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declarative control flocking dynamics usama mehmood department computer science stony brook university usa radu grosu oct systems group technische universitat wien austria ashish tiwari sri international usa nicola paoletti department computer science stony brook university usa lin department electrical computer engineering stony brook university usa junxing yang department computer science stony brook university usa abstract popularity flocking models reynolds classic flocking model raises question whether declarative flocking models possible question motivated observation declarative models generally simpler easier design understand analyze operational models introduce simple control law flocking based cost function capturing cohesion agents want stay together separation agents want get close refer declarative flocking use control mpc define controllers centralized distributed settings thorough performance comparison declarative flocking reynolds classic flocking model recent flocking models use mpc cost function based lattice structures demonstrate yields best cohesion least fragmentation maintains surprisingly good level geometric regularity still producing natural flock shapes similar produced reynolds model also show high resilience sensor noise acm reference format usama mehmood nicola paoletti dung phan radu grosu lin scott stoller ashish tiwari junxing yang scott smolka declarative control flocking dynamics proceedings symposium applied computing sac acm new york usa pages https introduction flocking collective behavior exhibited large number interacting agents possessing common group objective term commonly associated birds recently drones examples include foraging food executing predatoravoidance maneuver engaging migratory behavior permission make digital hard copies part work personal classroom use granted without fee provided copies made distributed profit commercial advantage copies bear notice full citation first page copyrights components work must honored uses contact sac april pau france copyright held acm isbn https dung phan department computer science stony brook university usa scott stoller department computer science stony brook university usa scott smolka department computer science stony brook university usa introduction reynolds model control became norm flocking community specifically model agent executes control law given terms weighted sum three competing forces determine next acceleration forces rule separation keep safe distance away neighbors cohesion move towards centroid neighbors alignment steer toward average heading neighbors descriptions suggest rules executed agent distributed environment sensing communication popularity reynolds model many variants raises question abstract declarative form control flocking question important declarative models generally simpler easier design understand analyze operational models analogous declarative programs functional programs logic programs easier write verify imperative programs show answer question indeed positive providing simple control law flocking based cost function comprising two main terms cohesion average squared distance pairs agents separation sum inverse squared distances except time pairs agents within sensing range example term representing velocity alignment needed cost function specifies want goal hence declarative contrast update rules reynolds model aim achieve implicit goal hence operational executing declarative control amounts finding right balance attracting repelling forces agents refer approach declarative flocking use mpc control define controllers refer approach define centralized version requires communication distributed version previous mpcs flocking exist mpcs designed conform model flocking proposed impose highly regular structure flocks neighboring agents distance apart specified constant kind structure seen settings beehives expected many natural engineered settings imposed reynolds model sac april pau france mehmood paper show via thorough performance evaluation centralized distributed compare reynolds rulebased approach approach variant zhan centralized mpc approach zhang distributed mpc approach consider performance measures capture multiple dimensions flocking behavior number flock fragmentation maximum diameter cohesion velocity convergence new measure geometric regularity formation experimental results demonstrate yields best cohesion least fragmentation produces natural flock shapes like produced reynolds model also distributed maintains surprisingly good level geometric regularity also analyze resiliency mpc approaches considering impact sensor noise results demonstrate remarkably high level resiliency part comparison approaches rest paper organized follows section presents mpc approaches mentioned section defines declarative flocking approach section defines performance measures flocking models section presents experimental results performance evaluation section discusses related work finally section offers concluding remarks directions future work models flocking behavior consider set dynamic agents move according following equation motion respectively position velocity acceleration agent space step time step consider physical constraints velocities accelerations described sets respectively defined limit allowed magnitude velocity acceleration vectors respectively flocking models agents update motion changing acceleration sense represents control input agent configuration agents described vector let vnt atn equation motion agents expressed local neighborhood agent defined set agents called neighbors within given distance mimicking agent visibility sphere interaction radius configuration set spatial neighbors agent given denotes euclidean norm figure examples quasi solid lines connect agents neighborhood distance dashed lines connect distance tolerance configuration define associated proximity net graph connects agents within interaction radius capture regular geometry flocks introduced notions configurations agent equally distant neighbors quasi configurations modulo small error distances scale parameter defines ideal distance definition configuration called scale tolerance configuration called quasi sensing noise extend classical equations motion eqs sensing noise affecting agent perceives positions velocities neighbors existing work put little focus flocking dynamics subject noise unfortunately unavoidable realistic natural engineered flocks actual positions velocities step let denote noisy counterparts sensed generic agent defined vectors independent identically distributed random variables position noise velocity noise distributed according gaussian distributions mean standard deviation respectively stress dependency noise variables independent across time steps centralized flocking algorithms agent decisions computed single controller information whole population use define noisy measurements distributed algorithms sensing noise independent agent denote noisy measurement agent positions velocities noisy agents except agent vit declarative control flocking dynamics sac april pau france defined per implicitly agent agent noise distribution sampled independently compute component reynolds model reynolds distributed model agents follow simple rules compute accelerations positions velocities neighbors rules illustrated figure explicitly specify desired flocking formation objective rather flocking emerges interaction rules specifically agent updates acceleration step considering following three components adapted include sensing noise alignment agents match velocities average velocity nearby agents cohesion agents move towards centroid agents local neighborhood aial aci separation agents move away nearby neighbors cohesion alignment rules help form maintain closely packed formation separation rule prevents agents coming close thus reducing crowding collisions constants weights acceleration component typically smaller interaction radius hence smaller neighborhood used separation rule significant agents close overall acceleration reynolds model given asi models model predictive control mpc control technique works follows time step computes optimal control sequence agents accelerations case minimizes given cost function respect predictive model controlled system finite prediction horizon length step first control input optimal sequence applied remainder sequence unused algorithm proceeds new iteration two main kinds flocking models exist centralized distributed centralized models assume information positions velocities agents available compute optimal accelerations formally time step solves following optimization problem min model flocking models interaction pair agents modeled potential field assumed agent point source potential field around exerts force equal gradient agents range influence potential field circular symmetry hence function distance source work potential function pair agents minimum desired distance desired outside interaction radius potential function constant potential field exerts force exact definition complicated definite integral action function product bump function uneven sigmoidal function control law computes agent acceleration based sum forces agents neighborhood velocity alignment term control input accelerations agents predicted time step starting step first term primary cost function controller seeks optimize within prediction horizon implicitly function predicted configurations prediction horizon time step second term standard mpc problems penalizes large control inputs weight distributed flocking models agent computes optimal acceleration based information neighbors agent solves optimization problem form aial aci asi min acceleration agent predicted time step starting step cost function agent distributed mpc agent way know current future control decisions neighbors needed make accurate predictions behavior address problem approaches allow agents communicate local control decisions future positions assume neighbors follow default motion law move constant velocities adopt second strategy require communication majority existing approaches flocking designed optimize regularity flock penalizing configurations neighboring agents exactly distance apart configurations differ call approaches mpc next describe representative centralized distributed mpc flocking models extend account sensing noise centralized model variant model zhan distributed model zhang centralized mpc flocking centralized mpc problem defined min sac april pau france mehmood alignment cohesion separation figure interaction rules flocking behavior reynolds model configuration system predicted time step starting step following dynamics initial state prediction window given noisy measurements configuration captures irregularity total deviation agent distances model inspired differs cost function also contains velocity alignment term authors removed subsequent work impulsive mpc used means agents directly control velocities instead accelerations abstraction allows physically unrealizable accelerations distributed mpc flocking distributed mpc flocking model zhang agent controls acceleration based position velocity measurements neighbors assumes constant velocity zero acceleration prediction horizon similarly set neighbors assumed invariant prediction horizon denote control law agent min predicted future dynamics determined neighbors constant velocity configuration defined similar way quantifies much neighborhood deviates declarative flocking section introduces centralized distributed versions declarative flocking model presents flocking algorithm based mpc formulation declarative consists two simple terms cohesion term based average squared distance pairs agents keep flock together separation term based inverse squared distances pairs agents avoid crowding two terms represent opposing forces agents causing agents move towards positions forces balanced unlike majority existing approaches designed optimize conformance design impose specific geometric structure centralized model cost function centralized model contains two terms described cohesion term considering pairs agents separation term considering pairs agents neighbors weight separation term provides control density flock control law equal distributed model cost function distributed model similar centralized one except terms limited consider pairs agents neighbors jid control law agent equal measures flocking performance introduce four key measures flocking performance single measure insufficient flocking indeed characterized multiple desirable properties aligned velocities cohesion introduces four main properties flocking informally described group agents stays connected unique flock fragmentation emerge group remains cohesive formation declarative control