern languages program design longman publishing boston usa active object object behavioral pattern concurrent programming odersky scala programming language http padovani programming submitted art science engineering programming preprint available https schmitt ulbrich dynamic frames java dynamic logic foveoos vol lncs springer tasharofi dinges johnson scala developers mix actor model concurrency models ecoop vol lncs springer toninho yoshida certifying data multiparty session types log algebr meth program soundness proof theorem similar proof theorem thus give sketch point proofs differ propagation first state correctness propagation let projection trace results replacing events issued lemma let prgm program type prgm traces produced prgm order invocation events every trace satisfies translation propagated type iff satisfies translation original type prgm prgm proof fix denote show induction number applications prp fixpoint induction base lemma holds trivially induction step induction hypothesis type prpn desired property holds make case distinction applied case definition prp last application case put put case show start execution method formula holds let method whose termination action put responsible assumption order invocation events fixed program typed thus trace invocation action another invocation event thus trace contains pairs form irev contains pair part subtrace following form fev irev every state change must executed process process would invocation event two events subtrace still holds invocation event exactly condition captured propagation case case put put definition set indices every trace every position invariant holds every pair consecutive positions subtrace satisfies put last event repetition thus first position pair fev process active regard traces produced prgm thus must invocation reaction event irev argument condition must hold process could changed note every local type starts receiving action repetition case case analogous case case case analogous case case put put definition set indices every trace every position invariant holds every pair consecutive positions repetition start receiving action ends termination chosen positions termination actions first action syntactic form guarantees position reduces case show propagation case holds thus technical details analogous case main theorem given global type say another possibly wellformed global prefix extend concatenating another global type similarly local types traces main lemma similar subject reduction semantics types connects types operational semantics language states step execution preserves property trace far prefix trace model type lemma let prgm program type prgm every prefix every trace prgm satisfies translation prefix prgm prgm property whole execution program satisfies translation whole type prefix follows global types theorem deadlock freedom main differences proof following traces defined assumed conditions hold point used distinguish following kinds assumed conditions precondition method start precondition conjunction resulting projection propagation holds follows lemma holds follows fact precondition projected formula global call fully proven caller checked rule condition equal projection caller selection condition passive choice must connect additional condition executes statement branch typed corresponding type branch follows directly two additional promises respect methods executed right order assume method executed object type executed way around generated trace depend assumption program typed rule checks however start causes admissibility check means executed lemma deadlock freedom deadlocked configuration configuration terminated yet continue execution first observe every deadlock caused processes blocking get statements single process process access future also stored heap call execution start would mean another method active store would violate condition methods executed right order shown assume would deadlock assume two processes involved executing executing blocks attempting read future belonging analysis include set possible method rule type checked mean edge corresponding termination corresponding read would added causality graph holds termination object type reading type thus path read termination resulting graph pictured contains cycle absence cycles however checked rule full formalization deadlocks causality graphs refer
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dec obtaining accurate probabilistic causal inference calibration fattaneh mahdi pakdaman gregory intelligent systems program university pittsburgh pittsburgh paulson school engineering applied sciences harvard university cambridge department biomedical informatics university pittsburgh pittsburgh pakdaman gfc abstract discovery accurate causal bayesian network structure observational data useful many areas science often discoveries made uncertainty expressed probabilities guide use discoveries including directing investigation important probabilities paper introduce novel framework derive calibrated probabilities causal relationships observational data framework consists three components approximate method generating initial probability estimates edge types pair variables availability relatively small number causal relationships network truth status known call calibration training set calibration method using approximate probability estimates calibration training set generate calibrated probabilities many remaining pairs variables also introduce new calibration method based shallow neural network experiments simulated data support proposed approach improves calibration causal edge predictions results also support approach often improves precision recall predictions introduction much science consists discovering modeling causal relationships nature increasingly scientists available multiple complex data large number samples enormous number measurements recorded thanks rapid advancements sophisticated measurement technology data often purely observational past years tremendous progress developing computational methods discovering causal knowledge observational data primary use methods analyze observational data generate novel causal hypotheses likely correct subjected experimental validation approach significantly increase efficiency causal discovery science make informed decisions novel causal hypotheses investigate experimentally scientists need know likely hypotheses true probabilistic terms means need probabilities hypotheses output causal discovery algorithm informally say probabilities events predicted occur probability occur fraction time general important use calibrated probabilities making decisions using decision theory paper focus discovery causal bayesian network cbn structure observational data particular focus discovery causal relationships edge types pairs measured variables causal arc novel important may worthwhile experimentally investigate extent worth depends part high nips workshop next long beach usa calibrated probability causal arc present introduce method calibrate edge type probabilities cbns thousands measured variables arbitrarily many latent variables method requires following components method generating initial probability estimates edge types pair variables general estimates need wellcalibrated truth status small unbiased sample causal relationships network call calibration training set calibration method using uncalibrated probability estimates calibration training set generate calibrated probabilities large number remaining pairs variables use bootstrapping method generating probability estimates edge types method resamples dataset times replacement learns model dataset particular dataset use really fast causal inference rfci algorithm estimate underlying generative network allowing possibility latent confounders given pair nodes probability given edge type estimated fraction edge type networks previously researchers successfully applied approach estimating probabilities edge types bayesian networks bootstrap estimates guaranteed represent calibrated posterior probabilities however even large sample limit number bootstrap samples key reason heuristic search practically necessary may get stuck local maxima thus need map estimates calibrated probabilities focus current paper bootstrapping approach described provides empirical estimates posterior probabilities fci rfci bayesian structure learning algorithms ges bayesian model averaging provides alternative approach estimating edge probabilities however bayesian methods typically applicable using datasets number random variables double digits exact search methods triple digits heuristic search methods contrast interested providing calibrated estimates edge probabilities datasets may contain thousands variables typically encountered modern biological data also note bayesian model averaging methods sensitive method applied heuristic search structure parameter priors used even consequently generated probabilities still subject possibly uncalibrated finally computationally tractable bayesian methods discovering cbns contain latent confounders contrast methods exist perform discovery cbns hundreds latent variables datasets thousands variables feasible amount time assume availability calibration training set allows induce mapping bootstrap probability estimates calibrated posterior probabilities training set contain truth status subset edge types domain biomedical applications truth status might come example results published literature emphasize calibration training set small relative number total node pairs experiments performed consists less node pairs using goal generate better calibrated probabilities remaining node pairs application using biomedical data example biomedical scientist chose experimentally test causal relationships high probabilities close could confident experiments would usually corroborate relationships introduce new calibration method uses calibration training set construct mapping bootstrap probability estimates calibrated posterior probabilities edge types node pairs cbn except used training apply mapping node pairs paper use simulated data investigate two main first calibrated probabilities edge types second calibrated probabilities produced calibration method given finite calibration training set latter method guaranteed always output perfectly calibrated probabilities either main hypothesis paper calibration method output probabilities better calibrated bootstrap probabilities least discriminative terms measures precision recall score note difficult obtain gold standards causal relationships among variables large observational datasets result use simulated data important commonly done evaluate causal discovery methods method section briefly describe rfci search bootstrap rfci calibration model overview rfci colombo developed algorithm called really fast causal inference rfci identifies causal structure process presence latent variables using partial ancestral graphs pags representation pag encodes markov equivalence class bayesian networks possibly latent variables exhibit conditional independence relationships rfci two stages adjacency search involves selective search dependencies among measured variables orientation phase orients endpoints among pairs nodes connected according first stage typical causal discovery algorithms rfci outputs single graph structure pag provide information uncertainty edges nodes structure bootstrap rfci considering pag generated rfci possible partition pairs nodes following seven classes edge directed edge means direct indirect cause similar edge type indicates either cause unmeasured confounder similar edge type expresses cause cause unmeasured confounder unmeasured confounder one two causal relationships holds edge represents presence unmeasured confounder bootstrap rfci brfci method apply three main steps first performs bootstrap sampling training data times experiments create different bootstrap training datasets second step runs rfci datasets obtain pags finally every pair nodes uses frequency counts edge class pair generated pags determine probability distribution seven possible edge classes mentioned bootstrap estimates guaranteed calibrated following section describe method map bootstrap probabilities calibrated probabilities calibration model pair nodes resulting output brfci method seven jointly exhaustive mutually exclusive class probabilities correspond seven classes described therefore need apply calibration method classification score case seven classes one simple approach devise calibration model use binary classifier calibration method isotonic regression averaging bayesian binning abb bayesian binning quantiles bbq corresponding output probabilities class separately performed fashion described major drawback approach binary calibration methods require considerable amount data produce probabilities however often expensive feasible obtain truth status large number node pairs real applications causal discovery consequently availability small calibration training set critical constraint design calibration approach resolve problem make simple extension platt method parametric binary classifier calibration approach platt method uses sigmoid transformation map output binary classifier calibrated probability uses logistic loss function learn two parameters model method two advantages two parameters make viable choice low sample size calibration datasets method runs test time thus fast natural extension platt method calibration task use combination softmax transfer function loss function instead sigmoid function logistic loss function respectively minimizing cross entropy equivalent minimizing empirical divergence estimated probabilities observed cal cal cal cal cal cal figure structure calibration method inputs left bootstrap probabilities seven edge types outputs right corresponding postprocessed probabilities intended better calibrated cal softmax output layer number nodes hidden layer degree figure parent size distribution simulated cbns nodes edges ones minimum achieved true probability distribution minimizing cross entropy function result finding closest distribution parameterized model observed distribution data model uses softmax transfer function optimizes cross entropy loss function called softmax regression softmax calibration model inherits desirable properties platt method however similar platt method mapping softmax calibration method learn restrictive since final separating boundaries pair classes always linear simple relaxation restriction use shallow neural network one hidden layer figure shows architecture shallow neural network model use probabilities experiments train different shallow neural networks setting number neurons hidden layer randomly test time use average different outputs generated models final calibrated probability estimates averaging helpful since reduces variance error predictions improves final performance probabilities use notation fcal denote mapping vector uncalibrated probabilities input vector calibrated probabilities output implemented model using scikit flow python uses tensorflow machine learning package used loss function adagrad optimization method learn parameters set learning rate batch size respectively experimental methods section describes experimental methods used evaluate performance calibrated network discovery method introduced evaluation involves following steps create random causal bayesian network nodes edges set also set average number edges per node construct first ordered nodes randomly added edges forward direction obtaining specified mean graph density process generates graph distribution number parents nodes many average number parents figure nodes correspond continuous random variables every pair nodes parametrize relation structural equation model sem gaussian noise terms linear coefficient experiments similar ramsey variances uniformly randomly chosen interval drawn uniformly randomly interval choice parameter values simulations implies average around half variance variables due error term makes structure learning difficult simulate dataset size subject constraints described https set percentage variables unobserved latent latent variables randomly chosen confounder variables common causes given set either generate bootstrap datasets bootstrap dataset learn pag using rfci method let designate pags rfci uses fisher test check conditional independence variables dataset set significance level independence judgments made node pair calculate probability distribution edge types using maximum likelihood estimates counts perform stratified random sampling node pairs obtain training samples calibration use rest data testing set obtaining samples used stratified random sampling select samples seven edge particular first sorted probability scores edges edge class according bootstrap probabilities partitioned instances bins based bootstrap probabilities finally sampled separately bin equal frequency learn calibration function fcal using calibration training data node pair test set derive pecal fcal compare performance pecal versus correctly predicting structure test set pairs manner well calibrated running evaluation procedure step time consuming part involves running rfci bootstrap datasets however still feasible due efficiency rfci method use parallel simulations used freely available software application coded java steps repeated randomly generated bns performance results averaged given node pair take predicted edge type pair one highest probability note although seven different edge classes consider five edge types performance evaluation directed edge types partially directed edge types first two evaluation measures precision recall compute measures edge type calculated four basic statistics true positives false positives true negatives false negatives types separately precision derived ratio recall derived ratio also report harmonic mean precision recall summary measure shows overall performance predictions terms precision recall also evaluated predictions terms maximum calibration error mce calculated mce edge type partitioning output space estimated probabilities interval bins randomly chosen instances estimated probability instance located one bins bin define associated calibration error absolute difference mean value predictions actual observed frequency positive instances mce calculates maximum calibration error bins lower value mce better calibration probability scores lowest possible value mce highest possible value also report overall micro averaged mce summary measure shows performance predictions terms calibration compute measure augmented probability distribution vectors test instances form aggregated vector pall also augmented corresponding binary labels form aggregated binary vector zall overall mce defined maximum calibration error calculated based pall zall using stratified random sampling crucial due severe class imbalance data pairs type running times experiments varied minutes compute node computationally feasible step needs done one time https experimental results section presents results experiments evaluating performance generated probabilities five edge types calibration use shallow neural network calibration method learn calibration function fcal calibration training data since purpose paper compare calibration methods report results experiments using calibration methods isoreg platt method rather report results calibration using neural network method found performs well relatively small calibration training sample sizes compared calibration methods tried set configurations report average results using randomly simulated cbns tables show results cbns nodes due page limit results experiments included similar results achieved tables boldface indicates results statistically significantly superior based wilcoxon signed rank test significance level tables indicate bootstrap probabilities improve overall performance terms discrimination calibration exception edge type lose discrimination happening original precision recall bootstrap probabilities low edge type consequently often obtain positive instances edge type calibration training set negatively affects performance predictions calibration note type report precision recall always close figure shows calibration diagram estimated probabilities calibration use calibration training instances emphasize observing calibration instances equivalent observing less node pairs cbn node pairs cbn nodes draw calibration diagrams partitioned output space estimated probabilities five bins bin draw average frequency positive class versus mean predictions located bin diagrams straight dashed line connecting represents perfectly calibrated model closer calibration curve line better calibrated prediction model figure shows proposed shallow neural network method often improves calibration performance predictions important edge type since one likely drive experimentation particular directed edges scientist considers high probability well novel important would prime candidates experimental validation furthermore high probability region arguably critical one making decisions directed arcs investigate false positive experimental investigations minimized also associated diagrams type figure show estimated probabilities pretty calibration interesting observation considering fact precision recall also always close edge type using calibration method results indicate calibrated probabilities indicate high probability edge pair variables nodes rarely directly causally related result provides confidence prioritizing experimental investigation node pairs direct causal relationships another interesting observation figure bootstrap probabilities highly overestimated results high false positive rate consequently increases false negative rate edge types note bootstrap probabilities generate high probabilities consequently red circle bins low probability bins appropriate edges output rfci seldom correct overall calibration diagrams figure show bootstrap probabilities using proposed network model generally improves calibration performance predictions key advantage shallow neural network approach estimated probabilities readily condition types features learning calibration mapping features extracted structure predicted pags rfci method indegree generating calibrated probability edge type conditioning local global features learned graph could potentially yield improvements calibrated probabilities area future research table simulation results represent number variables edges percentage hidden variables cbn respectively significance level used rfci method calibration training set size boldface indicates results significantly better based wilcoxon signed rank test significance level mce lower better method mce mce mce mce mce overall mce mce mce mce overall mce mce mce mce overall mce mce mce mce overall mce method mce mce method mce mce estimated probability estimated probability estimated probability estimated probability estimated probability estimated probability estimated probability estimated probability objective probability objective probability objective probability objective probability mce objective probability objective probability objective probability mce objective probability method objective probability objective probability estimated probability estimated probability figure calibration curves probabilities blue crosses red circles calibration closer predictions diagonal calibrated probabilities results calibration training set size percentage hidden variables significance level test independence rfci conclusion paper introduced new approach improving calibration cbn structure discovery used bootstrapping method obtain estimated probabilities causal relationships pair random variables although applied bootstrapping method rfci algorithm applied type network discovery method long method sufficiently fast run hundreds times dataset obtain bootstrap probability estimates calibrate bootstrap probabilities devised natural extension platt calibration method supports calibration using shallow neural network experiments wide range large simulated datasets show using small set instances gold standards training calibration model obtain substantial improvements terms precision recall calibration relative bootstrap probabilities future work plan expand range simulated experiments perform well evaluate method using real biomedical data truth status known literature relatively small subset variables acknowledgement thank members center causal discovery helpful feedback research reported publication supported grant awarded national human genome research institute funds provided big data knowledge initiative content solely responsibility authors necessarily represent official views national institutes health material also based upon work supported national science foundation grant opinions findings conclusions recommendations expressed material author necessarily reflect views national science foundation references abadi ashish agarwal tensorflow machine learning heterogeneous distributed systems david maxwell chickering optimal structure identification greedy search journal machine learning research diego colombo marloes maathuis markus kalisch thomas richardson learning directed acyclic graphs latent selection variables annals statistics morris degroot stephen fienberg comparison evaluation forecasters statistician pedro domingos useful things know machine learning communications acm john duchi elad hazan yoram singer adaptive subgradient methods online learning stochastic optimization journal machine learning research daniel eaton kevin murphy exact bayesian structure learning uncertain interventions international conference artificial intelligence statistics pages bradley efron robert tibshirani introduction bootstrap crc press nir friedman moises goldszmidt abraham wyner data analysis bayesian networks bootstrap approach proceedings fifteenth conference uncertainty artificial intelligence pages nir friedman daphne koller bayesian network structure bayesian approach structure discovery bayesian networks machine learning clark glymour gregory cooper computation causation discovery mit press phyllis illari federica russo jon williamson causality sciences oup oxford mikko koivisto advances exact bayesian structure discovery bayesian networks mikko koivisto kismat sood exact bayesian structure discovery bayesian networks journal machine learning research david madigan jeremy york denis allard bayesian graphical models discrete data international statistical review pages mahdi pakdaman naeini gregory cooper milos hauskrecht binary classifier calibration using bayesian approach proceedings siam international conference data mining pages mahdi pakdaman naeini gregory cooper milos hauskrecht obtaining well calibrated probabilities using bayesian binning aaai pages michael nielsen neural networks deep learning determination press judea pearl causality models reasoning inference econometric theory john platt probabilistic outputs support vector machines comparisons regularized likelihood methods advances large margin classifiers joseph ramsey scaling greedy equivalence search continuous variables ricardo silva richard scheines clark glymour peter spirtes learning structure linear latent variable models journal machine learning research peter spirtes introduction causal inference journal machine learning research peter spirtes clark glymour richard scheines causation prediction search mit press bianca zadrozny charles elkan transforming classifier scores accurate multiclass probability estimates proceedings acm sigkdd international conference knowledge discovery data mining pages
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feb partial correlation screening joint robust approach jialiang xiaochao department statistics applied probability national university singapore graduate medical school singapore eye research institute college science huazhong agricultural university wuhan china abstract screening ultrahigh dimensional features may encounter complicated issues outlying observations heteroscedasticity distribution multicollinearity confounding effects standard marginal screening methods may weak solution issues contribute novel robust joint screener safeguard outliers distribution response variable covariates account external variables screening step specifically introduce partial correlation cpc screener show empirical process estimated cpc converges weakly gaussian process establish sure screening property cpc screener mild technical conditions need require moment condition weaker existing alternatives literature moreover approach allows diverging number conditional variables theoretical point view extensive simulation studies two data applications included illustrate proposal keywords copula partial correlation outlier joint screening sure independent screening introduction arrival big data era ultrahigh dimensional data become readily available many business scientific research fields including medicine genetics finance economics massive data usually carry two common features number predictors features tremendous diverge infinity sample size data distribution likely heteroscedastic response covariates two features observed two real data sets investigated paper usually hoped screening identifies important predictors among numerous candidates note ultimately data scientists large scale data still need construct comprehensive joint model accurately predict future outcome thus purely marginal screening approach usually adopted literature may adequately serve model building purpose contribute new screening method addresses issues complements existing methodology variable screening serves fast efficient computing device abundant feature screening methods proposed recent decades including sure independence screening sis fan first established sure screening property gaussian linear model sure independent ranking screening sirs zhu kendall based screening distance correlation based screening screening qasis empirical likelihood screening chang censored rank independence screening lifetime data cris song screening method based quantile correlation conditional quantile screening yin survival impaction index screening sii nonparametric independence screening nis fan among many others screening tools might suffer following two drawbacks first almost methods evaluate marginal association response predictors without adjusting external variables therefore jointly important markers may incorrectly screened marginal signal strong spurious markers ranked list hand marginally important variables may jointly ineffective hence including multivariate model may lead less convincing prediction take account joint effects marginal feature screening usually followed iterative calculation iterative sis isis fan computationally expensive come theoretical guarantee secondly distribution response covariates may rather different symmetric normal distribution often outliers affecting computed screening indices aforementioned procedures address robustness response knowledge none existing work addresses robustness covariates yet harder problem higher dimension aim tackle two problems new screener specifically address first issue develop joint feature screening method incorporating additional information recently conditional feature screening methods proposed instance liu considered sure independence screening procedure via conditional pearson correlation coefficient kernel smoothing method employed handle ultrahigh dimensional feature variables investigated fan cheng well addition xia considered robust screening method based conditional quantile correlation generalized conception however authors considered single conditional variable extend multivariate conditional variables chu studied several confounding variables barut extended fan song approach allow portion predictors conditional variables work provides general framework ultrahigh dimensional markers low dimensional confounders jointly considered screening process second issue incorporate robust correlation partial correlation screening methods nonparametric copula summary measure naturally leads screener outliers distribution best knowledge works applying classical dependence concept setting xia proposed robust conditional feature screening approach however method robust response covariates another relevant recent work contribution paper summarized follows firstly proposed double robust correlation copula popular bivariate function model nonlinear dependence paired variates see nelsen introduction copula characterizes empirical dependence two random variables evaluated level pair invariant monotone transformation variables study asymptotic process properties marginal variable screening approach via performed achieves desired sure screening consistency fan secondly extending correlation partial correlation cpc construct general framework joint screening importance marker evaluated presence conditional variables provides fast way conditional feature screening avoids iterated computation cpc