flocking dynamics sac april pau france group moves coherent way unique body agents velocities aligned group maintains regular geometry sense introduce following four measures capture four requirements important concept definitions subflock set interacting agents far apart agents interact formally configuration corresponds connected component proximity net let set connected components proximity net number connected components proximity net quantifies equivalently flock fragmentation fragmentation exists fragmentation may temporary move different directions permanent maximum component diameter denoted quantifies cohesion defined max diameter connected component max note agents isolated domain max function equation empty singleton note consider maximum diameter order make measure independent connectedness instead considered overall diameter entire possibly fragmented flock flocking model poorly connectedness would also poorly measure velocity convergence measure adopted quantifies average discrepancy agent velocity average velocity flock particular extend measure average velocity convergence values across measure regularity geometric structure flock reflected spacing introduce parameterfree irregularity measure connected component defined sample standard deviation distances agent closest neighbor thus measure penalizes configurations dispersion distances imposing fixed distance unlike let set connected components isolated agents excluded equivalently agents isolated set irregularity optimal value reflects fact single point regular structure moreover configuration already highly penalized measure defined min standard deviation multiset samples sum operator disjoint union multisets see def optimal value since neighboring agents located distance leading zero standard deviation term shows captures regularity underlying concept introduce measure previous measures regularity irregularity measure deviations specified distance therefore inapplicable flocking models reynolds model models based specified target distance also irregularity measure flexible based gives optimal score configurations geometrically regular example consider configuration agents vertices grid edge length interaction radius equal length diagonal box grid configuration optimal value irregularity measure distance every agent nearest neighbor configuration hence nave optimal value irregularity measures used prior work performance evaluation compare performance models section newly introduced flocking models setting first set experiments section evaluate performance measures illustrated section second set experiments section analyze resilience algorithms sensor noise consistency experimental settings latticebased mpc problems solved using interior point method implemented matlab fmincon function problems solved using gradient descent optimization unless otherwise specified population size simulation length parameter values ones reported following settings opensteer project parameters reynolds model weight separation term centralized distributed initial positions initial velocities agents uniformly sampled respectively performance comparison flocking algorithms fig shows examples final formations flocking models particular chose configurations fragmentation occur observe formations mpc algorithms rigid structures consistent sac april pau france design objective maximizing regularity hand reynolds mpc models result natural flock shapes fig compare performance measures averaged runs flocking model regarding number connected components centralized registers best behavior rapidly stabilizing average component see plot distributed reynolds model comparable performance reaching average number mpcs instead lead constant fragmentation distributed mpc centralized mpc model ranking confirmed diameter measure plot centralized distributed reynolds model show best cohesion outperforming approaches recall measure indicates maximum diameter diameter entire population consequence fragmentation tends improve diameter values since produces fewer individuals explains distributed performs better measure centralized version similarly model smaller diameter measure centralized mpc turn smaller diameter measure distributed variant expected model mpcs good performance irregularity plot since designed achieve regular geometric formation surprisingly distributed performs almost well measure centralized reynolds model least regular formations velocity convergence plot find models perform comparably well able achieve flocks consistent velocities fairly quickly initial spike robustness sensing noise evaluate resiliency models sensor noise performed runs model noise levels noise levels numbered noise level performance metric averaged final values runs noise level results plotted fig six models model vulnerable sensing noise number model quickly increases nearly rendering metrics irrelevant mpc models also exhibit high fragmentation leading nominally good largely irrelevant values performance metrics distributed reynolds model best resiliency sensing noise models exhibiting similar profiles metrics irregularity velocity convergence measures increase noise level expected models remarkably maintain almost single connected component nearly constant component diameter noise levels achieving smaller diameter reynolds model related work reynolds introduced first approach simulation flocking behavior three simple rules model mehmood able capture complex flocking behaviors animals additional rules added model simulate specific behaviors leader following predator avoidance pearce present strategy flocking agents move maximize view flock cucker dong present rulebased flocking approach proofs convergence collision avoidance cucker smale introduced another popular flocking model model parameterized constant velocity convergence guaranteed velocity convergence also achieved conditions initial positions initial velocities agents ahn investigated effects multiplicative noise long term dynamics model erban extend model take account stochasticity imperfections agent behavior delay agents responses changes environment flocking models based potential fields proposed several papers tanner propose potential function given equation distance agents distances greater potential set constant value indicating zero force control law acceleration agent based sum neighbors gradient potential function similar potential function also proposed furthermore solutions extended additional behaviors obstacle avoidance leader following example ogren use motion leader guide motion flock leader motion independent influenced agents sheng propose extension model designed noisy environments addition terms found model control law contains feedback terms position velocity make agents tend stay close centroid neighborhood minimizing velocity mismatch neighbors show adding feedback terms control law helps bound error dynamics system conclusions paper presents abstract declarative form control flocking behavior results thorough comparison centralized distributed versions declarative flocking four flocking models simulation results demonstrate yields best cohesion least fragmentation produces natural flock shapes like produced reynolds model resiliency analysis shows distributed version highly robust sensor noise future work plan study resilience flocking models respect additional noisy scenarios actuation noise noise affecting acceleration faulty agents deviant behavior also plan investigate smoothing techniques increase resilience sensor noise declarative control flocking dynamics sac april pau france reynolds ized mpc tributed mpc centralized mpc distributed mpc figure examples final formations different flocking models red dots agent positions blue lines denote agent velocities line lengths proportional speeds lattice distributed reynolds centralized lattice centralized distributed time time number connected components irregularity time time max component diameter velocity convergence figure comparison performance measures obtained runs flocking algorithm time lattice distributed centralized distributed lattice centralized reynolds noise level number connected components noise level irregularity noise level noise level max component diameter velocity convergence figure comparison final values performance measures obtained runs flocking algorithm time noise level references shin ahn stochastic flocking dynamics model multiplicative white noises math phys https arxiv http camacho bordons model predictive control springer felipe cucker dong general flocking framework ieee trans automat control cucker smale emergent behavior flocks ieee trans automat control may https radek erban jan yongzheng sun model noise delay siam appl math july https sheng flocking control multiple agents noisy environments ieee international conference robotics automation https reza flocking dynamic systems algorithms theory ieee transactions automatic control daniel pearce adam miller george rowlands matthew turner role projection control bird flocks proceedings national academy sciences https arxiv http naomi ehrich leonard peter ogren cooperative control mobile sensor networks adaptive gradient climbing distributed environment ieee transactions automatic control john reif hongyan wang social potential fields distributed behavioral control autonomous robots robotics autonomous systems https craig reynolds opensteer steering behaviors autonomous characters http craig reynolds flocks herds schools distributed behavioral model siggraph comput graph https craig reynolds steering behaviors autonomous characters proceedings game developers conference sac april pau france tanner jadbabaie pappas stable flocking mobile agents part dynamic topology ieee international conference decision control ieee cat vol jingyuan zhan xiang flocking systems predictive mechanisms ifac proceedings volumes jingyuan zhan xiang flocking systems via model predictive control based measurements ieee transactions industrial informatics zhang zhaomeng cheng guanrong chen chunguang model predictive flocking control systems input constraints ieee transactions circuits systems regular papers lifeng zhou shaoyuan distributed model predictive control flocking via neighbor screening optimization international journal robust nonlinear control mehmood
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categorical time series models covariates oct konstantinos lionel truquet october abstract study problem stationarity ergodicity autoregressive multinomial logistic time series models possibly include latent process defined recursive equation improve considerably upon existing stationarity ergodicity conditions models proofs based theory developed chains complete connections useful coupling technique employed studying ergodicity infinite order stochastic processes generalize markov chains addition finite order markov chains discuss ergodicity properties model includes strongly exogenous necessarily bounded covariates mathematics subject classification primary secondary keywords phrases autoregression categorical data chains complete connection coupling covariates ergodicity markov chains university cyprus department mathematics statistics box nicosia cyprus email fokianos umr cnrs irmar university rennes campus beaulieu rennes cedex france ensai campus rue blaise pascal bruz cedex france email introduction goal article improve upon theoretical properties regression based models analysis categorical time series might include covariates assumed bounded binary time series particular cases categorical time series results obtain apply logistic autoregressive models take point view generalized linear models theory see mccullagh nelder conditional distribution categorical time series given past multinomial obviously belongs multivariate exponential family distributions theory generalized linear models applied modeling different types categorical data nominal interval scale mostly concerned nominal data therefore multinomial logistic model natural candidate model fitting see fahrmeir tutz kedem fokianos among references discussion modeling issues regarding categorical data emphasize finite state markov chains provide simple prominent model categorical time series lagged values response affect determination future states however markov modeling context categorical time series poses challenging problems indeed order markov chain increases number free parameters fact number free parameters increases exponentially fast furthermore markovian property requires simultaneous specification dynamics response possible covariates observed jointly specification might possible general studying models binary generally categorical time series infinite order driven latent process feedback mechanism type models quite analogous garch models bollerslev defined terms conditional instead conditional variances particular feedback models make possible low dimensional parametrization yet accommodate quite complicated data structures examples feedback models context binary categorical time series studied recently moysiadis fokianos fokianos moysiadis among others discuss results compare findings models inference binary time series topics studied several authors see kedem early treatment regression modeling context studied cox stern coe slud kedem among others see also kedem fokianos early references recently binary time series data increasingly popular various financial applications breen butler malaikah christoffersen diebold christoffersen startz nyberg kauppi cui also scientific fields previous results related theoretical properties binary time series models given jong woutersen related work categorical time series reported fahrmeir kaufmann kaufmann fokianos kedem russell engle proposed categorical time series model financial transactions data alternative classes models based probit link function autoregressive models considered zeger qaqish rydberg shephard kauppi saikkonen among others several classes models analysis categorical data studied see books joe macdonald zucchini articles biswas song prove theoretical results assuming contraction type condition conditions usually employed theoretical analysis time series models instance case count time series models see fokianos neumann doukhan however work closely related modeling approach suggested fokianos main idea essentially employ called canonical link process model observed data note jong woutersen shown near epoch dependence binary time series models authors different modeling point view likelihood based inference models study developed along lines previous references proof consistency asymptotic normality based standard arguments concerning convergence score function hessian matrix however mention work relaxes considerably previous results case model covariates improve upon kaufmann kedem fokianos avoid assumptions design covariates addition show obtain ergodicity unbounded covariates included model study maximum likelihood estimation requires existence appropriate moments covariate process though central limit theorems maximum likelihood estimators given previous references therefore give details article structured follows section discusses general categorical time series models allowing conditional probabilities depend whole past series addition giving result stationarity ergodicity chains complete connections results applied case infinite order autoregressive multinomial logistic model section discuss models might include latent process results obtained theorem improve results obtained moysiadis fokianos fokianos moysiadis finally section discuss inclusion exogenous covariates autoregressive multinomial logistic model theorem main result section discuss existence processes ergodic properties time series autoregressive models categorical data general approach let finite set simplicity assume nonnegative integer suppose observe process state space interested modeling dynamics instance consider modeling stock price change change positive change negative change see russell engle sleep state status see fokianos kedem towards goal define vector category observed time otherwise throughout work consider stochastic processes adapted filtration defined vector conditional success probabilities say words last category set ynt corresponding success probability pnt several possibilities autoregressive modeling processes take values finite space instance assuming vector matrices appropriate dimension consider following linear model studied russell engle qaqish model implies quite complex restrictions parameters element vector belong interval restrictions become even involved covariate process included avoid subtle technicalities adapt generalized linear models point view considering canonical link model see fokianos kedem instance define log suppose vector process determined infinite order model suitably defined function process satisfies takes values set canonical basis null vector furthermore measurable function conditional distribution given possibly depends infinite past useful example process past values given linear process vector sequence matrices comparison shows unnecessary restrictions unknown coefficients circumvented since vector furthermore covariates easily included including additional additive term categorical type autoregressive models considered widely used several applications case simple logistic regression model studied widely literature see cox snell early reference processes consider work particular examples general class processes called chains complete connections processes widely studied applied probability doeblin fortet harris iosifescu grigorescu following work bressaud discuss next coupling technique related chains complete connections results chains complete connection throughout section consider finite state space positive integer write consider probability kernel defined takes values satisfies following assumption assumption exists sequence decreases zero infm chain complete connections stationary process satisfying assumption consider chain znx satisfies znx addition given sequence let markov chain taking values defined define quantity plays crucial rule evaluating mixing coefficients chain following result given bressaud prop lemma proposition coupling process defined znx inf satisfies proposition proved defining iteratively pair unx vnx using maximal coupling conditional distributions coupling associated total variation distance conditional distributions proposition yields following corollary see also bressaud cor specific case following result corollary proof using proposition obtain znx proposition implies result corollary follows bounding along lines derivation bressaud pointed bressaud corollary implies existence uniqueness stationary chain complete connections satisfying assumption furthermore corollary yields bound controlling coefficients associated indeed recall two coefficients defined see doukhan instance sup coefficients random sequence given sup borel set infinite product approximated finite union cylinder sets infinite order stationary markov chain exists corollary mixing coefficients satisfying proposition suppose proof suppose denotes probability distribution znx last bound depend hence obtain last bound depend also upper bound remark shown bressaud prop moreover decreases exponentially hence result prop follows decreases zero exponentially fast note also property implies ergodicity process see bradley application categorical time series recall categorical time series model whose state space defined results previous subsection deduce following corollary denotes euclidian norm corollary assume model let function exist sequence satisfies exists unique stochastic process taking values exp exp moreover stationary proof denote probability kernel defined defined exp exp lipschitz assumption implies component exp say bounded hence exists moreover bounded set provided large enough choose hence exists using remark also corollary proposition exists unique stationary solution satisfying solution equivalent condition infinite order linear model corollary applies provided jka denotes note condition corresponding operator norm matrix particular obtain logistic autoregressive model infinite order stationary denotes real valued sequence categorical time series latent process section consider specific instances chains complete connections following methodology garch models see engle bollerslev book francq instance recalling notation introduced model latent process depend additionally past values statistical perspective parametrization yields parsimony allows flexible structures accommodate various forms autocorrelation specific suppose two positive integers let function say process solution problem satisfied general result define mapping defined main result section following theorem suppose exist integer kgy kgy following hold true let vector sequence elements limit lim exists depend let function defined lim function bounded moreover exist process solution problem chain complete connection associated function see corollary defined denotes first coordinates function defined previously exists unique strictly stationary solution equations moreover process geometrically decreasing mixing coefficients implies ergodicity joint process proof first part assertion straightforward consequence assumption omitted focus proof lipschitz property function set lim stated assumption obtain kgy hence setting letting obtain result first point theorem necessary condition follows easily let assume setting note continuity function gvt implies lim gvt gvt lim gvt proofs satisfies third point straightforward consequence two first results corollary moreover geometric decay coefficients discussed remarks made following proposition finally implies ergodicity process ergodicity process see samorodnitsky instance linear models let real matrices size assume model written alternatively gvt denotes identity matrix order assumptions theorem less unity norm less one satisfied spectral radius large enough also means roots polynomial det outside unit disc case result improves conditions proved moysiadis fokianos since require additional assumption coefficient addition results answers affirmative question posed case binary autoregressive model compared work fokianos moysiadis note case logistic autoregressive modeling binary data obtained conditions simplify considerably conditions models recall assume exists norm positive real numbers kxi kyi proved condition process taking values random mapping gzt contracting iteration indeed follows induction hence exists integer mapping gzt satisfies therefore assumption theorem satisfied note condition improves upon conditions obtained moysiadis fokianos fokianos moysiadis since require inclusion exogenous covariates section study problem including covariate process autoregressive categorical time series model assuming covariate process strongly exogenous unbounded assumption exogeneity implies time independent conditionally allows simple computation likelihood function indeed denotes conditional density given bounded measurable function dyn denotes reference measure model type exogeneity also called granger causality sims causality literature see instance monfort sec discussion different concepts restricting study case finite order markov chains parameter depend past values general case appears difficult tackle considered another communication aware result guarantees ergodicity model covariate even simple case markov chain see interesting parallel markov chains exogeneous covariates markov chains random environments studied probability theory proof theorem given use approach discussed cogburn showing ergodicity markov processes random environments general result finite state markov chains covariates discussing results concerning stationarity ergodicity finite state markov chain jointly observed covariate process follows denote stationary process values space process takes values finite set addition conditionally inhomogeneous markov chain precisely assume exist family transition matrices pzt throughout section assume following exists integer pzm process mixing ergodic theory sense elements lim denotes shift operator defined note assumption implies process satisfying also satisfies addition assumption stronger assuming ergodicity process weaker classical strong mixing condition usually employed literature large number useful stochastic processes mixing instance strong mixing processes bernoulli shifts defined measurable function sequence instance samorodnitsky discusses several properties different types mixing ergodic theory stationary processes assumption employed obtaining ergodicity shift operator implied ergodicity shift operator main result section given following theorem theorem suppose hold true exists unique stochastic process satisfying moreover process ergodic proof first show almost sure limit lim pzt exists depend markov chains condition comparable weak ergodicity notion limit taken backward sense see seneta several sufficient conditions ensuring weak ergodicity properties markov chains using ergodicity coefficients recall dobrushin contraction coefficient stochastic matrix defined sup two probability mesures finite set total variation distance defined well known contraction two stochastic matrices moreover min denotes cardinality set assumption ensures pzt let obtain setting pzt pzt pzt assumption covariate process mixing process also mixing indeed denote mappings defined respectively two borel sets aand get moreover observe operator ergodic indeed borel set obtain using assumption conclude shows ergodic using assumption log pzm ergodic theorem