also robust construction nonparametric estimation thus may reliable similar approach broader range application provide theoretical numerical support proposed screening method data analysis indicates final multivariate regression models built screening approach indeed predict outcome improved accuracy rest paper organized follows section presents empirical estimate asymptotic properties estimated functions established section well methodologies large sample properties cpc presented section implementation details different cases given section simulation studies two applications carried section section concludes paper technical proofs relegated appendix correlation variable screening consider two random variables let cumulative distribution function cdf assumed right continuous inf quantile joint cdf conditional distribution function given density use denote empirical versions respectively based sample size let banach space functions interval equipped uniform norm denotes collection bounded functions use denote convergence distribution correlation propose following correlation first term numerator copula function defined evaluated see corollary nelsen simple algebra follows cov indicator function since var indeed legitimate correlation coefficient lives like correlation measures equals independent measure nonlinear dependence thus incorporates kinds bivariate joint distribution addition indicator function unaffected outliers extreme values robust certain distribution note monotone transformation alter value given sample observations construct empirical estimate let following fix level write simplicity furthermore define weak convergence result established next theorem theorem let suppose marginal distributions continuously differentiable intervals positive derivatives respectively furthermore assume conditional density functions continuous product intervals denotes converge weakly gaussian process mean zero covariance function may write covariance function particular fixed independent producing null distribution used classical correlation studies compared result free moment conditions requires existence fourth order moment achieve convergence law order make statistical inference constructing confidence interval testing hypothesis like need estimate covariance function end denote use nonparametric approach like method obtain estimates respectively unknown replaced respectively therefore obtain estimates next give estimate denote obtain consistent estimate relevant paper consider single many candidates practice compare dependence strength two random variables may check difference particular may test hypothesis regarding difference given sample similarly define following theorem applied answer question theorem let suppose marginal distributions fxk continuously differentiable intervals positive derivatives fxk respectively furthermore assume conditional density functions fxk continuous product intervals gaussian process mean zero covariance function defined theorem follows theorem fixed pair except involved substituted mutually independent next estimate covariance function let given estimated variable screening suppose collect sample consisting independent copies response variable vector predictors number predictors exponential order sample size socalled ultrahigh dimension predictors irrelevant use screener identify sparse set informative predictors write instead emphasize dependence sample size empirical estimate given xij may select empirical active set threshold parameter controls size finally screened model using may lead sure independence screening sis property procedure abbreviated denote true active set write inf establish screening consistency need following conditions neighbourhood density uniformly bounded away zero infinity bounded derivative every neighbourhood density fxj uniformly bounded away zero infinity bounded derivative theorem screening property suppose condition holds constant exists positive constant sufficiently large exp max addition condition satisfied choosing exp sufficiently large cardinality set result implies select truly active predictors overwhelming probability dimensionality high exp similar many feature screening methods see yin example moreover result requires less condition predictors response due nonparametric nature fact condition involve moment assumption predictors response practice threshold parameter plays important role producing satisfied model small value result large number predictors screening turn leads many incorrect positives consider procedure determine threshold controlling false discovery rates fdr theorem covariate follows asymptotically use select varib small controls ables defined zhao following proposition fdr justifies fdr procedure condition proposition fdr property conditions condition theorem choose cdf standard normal variable number false positives tolerated constant partial correlation variable screening partial correlation cpc facilitate joint screening procedure define partial correlation cpc conditional random vector note implies parameters interpreted marginal increment conditional quantiles given respectively increasing unit cpc actually removing confounding effects linear partial correlation widely used regression diagnostics describes association response predictor conditional specifical values predictors unconditional value may spurious due lurking variables necessarily imply value conditional copula based version relatively robust real data analysis special case cpc constant sample observations obtain following estimate argmin zti let obtained quantile regression straightforwardly empirical estimator zti zti study asymptotic property denote zzt zzt zzt zzt zzt defined following asymptotic result cpc theorem let suppose uniformly positive definite matrices exists constant uniformly integrable uniformly bounded away zero infinity gaussian process mean zero covariance function another word conditional variable available asymptotic distribution theorem reduces theorem result implies fixed pair theorem used statistical inference find consistent estimate end let assume random vectors joint densities respectively denote marginal densities conditional densities given given respectively verified zzt zzt similarly zzt estimate quantities first calculate obtain corresponding quantile residb quantile regression estimates next provide estib zti uals zti mate estimators obtained similarly use nonparametric estimates used estimating section obtain estimates component using denote obtain data zti nonparametric kernel density estimate consistent regubased shown larity conditions unknown terms involved zti zti zti using approach consisi tent estimate thus obtained denoted next theorem used test whether two different random variables write let involved replaced respectively manner define accord ingly addition write zzt theorem let suppose matrices uniformly positive definite exists constant fxk fxk uniformly integrable uniformly bounded away zero infinity gaussian process mean zero covariance function given theorem fixed given sample observations asymptotic variance defined given estimated need estimate since rest obtain estimate unknown quantities involved estimated using previous methods end use following estimates zti zti zti zti argmin xik zti variable screening may propose joint robust screening using cpc two practical scenarios favor joint screening marginal screening first often may acquire variables addition ultrahigh dimensional covariates example studying relationship disease phenotype genetic variables may also patient demographical information environmental variables include consequently data set second even external variables may still necessary consider joint screening removing effects correlated components instance covariates xsj may closely correlated influence observed correlation indirectly subset also considered set referred conditional set relatively small size account scenarios may consider conditional variables xtsj paper allow conditional variables differ however simplicity presentation still use instead denote dimension principle need sure screening practice may select proper follows treat response predictors apply sensible marginal screening method pick top important predictors set conditional variables ultrahigh dimensional covariates xpn define cpc jth marker given way namely sample estimate given xij zti zti argmin xij zti cpc screening yields following empirical active set threshold parameter refer sure independence screening procedure clearly extends earlier conditional sure independence screening barut let true active set write inf simplicity still use denote underlying empirical cpc utilities respectively establish sure screening property need following conditions mild similarly imposed conditional density given satisfies lipschitz condition order neighborhood every conditional density fxj given satisfies lipschitz condition order fxj neighborhood exist finite constants max max max exist two positive finite constants cmin cmax cmin zzt zzt cmax zzt zzt stand minimum maximum eigenvalues zzt respectively theorem screening property suppose conditions hold constant exists positive constant sufficiently large exp max addition condition satisfied choosing exp sufficiently large conditional variables available proposed method handle dimensionality order exp dimension high order moreover proposed readily used ultrahigh dimensional data long section determine proper controlling fdr theorem covariate asymptotically select variables small following condition controls fdr proposition fdr property conditions condition theorem choose proposition constant implementation provide details implementation consider three practical types conditional variables following case available consider conditional variables namely xsj start empty active set step select confounding sets via partial correlation based consequential test step kth iteration log given update find variable index update step kth iteration set find update use final set selected covariates case available consider conditional variables target namely step compute cpc utility statistics step rank covariates terms decreasing order select top covariates final set selected covariates case available slightly modify algorithm case steps case except consider conditional variables xtsj iteration step case utilizes confounding information covariates case incorporates exogenous conditional information ignores confounding effect case flexible version incorporating types covariate information implement case real data analysis paper numerical studies simulation studies section conduct simulations examine finite sample performances two proposed correlations cpc well two screening procedures inference performance consider two simulation examples fixed dimension subsection illustrate practical performance estimated cpc respectively consider sample size set number repetitions examples example generate response two models exp exp standard bivariate normal distribution corr model covariates generated mixture distribution normal distribution probability cauchy distribution probability following standard bivariate normal distribution corr cauchy cauchy independent model error generated interest example test significance level various values consider different set implies holds true whereas rejected large probability values report empirical size power setup runs table observing table see proposed testing procedure based theorem performs satisfactorily across different quantile levels since empirical size close nominal level enlarging sample size generally tends improve performance also see runs away empirical power increases correlation low performance better example example generate conditional variables multivariate normal distribution method model model table empirical size power testing using across example generate model independent setups example example interest test significance level consider performance cpc corresponding empirical size power reported table similar conclusion drawn example numerical results empirically demonstrate theoretical result theorem valid screening performance throughout subsection adopt following simulation setup sample size covariate dimension number simulations cauchy table empirical size power testing using cpc example simulation setup moreover purpose comparison use three criteria evaluation first criterion minimum model size mms namely smallest number selected covariates contain active covariates robust standard deviation rsd second rank active covariates third proportion active covariates selected screening threshold specified log simulations report median mms example compare methods existing methods sis fan sirs zhu zhang example compare procedure aforementioned marginal screening methods well confounding effects arise covariates example employ algorithm case given section order compare example apply algorithm case compare example example used assess performance proposed let latent random vector normal distribution set correlation write component independent components standard cauchy distribution cauchy generate covariates mixture distribution simulate response data following three models sin exp sin exp log simulated two scenarios cauchy model set log bernoulli resulting screening results terms mms presented table eyeballing table make key observations models outperforms sis sirs case response covariates thus traditional linear correlation screening methods fail work methods also comparable nonparametric kendall achieves high accuracy slower due numerical integration implementation difficult case hard screen accurately case still performs much better methods example example designed evaluate performance proposed cpcsis generate response following two models models model consider observed covariates implying covariates outliers response fully dependent observed random noise setting desired expect performs better since normal method sis sirs sis sirs sis sirs cauchy mms rsd mms rsd model model model cauchy mms rsd mms rsd table simulation results example mms stands median minimum model size robust standard deviations rsd given parenthesis proportion screened sets cover active predictors screening parameter log model generate covariates mixture distribution element independent distributed cauchy model covariates contaminated outliers heterogenicity response stems merely random error addition model let except model set rest model models population level covariate marginally uncorrelated consider two cases simulation comparison tables report screening results regarding rank mms tables see marginal screening approaches fail pick covariate large values rank work much better marginal methods particularly model competitive performance model covariates highly correlated proposal best performance example example examine case since conditional variables selected simply employ proposed variable screening generate response model distribution example cauchy model error example simulation results given table expected see marginal screening procedures fail work since unable identify covariate proposed still performs better terms mms method sis sirs sis sirs mms rsd model cauchy mms rsd table simulation results example indicates median rank relevant predictors mms stands median minimum model size robust standard deviations rsd given parenthesis method sis sirs sis sirs mms rsd model cauchy mms rsd table simulation results example indicates median rank relevant predictors mms stands median minimum model size robust standard deviations rsd given parenthesis method sis sirs sis sirs mms rsd cauchy mms rsd table simulation results example indicates median rank relevant predictors mms stands median minimum model size robust standard deviations rsd given parenthesis real data applications rats data illustrate gene expression data male rats weeks old including expression measurements gene probes analyzed scheetz investigation gene regulation mammalian available ftp follow consider expression gene probe response variable since identified cause syndrome closely associated human hereditary disease retina chiang gene probes treated covariates first apply iglewicz hoaglin approach check outliers constructs mad denotes median absolute deviation recommends labeled outliers method quite popular real applications engineering consequently find gene probes one outliers figure displays two selected genes response one employs conventional screening method ignoring outliers would lead inappropriate results methods may thus robust situation data analysis sample rpn report overlaps top log selected genes various methods table see different methods select quite different genes low level agreement overlooked practice robust joint screening methods like propose paper lead entirely different set genes otherwise screened conventional marginal screening approaches notice couple overlaps partly conditional screening procedures able adjust confounder effects rats data breast cancer data name gene gene gene expressions gene expressions name gene gene gene gene gene gene figure response two randomly selected genes two datasets left panel rats data right panel breast cancer data table gives summary top gene probes different methods along pvalue resulted marginal use genes regressors build joint statistical models predict linear regression quantile regression considered purpose displays mean prediction errors random partitions partition ratio training sample test sample partition computed average ybi testing set ybi predicted value ith test data point using model constructed training sample genes table see proposed partial correlation screening performs best smallest prediction error nice prediction result may attributed fact cpc selects appropriate markers joint modelling addressing distribution heterogeneity conditional effects heterogeneity problem typically inflates variance purely marginal screener could introduce bias prediction error consists variance bias components thus much smaller employing cpc screening method sis sirs sis sirs kendall table overlaps selected genes using various approaches rats data screening threshold parameter set log method applies algorithm case rank sis sirs table summary top gene probes selected different screening methods rats data means selected gene computed cumulative distribution function standard normal random variable mean prediction errors random partitions partition ratio training sample testing sample prediction error defined average ybi testing set indicate ybi predicted value via fitting median regression model linear model respectively using top genes selected breast cancer data breast cancer second common cancer world leading cause women nearly million new cases diagnosed according worldwide meanwhile approximately new cases invasive breast cancer breast cancer deaths expected occur among women reported desantis although major progresses breast cancer treatment made also limited ability predict metastatic behavior tumor van veer first study breast cancer study involving lymph breast cancer patients years old younger developed distant metastases within years metastatic outcome coded remained metastases free least years metastatic outcome coded expression data set clinical variables well analyzed many papers classification boulesteix among others study removing genes missing values expression levels gene probes entering next analysis addition gene expression measurements data several clinical factors available well interest identify gene probes affect tumor size given clinical factors including age histological grade angioinvasion lymphocytic infiltration estrogen receptor progesterone receiptor status therefore data rpn analysis using method outlier detection find gene probes least outliers suggesting approximately three quarters overall gene probes contain extremely large values right panel figure displays empirical distribution response two typical covariates thus suitable apply robust joint screening approach proposed consider three cases http sis sirs kendall sis sirs table overlaps selected genes probes using various approaches breast cancer data screening threshold parameter set log method means case indicates case stands case discussed section denote methods respectively table gives overlaps selected genes various methods similar conclusion rats data analysis made furthermore table presents summary top gene probes selected various methods results table empirically verifies proposed case satisfactory performance prediction conclusion discussion propose correlation partial correlation facilitate robust marginal joint screening ultrahigh dimensional data sets large sample properties estimated correlation sure screening properties cpc screeners provided empirical studies including simulations two data applications show proposed outperform existing variable screening approaches outliers present covariates response therefore current proposals applicable ultrahigh dimensional heterogeneous data provide guideline carry variable screening follows response predictors normal without sis rank rank sirs table summary top gene probes selected different screening methods breast cancer data means selected gene computed cumulative distribution function standard normal random variable mean prediction errors random partitions partition ratio training sample testing sample prediction error defined average ybi testing set indicate ybi predicted value via fitting median regression model linear model respectively using top genes selected heteroscedastic variance predictors low correlation marginal screening methods sis sirs applied response contains outliers follows heavy tail distribution covariates normal robust screening methods kendall sis employed covariates highly correlated conditional variables available conditional screening procedures csis work data heteroscedastic response covariates covariates may highly correlated applied copula formulation may suggest many possible extensions methodology particular may consider censored survival time outcome framework estimation correlation partial correlation needs incorporate random censoring data need invoke complicated empirical process theories argue weak convergence results furthermore may even allow predictors censored see cheng fine cheng earlier cussion extension data requires development acknowledgement thank professors runze shujie sharing code implementation qpc approach appendix proofs following lemma result bivariate empirical process theory lemma lemma van der vaart wellner fix suppose distribution function marginal distribution functions continuously differentiable intervals positive derivatives respectively furthermore assume exist continuous product intervals map tangentially derivative given proof theorem definition employing lemma empirical process theory example van der vaart wellner able show means asymptotically equivalent given furthermore easily verified hence follows defined line theorem class indicator functions donsker thus donsker donsker theorem denotes converge weakly process mean zero covariance function cov given theorem proof theorem following arguments proof theorem show given theorem since donsker also donsker donsker theorem process mean zero covariance function cov given theorem proof theorem first prove first statement observe xij xij xij follows going bound tail probabilities respectively since xij var xij applying bernstein inequality lemma van der vaart wellner gives max exp next consider note xij xij say hand follows lemma serfling log almost surely let inf fxj positive condition event xij xij xij xij fxj xij fxj fxj xij fxj fxj itive constant condition since xij var xij xij applying bernstein inequality lemma van der vaart wellner yields xij fxj xij fxj exp implies xij xij xij xij fxj xij fxj max exp third inequality due similarly condition show max exp positive constants therefore combining obtain max max max max exp exp exp exp positive constants max provided max using basic inequality union bound probability follows max max max hence conjunction result proves desired result letting prove second assertion choice using condition min min min max min max exp thus completes proof proof proposition observe expected false discovery rate rewritten theorem result follows exists constant sup combining results obtain plugging inequality yields result proof theorem first observe using arguments koenker obtain fyi zti zti zti zti let zti zti follows nun zti zti nun say employing result empirical processes corollary van der vaart wellner obtain zti zti taylor expansion using zti finally combining recalling definitions zti zti implies checked donsker therefore donsker theorem conclude result proof theorem result proved using arguments proof theorem omit details prove theorem need following two lemmas verified using arguments proofs omitted lemma conditions given constant exist constant exp lemma conditions every given constant exist constant exp proof theorem prove first part result need prove following fact conditions every given constant exist positive constants xij xij zti zti exp end first make decompose xij xij zti zti zti xij zti zti xij zti zti xij zti zti xij xij zti next going find probability bounds let zti xij zti since thence var obtain exp applying bernstein inequality lemma van der vaart wellner zti follows observe zti zti zti let zti zti condition exist finite positive constant zti zti cmax cmax last inequality uses inequality analogously obtain positive constant notice hence applying bernstein inequality lemma van der vaart wellner yields exp letting using inequality obtain exp first probability right hand side bounded lemma results exp exp provided similarly using lemma conditions exp exp positive constant provided constant let max max combining results gives max exp exp exp provided max result follows letting max addition applying using similar arguments proof theorem conclude first part theorem second part result verify following proof theorem proof proposition proof directly follows proof proposition details omitted references bahadur note quantiles large samples annals mathematical statistics barut fan verhasselt conditional sure independence screening journal american statistical association boulesteix porzelius daumer classification clinical predictors combined classifiers additional predictive value bioinformatics chang tang marginal empirical likelihood sure independence feature screening annals statistics chang tang local independence feature screening nonparametric semiparametric models marginal empirical likelihood annals statistics cheng fine nonparametric estimation cross hazard ratio bivariate competing risks data biometrika cheng diagnostic accuracy analysis censored outcome censored predictor journal statistical planning inference cheng honda peng nonparametric independence screening structure identification dimensional longitudinal data annals statistics chiang beck yan homozygosity mapping snp arrays identifies ubiquitin ligase bardetbiedl syndrome gene proceedings national academy sciences chu reimherr feature screening coefficient models ultrahigh dimensional longitudinal data annals applied statistics desantis sauer newman jemal breast cancer statistics racial disparity mortality state cancer journal clinicians harvard fan sure independence screening ultrahigh dimensional feature space journal royal statistical society series fan feng song nonparametric independence screening sparse additive models journal american statistical association fan dai nonparametric independent screening sparse dimensional varying coefficient models journal american statistical association fan song sure independence screening generalized linear models annals statistics wang hong variable screening heterogeneous data annals statistics iglewicz hoaglin volume detect handle outliers asqc basic references quality control statistical techniques edward mykytka editor koenker quantile regression new york cambridge university press tsai quantile correlations quantile autoregressive modeling journal american statistical association peng zhang zhu robust rank correlation based screening annals statistics zheng peng huang survival impact index ultrahighdimensional screening survival outcomes biometrics zhong zhu feature screening via distance correlation learning journal american statistical association liu feature selection varying coefficient models covariates journal american statistical association zhang robust feature screening via quantile correlation journal multivariate analysis tsai variable screening via quantile partial correlation journal american statistical association nelsen introduction copulas springer science business media serfling approximation theorems mathematical statistics wiley new york scheetz kim swiderski regulation gene expression mammalian eye relevance eye disease proceedings national academy sciences united states america song jeng censored rank independence screening survival data biometrika van der vaart wellner weak convergence empirical processes application statistics springer new york vant veer dai van vijver gene expression profiling predicts clinical outcome breast cancer nature yin conditional qunatile screening heterogeneous data biometrika xia conditional quantile correlation learning ultrahigh dimensional varying coefficient models application survival analysis statistica sinica accepted adjusting confounders ranking biomarkers roc approach briefings bioinformatics zhao principled sure independence screening cox models covariates journal multivariate analysis zhu zhu feature screening ultrahigh dimensional data journal american statistical association