get exp log addition deduce pzt pzt pzt shows product matrices pzt converges stochastic matrix whose rows equal exists measurable function lim pzt setting random probability measure integer set kolmogorov extension theorem exists values unique measure marginals hence denotes probability distribution measure defined couple stationary process satisfying show uniqueness let another stochastic process satisfying distribution markov chain transitions pzt shown conditional distribution unique equal next show ergodicity process end use approach introduced cogburn study markov processes random environment type argument also used sinn poupart positive transition matrix pzt give general shorter proof approach used cogburn consists considering markov kernel defined denotes probability distribution takes values invariant process defined markov chain transition kernel let invariant set every using corollary hairer markov chain forms ergodic process every set measure case end first note get every complement hence obtain every also gives defined assumption moreover pzm write denotes symmetric difference sets assumption entries matrix pzm positive deduce almost every set let denote marginals first remark employing assumption get set stated conclude stationarity therefore every using assumption conclude every easily get exists using equality min finally obtain hence shown process ergodic process application multinomial logistic model covariates assume conditionally covariate process taking values process markov chain exp exp measurable functions let check assumption satisfied conditional markov chain defined conditionally process defines markov chains qzt since transition qzt takes positive values assumption follows taking assuming covariate process theorem applies guarantees ergodicity process results show approach simplifies conditions required obtain consistency asymptotic normality maximum likelihood estimator even case considering covariates however existence moments covariate process still required study large sample properties maximum likelihood estimator references biswas song arma processes statistics probability letters bollerslev generalized autoregressive conditional heteroskedasticity journal econometrics bradley introduction strong mixing conditions vol heber city kendrick press breen glosten jagannathan economic significance predictable variations stock index returns journal finance bressaud galves decay correlations dynamics coupling approach electron probab butler malaikah efficiency inefficiency thinly traded stock markets kuwait saudi arabia journal banking finance christoffersen diebold mariano tay tse forecasts asian equity markets based conditional variance skewness kurtosis dynamics evidence hong kong singapore journal financial forecasting christoffersen diebold financial asset returns forecasting volatility dynamics management science cogburn ergodic theory markov chains randon environments wahrscheinlichkeitstheory verw gebiete cox statistical analysis time series recent developments scand statist cox snell analysis binary data london chapman hall jong woutersen dynamic time series binary choice econometric theory doeblin fortet sur les liaisons bull soc math france doukhan mixing properties examples number lecture notes statistics new york doukhan fokianos weak dependence conditions poisson autoregressions statist probab lett engle autoregressive conditional heteroscedasticity estimates variance united kingdom inflation econometrica fahrmeir kaufmann regression models nonstationary categorical time series journal time series analysis fahrmeir tutz multivariate statistical modelling based generalized linear models second springer series statistics new york contributions wolfgang hennevogl fokianos kedem regression theory categorical time series statist sci fokianos moysiadis binary time series friven latent process econometrics statistics fokianos rahbek tjostheim poisson autoregression amer statist assoc fokianos poisson autoregression multivariate anal francq garch models stracture statistical inference financial applications united kingdom wiley monfort statistics econometric models volume cambridge university press hairer ergodic properties markov processes lecture notes available http harris chains infinite order pacific math iosifescu grigorescu dependence complete connections applications cambridge university press joe multivariate models dependence concepts london chapman hall kaufmann regression models nonstationary categorical time series asymptotic 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distributions simulating correlated binary variables specified marginal means correlations biometrika russell engle econometric analysis financial transactions data using new autoregressive conditional multinomial model ssrn elibrary russell engle model financial transactions prices times journal business economic statistics rydberg shephard dynamics price movements decomposition models journal financial econometrics samorodnitsky stochastic processes long range dependence springer seneta matrices markov chains springer series statistics springer new york revised reprint second edition new york sinn poupart asymptotic theory conditional random fields gordon dunson dudik eds proceedings fourteenth international conference artificial intelligence statistics volume proceedings machine learning research fort lauderdale usa slud kedem partial likelihood analysis logistic regression autoregression statist sinica startz binomial autoregressive moving average models application recessions journal business economic statistics stern coe model fitting analysis daily rainfall data journal royal statistical society series general rejoinder recent theory autoregressive count time series test generalized choice models categorical time series statist plann inference cui logit regression model binary time series journal time series analysis zeger qaqish markov regression models time series approach biometrics
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socrates system scalable graph analytics savkli chapman chee minch johns hopkins university applied physics laboratory mail stop laurel usa distributed semantic graph processing system provides locality control indexing graph query parallel processing capabilities presented semantic introduction graphs provide flexible data structure facilitates fusion disparate data sets popularity graphs shown steady growth development internet cyber social networks graphs provide flexible data structure processing large graphs remain challenging problem successful implementation graph analytics revolves around several key considerations rapid data ingest retrieval scalable storage parallel processing paper present graph analytics platform particularly focused facilitating large scale analysis semantic graphs recently nosql systems hadoop become popular storing big data however systems face several fundamental challenges make analyzing data difficult lack secondary indexing leads poor performance attribute queries flights seen moving faster lack locality control lead unnecessary movement data lack schema makes database maintenance challenging traditional relational databases rdbms problems face set challenges dealing big data table structures flexible enough support new kinds data easily poor parallelization scalability socrates provides solution combines key features two approaches avoids problems features several advances data management parallel computing scalable distributed storage graph api supports representation different kinds data using simple extensible database schema distributed storage processing functionality execute algorithms local machine results merged without moving data indexing every attribute indexed fast random access analytics distributed management nothing cluster centrally managed includes communication well locating graph elements locality control graph vertices placed specific machines feature essential minimizing data movement graph analytics platform independence built using java standards run heterogeneous configurations related work socrates designed attribute rich data many labels nodes edges systems twitter cassovary flockdb gps pregel pegasus fast processing structural graphs however leverage provide ability store use labels edges nodes besides edge weights current graph benchmarking tools hpc scalable graph analysis benchmark generate tuples data form startvertex endvertex weight attributes plays well strengths cassovary pregel however benchmarks tend ignore functionality specifically aim within work graph databases focus content tend model specific relationship types ontology databases include jena openlink virtuoso commercial offerings use resource description framework rdf originally designed represent metadata rdf expressions consist triples subject predicate object stored queried predicate triple represents relationship subject object intuitively general set rdf tuples considered graph however formally rdf defined mathematical concept graph titan hypergraphdb dex offer similar capabilities socrates socrates titan example make use blueprints api interacting graphs however socrates extended api offer enhanced graph processing functionality includes locality control additional graph methods facilitating graph analytics well parallel processing capability socrates built upon cluster sql databases similar facebook tao twitter deprecated flockdb facebook social graph currently stored tao data model api specifically designed social graphs facebook social graph served via tao tailor fit workload needs frequent reads fewer writes edge queries empty results node connectivity data size distributions long tails long tails enable efficient cache implementations example large portion reads potentially cache people interested current events time locality alternatively something becomes viral viewed many people tao enables relatively queries stores main data objects associations pairs similar socrates stores attributes utilizing many tables attribute stored value node edge key schema removes joins potential memory performance bottleneck tao developed specific constraints mind global scaling time based social graphs whereas socrates targeted general graph structures edge attributes stored machine edges originate fig jgraph model parallelism socrates users submit processing job run parallel node cluster given access subgraph stored node access data provided graph api users interact cluster using simple graph interface graph api includes implementation blueprints api besides standard blueprints methods socrates graph api provides additional methods take advantage architecture efficient implementation analytics one method involves adding vertices graph specific machines machine level access graph allows user partition graph minimize edge crossings machines following example built using brightkite data set illustrates benefit locality control iii design data management locality socrates uses following conventions graph vertex exists machine graph edge exist machines edges know ids vertices connect well machine nodes reside central management location information fig graph structure representation cluster attributes graph stored separately column tables attribute independently indexed queried approach avoids complexity typically associated table structure changes relational databases attributes graph vertices stored machine vertices reside fig ability partition graph cluster nodes minimizes movement data graph left represents node machine cluster data archived without locality control case probability neighbor resides machine reflected outcome quarter vertices neighbors local example right shows partial graph machine looks like archived using socrates ability place graph nodes particular machines also used automatically partition data attributes hashed generate machine partitioning graph based latitudelongitude attributes vertices socrates provides efficient implementation parallelized graph query matching particularly useful query involves finding joint neighbors pair vertices query key operation variety link discovery analysis efficiently implemented without moving data irrespective vertices located implementation joint neighbor finding relies data structure maintained machine identify joint neighbors database query rather iterating neighbors vertex client every vertex graph knows location unique vertices connected without query another machine example query provided figure figure triangle shown left represent graph query matching result illustrated right fig query example structure attribute constraints found larger graph parallel processing key feature handling large scale graphs ability process graph parallel without move data primary challenge parallel processing graphs associated fact nontrivial