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feb group alan logan abstract use group build framework construction tractable groups pathological outer automorphism groups apply framework strong form question outer automorphism groups finitely presented residually finite groups introduction matumoto proved every group realised outer automorphism group group many authors refined result placing restrictions groups asked following question problem question every countable group realised outer automorphism group finitely generated residually finite group main result theorem relates strong form question group taken finitely presented rather finitely generated almost nothing known general outer automorphism groups finitely presented residually finite groups group serre property every action tree global fixed point related property splitting free product amalgamation hnnextension theorem theorem suppose finitely presented group serre property exists finitely presented group embeds finite index moreover residually finite chosen residually finite mathematics subject classification key words phrases outer automorphism groups residually finite groups alan logan examples finitely presented groups serre property triangle groups finitely presented groups kazhdan property random groups sense gromov thompson groups groups proof theorem two main ingredients firstly theorem applies certain framework built paper explained secondly theorem applies version rips construction rips construction usually applied obtain finitely generated nonfinitely presentable groups pathological properties application finitely presented groups unusual framework paper principally consists analysis group group defined section subgroup arises naturally context serre theory defined section class tractable tractable mean easy class groups work possess nicer properties general section package technical results analysis framework construction tractable groups pathological outer automorphism groups framework applied prove theorem see also describing theorem striking special case framework described section theorem gives convenient description certain residually finite hnnextensions motivation comes question assumption residually finite completely natural functions write write aut define fix define aut subgroup aut theorem let aut assume fix finitely generated residually finite short exact sequence inn inn subgroup group theorem follows immediately theorems indeed theorem classifies conditions applicable omitted brevity note proof theorem requires general result theorem groups theorem necessarily residually finite note labels results previous paper cited certain results current paper labels results current paper subsequently changed record relevant changes theorem theorem lemma lemma lemma lemma proposition proposition outline paper section define explain tractability section define group pettet group section prove conditions implying group whole outer automorphism group main result theorem section give description group main result theorem section prove results relating structure jiang group commutativity base group main result theorem section package technical results paper framework construction tractable groups pathological outer automorphism groups prove theorem acknowledgements author would like thank stephen pride tara brendle many helpful discussions paper let group let isomorphism subgroups group relative presentation write hnnextension relative presentation alan logan aut isomorphism associated subgroups induced automorphism write throughout paper use letters general form embeddings two associated subgroups may completely different example may normal malnormal contrast form embeddings necessarily normal malnormal practice image group may disregarded example supposing finitely generated residually finite residually finite residually separable current paper theorems mention image group groups tam infinite cyclic group associated subgroups provide standard examples pathological properties example groups hopfian gcd hopfian finite residually finite note finitely generated residually finite group hopfian generated outer automorphism group example finitely generated however groups residually finite classification virtually cyclic outer automorphism group two paragraphs demonstrate tractable certain cases nicer properties general therefore increasing complexity base group maintains tractability allow pathological properties see example cubulated hyperbolic group serre property triangle group notation write write inner automorphism defined contrast notation used makes certain proofs readable mapping torus group see section different extends inner automorphism group following proposition underlies whole paper implies embeds aut paper represents effort obtain conditions implying close example isomorphic commensurable proposition let define map aut homomorphism kernel proof routine prove aut homomorphism kernel britton lemma section framework built paper explained section pathological properties inherited particular proposition residually finite finitely generated residually finite statement finite index see later theorem fix also embeds lemma gives version embedding thus properties certain sense bestowed upon note infinite cyclic hence group cyclic bestows pathological properties upon group paper analyses group automorphism induced proof theorem applies analysis section define group outh hnnextension also define pettet group outh applied one main technical results theorem pettet group lies group full outer automorphism group outh outh alan logan note groups may defined graphs groups general brevity define theory definitions section certain proofs section apply theory relevant definitions notation taken serre book acts standard way tree called tree precisely conjugates base group precisely conjugates associated group vertices write geodesic connecting define length denoted number positive edges group aut denotes coset containing group outh subgroup consisting representative helpful geometric view group auth full outh aut auth acts diagram figure commutes group fundamental group graph groups described levitt described group related investigation hyperbolic group raptis gave conditions implying generalised group theorem mirrors result pettet group pettet group outh subgroup consisting representative helpful geometric view pettet group auth full outh aut auth acts tree diagram figure commutes pettet studied auth implicitly pettet group focused conditions pettet group equal group related conditions theorem two examples give two examples demonstrate group auth auth aut aut figure diagram commutes auth full group tree canonical homomorphism inn figure diagram commutes auth full pettet group vertices tree canonical homomorphism inn general outh outh outh first example due levitt consider base group hai map automorphism clearly however shall write free group free basis consider base group clearly hence map automorphism however conjugate power group versus group outh interest full outer automorphism group outh section prove theorem gives conditions implying outh prove conditions use pettet group outh defined section throughout remainder paper assume unless explicitly state otherwise emphasize assumption necessary results lemma gives condition implying outh lemma gives conditions implying outh outh lemmas combine prove theorem alan logan conditions implying outh lemma proves serre property pettet group equal full outer automorphism group outh outh note similarity lemma comment pettet introduction principal difference pettet additionally assuming conditions implying outh outh require preliminary assumptions proof lemma shows conditions lemma base group always conjugacy maximal proof lemma may written purely algebraically used theory similar proofs section recall lemma let serre property outh proof prove aut sufficient representative let aut consider action tree serre property stabilises vertex hence similarly suppose shall find contradiction exist vertices contradiction stabilises geodesic particular stabilises initial edge therefore exists hence contradiction conditions implying outh outh work towards proving lemma gives sufficient conditions group equal pettet group outh outh conjugacy begin proof lemma lemma describes conjugacy word product form words write exactly word say contains subwords form group every element representative word britton lemma lemma let exists note dependent proof let arbitrary let vertices tree let word representing induct length geodesic result trivially holds induction step note decomposes also decomposition result holds required prove lemma corresponds condition lemma condition theorem lemma let suppose outh outh additionally suppose exist exist exists proof suppose outh outh let recall acts vertices tree let connected single edge suppose write consider geodesic auth note stabilises stabilises algebraically note also first edge geodesic stabiliser hconjugate either without loss generality assume either exists edge geodesic gej lemma therefore gej alan logan gej gej lemma recall contains one edge let adjacent gej noting therefore every element equal element form hence form fixed hence assumptions lemma required residual finiteness prove lemma provides conditions outh outh hold general lemma essentially lemma automorphisminduced conditions either lemmas hold residually finite gave conditions general residually finite conditions lemma strong form residual finiteness conditions subgroup characteristic aut lemma let isomorphism subgroups let finitely generated residually finite suppose positive integer elements exists characteristic subgroup finite index empty set maps maps outh outh proof note induces isomorphism onto write natural map induces homomorphism onto proof theorem note finite theorem suppose outh outh exists auth aut note either therefore pick elements appropriate pick characteristic group appropriately form corresponding group induces map prove aut firstly note aut characteristic homomorphisms hence homomorphism surjective generate injective finitely generated residually finite theorem hence hopfian auth auth aut word appropriate hence contradiction difference conditions lemma require subgroup characteristic rather normal suppose tretkoff instead assume normal finitely generated exists characteristic subgroup satisfies first condition however clear characteristic also satisfies second condition example subgroup observation gives following lemma corresponds condition lemma also compare lemma subgroup residually separable exists finite index normal subgroup written lemma let let finitely generated residually finite residually separable outh outh proof residually separable conditions lemma hold result follows alternative proof lemma follows combining lemma result shirvani lemma see conditions lemma imply result follows condition lemma proof condition independent proof condition proof lemma lemma combines lemma lemma alan logan results pettet subgroup group conjugacy maximal exist lemma let suppose one following holds conjugacy maximal exist exist finitely generated residually finite residually separable finitely generated residually finite outh outh proof holds result known hold lemma theorem note impossible hold simultaneously inequality outh held would hold lemma obtain outh contradiction hence holds result holds holds result holds lemma holds holds result follows group versus theorem gives conditions implying group full outer automorphism group theorem let aut serre property outh one following statements holds outh outh conjugacy maximal exist exist residually separable residually finite proof theorem follows immediately lemmas note theorem obtains outh two disjoint steps proves outh separately outh outh pettet proved similar result theorem involving conditions theorem obtain disjoint steps due general setting group theorem gives concrete method proving automorphisminduced residually finite exists auth auth outh outh residually finite lemma provides way proving inequality condition implying outh outh know conditions lemma necessary sufficient outh outh hold reading closely proof lemma one might suspect lemma demonstrates close conditions lemma necessary sufficient lemma demonstrates proving conditions lemma fail compatibly one another outh outh note two conditions fail replacing necessary exists compatibility require take lemma let suppose also outh outh proof replacing may assume note word prove map automorphism auth auth result follows homomorphism see surjection prove begin following note alan logan therefore surjective injective invertible inverse describing group section prove theorem gives sequence description group combined theorem allow description gave decomposition group outh general description took form filtration however description contains associated subgroups description theorem contains difference description kernel outh short exact sequence theorem outh formally defined specifically find outh outh general therefore composition outh different decomposition orem simpler description outh fundamental framework described section essential proving theorem results proof theorem based two results state translated results definitions setting hnnextensions first result classification elements group outh theorem theorem let aut group map automorphism aut map automorphism moreover every element outh representative auth form form second result apply requires certain definitions restrictions second equivalent definition outh follow theorem outh inn subgroup clearly index two outh exists automorphism index one otherwise second result apply initial two steps filtration decomposition outh theorem recall subgroup aut defined introduction theorem let surjection inn inn kernel outh either outh exists aut whence index two outh paper restriction theorem hnnextensions cite general theorem certain errors crept section alan logan subgroup outh wish describe kernel outh theorem combining description obtain theorem proves main result section theorem description given lemmas lemmas mention subgroup outh lemmas also apply following technical result lemma let let suppose following hold ytg proof begin writing word either empty begins assume form hzw chosen word hzw appropriate contradiction britton lemma section hence empty follows required proofs lemmas certain proofs section require calculations automorphisms form note following identities certain calculations written terms cosets follow immediately two identities lemma map outh surjective homomorphism kernel proof suppose map note pair defines automorphism theorem surjective group also homomorphism ker ker see map suppose sufficient prove note aut aut therefore identity becomes hence therefore conditions lemma hold two underlying identities hence required note subgroup outh outh lemma need obtain semidirect product statement theorem possible splitting map discussed section lemma define result follows begin proof prove note homomorphic image therefore prove inn clearly inn hand suppose inn see note conditions lemma hold following take therefore first identity implies second implies hence describing outh give description jiang group outh theorem let aut let alan logan normal subgroup diagram exact sequences outh inn inn either outh exists aut whence index two outh proof theorem sufficient prove outh decomposes statement theorem follows lemmas alternative view recall definition aut following isomorphism substituted theorem inn inn inn inn isomorphism holds inn inn inn subgroup inn clear definition impact commutativity reader might feel cheated decomposition outh theorem split subgroups defined section indeed outh outh however elements form contained subgroups decomposition outh theorem split general section prove theorem group assumes fix decomposition outh split also prove related results assume commutativity conditions splitting description theorem begin reversing map lemma allows determine splits lemma map outh homomorphism kernel fix proof map homomorphism note ker see ker suppose conditions lemma hold two underlying identities hence second identity implies fix therefore fix required hypotheses theorem theorem additional assumption fix vertical exact sequence theorem splits theorem immediately follow theorem subgroup corresponds factor corresponds factor action theorem let theorem additionally assume fix short exact sequence inn inn conditions index outh theorem proof fix lemma implies map lemma splits theorem follows theorem virtually embedding lemma applies lemma obtain conditions implying subgroup alan logan embeds basis framework described section see also discussion proposition lemma fundamental paper hypotheses lemma occur example hyperbolic contains elements finite order lemma fix embeds outh index dividing proof recall fix prove beds outh note map induces maps maps canonical maps sufficient prove second map suppose exists however contradiction hence embeds outn prove index outh divides first note considering map index outh clearly divides outh therefore prove outh divides routine construct isomorphism necessarily normal take ing outh outh divides divides lemma required describing description theorems used describe full aut however description necessarily useful lemma gives conditions allow complete description lemma applied paper recall lemma inn described follows inn group proof note outh inn theorem gives representatives generate subgroup aut contains inner automorphisms therefore consider set set contains representative coset inn hence inn inn inn therefore inn result follows proposition inn tractable groups pathological outer automorphism groups section package technical results paper framework construction tractable groups pathological outer automorphism groups apply framework prove theorem introduction applications framework found framework aforementioned framework pathological properties inherited framework follows obtain required properties stipulated group finite etc apply theorems lemma obtain description outh terms apply theorem obtain outh obtain description terms apply lemma obtain auth inn obtain description aut terms proof theorem prove theorem proof theorem suppose finitely presented serre property exist hyperbolic groups serre property hence exists hyperbolic group serre property finitely generated noncyclic normal subgroup form note residually finite residually finite proposition alan logan firstly note outh follows theorem normal conjugacy maximal serre property secondly note embeds finite index outh follows theorem hyperbolic finite hyperbolic serre property hence outh required references logan pride automata products groups preparation baumslag automorphism groups residually finite groups london math soc braun outer automorphisms locally finite algebra bass jiang automorphism groups tree actions graphs groups pure appl algebra belegradek osin rips construction kazhdan property groups geom dyn baumslag tretkoff residually finite hnn extensions comm algebra bumagin wise every group outer automorphism group finitely generated group pure appl algebra collins levin automorphisms hopficity certain groups arch math basel droste giraudet groups outer automorphism groups simple groups london math soc dahmani guirardel przytycki random groups split math ann farley proof thompson groups infinitely many relative ends group theory frigerio martelli countable groups mapping class groups hyperbolic math res lett gilbert howie metaftsis raptis tree actions automorphism groups group theory paras outer automorphism groups metabelian groups pure appl algebra group kato higher dimensional thompson groups serre property arxiv kojima isometry transformations hyperbolic topology appl levitt automorphisms hyperbolic groups graphs groups geom dedicata automorphism group generalized groups geometry topology logan outer automorphism groups finitely generated residually finite groups algebra question bumagin wise new york math every group outer automorphism group hnnextension fixed triangle group residual finiteness hyperbolic lyndon schupp combinatorial group theory springerverlag york ergebnisse der mathematik und ihrer grenzgebiete band matumoto group represented outerautomorphism group hiroshima math meskin nonresidually finite groups trans amer math soc minasyan groups finitely many conjugacy classes automorphisms comment math helv pettet automorphism group graph product groups comm algebra stephen pride finitely presented groups cohomological dimension two property pure appl algebra serre amalgames points fixes proceedings second international conference theory groups australian nat canberra springer berlin lecture notes vol trees springer shirvani residually finite arch math basel watatani property kazhdan implies property serre math japon school mathematics statistics university glasgow glasgow address
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partial consistency sparse incidental parameters jianqing runlong xiaofeng aug princeton university johns hopkins university march abstract penalized estimation principle fundamental problems literature extensively successfully applied various models structural parameters contrast paper apply penalization principle linear regression model vector structural parameters vector sparse incidental parameters estimators structural parameters derive consistency asymptotic normality reveals oracle property however penalized estimators incidental parameters possess partial selection consistency consistency interesting partial consistency phenomenon structural parameters consistently estimated incidental ones structural parameters also considered alternative penalized estimator fewer possible asymptotic distributions thus suitable statistical inferences extend methods results case dimension structural parameter vector diverges slower sample size approach selecting penalty regularization parameter provided performance penalized estimators structural parameters evaluated simulations real data set analyzed keywords structural parameters sparse incidental parameters penalized estimation partial consistency oracle property estimation confidence intervals introduction since pioneering papers tibshirani fan penalized estimation methodology exploiting sparsity studied extensively example zhao provides almost necessary sufficient condition namely irrepresentable condition lasso estimator strong sign consistent fan shows oracle property holds folded concave penalized estimator ultrahigh dimensionality overview topic see fan aforementioned papers consider models structural parameters related every data point contrast consider paper another type models structural parameters also incidental parameters related one data point specifically suppose data following linear model vector incidental parameters sparse vector structural parameters main interest covariate vector random error let model different data point depends different subset model arises working model estimation fan considers hypothesis testing problem arbitrary dependence test statistics principal factor approximation method proposed fan dependent test statistics decomposed ith row first unstandardized principal components denoted common factor drives dependence among test statistics realized unobserved factor critical false discovery proportion fdp estimation power improvements removing common factor bti test statistics hence important goal estimate given many applications hypothesis testing parameters sparse example association studies show expression level gene highly related phenotype syndrome interest test association millions snp gene expression level framework fan stands association ith snp associated gene expression level otherwise associated since snp associated gene expression level reasonable assume sparse replacing respectively obtain model formally interest study model independently simplifications although model emerges critical component estimating fdp fan stands interest example applications signals nonzero interesting learn reflects relationship covariates response another example nonzero might measurement recording errors responses cases model suitable modeling data contaminated responses method producing reliable estimator essentially robust replacement ordinary least squares estimate sensitive outliers several models structural incidental parameters first studied seminal paper neyman scott points inconsistency maximum likelihood estimators mle structural parameters presence large number incidental parameters provides modified mle however method work model due exploration sparsity incidental parameters kiefer wolfowitz shows consistency mle structural parameters incidental parameters assumed common distribution eliminate essential issue incidental parameters randomizing contrast paper considers deterministic incidental parameters handles issue penalization sparsity assumption basu considers elimination nuisance parameters via marginalizing conditioning methods moreira solves incidental parameter problem invariance principle review incidental parameter problems statistics economics see lancaster without loss generality suppose first incidental parameters nonvanishing remaining zero model written matrix form identity matrix generic block zeros although sparse problem matrix satisfy sufficient conditions theoretical results zhao fan due inconsistency estimation incidental parameters penalty simply placed parameters details see supplement paper investigate penalized estimator defined argmin penalty function regularization parameter since incidental parameters sparse penalty imposed iterative algorithm proposed numerically compute estimators estimator possesses consistency asymptotic normality oracle property hand nonvanishing elements consistently estimated even known partial consistency phenomenon penalized estimation method estimation also propose twostep method whose first step designed eliminate influence data large incidental parameters estimator method fewer possible asymptotic distributions thus suitable constructing confidence regions asymptotically equivalent estimator sizes nonzero incidental parameters small enough incidental parameters really sparse also method improves convergence rate efficiency method challenging situations large nonzero incidental parameters increase asymptotic covariance even reduce convergence rate method rest paper organized follows section model penalized estimation method formally introduced corresponding penalized estimators characterized section asymptotic properties penalized estimators derived penalized estimator proposed theoretical properties obtained also provide approach selecting regularization parameter section consider case number covariates grows slower sample size section present simulation results analyze read data set section concludes paper discussion proofs theoretical results relegated appendix supplements model method matrix form model given covariates independent identically distributed copies random vector mean zero covariance matrix independent random errors copies random variable mean zero variance denote respectively assumption covariates random errors assumption exist positive sequences max max stands norm suppose three types incidental parameters model simplicity indexes first incidental parameters large sense max next ones nonzero bounded last ones zero note unknown large bounded zero sparsity understood denote vectors three types incidental parameters respectively penalized estimation written argmin penalty function soft lasso hard scad general folded concave penalty function fan simplicity next consider soft penalty function cases hard scad penalties considered similar way lemma supplement necessary sufficient condition minimizer sign sign function numerically special structure suggests marginal decent algorithm minimization problem iteratively computes argmin argmin convergence advantage algorithm exist analytic solutions two minimization problems respectively estimators residuals ordinary estimator responses section next number covariates assumed fixed integer case diverging infinity considered section case finite make following assumption assumption regularization parameter satisfies min defined constant greater simplicity abbreviate probability going one stopping rule algorithm based successive difference proposition supplement iterative algorithm stops second iteration given initial estimator bounded suppose theoretical limit corresponding limit estimator solution following system nonlinear equations applied component follows sign returns maximum value input zero lemma supplement necessary sufficient condition minimizer solution equations hence minimizer also denoted note also minimizer profiled loss function function given interestingly profiled loss function criterion function equipped famous huber loss function see huber huber specifically profiled loss function expressed exactly huber loss function optimal minimax sense equivalence penalized estimation huber robust estimation indicates penalization principle versatile naturally produce important loss function robust statistics equivalence also provides formal endorsement least absolute deviation robust regression lad fan indicates better use data lad regression rather worthwhile note penalized estimation formally equal huber model considers deterministic sparse incidental parameters model huber works assumes random contamination kiefer wolfowitz recently appear papers robust regression settings see example chen zwald bean portnoy provide high level review literature robust statistics equations solution general problem following theoretical analysis based characterization end section provide analysis notations expansion let subset straightforward show index sets defined similarly except absolute operation omitted replaced defined similarly except replaced note index sets depend asymptotic properties section consider asymptotic properties penalized estimators assumption together assumption enables penalized estimation method distinguish large incidental parameters others thus simplifies asymptotic properties index sets sij sense become independent denote hypercube constant lemma index sets sij assumptions every every limit index sets lemma solution analytic expression derive asymptotic properties analysis needs following assumption assumption exists constant ekx diverges infinity following result shows existence unique consistent estimator theorem existence consistency assumptions either assumption holds every fixed exists unique estimator theorem two different kinds sufficient conditions size bounded incidental parameters assumption norms come different analysis approaches term one imply details see supplement specially consistent theorem next consider asymptotic distributions consistent estimator obtained orem without loss generality assume sizes index sets asymptotically equivalent constant similar theorem two different sets conditions corresponding two different analysis approaches denote asymptotic equivalence theorem asymptotic distributions assumptions suppose holds assumption hold main case every constant incidental parameters really sparse size large incidental parameters small size magnitude bounded incidental parameters also small conditions case tends hold case interest denote main case cases presented provide relatively complete picture asymptotic distributions fact theorem supplement shows possible asymptotic distributions note constant appear limit distributions theorem due cancelation consistency emerges case large case impact large incidental parameters big handled efficiently penalized estimation case one direction condition limit distribution become case direction increases approaches case boundary phenomenon spirit similar tang specially main case theorem remark oracle property suppose oracle tells true adjusted responses oracle estimator given limiting distribution comparing main case theorems follows penalized estimator enjoys oracle property although mainly interested estimation also obtain estimator sgn denote theorem partial selection consistency assumptions theorem shows indexes estimated correctly wrongly call partial selection consistency phenomenon estimation theorems shows penalized estimator multiple different limit distributions complicates application theorems practice addition convergence rate less optimal rate challenging cases impact large incidental parameters substantial address issues propose following estimation method firstly apply penalized estimation let secondly define estimator consists whose indexes consists corresponding following theorem shows consistent asymptotic gaussian theorem consistency asymptotic normality suppose assumptions hold either assumption holds either holds assumptions hold comparing theorem theorem see three possible asymptotic distributions one since conditions disappear method identifies removes large incidental parameters exploiting partial selection consistency property theorem estimator improves convergence rate optimal one estimator challenging case advantages suggest use method make statistical inferences incidental parameters sparse sense size magnitude bounded incidental parameters small follows theorem confidence region asymptotic confidence level given upper square root distribution degrees freedom component asymptotic confidence interval given square root entry upper confidence region interval involve unknown parameters estimated law large numbers consistent lemma supplement also consistent hence replacing confidence region interval resulting confidence region interval asymptotic confidence level theoretical regularization parameters assumption theoretical regularization parameter depends also crucial boundary conditions asymptotic properties penalized estimators assumption determined distributions respectively interest explicitly derive typical cases covariates errors covariates bounded random errors follow let dcx log satisfy assumption specification assumption becomes log min follow maximum diagonal elements take respectively denote log log becomes log min case exponentially tailed random variables considered supplement although theoretical specification guaranties desired asymptotic properties datadriven specification interest practice popular way specify use validation set however needs made little contaminated possible propose following procedure identify regularization parameter step training testing data sets ols apply ordinary least squares ols data obtain residuals ols ols identify set pure data corresponding npure smallest values compute updated ols estimator residuals pure data obtain updated identify updated pure data set corresponding npure smallest label remaining contaminated data set randomly select subset updated pure data set testing set merge remaining pure data set contaminated one training set step range regularization parameter compute standard deviation residuals pure data set set positive constants step regularization parameter grid point interval apply penalized method ing set obtain estimator train corresponding test error test testing set train identify regularization parameter minimizes test among grid points simple procedure certainly improved example step repeated times obtain better pure data set step two range also obtained quantiles quantiles also hybrid quantities based determine good performance regularization parameter demonstrated subsection diverging number structural parameters sections considered model assumption number covariates fixed integer however moderate large number covariates appropriate assume diverges infinity sample size section consider model assumption since number covariates grows orderly slower sample size chose continue use penalized estimation penalized estimation corresponding estimators still denoted keep mind dimensions diverge infinity characterizations lemmas still valid since results iteration algorithm also stops second iteration shown proposition supplement critical properly specify regularization parameter case diverging number covariates assumption regularization parameter satisfies defined comparing assumption assumption main difference formation changed fact also depends shown supplement difference caused assumption diverges lemma index sets sij assumptions conclusion lemma holds thus still valid crucial analytic expression derive theoretical properties similar previous section additional technical complexity caused diverging dimension denote frobenius norm average square root fourth marginal moments make following assumptions assumption bounded assumption bounded theorem existence consistency suppose assumptions hold exists sequence positive numbers depending every fixed exists unique estimator next consider asymptotic distribution since dimension diverges infinity following fan appropriate study linear maps let matrix fixed integer atn largest eigenvalue denote smallest eigenvalue max var min var max abbreviate respect wrt assume assumption bounded away zero implies assumption assumption bounded assumption kan bounded converges symmetric matrix wrt assumption max max min bounded min bounded away zero similar main case theorem properly scaled asymptotically gaussian theorem asymptotic distribution suppose assumptions hold log nan penalized estimator obtained partially consistent theorem partial selection consistency suppose assumptions hold consistent estimator wrt construct penalized estimator estimator consistent theorem supplement asymptotic distribution extension main case theorem given following theorem theorem asymptotic distribution suppose assumptions conditions orem hold except condition required nan theorems asymptotic confidence regions availabe example confidence region based asymptotic confidence level given nkgx since involves unknown estimate atn hand estimated plugging obtain lemma appendix consistency assured theorem appendix guarantees asymptotic validity confidence region numerical evaluations real data analysis section evaluate performance penalized estimation simulations use analyze real data set model simulations given simplicity deterministic sparse incidental parameters generated copies exp probabilities respectively uniform random variable takes values probabilities respectively exp follows exponential distribution mean note viewed contamination parameter larger contaminated data hand determines asymmetry incidental parameters regression coefficients exp toeplitz matrix constant used inflate covariance little covariates independent performance penalized methods following methods estimating evaluated oracle method oracle knows index set zero performance used benchmark ordinary least squares method ols thought zeros iii four penalized least squares pls methods namely pls soft penalty pls hard penalty pls soft penalty pls hard penalty specifically oracle estimator given hard penalty function see fan method evaluated square root empirical mean squared error rmse every penalized method evaluated grid values regularization parameter ranging sequence plot figure shows realized incidental parameters used data generation simulations scatter plot figure shows responses first covariate generated data set red stars stand contaminated sample points ones nonzero fifty covariates usually difficult graphically identify contaminated data points incidental parameters six methods evaluated simulations iteration number since toeplitz matrix equal diagonal elements asymptotic variances estimators different representative estimators report simulation results estimation figure shows rmse six estimators rmse similar expected oracle method smallest rmse ols largest rmse pls method function forms convex curve achieves minimal rmse significantly green line ols close cyan line specifically rmse achieves marginal data plot sequence plot incidental parameters index figure sequence plot left shows incidental parameters red scatter plot right shows responses first covariate data set generated incidental parameters red stars stand contaminated sample points ones nonzero minimal rmse around hand rmse decreases little till around increases stays rmse reflects fact large method usually causes bias similar performance similar performance small however becomes large moves closer similar estimation large minimal rmse slightly larger pls methods table depicts minimal rmse estimators corresponding optimal biases biases ignorable comparing withe rmse optimal pls methods around respectively indicates simple soft threshold method tends work best small due bias issue denote empirical relative efficiency ere estimator respect another estimator rmse estimation ere respect around respectively ere pls methods respect ols around respectively ere similar thus terms ere rmse pls methods perform closely significantly better ols rmse deterministic sparse inicidental parameters ols rmse lambda figure rmse oracle ols estimators incidental parameters shown figure top bottom solid horizontal lines show rmse ols respectively four horizontal lines indicate minimal rmse four pls methods corresponding four best shown vertical lines table also see rmse estimators always smaller first covariate less correlated others covariates second one order examine performance methods general incidental parameters figure generate randomly iteration iteration number simulation also figure shows rmse six estimators two settings plot figure presents similar pattern figure fixed rmse estimator increases contamination parameter increases indicates estimator performs worse data becomes contaminated however pls estimators robust ols sensitive change also done simulations rmse estimators similar corresponding plots similar figure words rmse estimators stable respect means ols lad bias rmse bias rmse table rmse oracle ols lad estimators minimal rmse penalized estimators corresponding optimal biases incidental parameters shown figure used method different iteration reported averages lines numbers emphasize numbers averages standard deviations respectively rmse random sparse incidental parameters rmse rmse ols lambda lambda figure similar figure plots show rmse oracle ols estimators randomly generated two settings rmse ols lad table similar table one shows rmse minimal rmse nine estimators eight settings randomly generated standard deviations different settings magnitudes nonzero incidental parameters matter signs also note penalized methods perform closely even outperform oracle one small showed plot happens ignores contaminated data points even light contamination penalized methods exploit information points table contains rmse estimators eight settings increases rmse multiplication factor almost constantly around ols increases pls ones grow confirms robustness pls estimators small respect variance random error data points slightly contaminated ols pls methods perform similar however large means data contaminated rmse ols become significantly larger pls methods perform still closely performance penalized methods previous simulations shown pls methods optimal good rmse comparing oracle ols ones practice however optimal unknown one approach obtain data driven introduced subsection since shown previous simulation results pls methods perform similarly pls methods latter studied simulations denoted respectively estimating size pure data set npure testing data set npure simulations first run deterministic sparse incidental parameters showed figure see table rmse estimators around slightly larger optimal values respectively however still significantly smaller rmse ols observations estimators similar also evaluate performance pls methods random sparse incidental parameters table shows given small rmse close optimal cases performs slightly better even better optimal oracle method hand given large rmse greater respectively still less ols cases rmse larger indicates bias issue soft threshold method thus regularization parameter works well penalized estimation data slightly contaminated soft penalty preferred otherwise hard penalty recommended tables also contain rmse least absolute deviation regression method lad used fan part sample points small residuals generally speaking deterministic random incidental parameter cases lad performs similarly pls methods specifically small outperform lad otherwise lad performs better cases lad nated observations confirm lad effective robustness method penalized methods make improvement confidence intervals next turn investigate performance asymptotic confidence interval based pls methods since based properties penalized estimator soft penalty focus regularization parameter choice subsection minimizing rmse usually longer suitable constructing confidence intervals since designed achieve minimal rmse propose first obtain procedure subsection simply set five times since tends underestimate usually large respect denote method plugging square root element replace theoretical obtain estimator compared based oracle ols methods specif ically denote oracle ols estimators corresponding given ols respectively ols ols number zero incidental parameters ols estimators ols methods respectively simulation settings previous ones deterministic sparse incidental parameters except following changes number covariates reduced nominal level even empirical coverage rate oracle confidence interval becomes close iteration number increased improve accuracy probabilities nonzero incidental parameters set contamination parameter increased order achieve good second order asymptotic approximation either increase sample size enlarge signal noise ratio adopt latter table reports empirical coverage rates average lengths ols methods three different settings incidental parameters oracle method three settings thus one set ols ols table coverage rates average length confidence intervals ols methods three settings deterministic sparse incidental parameters simulation results presented table shows methods settings close nominal level ols treats deterministic incidental parameters random ones achieves excellent however ols significantly larger especially incidental parameters hand slightly larger means excellent efficiency terms given excellent also note less asymptotic variance less covariance matrix toeplitz matrix simulations random incidental parameters settings also done results similar table slightly inflated ols due randomness incidental parameters real data analysis implement penalized estimation soft penalty method estimating false discovery proportion multiple testing procedure proposed fan investigating association expression level gene closely related syndrome phenotypes thousands snps data set consists three populations utah residents ceu japanese chinese jptchb yoruba yri details data set found fan testing procedure fan filtered least absolute deviation regression lad used estimate loading factors cases snps whose test statistics small thus resulting estimator statistically biased upgrade step estimated fdp lad estimated ceu estimated fdp estimated jptchb estimated fdp estimated yri figure discovery number estimated false discovery number estimated false discovery proportion fdp functions threshold populations ceu jtpchb yri population fdp lad fdp ceu jptchb yri table discovery numbers estimated false discover proportions fdp methods lad specific values threshold described subsections number false discoveries false discovery proportion fdp functions thresholding value figure shows number total discoveries fdp procedures using filtered lad clear fdp uniformly larger reasonably close filtered lad table contains fdp filtered lad several specific thresholds estimated fdps ceu yri slightly larger jptchb double filtered lad lad fdp suggests estimation fdp filtered lad might tend optimistic conclusion discussion paper considers estimation structural parameters finite diverging number covariates linear regression model presence sparse incidental parameters exploiting sparsity propose estimation method penalizing incidental parameters penalized estimator structural parameters consistent asymptotically gaussian achieves oracle property contrary penalized estimator incidental parameters possesses partial selection consistency consistency thus structural parameters consistently estimated incidental parameters presents partial consistency phenomenon order construct better confidence regions structural parameters propose estimator fewer possible asymptotic distributions asymptotically even efficient penalized estimator size magnitude nonzero incidental parameters substantially large simulation results show penalized methods best regularization parameters achieve significantly smaller mean square errors ordinary least squares method ignores incidental parameters also provided regularization parameter penalized estimators continue significantly outperform ordinary least squares incidental parameters large neglected terms average length together excellent coverage rates advantage confidence intervals based estimator alternative regularization parameter verified simulations data set genomewide association analyzed multiple testing procedure equipped penalized method false discovery proportions estimated econometrics fixed effect panel data model given yit tit unknown fixed effects diverges fixed effects consistently estimated finite greater equal although fixed effects longer consistently estimated removed transformation yit averages yit respectively equal however transformation fails note model becomes model proposed penalized estimations provide solution sparsity assumption fixed effects although paper illustrates partial consistency phenomenon penalized estimation method linear regression model phenomenon shall universally exist general parametric model contains structural parameter sparse incidental parameter example consider panel data logistic regression model yit exp tit finite fixed effects removed transformation panel data linear model however proposed penalized estimations still provide solution structural parameter dimension diverging faster sample size sparse expected partial consistency phenomenon continue appear sparsity penalty imposed structural incidental parameters acknowledgement authors thank editor associate editor three referees many helpful comments resulted significant improvements article research partially supported nsf grants nih grants appendix appendix provide proofs theoretical results section proofs results sections supplements denote let proof lemma first consider finally consider note let suffices show follows thus follows consider recall min note large let show denote event follows note denote event contains similarly show thus note disjoint union note consider denote large event contains thus note disjoint union proceeding proofs theorems denote var make following assumptions bounded assumption var assumption bounded assumption section implies assumptions inequality simplicity adopt notation means left hand side bounded constant times right constant affect related analysis three lemmas needed proving theorems proofs supplement suppose matrices matrix norm sequence random matrices deterministic matrix denote sample covariance matrix lemma stewart kik kkek lemma kkek kkek bounded convergence probability wrt lemma assumption holds wrt proof theorem proof lemma solution explicitly given dkti show bounded positive constant dkti consider lemma assumption condition lemma together assumption implies bounded positive constant consider consider thus asx sumption consider similarly next lemma needed proving theorem proof supplement suppose copies random vector mean zero denote max var min var max lemma suppose max max bounded max bounded zero log wrt proof theorem reuse notations proof theorems nan nan sufficient show consider ndkan assumption kan bounded lemmas assumption bounded thus nan consider nan first note one hand every ekz ekz ekz sumptions ekz hand cov assumption thus central limit thed orem see proposition van der vaart next consider note assumption kan log log lemmas log log lemma log log thus slutsky lemma consider first consider noting ndkan thus way proof theorem definition show converges one denote since consistent estimator wrt consider sufficient show converge zero similarly thus consider similarly proof theorem note nan nan nan defined proof theorem since similarly proof theorem therefore desired result follows slutsky lemma lemma consistency suppose assumptions conditions theorem hold proof lemma since assumptions conditions theorem hold penalized estimators consistent estimators wrt theorems supplement let occurs theorem note suffices show note clear thus sufficient show every assumption every ekx ekx theorem asymptotic distributions assumptions conditions theorem log similarly assumptions conditions theorem log note stronger requirement required handle theorem lemma needed proving theorem lemma wihler suppose symmetric matrices kpf bkf specifically bkf proof theorem show result since result obtained ilar way reuse notations proof theorems ngx theorem sufficient show wrt show converges zero probability finishes proof first establish inequality lemma note lemma lemma kan kan thus lemma kan since fixed teger follows kan kan consider note kan kan lemmas assumption kan bounded lemmas assumption bounded also note thus consider note kan log kan log lemmas log log assumption kan lemmas sumption bounded lemma log log thus consider kan kan thus way supplementary materials additional materials sections found file supplementary materials references debabrata basu elimination nuisance parameters journal american statistical association derek bean peter bickel noureddine karoui chinghway lim bin penalized robust regression louis chen larry goldstein shao normal approximation stein method jianqing fan runze variable selection via nonconcave penalized likelihood oracle properties journal american statistical association dec issn jianqing fan jinchi selective overview variable selection high dimensional feature space statistica sinica jianqing fan heng peng penalized likelihood diverging number parameters annals statistics jianqing fan yuan liao martina mincheva high dimensional covariance matrix estimation approximate factor models jianqing fan yingying fan emre barut adaptive robust variable selection jianqing fan yang feng xin tong road classification high dimensional space regularized optimal affine discriminant journal royal statistical society series peter huber robust estimation location parameter annals mathematical statistics peter huber robust regression asymptotics conjectures monte carlo annals statistics johannes jahn introduction theory nonlinear optimization springer berlin heidelberg kiefer wolfowitz consistency maximum likelihood estimator presence infinitely many incidental parameters annals mathematical statistics michael kosorok bootstrapping grenander estimator beyond parametrics interdisciplinary research festschrift honor professor pranab sen pages institute mathematical statistics hayward zwald robust regression huber criterion adaptive lasso penalty electronic journal statistics issn tony lancaster incidental parameter problem since journal econometrics marcelo moreira maximum likelihood method incidental parameter problem annals statistics issn jerzy neyman elizabeth scott consistent estimates based partially consistent observations econometrica stephen portnoy xuming robust journey new millennium journal american statistical association albert shiryaev probability second edition stewart continuity generalized inverse siam journal applied mathematics runlong tang moulinath banerjee michael kosorok likelihood based inference current status data grid boundary phenomenon adaptive inference procedure annals statistics robert tibshirani regression shrinkage selection via lasso royal statist soc aad van der vaart asymptotic statistics cambridge university press aad van der vaart jon wellner weak convergence empirical processes springer thomas wihler holder continuity matrix functions normal matrices journal inequalities pure applied mathematics peng zhao bin model selection consistency lasso journal machine learning research nov supplementary materials paper partial consistency sparse incidental parameters jianqing fan runlong tang xiaofeng shi supplement section supplement first show method proposed neyman scott work model explain assumptions conditions consistent results penalized methods zhao fan peng fan satisfied model although modified equations maximum likelihood method proposed neyman scott could handle number important cases incidental parameters unfortunately work model specifically consider simplest case model copies using notations neyman scott likelihood function exp function log log score functions log log log equation plugging replacing obtain thus depend structural parameters however means independent structural parameters consequently estimation equations degenerate two equations means modified equation maximum likelihood method work model next explicitly explain assumptions conditions consistent results penalized methods zhao fan peng fan valid model zhao derive strong sign consistency lasso estimator however consistency results theorems apply model since specific design matrix satisfy regularity condition page specifically model covariance matrix covariates means eigenvalues goes regularity condition positive constant hold thus consistency results theorems zhao applicable model fan peng show consistency euclidean metric penalized likelihood estimator dimension sparse parameter increases sample size theorem page framework function data point model random errors copies given log means proportional see functions different might different since might different different violates condition data points structural density assumption page violation might essential however since could consider function data directly consider log fisher information matrix given identity matrix fisher information one data point clear minimal eigenvalue violates condition minimal eigenvalue lower bounded assumption page thus consistency result theorem fan peng applied model fan consider variable selection problem nonpolynomial dimensionality context generalized linear models taking penalized likelihood approach penalties theorem page fan shows exists consistent estimator unknown parameters euclidean metric certain conditions condition page condition minimal eigenvalue min consists first columns design matrix model condition becomes matrix defined zhao positive constant since minwhere converges condition hold thus consistency imal eigenvalue result theorem fan applicable model supplement section supplement provide lemmas proposition proofs two graphs figures illustrating incidental parameters step updating responses iteration algorithm iii figure illustration three types large bounded zero negative half real line folded positive half convenience penalized least square method soft penalty function assumption fixed specification regularization parameter min figure illustration updating responses solid black line fitted regression line dashed black lines corresponding shifted regression lines circle diamond points original data points circle triangle points updated data points diamond points drawn onto shifted regression lines lemma necessary sufficient condition minimizer sign sign function proof lemma subdifferential calculus see example theorem jahn necessary sufficient condition minimizer zero subdifferential means thus conclusion lemma follows proposition suppose assumptions hold exist positive constants every remark prespecified critical value stopping rule proposition implies algorithm stops second iteration practice sample size might large enough estimator decent performance iterations usually needed activate stopping rule proposition iterations make distance small order small algorithm converges quickly verified simulations proof proposition first show bounded sij sij defined end section denote event defined beginning section sij lemma thus show ekx thus thus thus thus next consider since bounded lemma occurs thus follows means iteration algorithm stops second iteration finally repeat arguments least probability increases one lemma lemma necessary sufficient condition minimizer solution equations proof lemma first show solution satisfies necessary sufficient condition lemma denote solution exactly first condition lemma satisfies one three cases satisfies first case satisfies third condition lemma satisfies second case means second case satisfies second condition lemma similarly third case also satisfies second condition lemma thus satisfies necessary sufficient condition lemma direction suppose satisfies necessary sufficient condition lemma first condition lemma exactly satisfies one three cases satisfies first case satisfies first case satisfies second case vii means satisfies second case similarly satisfies third case satisfies third case thus satisfies supplement section supplement provide proofs theoretical results section point two different sufficient conditions theorem come different analysis term two different sufficient conditions imply specifically one hand suppose absolute values equal thus assumption holds automatically since means assumption holds least absolute magnitudes similar case still exists consistent estimator even hand suppose equal constant previous two terms asymptotically equivalent thus assumption fails sufficient condition holds proof lemma proof similar lemma omitted proof theorems lemma solution explicitly given show frobenius norm euclidean norm thus slutsky lemma see example lemma page van der vaart consistent estimator law large number continuous mapping theorem approach one suppose viii approach two assumption follows fact assumption implies lyapunov condition sequence random vectors see proposition page van der vaart specifically recall lyapunov condition exists constant assumption minimum eigenvalue law large number noting thus way show holds theorem one condition turns consider conditions derive possible asymptotic distributions theorem asymptotic distributions cases assumptions constants main case letting min theorem groups results according asymptotic magnitude given upper bound diverging speed alternatively theorem groups results according asymptotic magnitudes since three cases theorem basically contains nine cases last case three cases relationship theorem first case theorem denoted main case since case incidental parameters sparse sense size magnitude nonzero incidental parameters well controlled note implies means assumption cases theorem actually imply three results theorem theorem convergence rate becomes less size magnitude nonzero incidental parameters large boundary phenomenon also appears proof theorems sufficient provide proof case sizes asymptotically index sets proof theorems let sequence going infinity defined proof theorem next derive asymptotic properties desired results follow slutsky lemma proof theorem approach one krn approach two thus three cases means fast let thus first consider denote size function three cases note note equivalent note equivalent means large let rate note equivalent way analyzed parallel results obtained proof theorem proof similar theorem omitted supplement subsection following theorem implies theorem since contains details theorem consistency asymptotic normality suppose assumptions hold either assumption holds hand assumption main case every constant proof theorem denote note ensures consistent theorem theorem goes defined proof theorem proof consistency similar theorem omitted next show asymptotic normality defined proof theorem since similarly analysis proof theorem asymptotic distributions follows slutsky lemma lemma consistency suppose assumptions hold either assumption holds proof lemma assumption holds penalized estimators consistent estimators theorems denote theorem occurs sufficient show straightforward show condition noting consistent estimator supplement subsection supplement consider special case exponentially tailed covariates errors convenience first introduce definition orlicz norm related inequalities strictly increasing convex function orlicz norm random variable respect defined inf xii see page van der vaart wellner next introduce lemma orlicz norm suppose sequence random variables sequence random vectors zid lemma page kosorok following extension lemma exp exp max log log max log log universal constant independent proof lemma proof random variables proof lemma page kosorok random vectors max exp result random variables desired result random vectors follows suppose every exp exp lemma follows max log log max log log thus inequality let log otherwise let log similarly let log otherwise let log satisfy condition suppose positive means xij exponential tails set log case regularization parameter specification becomes log min xiii end supplement simply list explicit expressions different assumptions covariates case diverging number covariates extension results section magnitude becomes larger case fixed keeps specifically bounded dcx log log follows gaussian distribution orlicz norm exists average bounded instance log finally data satisfies right inequality component tailed log worthwhile note expressions depend factor involving diverging number covariates influence specification regularization parameter sufficient conditions theoretical results section supplement section supplement provide proposition proof proofs lemmas appendix additional results first extend proposition case list two simple lemmas diverging suppose sequence copies random vector mean zero denote var lemma suppose bounded lemma suppose bounded suppose specification regularization parameter given constant greater proposition suppose assumptions hold regularization parameter satisfies suppose exist constants xiv regularization parameter satisfies every least probability increases one specifically iterative algorithm stops second iteration proof proposition reuse notations proof lemma first show since regularization parameter satisfies easy check conclusion lemma continues hold implies thus show thus assumption thus thus assumption assumptions log log log log thus assumption thus assumption kss next consider since conclusion lemma holds implies occurs thus xvi thus thus means iteration algorithm stops second iteration repeating arguments least probability increases one completes proof next provide proofs lemmas appendix proof lemma let note kekf implies kekf thus kekf bounded constant lemma kekf kekf ckekf kekf therefore kekf ckekf completes proof proof lemma xik xil assumption thus thus consistent estimator wrt proof lemma let log max xvii standard deviation berry esseen theorem see example shiryaev exists constant noting max max max max max min therefore next result consistency penalized estimator theorem consistency suppose assumptions conditions theorem hold wrt proof theorems theorem wrt theorem defined proof theorem since similarly proof theorem bounded dkti thus wrt finally provide additional results asymptotic distributions different scaling specifically scaling section nan next consider another natural scaling nan xviii theorem asymptotic distribution suppose assumptions hold log nan theorem asymptotic distribution suppose assumptions conditions rem hold except condition nan theorems confidence regions constructed order validate confidence regions estimated need lemma following result theorem asymptotic distributions suppose assumptions conditions theorem hold log nan similarly suppose assumptions conditions theorem hold log nan remark comparison assumptions conditions theorem theorems reveals much stronger requirement needed ensure good estimator precisely former require log latter log stronger requirement price paid estimating remark condition dimension theorems log slightly weaker condition log theorems accordingly condition dimension theorem log slightly weaker condition log theorem means scaling nan slightly better scaling nan terms condition former scaling suitable constructing confidence regions entries end supplement provide proofs theorems proof theorems reuse notations proof theorems nan xix nan show desired result follows applying slutsky lemma ndkan assumption kan bounded assumption bounded lemmas assumption bounded means left side bounded constant times right side noted beginning appendix thus nan first consider nan one hand every ekz ekz atn ekz thus assumptions ekz hand cov atn thus central limit theorem see example proposition van der vaart next consider kan log log assumption kan assumption lemmas log log lemma together assumption log log thus slutsky lemma first consider noting nkan thus way completes proof proof theorem proof theorem nan nan nan defined proofs theorems since similarly proof theorem thus asymptotic distribution gaussian slutsky lemma proof theorem show result since result obtained similar way reuse definitions proof theorems nan nan nan theorem sufficient show wrt nan show converges zero probability finishes proof lemma nkan ndkan assumption kan bounded lemma lemmas assumption bounded thus kan log log log log assumption kan lemma log log lemmas log log lemma log log thus xxi first consider noting nkan thus way references jianqing fan jinchi penalized likelihood ieee transactions information theory jianqing fan heng peng penalized likelihood diverging number parameters annals statistics michael kosorok introduction empirical processes semiparametric inference springer new york jerzy neyman elizabeth scott consistent estimates based partially consistent observations econometrica albert shiryaev probability second edition aad van der vaart asymptotic statistics cambridge university press peng zhao bin model selection consistency lasso journal machine learning research nov xxii