problems analysis machine cluster requires access data machines able produce result area ability control partitioning graph cluster becomes critical socrates supports three models parallel processing distributed graph dgraph jgraph neighborhood type provides different tradeoff ease algorithm implementation parallelism client code network communication dgraph first model socrates provides clients access dgraph class implements blueprints api abstracts away distributed nature underlying graph methods dgraph class implemented parallel calls underlying database possible results sent back client machine client code runs socrates cluster model parallelism suitable developing analytics need global view graph benefit parallelized across entire cluster examples jgraph second model socrates allows clients create processing jobs submitted cluster run parallel node job given access jgraph local node run see fig jgraph another implementation blueprints api represents partial graph stored local machine vertex iterators used parallel jobs jgraphs iterate vertices local machine however questions asked local vertices getneighbors operation retrieves matching results independent located therefore implementation parallel jobs quite similar regular standalone program user clear view boundaries local graph limit operations local graph based information model parallelism suitable developing analytics make use wide view graph well benefit parallelism isomorphism also useful graph partitioned disjoint small enough fit one machine case trivial use socrates locality control features make sure entire placed machine allowing algorithm runs dgraph previous model parallelized neighborhood final model parallelism supported socrates intended algorithms perform local computation centrality measurements pagerank socrates provides interface allows clients define function run batch every vertex graph function called input tinkergraph implementation blueprints contains one vertex labeled root may contain elements client specifies function processing job submitted client able specify whether tinkergraph contain root vertex immediate neighbors neighbors case directed graph edges root well properties vertices edges fetched client function able write new property values root node neighboring edges future implementations allow new neighboring vertices edges added well model parallelism intended make easy write graph analytics take full advantage computing power cluster hood socrates takes care running client function parallel node using many threads hardware node support optimizing sql queries inserts caching values minimizing network communication nodes communication parallel processing provided java messaging service jms using method order eliminate centralized communication potential bottlenecks machine operates message broker every machine cluster therefore equal footing parallelizing message handling also eliminates potential single point failure system jobs executed parallel machine results optionally returned back client submitting job request performance section provide performance benchmarks demonstrating scalability socrates terms ingest parallelized analytics cluster configuration cluster consists servers equipped intel xeon processors ram two seagate constellation hdds raid servers running centos socrates using mysql tokudb storage engine data store http place btrees seems avoid problem figure shows ingest speeds logarithmic scale varying cluster sizes ingesting graph million vertices million edges ingest measure ingest speeds insert large randomlygenerated graphs http socrates cluster external machine graphs use consist connected components average edges however ingest speed graphs depends number vertices edges underlying structure graph insert graphs total size varying vertices one million edges million vertices one billion edges order see well ingest speed scales size cluster grows repeated ingest benchmarks using nodes addition full figure shows ingest speeds expressed elements inserted per second graphs clusters varying sizes fig socrates insertion speeds graph average edges per vertex varying cluster sizes see although cluster time reach top speed ingesting smallest graph tested takes seconds espectively ingest speeds hold steady even input graphs grow billion elements part reason use tokudb storage engine mysql previous versions socrates used standard innodb engine suffered severe bottlenecks graph grew roughly million edges likely due innodb longer able fit interior nodes buffer pool tokudb uses lookahead arrays http fig socrates insertion speeds logarithmic scale graph million vertices million edges varying cluster sizes cluster socrates exhibited approximately linear ingestion nodes added see socrates ingest speed scales linearly least nodes parallel processing measure parallel processing capability use naive connected component algorithm implemented neighborhood parallelism model described initial iteration algorithm assigns vertex component attribute equal smallest vertex among neighbors subsequent iterations algorithm examines component attribute neighbors updates component smallest value examined set algorithm terminates vertex component changes iteration benchmark necessarily fastest method computing connected components however useful benchmark socrates must fetch vertex neighbors along property information node thus speed iteration algorithm runs give idea batch processing capability neighborhood parallelism model run algorithm graphs ingested previous experiment graphs consisting connected components vertices average edges varying total size million billion elements repeated experiments using nodes cluster addition full figures present results experiments number vertices processed per second averaged iterations algorithm initial iteration note processing single vertex involves fetching immediate neighborhood average edges vertices well component property vertex neighborhood visualizations paper produced using pointillist graph visualization software developed cohen acknowledge silberberg distefano providing feedback support development socrates references fig average processing speeds iteration connected component algorithm graph average edges per vertex results qualitatively similar ingest results figure demonstrates graph large enough allow clusters reach full speed processing speed holds steady graphs billion elements fig average processing speeds logarithmic scale graph million vertices million edges varying cluster sizes see approximately linear processing speed nodes added figure demonstrates processing speeds also scale approximately linear fashion cluster nodes acknowledgment gupta goel lin sharma wang zadeh wtf follow service twitter proceedings international conference world wide web republic canton geneva switzerland salihoglu widom gps graph processing system proceedings international conference scientific statistical database management new york usa malewicz austern bik dehnert horn leiser czajkowski pregel system graph processing proceedings acm sigmod international conference management data new york usa kang tsourakakis faloutsos pegasus petascale graph mining system implementation observations ninth ieee international conference data mining icdm kalmegh navathe graph database design challenges using hpc platforms high performance computing networking storage analysis scc companion survey graph database performance hpc scalable graph analysis benchmark proceedings international conference information management berlin heidelberg candan framework ranked path queries weighted rdf graphs proceedings international conference web intelligence mining semantics new york usa angles gutierrez survey graph database models acm comput surv vol hayes gutierrez bipartite graphs intermediate model rdf semantic web iswc mcilraith plexousakis van harmelen eds springer berlin heidelberg jouili vansteenberghe empirical comparison graph databases international conference social computing socialcom bronson amsden cabrera iii chakka dimov ding ferris giardullo kulkarni marchukov petrov puzar song venkataramani tao facebook distributed data store social graph usenix annu tech conf atc
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method text line detection historical documents tobias gundram leiferta tobias roger labahna computational intelligence technology lab institute mathematics university rostock rostock germany feb abstract work presents text line detection method historical documents first stage deep neural network called labels pixels belong one three classes baseline separator separator class marks beginning end text line trainable scratch manageably manually annotated example images less achieved utilizing data augmentation strategies network predictions used input second stage performs clustering build baselines developed method capable handling complex layouts well curved arbitrarily oriented text lines substantially outperforms current approaches example complex track cbad competiton baseline detection increased framework train run open source keywords baseline detection text line detection layout analysis historical documents pixel labeling semantic segmentation state estimation introduction accessibility valuable cultural heritage historical documents important concern archives libraries well certain companies specialized genealogy years digitization industrial scale protect preserve valuable goods millions millions scanned pages stored servers world generic next step make enormous amount content document images accessible enable humanists historians genealogists well ordinary people efficiently work documents besides process manually annotating volumes subject current research scientific discussion automate process since tremendous progress field automated text atr well keyword spotting kws achieved performance systems reaches corresponding author email addresses tobias gundram leifert tobias roger labahn optical character recognition handwritten text recognition preprint submitted elsevier february character error rates atr mean average precisions kws complex handwritten documents although efforts made develop systems working solely rough input image without segmentation best performing recognition systems reference recently hosted competitions rely segmented words text lines input entirely approaches suffer either enormous time demonstrate applicability competitive quality challenging datasets hence workflow involves text line extraction followed transformation pixel information textual information widely used standard work deals first step information retrieval pipeline namely text line extraction mandatory step since errors directly effect performance overall information retrieval process text line extraction still unsolved certain extent historical documents due difficulties physical degradations faded away characters heterogeneous stroke intensity image capture conditions scan curve illumination issues complex layouts structured documents marginalia multicolumn layouts varying font sizes arbitrary orientations curved text lines results achieved approaches satisfying especially dealing heterogeneous data therefore work focuses extraction text lines arbitrary historical documents since different systems necessitate different text line representations bounding boxes areas precise polygonal representations following ascenders descenders one correct text line representation therefore limit towards text line detection task representing text line baseline detected baselines allow extraction text lines appropriate respect following method way problem extracting text line given baseline tackled applying histogram approaches estimate utilizing dynamic programming calculate separating seams besides classical image processing based approaches deep learning based methods became omnipresent document analysis community within last years techniques recently used solve several different problems binarization page boundary extraction page segmentation text line detection presented work knowledge first uses method combining deep learning strategies image processing based techniques propose extension fully convolutional extended incorporating residual blocks increase representative power furthermore spatial attention mechanism developed allows focus image content different positions scales network designed processes entire image take account spatial context universal way could used tackle pixel labeling task work trained fully supervised fashion classify pixel belong one following classes baseline separator separator class introduced explicitly predict beginning end text line rely information implicitly given baseline class advantageous text lines close together separated belonging different columns network output serves input image processing based clustering approach approach utilizes states superpixels encode local text orientation interline distances second stage allows error correction network output incorporating domain knowledge based assumptions hold text lines general see sec additionally easily possible incorporate separator information allows handling documents complex layouts images containing tables marginalia method relying supervised deep learning therefore relying training data suffer need enormous amount labeled training data demonstrate presented approach achieves high quality results bozen dataset less training samples using data augmentation strategies along annotating effort minutes per page adaptation proposed method easy cheap demonstrate applicability proposed method images arbitrarily oriented well curved text lines achieving nearly good results straight oriented text lines finally show presented approach outperforms methods three different datasets relative error gap reduction least achieved cbad dataset dataset composed images nine different archives libraries europe therefore opinion authors representative heterogeneous freely available dataset especially complex track contains mostly documents complex layouts average increased main contributions work introduction newly designed deep neural network pixel labeling along meaningful parametrization training framework open introduction new concept learned separators handle complex layouts instead page segmentation calculation introduction workflow combines deep learning image processing techniques entire workflow freely usable via transkribus related work comprehensive survey approaches text line extraction historical documents given section focus approaches relevant work https https principle dynamic programming utilized calculate cost optimal paths passing image left right separate different text lines methods basically differ way images definition cost function garz propose method based clustering interest points another name call superpixel using standard clustering technique interest points area exceeds certain density clustered form word clusters word clusters separated segments finally grouped build text lines ryu propose algorithm uses certain characteristics states extracted connected components assign costs certain clustering results states encode local text orientation interline distances introduced def subsequently using four different operations merge split initial coarse clustering costs minimized obtain optimal clustering leads final text line segmentation ahn improve approach introduction newly developed binarization method improved clustering process extended approach ryu applicable general superpixels newly introduced clustering procedure rely coarse initial clustering besides classical approaches based image processing techniques methods based machine learning gained importance within last two years moysset propose method based recurrent neural network network trained given number lines image utilizing connectionist temporal classification introduced train networks handwriting text recognition allows ground truth data without alignment trained neural network predicts confidences vertical coordinates image belong either classes line interline neural network output performed detect text lines works formulated problem regression problem recurrent neural network directly predicts bounding boxes well start text line respectively besides regression based approach classification based approaches proposed recently contrast approach moysset methods perform pixel labeling classify image pixel instead classifying rows pixels instance renton propose fully convolutional network fcn based dilated atrous convolutions classify pixels text line main body classification results utilized extract text line information techniques currently popular four five participants cbad competition baseline detection use methods relying fcns methodology section introduce method baseline detection see fig first stage relies deep neural network performs pixel labeling pixel labeling seen kind binarization instead detecting foreground elements stage superpixel clustering superpixel calculation stage state estimation input output figure workflow detect baselines first stage utilizes deep hierarchical neural network perform pixel labeling result stage input image processing based method stage method clusters superpixel build baselines image sampled cbad complex test set restricts elements interest specific task second stage performs superpixel extraction first stage output sps clustered build baselines following problem baseline detection formulated afterwards detailed description proposed given finally extraction clustering approach described problem statement introduce problem baseline detection formal way defining necessary termini notation within work follow definition baseline given definition baseline baseline defined typographical sense virtual line characters rest upon descenders extend definition image pixel intensity matrix called image height width pair called pixel matrix value row column called intensity pixel image means image rest work denotes height image denotes width analogously visualization purposes pixel intensity value means white means black colored image available usually use one visualization even though converted version calculations definition coordinate let pixel denote elements first second dimension called coordinates definition image space set possible images called image space definition polygonal chain closed polygonal chain length pixels polygonal chain called closed iff holds taking account def baseline represented polygonal chain definition polygonal chain space infinite set possible polygonal chains called polygonal chain space definition baseline detector baseline hypothesis call function maps image subset baseline detector set baseline detectors denoted output certain image called baseline hypothesis definition baseline ground truth set polygonal chains representing baselines image possibly annotated human operator called baseline ground truth image def allows baseline variety hence one unique correct ground truth image therefore ground truth information always biased creator taken account evaluation process well baseline detector design definition similarity score function assigning scalar value pair baseline ground truth baseline hypothesis polygonal chain sets called similarity score value indicates two polygonal chains regarded equal within work follow similarity score introduced measure accuracy baseline detector terms see detailed introduction problem tackled work formulated follows suppose two sets images along baseline ground truth information ttrain ttest aim design baseline detector given ttrain solves arg max hgi design phase set ttest unknown one allowed use solely ttrain hence one ensure generalizes well ttrain ttest since proposed design consists two stages first stage relies deep learning techniques adaptation differently biased ground truth produced different annotator done easily retraining first stage without fine tuning done experts stage typically layout analysis algorithms directly work input image binarized version instead employ transformation input image utilizing neural network trained supervised manner assign certain class pixel like often referred pixel labeling semantic segmentation introduce problem pixel labeling utilizing hierarchical neural networks followed description proposed architecture pixel labeling problem formulation definition neural pixel labeler neural pixel labeler npl classes hierarchical neural network npl parametrized performs prediction pixels possible classes subject denotes image encodes prediction probability class definition pixel ground truth cartesian product called pixel ground truth image assigns exactly one class pixel following problem formulation section aim npl tuned training set optimally performs test set assume training test sets style pixel ground truth information instead baseline ground truth information tetrain tetest performance npl evaluated terms predicted ground truth distribution also motivated maximum likelihood estimation results loss function definition loss function let set images along pixel ground truth npl performance evaluated terms loss function improve performance npl training set one calculate loss function gradient respect model parameters using technique backpropagation gradient used adapt model parameters gradient descent tetrain learning rate repeated successively adapt npl process adapting model minimizing loss called training since one aim minimization loss training set system generalize achieve high quality results test set well stabilize training avoid improve generalization dozens techniques improve simple rule introduced within last years since introduction beyond scope work refer details techniques used within work given sec architecture special form npl described section omit formal introduction used neural network components concepts refer mentioned literature within last years different architectures proposed pixel labeling task based convolutional neural networks cnns direct application cnns semantic segmentation presented presented fully convolutional network fcn combines local features produce meaningful high level features using pooling layers pooling reduces spatial dimension thus result suffers coarse resolution noh tackle problem applying deconvolutional network subsampled output fcn proposed furthermore introduces shortcuts layers spatial dimension allows easier combination local low level features global features additionally error propagation deep structures facilitated vanishing gradient problems reduced basis proposed extend two key concepts spatial attention depth residual structure described remarkably contrast proposed perform border padding hence spatial dimensions scale space see fig schematic representation output thus feature map features fig spatial dimension input hence becomes npl defined def adding convolutional get predictions softmax classifier top distinguishes different classes remark presented architectures used pixel labeling task implicitly assumed classifier always added generate per class probabilities pixel level identity output input identity identity iii max pool layer cnn block figure input image arbitrary spatial dimension act activation function thus rectangles represent sets activation maps rectangle represents array within scale space roman numbers feature map widths heights constant encoded height rectangles number feature maps pictured width rectangles adjacent scale spaces spatial dimension decreases certain factor figure representative depth number feature maps increases factor introduce deep neural networks still trainable yield results achieved using residual blocks residual blocks introduce shortcuts enable error backpropagation identity propagation even deep structures hence vanishing gradient problems reduced various different forms residual blocks one used within work depicted fig definition residual blocks means layer cnn blocks fig replaced residual block fig explicitly incorporate potential handle various font sizes especially mixed font sizes single page introduce spatial attention mechanism purpose introduce conv identity act output input logits figure residual block input convolved resulting array maps passed acitvation function referred logits used twice first branch passed activation function processed several convolution layers second branch directly fed summation node summation two logit maps activation function applied shortcut enables easy identity propagation error backpropagation arbitrarily many inner layers possible attention network cnn generates single output feature map applied along different scales network weights used scales weight sharing specially scale pyramid built downscaling input image several times resulting scaled images subscripts denote scaling factors fed trainable deconvolutional layers corresponding scales applied outputs obtain feature maps spatial dimensions equal inputs denote feature maps rus respectively applying softmax normalization attention maps exp exp sum one feature maps rui combined following normalized attention maps aru rui hadamard product multiplication aru input classifier build npl see rem definition incorporating described spatial attention mechanism called see fig multiplication combined attention maps allow pay attention different scales different positions image fig one see behavior indeed learned network seems like specialized certain font size distinguishes areas different font sizes bright dark areas introduced used pixel labeling task binarization page detection classification softmax deconv deconv figure input image downscaled versions fed accross different scales results lower resolutions deconvolved attention maps softmax