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remix automated exploration interactive outlier detection yanjie charu srinivasan deepak hui may missouri science technology missouri usa ibm watson research center usa rutgers university usa may abstract outlier detection identification points dataset conform norm outlier detection highly sensitive choice detection algorithm feature subspace used algorithm extracting insights outliers needs systematic exploration choices since diverse outlier sets could lead complementary insights challenge especially acute interactive setting choices must explored manner work present remix first system address problem outlier detection interactive setting remix uses novel mixed integer programming mip formulation automatically selecting executing diverse set outlier detectors within time limit formulation incorporates multiple aspects upper limit total execution time detectors diversity space algorithms features iii evaluating cost utility detectors remix provides two distinct ways analyst consume results partitioning detectors explored remix perspectives matrix factorization perspective easily visualized intuitive heatmap experiments versus outliers ensembled set outliers combines outlier scores detectors demonstrate benefits remix extensive empirical validation data introduction outlier detection identication points dataset conform norm critical task data analysis widely used many applications financial fraud detection internet traffic monitoring cyber security outlier detection highly sensitive choice detection algorithm feature subspace used algorithm outlier detection often performed high dimensional data unsupervised manner without data labels distinct sets outliers discovered different algorithmic choices could reveal complementary insights application domain thus unsupervised outlier detection data analysis task inherently requires principled exploration diverse algorithmic choices available recent advances interactive data exploration show much promise towards automated discovery advanced statistical insights complex datasets minimizing burden exploration analyst study unsupervised outlier detection interactive setting consider three practical design requirements automated exploration system automatically enumerate assess select execute diverse set outlier detectors exploration strategy guarantee coverage fuyan charu spartha turaga hxiong space features algorithm parameters predictable response time system conduct exploration within specified time limit implies exploration strategy sensitive execution time cost detectors visual interpretability system enable user easily navigate results automated exploration grouping together detectors similar terms data points identify outliers exploratory action data point perspective exploratory action data point perspective figure factorization outlier matrix two perspectives perspective heatmapped matrix whose columns data points rows detectors intensity cell perspective corresponds extent detector identifies point outlier perspective clearly identifies distinct set outliers key contributions present remix modular framework automated outlier exploration first address outlier detection problem interactive setting best knowledge none existing outlier detection systems support interactivity manner outlined particular aware system automatically explores diverse algorithmic choices outlier detection within given time limit following key contributions work mip based automated exploration remix systematically enumerates candidate outlier detectors formulates exploration problem mixed integer program mip solution mip yields subset candidates executed remix mip maximizes novel aggregate utility measure trades total utility selected candidates mip also enforces upper limit total cost budget selected candidates set diversity constraints ensure detection algorithm certain prioritized feature subspaces get least certain minimum share exploration budget cost utility remix mip requires estimate cost utility candidate detector order estimate cost remix trains model algorithm uses number data points size feature subspace various product terms derived significantly harder estimate even define utility remix handles defining utility detector proxy accuracy given data remix estimates training model algorithm uses various statistical descriptors detector feature subspace model trained corpus outlier detection datasets labeled domain experts perspective factorization diversity feature subspaces algorithms explored remix result different detectors marking distinct set data points outliers remix provides succinct way data analyst visualize results heatmaps consider outlier matrix normalized ranging outlier score assigned detector data point remix uses low rank matrix factorization scheme small number perspectives consensus among detectors within perspective idea outlier perspectives generalization idea outlier ensembles notable special case occurs number perspectives equals one results detectors ensembled single set outliers remix used two modes simple analyst gain rapid visual understanding outlier space providing two simple inputs exploration budget number perspectives yields heatmaps outliers easily interpretable figure advanced discussed section advanced user also specific modules within remix utility estimation enumeration prioritized feature subspaces would steer remix mip towards alternate optimization goals still guaranteeing exploration within given budget providing factorized heatmap visualizations diverse outlier sets rest paper organized follows survey related work section provide background definitions overview section sections present details feature subspace candidate detector enumeration cost utility estimation mip automated exploration perspective factorization present evaluation remix datasets section conclude section related work outlier ensembles combine multiple outlier results obtain robust set outliers model centered ensembles combine results different base detectors ensembles explore different horizontal vertical samples dataset combine results research area subspace mining focuses exploring diverse family feature subspaces interesting terms ability reveal outliers work introduces new ensemble model uses detector explanations choose base detectors selectively remain robust errors detectors remix related ensemble outlier detection since setting number perspectives one remix leads ensembling results base detectors however remix notable distinctions idea perspectives remix generalizes notion ensembles setting number perspectives number greater one possible remix results complementary views outlier space visualized heatmaps remix seeks guarantee coverage space features feature subspaces also available algorithms subject budget constraint total time available exploration none existing approaches literature provide guarantee exploration time multiview outlier detection deals setting input consists multiple datasets views view provides distinct set features characterizing objects domain algorithms setting aim detect objects outliers view objects normal show inconsistent class clustering characteristics across different views remix related multiview outlier detection matrix factorization nmf often used creating combined outlier score objects remix nmf used ensembling detector scores also factorizing detector results multiple heatmap visualizations automated exploration growing trend world commercial data science systems well machine learning statistical research focuses model selection focus exploration regression models deal algorithm exploration classification models deals automated feature generation classification regression budgeted allocations recommendations also studied context problems outlier analysis preliminaries baseline algorithms implementation remix uses following set five baseline outlier detection algorithms local outlier factor lof finds outliers measuring local deviation point neighbors mahalanobis distance detects outliers computing mahalanobis distance point center entire dataset outlier score outlier detection abod identifies outliers considering variances angles difference vectors data points robust distance space abod robust yet outlier detection fbod ensemble method based results local outlier factor lof iteration random feature subspace selected lof applied calculate lof scores based selected data subset final score fbod cumulative sum iteration subspace outlier detection sod sod aims detect outliers varying subspaces high dimensional feature space specifically point dataset sod explores subspace spanned neighbors determines much point deviates neighbors subspace interactive outlier exploration dataset remix matrix columns rows feature subspace subset columns outlier detector simply combination algorithm feature subspace cost utility values positive values associated intended estimates execution cost accuracy respectively remix enumerates candidate outlier detectors based family prioritized feature subspaces family randomly constructed feature subspaces interactive outlier exploration problem budgeted optimization problem selects subset detectors maximization objective linear combination two quantities total utility selected detectors total utility selected detectors subject following budget diversity constraints total cost selected detectors exceed budget ttotal algorithm gets guaranteed share exploration budget iii prioritized feature gets guaranteed share exploration budget remix framework describe five components remix shown figure starting feature subspace enumeration feature subspace candidate detector enumeration algorithm describes feature subspace enumeration three parts creating feature bag fnr lines creating prioritized family subspaces lines using feature ranking algorithm feature subspace enumeration input data matrix output feature subspace families fnr initialize feature bag bag least features mean correlation fnr arg max max high max correlation greater average correlation fnr fnr drop else fnr fnr drop end else break break loop end end initialize null set features fnr ranked laplacian scores add prioritized subspace end initialize null set random subspace feature sampled independently random without replacement fnr probability add random subspace end feature subspace enumeration perspective factorization ensembling candidate detector enumeration mixed integer program based exploration cost utility estimation figure remix components remix starts enumerating multiple feature subspaces given dataset turn used enumerate candidate detectors next remix evaluates cost utility enumerated detectors components training cost utility models shown figure cost utility values used part mixed integer program mip selects subset candidates execution based utility maximization objective budget diversity constraints outlier results detectors executed remix factorized perspectives also ensembled single set results approach iii creating randomized family subspaces lines used maximizing coverage diversity exploration redundant features fnr part subspaces algorithm begins initializing fnr features line iteratively looks member fnr considered redundant hence dropped fnr order feature fnr considered redundant needs maximally correlated feature pair fnr line correlation coefficient must remix redundancy threshold line iii two features pair must mean correlation features fnr mean correlation line set default value remix based experimental evaluation easy see end line algorithm yields feature bag fnr following property observation pair features fnr feature fnr fnr family randomized feature subspaces created purpose guaranteeing feature coverage diversity exploration particular select subspaces subspace consists features selected independently random without replacement probability fnr set default value remix balance size two families family prioritized feature subspaces created follows features fnr first sorted according laplacian scores add subspaces subspace set features fnr ranked laplacian scores provide brief justification use laplacian score due lack space refer reader exact details laplacian computation laplacian score computed way reflect feature ability preserve locality particular consider projection points onto subspace fnr consider neighborhood points features respect neighborhood provide better separation inlier class outlier class within data note experimental evaluation section consistently demonstrates improved outlier detection accuracy opposed purely also discuss potential alternatives prioritized feature subspace construction section enumerate candidate detectors cartesian product set baseline detection algorithms cost estimation cost candidate detector modeled function algorithm well size feature subspace runtime complexity algorithms notation provides asymptotic relationship cost input size however remix seek refined model accounts lower order terms constants hidden training multivariate linear regression model algorithm specifically consider following polynomial log log distinct terms expansion polynomial derived exactly given feature subspace form explanatory variables cost dependent variable linear regression model utility estimation given detector utility estimation algorithm algorithm first normalizes features computes variety statistics feature statistics include laplacian score useful measure unsupervised feature selection section standard deviation skewness measures assymetry feature distribution kurtosis measures extent feature distribution entropy measure information content feature note computations done feature fnr reused given detector next algorithm computes vector statistics combining statistics instance consider laplacian scores features mean median median absolute deviation min max standard deviation laplacian scores provide distinct components step finally uses algorithm specific utility model estimate utility remix trains five distinct linear regression models ulof uabod ulof usod corresponding algorithm models trained based distinct expert labeled datasets outlier dataset repository explanatory variables linear regression statistics described algorithm dependent variable detection accuracy measured fraction data points detector expert labeled ground truth agree outlier characterization algorithm utility estimation input candidate detector output utility detector normalize one time procedure applied fnr extract following statistics laplacian score standard deviation skewness kurtosis entropy one time procedure applied fnr extract following statistics mean median median absolute deviation min max standard deviation statistics extracted step let contain feature statistics return utility estimation model learnt algorithm discussion estimating even defining utility detector significantly harder estimating cost goal utility estimation remix learn perfect model rather goal merely learn utility model effectively steer solution mixed integer program mip used remix exploration section experiments section demonstrate indeed case remix remix intended flexible framework alternative mechanisms utility estimation plugged instance consider cumulative mutual information cmi metric feature subspace search algorithm presented subspace outlier detection remix use cmi detector utility algorithm alternative enumerating prioritized feature subspaces chose approach implementation since approach algorithm agnostic hence generalized easily exponent suffices model cost known outlier detection algorithms mip based exploration remix executes subset detectors enumerated detector enumeration component section subset determined solving following mixed integer program mip max utility detectors utility detectors ttotal ttotal ttotal given algorithm feature subspace binary indicator variable mip determines detector chosen execution mip solution recall denotes estimated cost observe following observation constraint guarantees feasible solution mip total cost selected detectors ttotal value total exploration budget ttotal provided analyst part interaction remix observation constraint guarantees feasible solution mip algorithm explored total estimated duration time observation constraint guarantees feasible solution mip prioritized feature ttotal explored estimated duration time also implies exploration focuses least half total time prioritized features consider binary indicator variable corresponding detector constraint ensures feasible solution mip chosen solution constraint ensures detectors chosen solution set consider optimal solution pthe mip since utility values detectors first part objective function maximized exactly detectors set fewer detectors whose set ones highest utilities amongst selected detectors leads following guarantee theorem optimal solution mip maximizes sum utilities detectors highest utilities total utility selected detectors scaled factor set implementation remix balances utility detectors total utility selected detectors perspective factorization detector executed remix provides outlier score data point results different detectors could potentially divergent present factorization technique called nmfe nonnegative matrix factorization ensembling detection results succinct perspectives detectors within perspective agree characterize outliers although could disagreement across perspectives let outlier score assigned detector data point normalized across data points range consider matrix outlier scores number detectors executed remix number data points perform matrix factorization minimizing divergence min log matrix definition expressed sum matrices whose rows columns correspond detectors data points respectively matrices every row column scaled multiple row column every entry properties make possible matrices visualized perspectives heatmaps intensity heatmap cell value corresponding entry perspective shown figure number perspectives specified user input remix setting simply results direct averaging ensembling detector results single perspective experiments section present evaluation remix data data collection chose datasets study outlier dataset repository delft university technology corpus contains outlier detection datasets labeled domain experts datasets span varied domains flowers breast cancers heart diseases sonar signals genetic diseases arrhythmia abnormals hepatitis diabetes leukemia vehicles housing satellite images dataset benchmark outlier labels entry representing outliers representing normal points figure illustrate statistics data sets specifically figure shows numbers features dataset sorted descending order figure observe datasets contains less features small portion datasets features figure shows outlier ratio total number data points dataset sorted descending order figure find outlier ratios datasets less outlier ratio feature number rankings data sets respect outlier ratio rankings data sets respect number features distribution feature sizes distribution outlier ratios figure statistics experimental datasets cost utility estimation prediction prediction among datasets used datasets train test cost utility estimation models baseline algorithms remix implementation split train test figures present performance cost utility models lof algorithm specific run recall utility estimates fraction data points detector expert labels agree outlier characterization consider hamming distance outlier bit vector outlier otherwise created detector outlier bit vector created expert labels clearly hamming distance equals number data points figure plots estimated hamming distance actual distance observed running detector red lines figures represent ideal predictors expected cost estimator clearly performs better utility estimator however mentioned section goal utility estimation perfect predictive model merely steer mip towards better solutions demonstrate case next set experiments groundtruth groundtruth cost estimation utility estimation figure cost utility estimation detection accuracy cost lof abod fbod sod lof abod fbod sod lof abod fbod sod precision recall fmeasure figure comparison remix baseline algorithms cardiac arrhythmia dataset present detailed study detection accuracy cost remix two datasets corpus first dataset cardiac arrhythmia irregularity heart beat may harmless life threatening dataset contains medical records like age weight patient electrocardiograph related data arrhythmia patients outlier healthy people task spot outlier healthy people arrhythmia patients second dataset sonar signals bounced metal cylinder dataset contains outlier sonar signals bounced roughly cylindrical rock task separate outlier rock related sonar signals sonar signals evaluation metrics recall remix perspective factorization scheme used outlier ensembling technique simply setting number perspectives use remix ensembling mode rest lof abod fbod sod lof abod fbod sod lof abod fbod sod precision recall fmeasure figure comparison remix baseline algorithms sonar signals dataset rsr nmfe rsr nmfe rsr nmfe precision recall fmeasure figure comparison remix exploration ensemble strategies cardiac arrhythmia dataset experiments define precision recall use compare detection accuracy remix various approaches let denote outlier label denote label normal points expert labeled data precision given list data points sorted descending order predicted outlier scores precision defined precision data points expert outlier label recall given list data points sorted descending order predicted outlier scores recall defined recall outlier data points label incorporates precision recall single metric taking harmonic mean precision baseline algorithms report performance comparison remix baseline algorithms sonar signals dataset cardiac arrhythmia dataset terms precision recall experiment provide sufficient time baseline algorithms lof abod bod sod complete executions meanwhile set limited exploration time budget second remix results effectiveness comparison figure shows cardiac arrhythmia dataset remix outperforms five baseline algorithms terms precision recall fmeasure figure shows sonar signal dataset method consistently better baseline algorithms terms precision recall fmeasure results efficiency comparison figure jointly show cardiac arrhythmia dataset sonar signal dataset mahalanobis distance takes least time outlier detection abod takes time angle expensive compute remix falls middle cost spectrum rsr nmfe rsr nmfe rsr nmfe precision recall fmeasure figure comparison remix exploration ensemble strategies sonar signals dataset lof abod fbod sod lof abod fbod sod times costs cardiac times costs sonar arrhythmia dataset signal dataset figure comparison execution costs remix baseline algorithms effectiveness efficiency remix nmfe study effectiveness remix nmfe ensembling strategy used nmf factorize outlier score matrix treat predicted ensemble outlier scores let denote number detectors selected remix mip solution let set candidate detectors enumerated remix compared remix following exploration ensembling strategies exhaustive ensemble execute detectors averaged outlier scores randomly select detectors rsr randomly select execute detectors average outlier scores detectors randomly select detector randomly select one detector use results randomly select detector mip solution randomly select one detector mip solution use results experiment provide sufficient time strategies complete execution set limited time budget seconds remix results effectiveness comparison figure shows cardiac arrhythmia dataset strategy outperforms exploration ensembling strategies terms precision recall fmeasure figure shows sonar signal dataset remix consistently better strategies terms precision recall fmeasure results efficiency comparison figure shows cardiac arrhythmia dataset remix takes second rsr require much time takes second detection accuracy lower figure shows sonar signal dataset strategy takes second much less time costs rsr takes second method outperforms respect detection accuracy conclusions future directions paper presented remix modular framework outlier exploration first study problem outlier detection interactive setting heart remix optimization rsr nmfe rsr nmfe times costs cardiac times costs sonar arrhythmia dataset signal dataset figure comparison execution costs remix exploration strategies approach systematically steers selection base outlier detectors manner sensitive execution costs maximizing aggregate utility function solution also ensuring diversity across algorithmic choice points present exploration data analysts naturally interested extracting interpreting outliers multiple detection mechanisms since distinct outliers could lead distinct actionable insights within application domain remix facilitates understanding practical manner shifting burden exploration away analyst automation summarizing results automated exploration coherent heatmap visualizations called perspectives believe many techniques presented paper could independent interest machine learning problems interested extending remix exploratory approach beyond outlier detection clustering data another interesting direction research sequential recommendations visual outlier exploration particular interested approaches presenting outlier insights approximately sequence visualizations like heatmaps length sequence well perceptual error involved across visualizations minimized coverage across various exploratory choice points maximized also study outlier aspect mining interactive setting deals inverse problem finding explanatory features characterize given set points outliers cost sensitive manner presents new interesting direction research references data robot http outlier detection datasets http sytree http charu aggarwal outlier analysis springer science business media charu aggarwal saket sathe theoretical foundations algorithms outlier ensembles acm sigkdd explorations newsletter alain biem maria butrico mark feblowitz tim klinger yuri malitsky kenney adam perer chandra reddy anton riabov horst samulowitz daby sow gerald tesauro deepak turaga towards cognitive automation data science proceedings aaai conference artificial intelligence january austin texas pages markus breunig kriegel raymond sander lof identifying local outliers acm sigmod record volume pages acm klemens bhm fabian keller emmanuel mller hoang nguyen jilles vreeken cmi contrast measure enhancing subspace cluster outlier detection sdm pages siam santanu das bryan matthews ashok srivastava nikunj oza multiple kernel learning heterogeneous anomaly detection algorithm aviation safety case study proceedings acm sigkdd international conference knowledge discovery data mining pages acm kyriaki dimitriadou olga papaemmanouil yanlei diao automatic query steering framework interactive data exploration proceedings acm sigmod international conference management data sigmod pages new york usa acm chris ding tao michael jordan convex matrix factorizations ieee trans pattern anal mach january david duvenaud james robert lloyd roger grosse joshua tenenbaum zoubin ghahramani structure discovery nonparametric regression compositional kernel search proceedings international conference machine learning june jing gao wei fan deepak turaga srinivasan parthasarathy jiawei han spectral framework detecting inconsistency across object relationships data mining icdm ieee international conference pages ieee grosse salakhutdinov freeman tenenbaum exploiting compositionality explore large space model structures uncertainty artificial intelligence xiaofei deng cai partha niyogi laplacian score feature selection advances neural information processing systems pages victoria hodge jim austin survey outlier detection methodologies artificial intelligence review alexander kalinin ugur cetintemel stan zdonik interactive data exploration using semantic 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shebuti rayana leman akoglu less building selective anomaly ensembles arxiv preprint ashish sabharwal horst samulowitz gerald tesauro selecting learners via incremental data allocation proceedings thirtieth aaai conference artificial intelligence february phoenix arizona pages tarique siddiqui albert kim john lee karrie karahalios aditya parameswaran effortless data exploration zenvisage expressive interactive visual analytics system proc vldb november abdul wasay manos athanassoulis stratos idreos queriosity automated data exploration barbara carminati latifur khan editors bigdata congress pages ieee arthur zimek ricardo campello sander ensembles unsupervised outlier detection challenges research questions position paper sigkdd explor march