normalization brighter map certain position attention paid corresponding scale attention maps muliplied feature maps summed classification performed weight sharing passed position results page segmentation purpose defined fixed number classes ground truth data provided training work limit baseline detection problem introduced sec purpose introduce three different classes baseline separator sep separators mark beginning end text line although separator information implicitly encoded baselines advantageous explicitly introduce possible classification result especially baselines close together belonging two adjacent columns approach helps avoid segmentation errors pixel ground truth classes sep automatically generated alg given baseline ground truth sample image baseline ground truth along generated pixel ground truth depicted fig prediction trained sample image shown fig stage baseline estimation subsection describes second stage proposed approach baselines estimated given output task consists three steps superpixel calculation state estimation superpixel clustering described following trained generates output image following denotes image encoding confidence pixel belonging baseline separator image see superpixel calculation number pixels image often exceeds several millions reduce dimensionality problem number pixels regarded baseline estimation limit algorithm pixel ground truth generation input image corresponding baseline ground truth output pixel ground truth dimension local text orientation see def interline distance see def polygonal chain length orient centered polygonal chain length orient centered draw draw follow chain set pixel values draw matrix ones return morphological dilation subset pixels definition superpixel let subset image pixels typically holds element called superpixel basically definition superpixel introduce new concept normal pixel somehow regarded certain importance since frequently used term decided introduce via definition easy see choice set sps crucial overall performance sps baseline baseline missed calculate suitable set sps utilize baseline map generated first step binarized comparison confidence threshold morphological skeleton ske calculated following formula foreground pixels pixels intensity build initial set pixels elements sorted descending order baseline confidences finally set iteratively adding pixels sorted list beginning first pixel keep number sps small new pixel added holds otherwise skipped fig set resulting sps shown sps build basis clustering remark experiments chosen fixed values binarization threshold baseline ground truth baselines described red dots better clarity dots baseline connected pixel ground truth produced alg green encodes separator class red baseline class black class figure baseline pixel ground truth shown top snippet image fig demonstrated well suited wide range different scenarios hence regarded free parameters system tuned also holds parameters fixed rem superpixel state estimation assume assign certain text line state encode meaningful characteristics text line characteristics defined combined build state work based previous work adapted characteristics sps extracted given output easier calculation local text orientation well different smoothing cost formulation definition local text orientation local text orientation slope text line baseline coordinates closest euclidean distance definition interline distance interline distance distance text line baseline nearest baseline distance means distance orthogonal local text direction definition state state pair local text orientation interline distance following describe method estimate states sps local text orientation calculated straightforward way utilizing solely baseline image local information output estimated baselines blue separators cyan shown superpixel neighborhood system calculated sps blue shown along resulting delaunay neighborhood system yellow figure baseline detection process two intermediate steps shown top snippet image fig hand estimation interline distances combines local information text line periodicity global assumption nearby sps tend similar interline distances approaches concepts neighborhood connectivity mandatory introduced definition neighborhood system edge adjacent call subset neighborhood system element called edge denoted directed two sps adjacent denotes remark following neighborhood system set sps always calculated delaunay triangulation definition connectivity function line segment defined connects two pixels edge function defined called connectivity function denotes intensity pixel closest euclidean distance coordinates connectivity function calculates average intensity given image along shortest path connecting two pixels local text orientation estimated lto utilizing baseline image see alg lto algorithm picks two neighbors largest baseline connectivity determines slope line passing neighbors algorithm local text orientation input neighborhood system baseline image output local text orientation sorted list sorted means denotes element else return arctan rxy periodicity text lines document images utilized calculate interline distances determine interline distance evaluating regional periodicity around follows circular region diameter around projection direction determined local text orientation let projection profile respect see fig calculation sps distance less taken account remark projection profile calculated efficiently utilizing cross product orientation vector cos sin vectors extract regional periodicity inherent projection profile discrete fourier transformation dft applied resulting coefficients hdp coefficient hkp corresponds portion signal period signal simplest case index dominant coefficient distance entire determines interline however may forced assign different value due additional constraints discussed moment therefore introduce data energy value possible value interline distance energy derive data cost used within cost minimization framework finding optimal interline distance definition data energy data cost data energy interline distance given figure interline distance estimation illustration several projection profiles certain red point profiles different diameters orientation shown green winning period interline distance drawn yellow curve blue histogram wrong orientation shown hkp corresponding data cost calculated log remarkably data energy normalized sums arbitrary cover suitable range different interline distances well robust disturbances due text regions different style projection profiles dfts calculated different diameters choice values application driven results reasonable interline distances sorted list following write assigned interline distance say labeled labeling assigns interline distance following greedy labeling strategy assigning interline distance highest energy defined leads noisy result see fig reduce noise effects influence sps taken account reasonable expect neighboring sps tend similar interline distances expectation encoded via smoothing cost defined adjacent sps definition smoothing cost assigned interline distances greedy states states greedy labeling using highest energy shown smoothed states states final labeling minimizing shown figure sps assinged states local text orientation visualized orientation green lines rotated length lines encode interline distance corresponding smoothing cost defined hsp else hsp index difference sorted list thus smoothing cost becomes large interline distances different size assigned adjacent sps maximum cost value used huge differences interline distances setting large value prevents neighboring sps differ much interline distances definition labeling cost labeling cost given data cost smoothing costs weighted respectively form labeling cost graphcut algorithm utilized minimize final labeling shown fig remark experiments chosen fixed values superpixel clustering previous subsections calculation sps enrichment state information described final step state information utilized cluster sps build baselines assignment clusters baselines following call set sps cluster subsection formulate clustering problem introduce greedy clustering procedure solve problem two assumptions hold baselines general constitute conditions clustering problem baselines exceed certain curvilinearity value within interline distance baseline baselines basically assumption claims baseline approximated polynomial function certain degree see assumption remark following denotes average orientation average interline distance sps definition curvilinearity value let deg set sps assume deg polynomial solves linear regression problem monomials tdeg rotated pixels cos sin sin cos regression error normalized called curvilinearity value denoted cur deg remark fix deg omit following def allows easy evaluation test introduce distance two clusters remarkably distances orthogonal text orientation taken account first orthogonal component distance two sps introduced afterwards generalized two clusters sps definition distance given two sps orientation distance length component orthogonal denoted remark distance efficiently calculated sin cos calculating minimal pairwise distance sps two clusters could result cluster distance distorted outliers therefore sps cluster projected onto corresponding regression curve obtained regression problem def taking pairwise distances definition regression curve let def spatial given tmin min tmax max curve results rotating graph tmin tmax called regression curve sps projected onto direction resulting projected sps denoted achieve robust distance estimates even curved differently slanted text lines focus sps different clusters quite close furthermore take account slope regression curve specific positions instead averaging entire text line definition cluster distance assume two clusters regression curves projected sps cluster distance defined min average slope corresponding regression curves respectively since possible evaluate conditions use introduce feasible sets clusters purpose limit partitions special kind cluster sets require baseline clusters definition partition let set call set subsets partition iff set partitions denoted par definition let cluster neighborhood system iff epi holds definition feasible set sps neighborhood system call set clusters feasible iff par conditions hold cur max set feasible sets clusters denoted easn clusters identify baselines constitutes clutter cluster containing sps belonging baseline identify baseline corresponding polygonal chain projected sps sic follow regression curve csi see fig number baselines unknown following incorporate domain knowledge promote sps belonging different baselines hence clusterings erroneously connected baselines feasible anymore done modification neighborhood system since baselines different text orientations contribute cluster adjust initial neighborhood system removing edges sps substantially different local orientations mod addition ease incorporate layout information adjusting layout information encoded separator image fig incorporated taking account connectivity sps edges separator crossed def holds removed see fig finally common scenario baseline detection given text regions assume text regions represented closed polygonal chains additional layout information available easy integrate edges contains holds removed roughly speaking closed polygonal chain contains ways image border one cross polygonal chain hence sps part different nonoverlapping text regions thus baseline entirely contained one text region feasible sets resulting neighborhood system still denoted remark experiments chosen fixed values def reducing neighborhood system introduce total baseline energy assign energy feasible sets aim optimal one allows formulation clustering problem solved definition total baseline energy let baseline image neighborhood system set clusters total baseline without separator information entire neighborhood system yellow shown separator information neighborhood system reduced removing edges cyan high separator connectivity corresponding separator information illustrated fig figure influence separator information resulting baselines blue lines without taking account separator information shown energy defined finally clustering problem formulated arg max easn could huge number feasible sets clusters large introduce greedy clustering algorithm solve proposed algorithm clusters edges instead clustering sps edge assigned cluster set edges assign corresponding sps corresponding cluster sps first step set edges sorted decreasing order sorted list denoted takes account value edge discounts rather orthogonal discounted edges less likely part baseline therefore sorted end list avoids edges falsely assigned baseline clusters composed correct edges statistics cluster reliable yet given proposed clustering process