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functional units natural numbers oct bergstra middelburg informatics institute faculty science university amsterdam science park amsterdam netherlands abstract interaction services provided execution environment forms part behaviours exhibited instruction sequences execution mechanisms related kind interaction question proposed setting thread algebra like thread service abstract behavioural concept concept functional unit similar concept service concrete state space inherent concept functional unit whereas inherent concept service paper establish existence universal computable functional unit natural numbers related results keywords functional unit instruction sequence acm computing classification introduction take view sequential programs essence sequences instructions interaction services provided execution environment forms part behaviours exhibited instruction sequences execution see interaction question concerned processing instructions earlier work mechanisms direct bearing kind interaction proposed setting basic thread algebra see thread service abstract behavioural concepts experienced recently limitations concept service state space inherent concept forms greater part motivation introducing studying concept functional unit paper concept similar concept service lower level abstraction concept functional unit state space inherent rather first considering functional units general arbitrary state space first consider special case state space set natural numbers case arguably simplest significant case establish general results concerning functional units natural numbers main result existence universal computable functional unit natural numbers results like one outside scope concept service work presented paper belongs line research whose working hypothesis instruction sequence central notion computer science line research program algebra setting used investigating issues instruction sequences involved instruction sequences also involved issues concerning functional units investigated paper program algebra perception program instruction sequence finite infinite sequence instructions instruction executed dropped executed jumped perception simple appealing links practice moreover basic thread algebra setting used modelling behaviours exhibited instruction sequences paper use program notation rooted program algebra instead program algebra paper organized follows first give survey program notation used paper section define semantics using basic thread algebra section next extend basic thread algebra operators related processing instructions services section introduce concept functional unit related concepts section investigate functional units natural numbers section also make remarks functional units finite state spaces section finally make concluding remarks section pglb boolean termination section give survey program notation pglbbt program notation variant program notation pglb belongs hierarchy program notations rooted program algebra presented pglbbt pglb boolean termination instructions instead termination instruction pglb pglbbt close existing assembly languages relative jump instructions pglbbt assumed fixed arbitrary finite set basic instructions given intuition execution basic instruction may modify state produces completion pglbbt following primitive instructions plain basic instruction positive test instruction negative test instruction forward jump instruction backward jump instruction positive termination instruction negative termination instruction pglbbt instruction sequences form primitive instructions pglbbt execution pglbbt instruction sequence primitive instructions following effects basic thread algebra introduced name basic polarized process algebra effect positive test instruction basic instruction executed execution proceeds next primitive instruction produced otherwise next primitive instruction skipped execution proceeds primitive instruction following skipped one primitive instruction proceed deadlock occurs effect negative test instruction effect role value produced reversed effect plain basic instruction effect execution always proceeds produced effect forward jump instruction execution proceeds lth next primitive instruction equals primitive instruction proceed deadlock occurs effect backward jump instruction execution proceeds lth previous primitive instruction equals primitive instruction proceed deadlock occurs effect positive termination instruction execution terminates delivers boolean value effect negative termination instruction execution terminates delivers boolean value thread extraction section make precise setting btabt basic thread algebra boolean termination behaviours exhibited execution pglbbt instruction sequences start reviewing btabt btabt assumed fixed arbitrary finite set basic actions tau given write atau tau members atau referred actions thread behaviour consists performing actions sequential fashion upon basic action performed reply execution environment determines proceeds possible replies boolean values standing true standing false performing action tau leads always reply btabt one sort sort threads make sort explicit extend btabt additional sorts section build terms sort btabt following constants operators deadlock constant positive termination constant negative termination constant atau binary postconditional composition operator assume countably infinite set variables sort includes terms sort built usual use infix notation table axiom btabt tau tau table approximation induction principle aip postconditional composition introduce action prefixing abbreviation term sort abbreviates thread denoted closed term form first perform proceed thread denoted reply execution environment proceed thread denoted reply execution environment threads denoted become inactive terminate boolean value terminate boolean value respectively btabt one axiom axiom given table closed btabt term sort denotes thread become inactive terminate performed finitely many actions infinite threads described linear recursion linear recursive specification btabt set recursion equations set variables sort btabt term form interested models btabt linear recursive specifications unique solutions regular threads threads finite number states solutions finite linear recursive specifications reason infinite threads assume infinitary conditional equation aip approximation induction principle aip based view two threads identical approximations finite depth identical approximation depth thread obtained cutting performed actions aip approximation depth phrased terms unary projection operator aip axioms projection operators given table table stands arbitrary action atau stands arbitrary natural number behaviours exhibited execution pglbbt instruction sequences considered regular threads basic instructions taken basic actions thread extraction operation defines pglbbt instruction sequence behaviour exhibited execution pglbbt instruction sequence thread extraction operation defined auxiliary operation defined equations given table defining equations thread extraction operation table rule beginning infinite jump interaction threads services thread may perform basic action purpose requesting named service process method return reply value completion processing method section extend btabt kind interaction threads services dealt resulting tatsi involves introduction service families collections named services assumed fixed arbitrary finite set methods given methods play role commands service able process certain methods processing method service may involve change state service completion processing method service produces reply value set reply values set algebraic theory service families introduced following assumed respect services set services given together total function total function satisfying condition exists unique signature given includes following sort sort services following constant operators empty service constant derived service operator service denoted closed term sort rule formalized constant denotes unique denotes service closed term denotes service request made service process method processes produces reply next proceeds rejects request process method unique service called empty service service unable process method also assumed fixed arbitrary finite set foci given foci play role names services service family offered execution environment service family set named services name occurs sorts constants operators addition following sort sort service families following constant operators empty service family constant unary singleton service family operator binary service family composition operator unary encapsulation operator assume countably infinite set variables sort includes terms built usual case see use prefix notation singleton service family operators infix notation service family composition operator service family denoted empty service family service family denoted closed term form consists one named service service concerned service denoted name service service family denoted closed term form consists named services belong either service family denoted service family denoted case named service service family denoted named service service family denoted name collapse empty service name concerned service family denoted closed term form consists named services name belong service family denoted service family composition operator takes place combination operator suggested name service family composition composition service families combination composition services disadvantage usefulness rather limited without additional renaming mechanism table axioms axioms given table table stands arbitrary focus stand arbitrary closed terms sort axioms simply formalize informal explanation given introduce two operators related interaction threads services called apply operator reply operator apply operator concerned effects threads service families therefore produces service families reply operator concerned effects service families boolean values threads deliver termination reply operator produce boolean values produces special value cases termination takes place set basic actions take set operators mentioned relate processing methods services service family pursuance basic actions performed thread service involved processing method service whose name focus basic action question tatsi sorts constants operators btabt addition following sort sort replies following constants operators reply constants binary apply operator binary reply operator use infix notation apply reply operators service family denoted closed term form reply denoted closed term form service family reply respectively result processing method basic action focus service family denoted thread denoted performs processing done service service family focus basic action name method basic action performed thread processed service service changes accordance method concerned affects thread follows two ways proceed reduces one basis reply value produced service reply boolean value thread denoted delivers termination terminates value standing divergent terminate table axioms apply operator tau table axioms reply operator tau axioms tatsi axioms btabt axioms axioms given tables tables stands arbitrary focus stands arbitrary method stands arbitrary term sort stands arbitrary natural number axioms simply formalize informal explanation given addition stipulate result apply reply inappropriate foci methods involved axioms allow reasoning infinite threads contexts apply reply respectively functional units section introduce concept functional unit related concepts functional unit degree assumed set states given assumed finite set methods given however setting functional units methods serve names operations state space reason members henceforth called method names method operation total function partial method operation partial function write set method operations write unique functions respectively functional unit finite subset implies write set functional units write set write unique look upon set interface looks convenient notation restriction functional unit subset interface write functional unit let extension following simple illustration use functional units unbounded counter modelled functional unit method operations set zero increment one decrement one test zero according definition functional unit unique functional unit empty interface interesting however considering services behave according functional units exactly functional unit according empty service service able process method behaves method names attached method operations functional units confused names used denote specific method operations describing functional units therefore comply convention use names beginning letter former case names beginning letter latter case use pglbbt instruction sequences derive partial method operations method operations functional unit write set pglbbt instruction sequences taking set set basic instructions derivation partial method operations method operations functional unit involves services whose processing methods amounts replies service changes according corresponding method operations functional unit concerned services viewed behaviours machine processing question takes place different states take set set services write service functions defined follows meh mrh fixed arbitrary state assume denoted closed term sort connection use following notational convention write arbitrary closed term sort denotes ambiguity thus introduced could obviated decorating wherever stands closed term however paper always immediately clear context whether stands closed term moreover believe decorations often distracting therefore leave reader make decorations mentally wherever appropriate let let instruction sequence produces partial method operation follows undefined unique total called derived method operation binary relation defined iff derived method operation binary relation defined iff theorem transitive equivalence relation proof property prove implies sufficient show obtain instruction sequences produce method operations instruction sequences produce method operations instruction sequences produce method operations without loss generality may assume instruction sequences form positive test instruction forward jump instruction backward jump instruction let let let suppose let let uiki consider obtained follows first increase jump leftmost occurrence next replace instruction uiki repeat previous step long occurrences easy see property follows immediately definition symmetric definition reflexive properties property definition follows immediately symmetric reflexive transitive members quotient set called functional unit degrees let functional unit degree exists functional units natural numbers section investigate functional units natural numbers main consequences considering special case state space following infinite notion computability known used without preparations example functional unit unbounded counter method names involved setzero succ pred iszero method operations involved functions setzero succ pred iszero defined follows setzero pred iszero succ functional unit counter defined follows counter setzero setzero succ succ pred pred iszero iszero proposition infinitely many functional unit degrees pred iszero counter proof define functional unit pred iszero counter follows pred pred iszero iszero pred let pred however exist pred hence method operation computable exist computable functions inductively defined functional unit computable computable theorem let computable computable proof show derived method operations computable take arbitrary derived method operations follows immediately definition thread extraction solution finite linear recursive specification btabt finite guarded recursive specification btabt side equation btabt term form variables sort let finite linear recursive specification btabt solution total may assumed without loss generality occur side equation suppose set equations using relevant axioms definitions obtain set equations solution every function functions defined usual follows way set equations obtained fact computable fact computable set equations equivalent set equations defined recursively sense kleene see means general recursive hence computable similar way proved computable computable universal computable exists universal computable functional unit natural numbers theorem exists computable universal proof show exists computable property computable derived method operation corollary theorem computable computed means register machine six registers say registers used follows input register output register output output register output auxiliary registers content represents boolean output follows represents natural numbers represent register incremented one decremented one tested zero means instructions respectively write set pglbbt instruction sequences taking set set basic instructions clearly adequate represent register machine programs using six registers define computable functional unit whose method operations simulate effects register machine instructions encoding register machine states natural numbers contents registers reconstructed prime factorization functional unit defined follows succ succ pred pred iszero iszero method operations defined follows max succ pred iszero prime number define function gives instruction sequence instruction sequence effect produced register machine six registers simulated function defined follows iszero theorem looked upon corollary theorem usual write divisible jump termination instruction succ pred iszero take arbitrary computable exist instruction sequence computes take arbitrary computes hence derived method operation universal computable functional unit defined proof theorem method operations however three method operations suffice theorem exists computable three method operations universal proof know proof theorem exists computable method operations say show exists computable three method operations define computable functional unit three method operations follows method operations defined follows max max hence derived method operations universal computable functional unit defined proof theorem three method operations show one method operation suffice theorem exist computable one method operation universal proof show exist computable one method operation counter counter functional unit introduced beginning section assume exists computable one method operation counter let one method operation counter let unique method name take arbitrary succ pred instruction processed least applied applied let number times instruction processed application let number times instruction processed application irrespective replies state state reached processed times thus repeated application different states reached contradicts succ hence exist computable one method operation counter open problem whether two method operations suffice functional units finite state spaces short section make remarks functional units finite state spaces special case state space state space consists two states four possible unary functions precisely method operations principle different functional units useless include method operation different names functional unit means upper bound number functional unit degrees however straightforward show different functional unit degrees general case finite state space consisting states say principle different functional units already becomes unclear whether number functional unit degrees determined manually actually know moment whether determined computer support either primitive instruction instruction sequence defined induction follows concluding remarks defined concept functional unit state space established general results concerning functional units natural numbers main result existence universal computable functional unit natural numbers case state space set natural numbers arguably simplest significant case yet investigated significant cases interesting case one state space set pairs sequences alphabet tape turing machine modelled functional unit state space turing machine simulated means functional unit corresponds tape turing machine pglbbt instruction sequence corresponds finite control turing machine variations turing machine theme dealt way well thus functional units allows many computability issues viewed issues programs rather machines introduce extension program algebra boolean termination instructions called pgabt define thread extraction operation pglbbt instruction sequences translated closed pgabt terms thread extraction pglbbt yields behaviours translation followed thread extraction pgabt also introduce extension basic thread algebra similar tatsi addition constants operators tatsi extension constant termination without delivery boolean value operator concerned effects service families threads therefore produces threads references bergstra loots program algebra sequential code journal logic algebraic programming bergstra middelburg program algebra instruction journal applied logic bergstra middelburg instruction sequence processing operators bergstra ponse combining programs state machines journal logic algebraic programming kleene general recursive functions natural numbers mathematische annalen minsky recursive unsolvability post problem tag topics theory turing machines annals mathematics ponse van der zwaag introduction program thread algebra beckmann eds cie lecture notes computer science vol sannella tarlecki algebraic preliminaries astesiano kreowski eds algebraic foundations systems specification berlin shepherdson sturgis computability recursive functions journal acm wirsing algebraic specification van leeuwen handbook theoretical computer science vol elsevier amsterdam
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evolution representation simple cognitive networks lars arend christoph aug department cognitive science macquarie university sydney australia microbiology molecular genetics michigan state university east lansing beacon center study evolution action michigan state university east lansing computer science engineering michigan state university east lansing keywords information representation cognition evolution abstract representations internal models environment provide guidance behaving agent even absence sensory information clear representations developed whether necessary even essential intelligent behavior argue ability represent relevant features environment expected consequence adaptive process give formal definition representation based information theory quantify measure measure changes time evolve two types artificial neural network network hidden markov solve categorization task using genetic algorithm find capacity represent increases evolutionary adaptation agents form representations environment lifetime ability allows agents act sensorial inputs context acquired representations enables complex behavior examine concepts features environment networks representing representations logically encoded networks form agent behaves solve task conclude able quantify authors contributed equally representations within cognitive system predictive agent adaptive success introduction notion representation old cognitive science see chomsky newell simon fodor wason marr pinker pitt usefulness artificial intelligence research doubted brooks widely cited article intelligence without representation brooks argued instead subsumption architecture autonomous behavior producing components layers cognitive system directly interface world rather central symbol processor dealing explicit representations environment particular inspired biological path intelligence brooks argued research needs rooted mobile autonomous robotics direct interaction action perception echoing moravec asserted necessary elements development intelligence mobility acute vision ability behave appropriately dynamic environment brooks architecture achieved intelligence brooks argued path higher level could forged incrementally increasing complexity subsumption architecture however years advocating radical departure classical approach subsumption approach seems stalled well believe reason lack progress lie attempt base research mobile autonomous robots instead representations also sometimes called internal models craik wolpert kawato key complex adaptive behavior indeed robotics made important strides nolfi limited problems representation hungry clark problems require past information additional external knowledge current context addition technical difficulty developing subsumption architecture increases number layers subsystems problem subsumption architecture mirrors difficulties classic representational approaches build accurate appropriate models world alternative approach engineering cognitive architectures internal models evolutionary robotics nolfi floreano instead designing structure functions control architecture principles darwinian evolution used create complex networks interface perception action often surprising ways structures give rise complex representations environment hard engineer equally hard analyze see floreano evolved representations provide context flexible readjusted given new stimuli contradict current assumptions representations updated lifetime course evolution thus able handle even new sensory input bongard argue robots evolve behave appropriately survive dynamic noisy world representations environment emerge within cognitive apparatus integrated perceived sensory data create intelligent current state environment crucially taking account historical data memory well test hypothesis make internal representations evolved systems accessible analysis propose new measure degree embodied agent represents environment within internal states show capacity represent environmental features emerges thousands generations simulated evolution main idea representations encode environmental features relevance cognitive system question clark toribio hence purposes representations symbolic neural states long physical basis long encoded measurable internal states however distinguish representations sensorial input sensor inputs provide past external context internal states thus explicitly define representations information relevant features environment encoded internal states organism goes beyond information present sensors haugeland clark particular implies representations time misrepresent haugeland information present sensors always truthfully correlates environment illuminate functioning evolved cognitive systems show principle possible determine representation representations form lifetime agent argue measure provides valuable tool investigate organization evolved cognitive systems especially cases internal representations epistemically opaque methods measure representation information theory used previously quantify context modulate decisions based sensory input phillipps phillips singer kay phillips present construction explicitly takes entropy environmental states account quantify representation first define relationship representing system represented environment terms information shared mutual entropy information measures correlation two random variables entropy measure uncertainty random variable absence information uncertainty therefore potential information random variable take states probabilities entropy given shannon log number possible states take information two random variables characterizes much degree order one variables predictive regularity variable defined using entropy difference sum entropies two random variables written joint entropy written log probability distributions random variables respectively joint probability distribution joint random variable shared entropy also written terms difference unconditional conditional entropies definition reminds information reduces uncertainty system words allows make predictions system accuracy higher information introduced concept conditional entropy shannon example read given entropy state variable known calculated log using conditional probability general information able detect arbitrary correlations signals sets events assume correlations instantiate semiotic information relationships representing represented use mutual information measure correlation network internal states environment see also marstaller example could imagine stands states environment whereas variable represents states environment need careful however exclude possible representational variables mere images environment trace world leaves agent sensors indeed mere correlations internal states environment sufficient treated representational could due behavior entirely reactive clark haugeland example understands representation something stands something environment longer reflected perceptual system agent haugeland indeed representation different mere translation consider digital camera relationship environment photo chip guarantees mapping environment structure camera state patterns camera able adapt environment taking picture camera learned anything environment affect future state simply stores received inputs without extracting information camera internal states fully determined sensor inputs representation goes beyond mere translation content feature environment represented depends goals system everything represented way camera functional specification internal states rule trivial representations like camera internal states define representation shared entropy environment states internal states given sensor states conditioned sensors thus representation part shared entropy environment states internal states goes beyond seen sensors see fig following take given probability distribution random variable describe environmental states describes sensor states internal states agent hidden output states characterized random variable probability distribution define representation earlier version see marstaller hcorr correlation entropy hcorr three variables also called total correlation watanabe mcgill schneidman amount information three share hcorr introduced shared conditional entropy three variables defined difference information unshared one shared third system thus representation world within internal states total correlation three without reflected respectively measured relationship entropies three variables conveniently summarized entropy venn diagram fig diagrams circle quantitative measure entropy associated variable shared entropy two variables represented intersection variables see cover definition representation carries discrete variables continuous variables unchanged seen follows let random variables defined normalized probability density functions differential entropy cover thomas defined log support random variable related discretized version noting log introduced discretization order define discretized shannon entropy log implies quantization continuous random variable approximately cover thomas let assume variables quantized bits respectively quantized bits follows hcorr hcorr continuous discrete variable correlation entropies limit sufficiently small approximately discretization correction cancels true informations correlation entropies two variables thus differential entropy version stress exact identity discrete continuous variable definitions ensured limit vanishing discretization cancellation correction terms log implies discrete version biased respect continuous version defines relation network activity patterns environment result information processing yields positive quantity measured bits logarithms taken base order show measure representation reflects functional purpose clark evolve cognitive systems networks control behavior embodied agent show fitness measure agent functional prowess correlated words show environment task complex enough agents react challenge evolving representations environment figure venn diagram entropies informations three random variable describing states environment sensors agent internal degrees freedom representation shaded evolution active categorical perception study evolution agent solves active categorical perception acp task beer modifications suggested van dartel see also van dartel categorization thought one key elements cognition see harnad cohen lefebvre categorical perception agent partition objects world different discrete categories based visual appearance active categorical perception agent takes active role aspects object view perception intimately linked action whether task requires internal representations may depend specific nature task general simple determine whether agent uses internal states represent environment particular features world represented computational units beer ward ward task studied beer agent discriminate circles diamonds falling vertically towards agent could move laterally change perception object version study agent sensor states figure simulation large small blocks fall diagonally towards bottom row world agent bottom row purpose illustrating task large brick avoided falling left small brick caught falling right simulations one block falling time small large bricks fall either left right depiction agent neurons bottom left triangles depict sensors circles illustrate brain neurons trapezoids denote actuators sequence activity patterns agent retina right large brick falls right discriminate large small blocks catching small blocks avoiding large ones order create visual ambiguity agent four upwardlooking sensors blind spot two units furthermore blocks falling diagonally left right right left agent categorize also predict see fig evolve active categorical perception two experiments using two different control architectures one artificial neural network ann one markov network specifically network hidden markov gates hmgs edlund described detail section agent located bottom row world periodic boundary conditions able move horizontally one unit per time step see fig note compared van dartel doubled vertical size arena order give agent time assess direction falling block agent four sensors state sensor block detected block seen arrangement agent sensors allow unambiguously identify falling block three eight possible input states counting input sensors active input states unambiguous first classifies large block positioned right agent second third sensor state remaining two units block positioned two units two patterns identify similarly positioned small blocks five input states created either small block big block block see fig trial block either small two units large size four units falls top bottom time steps blocks move continuously downwards sideways one unit per time step blocks either always move right left object caught position block units agent units time step overlap least one unit characterization correlations assign probabilities possible states world theoretically falling block different starting positions large small falling left right giving rise possible experimental initial conditions agent initial positions periodic boundary conditions ensure equivalent given initial positions falling block time steps block reaches bottom row total possible different states world expect states discriminated agent instead introduce world introducing four bits believe capture salient aspects world define environmental joint variable take states defined table course world state world character sensor activated least one sensor activated block left agent block right agent block two units small block four units large block moving left block moving right table world states four bits note could ambiguous case block centered agent exactly units away resolve ambiguity setting block centered agent exactly units away encoding reveals bias experimenters believe salient states world certainly underestimates amount discoverable entropy however hindsight appears sufficient capture essential variations world furthermore lends study aspects world represented within agent network controller defining representations different aspects world representation thus study four representations rhit rlr represent whether sensor activated whether block left right agent block large size small size whether block moving left right also measure much measured bits binary concept represented particular variable example rlr node measures much block left right concept encoded variable two architectures cognitive systems agent controlled cognitive system composed computational units loosely referred neurons map sensor inputs motor outputs cognitive system also neurons internal hidden layer neurons part input output layer define sensor neurons neurons directly process input input layer define output neurons units map units network output layer artificial neural networks ann evolvable topology first experiment robot movements controlled artificial neural network consists nodes four input units one sensor two output units ten hidden units states input units discrete values specifying whether object detected states output units actuators discrete integer values encoding one three possible actions move one unit right left move stand still move right move left stand still hidden units states continuous values evaluating states calculate discretize binary values become every value becomes discussed earlier discretization introduce bias value usually classic artificial neuronal networks fixed topology one layers connections defined associated weight previous experiment found fixed topologies lead approximately constant even fitness agent increases data shown one way increase complexity network information represents evolve network topology well connection weights make network topology evolvable beyond searching connection weights using neuronal gates arbitrarily connect nodes type input hidden output nodes without fixed layered topology classic anns connection associated certain weight calculates sum values set incoming nodes via gate multiplied associated weight applies sigmoid function calculate output tanh sum runs neurons feed gate value propagated every node connected apply genetic algorithm system ann encoded genome follows start codon two loci mark beginning subsequent two loci encode number inputs outputs two loci specify origin inputs neurons feed gate outputs writes information followed encoding weights see fig number gates network change evolves determined number start codons genome genomes encoding anns undergo mutational changes described later section respect evolving anns similar using evolutionary algorithms neat see stanley miikkulainen evolve neural networks augmented topology markov brains second experiment agent controlled network nodes four input two output ten internal nodes types number nodes anns connected via hidden markov gates hmgs see edlund networks hmgs markov brains mbs short type stochastic markov network see koller friedman related hierarchical temporal memory model neocortical function hawkins blakeslee george hawkins hmax algorithm riesenhuber poggio except markov brains need organized strictly hierarchical manner connectivity evolved rather designed hmg understood machine defined hidden neuronal gate neuronal gate figure artificial neural network four input neurons orange ten hidden neurons blue two motor neurons green nodes connected via two neuronal gates connects four arbitrary input nodes weight four output nodes figure illustrates nodes become updated time point two ngs write node outputs added indicated sigmoid function applied structure fig state transition table fig nodes markov brains binary principle hmgs stochastic output nodes fire set state probability determined transition table hmg receive four inputs distribute signals nodes minimum one input one output node settings configurable evolution acp task consider deterministic hmgs row transition table contains one value transitions probability turning hidden markov gates classical logic gates order apply evolutionary algorithm hmg encoded similar way ngs using genome specifies network whole locus genome integer variable following start codon marking beginning gene gene encodes single hmg next two loci encode number inputs outputs gate respectively followed specification origin inputs identity nodes written example hmg depicted fig loci following start codon would specify inputs outputs read write information followed encoding probabilities state transition table see supplementary fig edlund details example given fig particular hmg specified circular genome loci counting start start codon universally arbitrarily chosen consecutive loci combination occurs chance every pairs loci making start codons rare insert four start codons arbitrary positions loci initial genome jump start evolution thus ancestral genomes experiments markov brains encode least hmgs set hmgs encoded manner uniquely specifies markov brain encoding robust sense mutations change structure hmg leave probability table intact either adding removing parts table flexibility also implies considerable neutrality genome gene loci reserved probability table even many fewer loci used mbs anns differ respect gates connecting nodes network anns use weights sums tanh function together continuous variables compute actions contrast mbs use discrete states boolean logic perform computation using similar encoding topology means mutations similar effect topology systems different effects computations system performs figure single hmg three inputs two outputs reads nodes writes nodes updating states nodes process output states determined set probabilities denoted pxy decimal equivalent binary pattern input output respectively example probability pattern fire input evolutionary algorithm evolve two types networks anns mbs using genetic algorithm find solutions problems using evolutionary search see michalewicz operates specific genetic encoding networks structure genotype iterating cycle assessing network fitness population candidates selecting successful ones differential replication finally mutating new candidate pool testing network performance controlling agent network faced possible initial conditions world take fitness calculated fraction successful actions number large blocks avoided plus number small blocks caught tests number zero purpose selection use exponential fitness measure multiplies score factor every successful action divides score every unsuccessful action fitness assessment genotypes ranked according placed next generation probability proportional fitness roulette wheel selection without elite replication genotypes mutated implemented three different mutational mechanisms occur replication different probabiities point mutation happens probability per locus causes value locus replaced uniform random number drawn interval chance delete sequence adjacent loci ranging size chance stretch adjacent loci duplicated size sequence deleted duplicated unformly distributed range given duplicated stretch randomly inserted two loci genome duplications deletions contrained genome allowed shrink sites genomes grow beyond sites insertions likely deletions tendency genomes grow size evolution evolve networks generations run replicates experiment note type gates different anns neuronal mbs logic rate evolution two networks compared directly mutations vastly different effects respect function gates thus optimal mutation rate differs among networks orr end evolutionary run reconstruct evolutionary line descent lenski experiment following lineage successful agent end generations backwards way random ancestor used seed experiment possible use genotypes line descent given temporally ordered sequence genotypes recapitulates unfolding evolutionary process mutation mutation ancestor evolved agent high fitness captures essence particular evolutionary history organisms lines descent particular experiment calculate number quantities among much world agent represents brain using equation extracting probabilities behavior inputs measure representation probabilities observe particular state well joint probabilities describing probability observe state time another variable takes state representation defined sensor internal environment variables distinguished particular organism agent performs acp task evolved controller measured point evolution placing organism simulated world concurrently recording time series data states controller nodes states environment recordings used calculate frequency states based frequencies states probabilities relevant quantities calculated including take account temporal order events example probability variable takes state variable takes state particular state combination states never occurs probability zero recorded entry even though principle state combination states could occur quantities introduced section calculated organisms evolutionary line descent making possible follow evolutionary trajectory random ancestor adapted agent anns internal nodes continuous rather binary states mapping intervals applied calculation probabilities results establish baseline random controllers two network types created distributions fitness values obtained baseline served two purposes shows well randomly generated unevolved networks perform much information world represent chance well providing information distribution values random anns mbs created way randomly drawing values uniform probability distribution integers genome loci genome sprinkled start codons arbitrary positions within genome fig shows distribution fitness scores random anns fig shows distribution representation respecitve distributions mbs shown fig fitness scores fig representation systems use different types gates anns mbs differ respect initial fitness representation distributions shows random genomes high fitness rare need evolve functionality order perform optimally ann ann figure probability distribution fitnesses representation scores random machines probability distribution fitnesses fraction successful actions random anns probability distribution representation variable random anns distribution fitnesses random markov brains distribution representation networks order compare two network architectures evolutionary trajectories fitness representation analyzed evolutionary line descent lod described section different lods obtained following back member final population quickly coalesce single line hence lod effectively recapitulates genetic changes led random networks proficient ones development fitness representation evolutionary time fig averaged independent replicates anns mbs average low fitness begin evolutionary run seen fig mbs become significantly fit anns generations find runs mbs evolved perfect fitness none anns reached level best anns correctly make decisions time see fitness anns generations stagnating suggests runtime allow improvement previously mentioned rate fitness achieved evolutionary time compared across architectures mutations affect function gates differently tentatively explain difference performance anns mbs difference representing world anticipate different network architectures also solve categorization task differently understand information dynamics strategies employed detail measured number informationtheoretic measures besides see sections evolutionary trajectory representation see fig similar evolution fitness see fig mbs evolve significantly higher value attribute difference difference fitness two types networks discretization continuous ann variables introduce bias thus appears increased representation world within agent network controller correlates fitness test correlation fitness representation end run replicates mbs anns find fitness significantly correlated spearman mbs anns speculate anns forced compute using sigmoid function effectively implementing multipleand gate mbs use arbitrary logic operations process data anns struggle internalize represent environmental states words appears ease crucial forming representations efficiently transformed fit decisions mbs edlund generations generations figure fitness representation bits along line descent function evolutionary generations averaged independent evolutionary lines evolved networks anns black mbs blue analysis network structures strategies order successful task described agent perform active categorical perception followed prediction implementation acp task beer prediction achieved without memory network entered attractor representing category prediction move away stay directly coupled attractor task used achieved using memory data shown requires agent perform categorical perception comparing sensory inputs least two different time points also allows prediction object going land order analyze information processed calculated predictive information bialek evolved networks given mutual shannon information network inputs time outputs time predictive information defined way measures much entropy outputs firings motor neurons control agent understood terms signals appeared agent sensors prior action indirectly predictive information therefore also indicates contribution hidden nodes network high predictive information would show hidden nodes contribute much computations performed mainly input output neurons using variable sensor states actuator states predictive information written terms shared entropy sensor states time motor states time ipred log probability observe variable state probability observe variable state etc note joint random variables created variables node implying take different states take possible states probabilities extracted time series data described section figure shows course evolution predictive information ipred decreases mbs initial increase increases slightly overall anns drop predictive information mbs indicates actions become less dependent ipred hunpred generations siatom figure different measures information processing integration along lod types network architectures anns black markov brains blue predictive information unpredicted entropy network motor variables information integration siatom based sensor inputs driven hidden neurons anns actions remain predictable sensor inputs test whether internal states increasingly guide agent predictive information subtracted entropy output states maximally two bits calculate unpredicted entropy outputs much motor outputs uncorrelated signals input hunpred ipred figure shows hunpred increases course evolution suggesting indeed signals sensor readings guiding motors principle increase could due increase motor neuron entropy however latter stays fairly constant conclude network adapts environment less outputs determined inputs internal states effect stronger mbs anns suggests indeed internal states encode representations drive network behavior also possible motors evolve react sensor signals back time sensor neurons store information delayed response also processed via internal states absolute value predicted information unpredicted entropy depend time delay expect overall trend decreasing ipred coupled increasing hunpred predictive information sensorial signal stream temporal correlations quantify synergy network calculated measure information integration called synergistic information roughly speaking synergistic information measures amount information processed network whole understood terms individual node measures extent whole network sum parts edlund siatom xti measures amount information processed across time whole network joint random variable composed node variables whereas xti measures much processed node negative used quantify redundancy information processing neural network atick nadal parga schneidman siatom special case information integration measure tononi balduzzi tononi computationally far complex siatom relies computing information integration across possible partitions network siatom instead calculates information integration across atomic partition partition node part figure shows siatom increases markov brains well anns indicates architectures evolve ability integrate information perform task hand see marginal difference mbs anns ability integrate information time mbs dependent internal states ultimately perform better suggests measuring integrated information terms allow inferences system capacity memorize summary observe mbs evolve become less dependent sensorial inputs anns addition actions mbs become dependent internal states anns thus conclude network properties measures indeed representations networks create adaptive strategy networks represent environments features world represented form successful strategies epistemically opaque strategies order analyze markov brain function number different tools available first causal diagram generated drawing edge two neurons connected via hmg edges directed note edge principal perform different computation creating causal diagram nodes never written nodes removed computationally inert remain default state nodes also identified via procedure input node forced either state individually procedure effect fitness node inert fig shows causal diagram evolved markov brain solves figure causal diagram markov brain perfect fitness correctly catching small blocks avoiding large ones nodes colored red sensors motor variables green double arrows represent two causal connections one way nodes arrows point write output back input may work memory nodes return default set otherwise update state information maintained via internal nodes read motors giving rise proprioception precisely kinesthesia ability sense one motion motors used memory sification task perfectly one see network uses inputs sensors motors memory simultaneously decisions fusing different modalities intelligently murphy causal diagram however reveal function achieved network hmg present instantiation represents deterministic logic gate generally stochastic possible determine logical rules network transitions state state feeding transition table logical analyzer rickmann analyzer converts state transition table minimal description functions boolean logic using functions exactly describe node logical influence nodes possibly network depicted fig logic given numeral represents node index state time subsequent time point overbar stands note logical representation network dynamics optimized contribution inert nodes removed general possible determine minimal logic network based transition information finding minimal logic believed computationally intractable problem kabanets cai consequence possible capture network function terms set logical rules surprised evolution delivers epistemically opaque designs humphreys designs understand fundamental level however strength measuring representations measure able capture representations even highly distributed take part complex computations provides valuable tool analyze evolved neural networks seen node node node node node node rhit rlr gener gener gener gener gener gener figure representation environmental properties concepts defined eqs function time within nodes network evolved become one depicted fig representation measured bits along temporal genetic line descent measured generations concepts memory understand representations acquired representations concepts calculated property environment defined within key nodes example network shown fig network fig shows nodes prefer represent given single features concepts others represent several features time addition degree node represents certain property changes course evolution looking representation within individual node however tells part story clear representations generally smeared several nodes case pair nodes example represent feature sum representations node variables represent synergistically order discover combination nodes represents feature accurately search partitions network would performed much like search partition minimum information processing calculation network synergistic information processing balduzzi tononi one also ask whether brain states represent environment time represented whether represents environment past state words ask whether representations distant proximal events answer define temporal representations including temporal index markov variables example representation time point defined implicit representation events one update prior defined shared entropy internal variables time environmental states time given sensor states time naturally one define temporal representations distant events manner calculated averaged experiments generation anns mbs course evolution figure shows systems larger larger still note smaller data shown values increase evolutionary time similar see fig suggest networks evolve form memory reaches back one update suggest peak representation time difference two updates implied fig explained hierarchical structure networks process sensorial information least two time steps reach decision takes least two time steps order assess direction motion block decisions made shortly thereafter however order move generations generations figure representation function evolutionary time three different time intervals representation anns red green blue representation markov brains colors red curves fig shown purpose comparison agent correct location time strengthens view representations evolved furthermore build agent lifetime memory past events shape agent decisions conclusions defined quantitative measure representation terms information theory shared entropy states environment internal brain states given states sensors internal states necessary order encode internal models internal states representations indeed representational information environment subset information stored internal states information state past internal states count towards neither would information imagined worlds example testing nodes system contain representations aspect environment helps distinguish information present internal states information specifically used representation applied measure two types networks evolved control simulated agent active categorical peception task experiments showed achieved increases fitness evolution independently system used also showed algorithmic function artificial neural networks markov networks difficult understand deterministic markov networks reduced sets boolean logic functions logic however may epistemically opaque representation increases networks evolutionary time neuron represent parts individual concepts features environment however often concepts distributed several neurons represent synergistically addition representations also form lifetime agent increasing agent integrates information different concepts reach decision thus evolves markov artificial neural networks via darwinian processes representations rather evolves capacity represent environment representations formed agent observes interacts environment argued measured continuous artificial neural networks well discrete systems markov networks suggests measure used complex natural systems found implementation used markov networks able evolve ability form representations easily artificial neural networks future investigations show kind system powerful make intelligent decisions using representations acknowledgements thank koch tononi extensive discussions representations information integration qualia research supported part german federal ministry education research paul allen family foundation national science foundation frontiers integrative biological research grant nsf beacon center study evolution action contract well agriculture food research initiative competitive grant usda national institute food agriculture wish acknowledge support berlinbrandenburg academy sciences michigan state university high performance computing center institute cyber enabled research references atick could information theory provide ecological theory sensory processing network bertschinger der guettler olbrich predictive information explorative behavior autonomous robots eur phys balduzzi tononi integrated information discrete dynamical systems motivation theoretical framework plos comput biol beer toward evolution dynamical neural networks minimally cognitive behavior maes mataric meyer pollack wilson editors animals animats proceedings fourth international conference simulation adaptive behavior pages cambridge mit press beer dynamics active categorical perception evolved model agent adaptive behavior bialek nemenman tishby predictability complexity learning neural computation bongard zykov lipson resilient machines continuous science brooks intelligence without representation artificial intelligence chomsky aspects theory syntax cambridge mass mit press clark dynamical challenge cognitive science clark toribio without representing synthese cohen lefebvre editors handbook categorization cognitive science amsterdam netherlands elsevier cover thomas elements information theory john wiley new york craik nature explanation cambridge university press cambridge edlund chaumont hintze koch tononi adami integrated information increases fitness simulated evolution autonomous agents plos comput biol floreano mondada evolution homing navigation real mobile robot systems man cybernetics part cybernetics ieee transactions fodor language thought new york crowell george hawkins hierarchical bayesian model invariant pattern recognition visual cortex prokhorov editor proceedings international joint conference neural networks ijcnn volume pages ieee george hawkins towards mathematical theory cortical microcircuits plos comput biol harnad editor categorical perception groundwork cognition cambridge cambridge university press haugeland representational genera ramsey stich rumelhart editors philosophy connectionist theory pages hillsdale lawrence erlbaum hawkins blakeslee intelligence henry holt new york humphreys philosophical novelty computer simulation methods synthese wason thinking readings cognitive science cambridge university press kabanets cai circuit minimization problem yao luks editors proc symposium theory computing pages kawato internal models motor control trajectory planning current opinion neurobiology kay phillips coherent infomax computational goal neural systems bull math biol koller friedman probabilistic graphical models mit press cambridge lenski ofria pennock adami evolutionary origin complex features nature marr vision computational investigation human representation processing visual information henry holt new york usa marstaller hintze adami measuring representation christensen schier sutton editors proceedings conference australasian society cognitive science pages sydney macquarie centre cognitive science mcgill multivariate information transmission psychometrika michalewicz genetic algorithms data strucures evolution programs springer verlag new york moravec locomotion vision intelligence brady paul editors robotics research pages murphy biological cognitive foundations intelligent sensor fusion ieee transactions systems man cybernetics part humans nadal parga nonlinear neurons low noise limit factorial code maximizes information transfer network newell simon human problem solving englewood cliffs prentice hall nolfi power limits reactive agents neurocomputing nolfi floreano evolutionary robotics mit press cambridge orr rate adaptation asexuals genetics phillipps kay smyth local cortical processors maximize coherent variation could lay foundations representation proper smith hancock editors neural computation psychology pages new york springer verlag phillips singer search common foundations cortical computation behavioral brain sciences pinker learnability cognition cambridge mass mit press pitt mental representation zalta editor stanford encyclopedia philosophy metaphysics research lab fall edition rickmann logic friday version computer software retrieved http riesenhuber poggio hierarchical models object recognition cortex nature neuroscience schneidman still berry bialek network information connected correlations phys rev lett shannon mathematical theory communication bell system technical journal stanley miikkulainen evolving neural networks augmenting topologies evol comput tononi consciousness integrated information provisional manifesto biol bull van dartel postma van den herik reactive agents perceptual ambiguity adaptive behavior van dartel situated representation phd thesis maastricht university ward ward representation dynamical agents neural networks watanabe information theoretical analysis multivariate correlation ibm journal research development wolpert ghahramani jordan internal model sensorimotor 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fusing video inertial sensor data walking person identification yuehong huang tseng feb department computer science national chiao tung university taiwan emails yctseng abstract autonomous computer system robot typically needs identify locate track persons appearing sight however solutions limitations regarding efficiency practicability environmental constraints paper propose effective practical system combines video inertial sensors person identification pid persons different activities easy identify show robustness potential system propose walking person identification wpid method identify persons walking time comparing features derived video inertial sensor data associate sensors smartphones human objects videos results show correctly identified rate wpid method seconds index artificial intelligence computer vision gait analysis inertial sensor walking person identification introduction human navigates world five senses including taste touch smell hearing sight sometimes rely one sense sometimes multiple senses computer systems optical sensor perhaps essential sensor captures information like human eyes cameras widely used public safety services hospitals shopping malls streets etc hand booming use sensors seen many iot applications due advances wireless communications mems work like raise one fundamental question improve perceptivity computer systems integrating multiple sensors specifically interested fusing video inertial sensor data achieve person identification pid shown fig efficient pid first step toward surveillance home security person tracking checkout supermarkets conversation traditional pid technologies usually based capturing biological features like face voice tooth fingerprint dna iris however techniques require intimate information users cumbersome registration training process user cooperation also fig scenes biological features difficult extract relying optical sensors implies high environmental dependency lighting obstacle resolution view angle etc thus suitable public sites scene captured construction site shown fig workers must wear helmets masks protect falling objects toxic gases top view courtyard shown fig clearly recognizing biological features difficult scenarios recognition approaches based wireless signals require active participation users method proposed integrates computer vision via depth camera uhf rfid capable recognizing individuals walking groups wearing rfid tags thus enabling interaction however method handicapped short range users need carry extra rfid tags work propose practical effective convenient pid system combining computer vision inertial sensor data almost everyone carries smartphone almost every smartphone inertial sensors inside main workflow pid system shown fig video data set human objects comparable features retrieved similarly inertial sensors set inertial data comparable features retrieved similarity score calculated analyzing similarity scores pairing derived leads pid result inertial sensors widely used data collector video camera inertial sensor video feature extraction acc feature extraction human object retrieval remove direction butter worth filter step feature extraction sampling matching result fig data fusion workflow rive carrier motions paths physical activities standard modules current smartphones hand get motions traces physical activities people videos persons different types activities easy pair object sensor people activity time difficult identify persons work discusses situation people camera walking contributions work follows first develop practical robust pid system second solution integrates two types popular sensors third work matching method focuses wpid show robust pid system combines video inertial sensor data together rest paper structured follows section introduces pid system wpid method performance evaluation results presented section conclusions drawn section proposed walking person identification consider environment fig video camera multiple users data collected camera smartphones sent server pid purpose pid system four software modules shown fig video feature extraction module retrieves human objects walking traces sequence video frames acceleration acc feature extraction module retrieves walking information acceleration data similarity scoring module compares walking features data sources assigns similarity scores pairing module couples human objects smartphones based similarity scores video feature extraction module human object retrieval processes frame extract objects recognized human directly realized yolo frame yolo outputs set human objects represented bounding boxes bounding box rectangle inside yolo features features similarity scoring matching matrix pairing person ids video objects human pairs fig pid architecture frame number person person fig changes bounding boxes walking recognizes human object ith bounding box denoted center width height denoted respectively examples shown fig connect human objects adjacent video frames form continuous traces trace sequence human objects regarded person efficient object tracking algorithms available design lightweight tracing method based movement limitation generally human running speed less assuming frame rate frames per second fps cases person move height two frames based assumption trace search range find human object next frame results traces connecting human objects continuous frames step feature extraction retrieves features trace fig shows two sequences frames two human objects suppose camera downward viewing angle person walks along vertical line steps forward bounding box becomes taller closes feet bounding box becomes shorter person walks along horizontal line steps forward bounding box becomes process transforms ternary sequences tei aej similarity score tei aej defined sim tei aej dif aej stride stride person person stride step patterns stride stride stride stride stride stride stride stride frame number fig walking traces wider closes feet bounding box narrows result changes time regarded step patterns use denote ith trace fig shows extracted two persons make strides frames respectively also mark ground truth strides graph seen well present human step patterns acc feature extraction module work user carries smartphone installed application app put pockets hand software collects acceleration inertial sensor since activity recognition inertial sensor data intensively studied simply adopt existing solutions sensor data ith device sequence acceleration magnitudes removing direction since energy captured accelerometer associated human movements remove components lowpass filtered order butterworth filter frequency since frame rate fps simple frequency decreased per second steps get step feature similarity scoring module retrieving step features video data sensor data want answer following question similar sequence acc sequence similarity denoted sim work try match extremum positions two sequences ignoring exact values first conduct extremum detection find local points window length example set traverse points adjacent points value point adjacent points point recorded point experiments set maximum point marked minimum point marked rest marked number extremums tei dif tei aej defined tei dif aej exists otherwise scan binary value tei tei tei dif tei aej returns tei dif tei aej traverse aej range position search range nearest position aej tei exists search range dif tei aej returns returned dividing sum differences gives similarity score tei aej pairing module similarity scoring get sim sim array recording similarity scores frame let pairing result frame pairing problem formulated different expression linear sum assignment problem lsap max simij pij simij similarity score ith human object jth sensor assignment constraints pij pij pij use hungarian algorithm solve problem pij means human object paired sensor pij means human object paired sensor work frame pairing result call pairing result stage raw pair stage result however practice identification result unstable base raw pair stage result example may identify one person sansa one frame comes may identify person jack next frame comes person may identified lucy frame next frame comes problem especially serious trace person still short makes result rough hard see consideration propose refined pair stage refined pair stage identification result trace depends let rpf result generated refined pair stage frame let rsim twodimensional array value rsimij number times object paired sensor refined pairing problem formulated lsap max rpij rsimij subject rpij rpij fig correctly identified results different viewing angles different spaces table correctly identified rates different rpij different simij rsimij number pairing times simij small rsimij large trace human object long logarithmic function shown used weaken impact length traces pairing pij rpij means human object paired sensor rpij means human object paired sensor performance evaluation developed prototype system one video camera multiple mobile devices camera logitech webcam resolution prove solution tried different models smartphones including redmi note asus zenfone htc evo server personal computer intel core cpu nvidia geforce graphics card devices used system synchronized network time server conduct number experiments wpid method average speed tracing wpid method different pairing stages around fps apparently wpid method two different stages consumes server resources show robustness wpid method experiments carried different areas viewing angles downward viewing angle outdoor area set shown fig horizontal viewing angle indoor area set shown fig experiments persons carry smartphones pockets hands wander around freely styles shown fig wpid method work different areas different view angles different ways carrying smartphones different walking styles following statistics cases two persons result condition generated least continuous frames measure accuracy wpid method let number persons shown front camera stage ppp raw refined latest frame let niid number frames ith person identified program nicd number frames ith person correctly identified program define correctly identification rate rcd nicd rcd let length time person continuously detected yolo small sequences extracted short considered step pattern result set threshold lengths two sequences larger matching processes setting seconds table shows rcd two stages table increase leads increase rcd cases however increase rcd obvious also refined stage achieves better performance raw stage especially visual performance conclusions propose new pid system combining optical inertial sensors design light tracking algorithm wpid method two pairing stages people different activities easy identify persons comparing behaviors extracted video inertial sensor data complex part system identify persons activities time work design wpid method identify walking persons show robustness potential pid system conduct extensive experiments lot discussions validate claims results show correct identification rate wpid method seconds references taigman yang ranzato wolf deepface closing gap performance face verification ieee conf comput vision pattern recognition bae yoon robust online tracking based tracklet confidence online discriminative appearance learning ieee conf comput vision pattern recognition june dorothy lunt identification tooth morphology forensic sci vol apr min yang yunde jia temporal dynamic appearance modeling online tracking comput vis image vol ruizalbacete iris recognition based sift features first ieee int conf biometrics identity security sept xiang alahi savarese learning track online tracking decision making ieee int conf comput vision iccv dec mahsan rofouei andrew wilson brush stewart tansley phone mine fusing body touch device sensing interaction proc sigchi conf human factors computing new york usa chi acm shane colton balance filter simple solution integrating accelerometer gyroscope measurements balancing platform sherry hsi holly fait rfid enhances visitors museum experience exploratorium commun acm vol hanchuan peijin zhang samer moubayed shwetak patel alanson sample hybrid computer vision rfid system recognizing individuals groups chi conf extended abstracts human factors comput syst acm joseph redmon santosh divvala ross girshick ali farhadi look unified object detection ieee conf comput vision pattern recognition cvpr joseph redmon ali farhadi better faster stronger ieee conf comput vision pattern recognition cvpr ross girshick jeff donahue trevor darrell jitendra malik rich feature hierarchies accurate object detection semantic segmentation ieee conf comput vision pattern recognition cvpr washington usa alex bewley zongyuan lionel ott fabio ramos ben upcroft simple online realtime tracking ieee int conf image yoon yang lim yoon bayesian tracking using motion context multiple objects ieee winter conf applicat comput vision jan muhammad shoaib stephan bosch ozlem durmaz incel hans scholten paul havinga fusion smartphone motion sensors physical activity recognition sensors vol mathie monitoring interpreting human movement patterns using triaxial accelerometer university new south wales melania susi valrie renaudin grard lachapelle motion mode recognition step detection algorithms mobile phone users sensors vol rainer burkard eranda linear assignment problems extensions springer boston
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pricing ramping reserve capacity reserve real time markets dec hongxing student member ieee zuyi senior member ieee increasing penetration renewable energy recent years led uncertainties power systems order maintain system reliability security electricity market operators need keep certain reserves securityconstrained economic dispatch sced problems new concept deliverable generation ramping reserve proposed paper prices generation ramping reserves generation capacity reserves derived affine adjustable robust optimization framework help prices valuable reserves identified among available reserves prices provide crucial information values reserve resources critical flexibility investment market equilibrium based prices analyzed simulations system ieee system performed illustrate concept ramping reserve price capacity reserve price impacts reserve credit market participants discussed index reserve capacity reserve marginal price uncertainties affinely adjustable robust optimization omenclature indices index unit uncertainty line time index bus functions sets expectation function trace transpose row matrix cost function adjustment matrix lagrangian function optimal value variable set units located bus constants number buses units uncertainty constraints number lines time intervals cost coefficients unit rnt unity matrix rnt cost related unit commitment decision aggregated equivalent load demand bus dnd dnd branch flow limit abstract vector shift factor line bus pimin pimax minimum maximum generation outputs unit shutdown indicators authors robert galvin center electricity innovation illinois institute technology chicago usa lizu riu rid variables unit ramping limits incidence matrix rnt abstract matrix vector abstract shift factor matrix unit load polyhedron uncertainty set matrix vector generation adjustment matrix rnt prices energy lmp unit rnt reserve price vector unit marginal price affine adjustment rnt generation output generation vector rnt generation vector rnt generation ramping reserve qru generation capacity reserve uncertainty vector rnt lagrangian multipliers constraints lagrangian multipliers constraints credits uncertainty mitigators ntroduction renewable energy sources res wind power generation demand response attracted lot attentions recently total installed capacity wind power reached end several pjm iso new england nyiso caiso initiated programs markets essential objective use renewable energy initiate programs electricity markets maximize total social warfare well protect environment however also pose new challenges system operators electricity markets due intrinsic characteristics amount available renewable energy sometimes hard predict instance wind production varies installed wind capacity denmark hourly basis prediction error wind farms aggregated output existing method may fall range total installed capacity meantime amount loads also increases wholesale market forecasting loads relies forecasted price input significant error surviving uncertainties fundamentally important reliable secure operation power system system schedule accommodate deviation wind power load forecasted values system operator may curtail wind energy shed load market rtm order keep certain level reliability security increase ramping capability system compensate variations wind energy load short time efficient reliable methods required determine optimal reserves uncertainty level high recently stochastic robust approaches successfully applied researchers address issues related uncertainties electricity markets time market designers also seeking effective market mechanism address uncertainty issues instance intraday market idm established market dam rtm european countries since uncertainties intraday level significantly smaller compared level scheduling process hasp employed california iso typical approaches solving stochastic scuc scenario based basic idea generate enough samples uncertain parameters assumption probability distribution function pdf known samples modeled mixed integer programming problem two main drawbacks approaches pdf hard obtain circumstances uncertainty accommodation guaranteed fact problem becomes intractable sample size large comparing stochastic optimization two largest merits robust optimization solution immunized uncertainties pdf required robust scuc problem solved two stages first stage determine unit commitment solution immunized worst case lowest cost second stage feasible solution sced obtained affinely adjustable robust optimization aaro models proposed recently employ affine function adjust generation output following load deviation recently propose robust scuc model fast solution approach although applying robust techniques receives lot attentions researchers still remains big challenge credit flexibilities electricity markets existing ancillary service market reserves determined advance amount required reserves generally extracted larger number monte carlo simulations contingencies help aaro reserves determined one shot based uncertainty information critical issue price reserves explicit reserve requirement constraints hand reserves free byproducts approach kept market participants want get energy profits moreover reserves scarce resources due deliverability observations indicate available reserves system valuable system operator point view many countries electricity markets still evolving challenges uncertain energy resource load ramping products proposed california iso accommodate uncertainties emphasized delivery considered ramping capacity applying robust optimization sced real market corresponding pricing theory imperative paper tries propose new ideas clear obstacle three major contributions paper listed follows new concept deliverable generation ramping reserve proposed within aaro sced framework generation ramping reserve additional ramping capability generator part locked sced schedule prices generation ramping reserves well generation capacity reserves derived within robust framework help price information valuable reserves easily identified among available reserves market equilibrium characterized proposed prices dispatch instructions market participants get maximal profit following iso dispatch instruction price signals rest paper organized follows derivation reserve prices presented section based aaro sced market mechanism credit flexibilities discussed case studies ieee systems presented section iii section concludes paper aaro sced rices electricity markets normally operate two markets including dam rtm balancing market majority trades cleared dam via scuc sced normally performed periodically rtm paper mainly focuses rtm standard sced problem unit generation output subject following constraints include unit capacity limits unit ramping limits pimin pimax riu pimin rid pimin equations show unit operate minimum capacity two cases right turned right turned implies unit provide reserve two cases notation brevity use matrix vector replace formulations sced problem formulated min api stands operation cost denotes load balance constraints compact form represent transmission constraints due forecasting errors renewable power output load need run sced rolling basis real time balance system recently approaches successfully applied problem address uncertainty issues caused variations load renewables stochastic robust studied intensively considering uncertainties best knowledge paper represents first work pricing reserves robust optimization framework hence following assumptions made focus concept transmission loss ignored sced problem proposed approach ante dispatch ante pricing assumed units dispatched according instructions scheduling intervals units bid energy price reserve bid zero uncertainty comes loads renewables treated negative loads uncertainties contingencies discussed paper uncertainty set information available expectation covariance uncertainties obtained historical data however pdf information specified uncertainty hard obtain uncertain sources res rop formulated rop min rnt affine adjustment matrix rnt rnt base dispatch adjusted dispatch respectively rnd uncertainty vector deviation loads forecasted values new unit dispatch regulated based load deviation noted rnk rnk polyhedron includes lower upper bounds uncertainty noted entry considered uncertainty level positive aaro sced denoted matrix diagonal entry objective function minimize expected cost without loss generality assume objective function rewritten assumed covariance matrix available pdf unavailable problem rop converted computationally tractable problem follows constraints including uncertain parameters exactly reformulated min api basic idea aaro optimization originally paper linear feedback control theory used adjust dispatch realization load authors applied solve sced problem section generation output affinely adjusted according uncertainties affinely adjustable robust sced agi also variables derived obtained strong duality problem convex solved efficiently commercial solvers cplex gurobi observed pdf information required solve different standard sced extra terms added inequality constraints respectively problem indicates certain unit transmission constraints standard sced replaced stronger constraints robust framework thus system actually keeps certain flexibilities uncertainty accommodation following sections analyze flexibilities also called reserves generation ramping reserve capacity reserve uncertainty mitigator refers flexible resource provider participates management uncertainties marginal prices power ramping rate limit time interval fig illustration upward ramping reserve locked ramping ramping reserve flexible resources include generators available ramping capabilities adjustable loads ums keep certain reserves order accommodate uncertainties compared constraint may binding even scheduled dispatch reach capacity limits ramping limits based optimal solution reserves calculated generation ramping reserve defined unused unit ramping capability value slack variable pimin qru pimin qrd qru upward downward ramping reserves respectively fig illustrate concept upward ramping reserve scheduled generation outputs locked ramping used unit ramping capability generation ramping reserve available ramping rate less locked ramping ramping reserve guarantees uncertainty mitigator still additional ramping capability scheduled ramping process existing market locked ramping normally ignored spinning reserves rtm time resolution minutes minutes ignore ramping process chance system provide enough ramping capability accommodate uncertainties contrast time resolution one hour dam units enough time redispatch even locked ramping ignored generation capacity reserve defined unused unit generation capacity constraint slack max qcu min qcd lower upper generation capacity reserves respectively system point view total capacity reserves fixed unit commitment load level determined generation reserve approaches market explicit reserve requirement constraint modeled shadow prices type constraint employed derive reserve price reflects coupled effects generation reserve instead setting reserve manually heuristically based monte carlo simulations reserves aaro sced determined automatically one shot although obvious advantages traditional reserve determination also poses new challenges reserve price derivation existing pricing approaches used directly due lack explicit reserve requirement constraints amount reserves calculated according question set prices one hand well known reserves deliverable uncertainty accommodation due network constraints hand generation reserve coupled together rtm even reserve bid price zero market clearing price reserve certain relations energy paper call reserve obtained available reserve small increment decrement reserve amount causes change expected operation cost type reserve called valuable reserve determine exact value reserve derive marginal prices reserve according lagrangian function follows lagrangian function api agi ramping reserve price defined marginal expected cost due unit decrement generation ramping rate capacity reserve price defined marginal expected cost due unit decrement generation capacity obtained lagrangian function hence reserve provided valuable note consists lagrangian multipliers unit ramping limits unit capacity limits lmp aaro sced obtained based definition marginal expected cost due unit increment load generator formulated rnt also consists energy component congestion component shown affine adjustment price obtained rnt represents marginal value adjustment coefficient stated payment noted unit participant factor credit uncertainty mitigators within aaro framework ums help system withstand load deviation future generation reserve coupled together credit reserve reflect coupling effect ums provide valuable reserves entitled credits price flexible resources total credit allocated product reserve price reserve amount fact reserve price reflects much value reserve reflects available reserve reserve quantity time interval reserve valuable reserve get reserve credit otherwise credit entitled zero even available reserve provides similar phenomenons traditional reserve market example reserve price specific zone occurs cleared system reserve higher required amount case section market equilibrium source obtain maximum profit following iso dispatch instruction objective function strictly convex unique choice get maximal profit price signal associated credit provide incentives dispatch power output supplies load maintains ramping capacity reserves addition price signal provides incentive follow adjustment instruction given optimality condition problem satisfied dispatch signal price signal constitute competitive partial equilibrium assumption made equilibrium model market participant uses affine policy adjust generation dispatch obtained aaro sced solution consequently equilibrium conditions also market participants alternative way get partial market equilibrium lump reserve price lmp however serious incentive issues integrated lmp written api new credit receives energy reserve observed reserve credit negative shadow price upper bound constraint nonzero means flexible resources provides fewer profit gets provided negative incentives flexible resources contrast reserve credit defined always nonnegative iii ase tudy system modified ieee system studied section illustrate concepts reserves associated prices well impacts market participants simulations carried cplex intel ram partial market equilibrium model assumed market participants price takers assumption popular electricity market expected profit maximization problem source formulated pmpi max system system consisted two units one wind farm two loads three lines please refer http decision variables source bus detailed data simplicity given price signal credit three time intervals studied increi affine adjustment matrix stated mental cost employed represent fuel cost noted constraint mitigator time resolution minutes assumed units incentive necessary follow dispatch system committed determined dam instruction objective function converted load current interval time assumed accurate loads time time forecasted based api current available information forecasting errors may qpi exist assume expectation uncertainties time observed portion lagrangian time probability distribution unknown total expected cost calculated based aaro function optimal solution pmpi function since problem convex sced higher standard sced cost slater condition satisfied strong duality holds immuned uncertainties fore saddle point optimal solution indicates expensive supplies loads within also optimal solution pmpi consequently aaro sced standard sced reason table base lmp ystem ase load time lmp time interval fig net load table pward eserves rices ystem ase time rrp crp rrp crp ramping reserve rrp ramping reserve price capacity reserve crp capacity reserve price table iii redits ntitled nits based eserve rices ase time dispatching extra generation operator needs additional deliverable reserves accommodate uncertainties minimizing total cost entries entries loads bus bus interval increased respectively units based adjustment matrix question whether units incentives maintain reserves first consider scenario without reserve credits lmp bus time larger marginal cost therefore inclined supply loads time increase output shrink reserve provide fact negative profit time uplift issue beyond scope paper scenario market participants would ignore uncertainties game market consider second scenario reserve credits upward reserves provided ums shown table observed capacity reserve scarce resource online capacity adequate available reserves contrast prices upward ramping reserve time time according definition paper valuable reserves scenario ums entitled certain credits based contribution shown table iii uncertainty mitigator entitled time time entries ramping constraints time time respectively ramping reserves times respectively credit calculated based noted credits ramping reserve calculated according difference power outputs two intervals next give example show whether inclined deviate base dispatch instruction credits lmps follows instruction shown table total credit associated lmps consider possible base dispatch generates less time profit negative time also generates less time maintain ramping reserve new credit ramping reserve time decreased credit lower following saved fuel cost supplying smaller load therefore get profit using dispatch example illustrates gets credit associated lmps inclined deviate dispatch instruction rigorous mathematical analysis market equilibrium shown section modified ieee system traditional units branches modified ieee system scheduling period hours time interval minutes loads depicted fig ucs determined advance solution robust scuc problem reserves five wind farms introduced system located buses respectively denote set buses uncertainty assumed forecasted power output nominal output installed capacity wind farm respectively uncertainties case res uncertainty satisfies reflects forecast error confidence interval single wind farm reflects forecast error confidence interval aggregated wind output indicates aggregated forecast error confidence interval smaller sum five single intervals experiment forecast error increases time intervals table eserves ncreasing ncertainty evels ixed ominal ind ower fig total reserve credit uncertainty levels fixed nominal wind power detailed data including unit parameters uncertainty correlation matrix line reactance ratings net load profiles found http consider interval bounds uncertainties perform sensitivity analysis respect fig shows reserve credits ums receive change reserve credit sum ramping reserve credit capacity reserve credit ums entitled high ums also entitled high credits shown reserve credits sum products amount valuable reserve price valuable reserve analyzed follows table presents available reserves valuable reserves time increasing forecast errors fixed normal wind power output fixed error percentage installed capacity observed available reserves remain valuable reserve change dramatically forecast errors shown table upward available ramping reserve remains capacity reserve remains main reason unit commitment load demand fixed time system contrast upward valuable reserve indicates opportunity cost keeping ramping reserve zero recover profit energy credit valuable ramping reserve around valuable capacity reserve around respectively ums entitled credits keeping reserves also shown fig suggests opportunity cost keeping reserve nonzero ums get profits deviating dispatch instruction entitled reserve credits increased amount valuable ramping reserve jumps means available reserves become valuable uncertainty level high valuable capacity reserve also increases case similar tendency also observed downward reserves shown table emphasized amount valuable reserve change monotonically uncertainty level instead revealed paper trend upward available ramping reserve price unit depicted different time intervals fig shown fig time unit provides available reserve valuable reserve downward reserve avaramp available ramping reserve valramp valuable ramping reserve avacap available capacity reserve valcap valuable capacity reserve ramping reserve ramping reserve ramping reserve price time interval fig upward ramping reserve price unit price zero observed ramping reserve price reaches highest point time also peak load interval contrast ramping reserve price low time although load demand time still relatively high compared intervals observed load climbs time time falls time time shown fig indicates ramping reserve price related load demand also load change case upward ramping reserve scarce resource time opportunity cost keeping also high contrast upward ramping reserve relatively cheap load demand falling time fig depicts available capacity reserve price capacity reserve unit although reserve amount unit keeps time interval price different observed capacity reserve price case similar trend system load level shown fig example capacity reserve capacity reserve capacity reserve upward reserve avaramp valramp avacap valcap avaramp valramp avacap valcap ramping reserve price capacity reserve price capacity reserve price reserve credit time interval fig upward capacity reserve price unit eoc eoc fig reserve credit expected operation cost eoc different expensive time peak load occurs reason upward capacity reserve online installed capacity fixed less load level load level high upward capacity reserve small becomes scarce resource system reserve credits respect different presented fig shows decrease also leads lower payments related reserve example decreases total decreases around expected operation cost also decreases indicates shrinking uncertainty set actually increases feasible set robust dispatches numerical results part indicate reserve payment proposed paper helps maximize social warfare uncertainty level high payment related reserve also high may attract investment flexible resources flexible resources also mean system capabilities handle uncertainties system accommodate higher res penetrations onclusions paper proposes new concept ramping reserve within aaro sced framework aaro sced effective tool rtm address uncertainty issue although solution may flexibilities paper include generation ramping reserve generation capacity reserve prices ramping reserve capacity reserve also derived based lagrangian function opportunity costs uncertainty mitigators keep reserves flexibilities help prices reserves classified two categories available reserves valuable reserves case studies explain concept reserves impacts behaviors market participants many researches topic open future increasing res penetration power system flexibilities play crucial role uncertainty accommodation prices derived paper provide option provide reserve signals within robust optimization framework reserve credits ums may attract investment flexible resources long term return new flexibility investment allows system accommodate higher res level pointed reserve prices unitspecified admitted perfect ideal resources bus price extension proposed reserve prices set maximum ramping capacity reserve prices units located node nodal price way nodal reserve prices determined expensive opportunity cost reserve node however due affine policy optimality aaro sced may deteriorate partial market equilibrium certain extent eferences wiser bolinger wind technologies market report lawrence berkeley national laboratory tech economic demand response performance report pjm interconnection tech online available http holttinen impact hourly wind power variations system operation nordic countries wind energy vol hodge milligan wind power forecasting error distributions multiple timescales power energy society general meeting ieee ieee integration renewable resources california iso tech integration wind system dispatch new york iso tech borggrefe neuhoff balancing intraday market design options wind integration discussion papers german institute economic research diw berlin tech california iso caiso scheduling takriti birge long stochastic model unit commitment problem ieee trans power vol shahidehpour comparison interval optimization approaches stochastic scuc ieee trans power vol jiang zhang guan network constrained robust unit commitment problem european journal operational research vol bertsimas litvinov sun zhao zheng adaptive robust optimization security constrained unit commitment problem ieee trans power vol warrington goulart mariethoz morari reserves power systems ieee trans power vol jabr adjustable robust opf renewable energy sources ieee trans power vol goryashko guslitzer nemirovski adjustable robust solutions uncertain linear programs math ser vol mar robust unit commitment recourse cost requirement power energy society general meeting ieee july zheng litvinov zonal reserve modeling pricing energy reserve market ieee transactions power systems vol may wang shahidehpour reserve requirements joint energy ancillary services auction power systems ieee transactions vol weber adequate intraday market design enable integration wind energy european power systems energy policy vol iso flexible ramping technical appendix http online accessed ott experience pjm market operation system design implementation ieee trans power vol zhang wang adjustable robust realtime power dispatch wind power integration ieee transactions sustainable energy vol shahidehpour unit commitment simultaneous clearing energy ancillary services markets ieee trans power vol ellison tesfatsion loose byrne project report survey operating reserve markets electric energy regions sandia natl labs publications zheng litvinov post pricing electricity market ieee trans power vol iso new england manual market operations manual revision iso new england access may online available http isone market operations revision whinston green microeconomic theory oxford university press new york vol shahidehpour yamin market operations electric power systems forecasting scheduling risk management press hogan ring pricing electricity markets electricity policy group
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