shown alg algorithm clustering input set sps sorted list edges output optimized partition four possible cases dependent sps case add edge existing cluster else case create new cluster rem else case cur extend cluster else case merge clusters cur min return experiments experiment section divided subsections first investigate influence training set size well influence different data augmentation strategies followed investigation performance proposed method applied images curved arbitrarily oriented text lines third subsection presents compares results different versions proposed npl architectures heterogeneous challenging cbad dataset perform statistical tests show statistical significance stated conclusion superiority proposed workflow architectures workflow finally compare proposed method methods datasets recently hosted competitions mentioned sec follow similarity score measure quality baseline detection configuration experiments including hyperparameters network architecture well training summarized tab configuration result extensive search hyperparameter space results impressive results various since early stopping based loss validation set used train entire training set workflow training inference tensorflow code well trained network table hyperparameters architecture training configuration used work described image input image downscaled factor max max followed normalization mean variance pixel intensity level architecture see fig number scale spaces initial feature depth residual depth activated layers residual block feature increasing spatial decreasing factor activation function relu kernel size stride architecture layer cnn activation function relu kernel size stride maxpooling size convolution feature number architecture see fig number image scales classifier convolution layer softmax activation training weight initialization xavier optimizer rmsprop learning rate learning rate decay per epoch weight decay norm exponential moving average model weights mini batch size due memory limitations gpu early stopping none trained fixed number epochs freely training takes scratch dependent number epochs samples per epoch titan gpu inference time per image ranges per image dual core laptop intel core ram reduces running titan influence training sample number data augmentation major drawback approaches sec need extensive expert tuning confronted scenarios already covered eligibility usage industrial scale depends possibility easily adapt reasonable cost approaches relying machine learning reduces two questions amount ground truth needed effort ground truth production concerning second question refer alg annotation baselines document image quite easy need remarkable expert knowledge compared ground truth production atr systems historical handwritings even text line annotation surrounding polygon level effort reduced several minutes per page using platforms following want examine first question influence training dataset size along different data augmentation strategies investigated freely available bozen see fig dataset subset documents https https https ratsprotokolle collection bozen composed minutes council meetings held consists pages written early modern german baseline ground truth information available form xml dataset quite challenging concerning layout analysis issues pages consist single main text region many difficulties line detection extraction bleed touching text lines marginalia following experiments randomly divided bozen set set training samples size test set size first step randomly set chain subsets contains training samples pages pixel ground truth since expect influence choice training samples sorting repeat mentioned procedure times notably test set remains untouched finally got training sets five quantity set trained epochs images per epoch therefore randomly choose samples training set remove set element training set used training start initial training set hence matter whether number training samples per epoch exceeds size training set procedure guarantees amount training samples shown networks training independent size training set chosen instead homogeneity bozen dataset concerning font size resolution trained scratch sets different scenarios training purposes image mentioned tab disabled instead training samples following one four strategies subsampled constant factor data augmentation one training sample per element training set randomly subsampled factor random affine transformation three corner points image randomly shifted within circle diameter max around original position elastic transformation test set images constant factor scenarios results experiments shown fig one see data augmentation strategies significantly improve performance compared base strategy notably small numbers training samples difference much larger higher number training samples hence http test set number training samples figure influence number training samples different data augmentation strategies bar height represents mean error bars encode values experiments standard deviation dashed green line marks maximum mean value achieved trainings samples description different augemantation strategies see main text training samples available choice importance best mean achieved training samples strategy nevertheless negligible loss performance training samples even training samples achieved strategy sufficient applications see fig results quite acceptable effort ground truth production making presented approach interesting even industrial production data augmentation strategy default rest work course presented numbers directly transferable collections pages entirely different scenarios census tables mixed postal cards mixed one would expect training samples necessary kind scenario nevertheless presented experiment reflects common situation one robust baseline detector trained heterogeneous data see sec detector work satisfyingly well certain cases quite homogeneous collection numbers presented give hint concerning effort ground truth production necessary scenario curved oriented text lines subsection demonstrate ability introduced approach handle curved arbitrarily oriented text lines first experiment test set bozen dataset deformed contain arbitrarily curved text lines purpose utilized trigonometric functions random period simulate curved text lines test phase see fig trained times epochs samples per epoch bozen training set using augmentation strategy strong elastic deformations choose elastic transformations training simulate curves different amplitudes frequencies image furthermore increased polynomial degree def enable system handle curvatures present test set remark different methods used deform images training test phases hence system learn concept curved text lines instead inversion image degradation method used training phase second experiment trained times arbitrarily oriented samples bozen training set evaluated resulting networks oriented pages test set results shown tab sample images shown fig curved scenario results good base scenario case oriented scenario results slightly worse still excellent demonstrates applicability images curved oriented text lines without remarkable adaptation workflow finally trained five models degradations affine elastic rotation evaluated model three different scenarios corresponding depicted tab system worse experts base curved scenarios oriented scenario even benefits additional elastic transformations table results bozen test set results base curved oriented scenario depicted pand strongly related well known precision recall measures see finally results single system trained degradations shown scenario min max base curved oriented base curved oriented combined workflow sec introduced workflow section investigate superiority classical well workflow purpose trained times random weight initialization random training sample order recently introduced cbad separable layer lowest resolution incorporate full spatial context details dataset described opinion challenging freely available dataset https separable mdlstm layer concatenation two blstm layers table results cbad test set results different neural network architectures workflow without stage shown architecture trained times cbad train set results sorted respect computational effort last two columns indicate whether architecture superior mentioned ones terms disjunct confidence intervals test method simple track complex track aru aru laru workflow baseline estimation basic image processing methods binarization followed analysis usage moment trained network epochs training samples epoch using data augmentation strategy assure statistical significance posed superiority newly introduced architecture follow provide results statistical analysis choice appropriate statistical tests quite limited since make assumptions regarding underlying distribution utilize confidence intervals provided bootstrapping well test level significance results obtained summarized tab performs significantly last two columns better architectures less computational effort could prove superiority therefore dismissed furthermore results show introduction second stage beneficial overall performance hence together workflow shown superiority statistically significant systems used following mentioned comparison fair concerning number trainable parameters aru laru millions concerning training even inference time comparison different architectures theoretically different capabilities whether make good use instance capable incorporating detailed spatial context fact benefit settings capability comparison state art subsection compare proposed framework state art chosen recent competitions text line detection historical documents namely icdar competition text line detection historical documents competition layout analysis challenging medieval manuscripts task cbad competition baseline detection introduce datasets metrics used refer competition papers icdar competition text line detection historical documents trained cbad training competition aims origin point detection roughly spoken lower left corner text line hence calculate left point detected baseline output system competition achieved results shown tab since trained original training data hard compare results ones nevertheless would like stress fact trained systems usually perform better training set test set sampled distribution trained cbad training set achieves average bozen test set worse system trained solely bozen training set see tab indicates prove superiority presented method methods tab table origing point detection results test set results dataset shown means number detection failures detected system means number detection misses detected far away ground truth means number false positives method hyp cor avg cost unifr snu proposed according extension competition layout analysis challenging medieval manuscripts task trained epochs samples per epoch competition training provided competition organizers allows entirely fair comparison participant results see tab proposed method substantially outperforms winning one reduces error gap relatively specialty competition methods focus special kind text comments annotated text hence learn distinguish different types text output detected baselines sample image subset test set shown fig one see entirely ignores text entities regarded competition main text remarkably information besides input image provided competition training data available authors http table results competition layout analysis challenging medieval manuscripts task participants proposed method shown different subsets test set method overall cvml byu citlab proposed figure results image subset test set original image main text lines ground truthed baseline image generated trained baselines detected proposed method shown left right cbad competition baseline detection compare average result see tab results presented see tab method performs considerably better tracks compared submissions especially increase performance complex track massive remarkably winning team uses unet based system task specific postprocessing indicates newly introduced concepts parametrization presented work significantly improve capability classical results chosen images cbad test set shown fig notably information besides input image text region information simple track provided second stage workflow inference conclusion work presented machine learning based method text line detection historical documents text lines represented baselines problem proposed method table results cbad test set participants proposed method simple complex track cbad competition baseline detection shown method simple track complex track litis upvlc byu dmrz proposed method based work presented introduced thoroughly proposed universal pixel labeling approach trained predict baseline position beginning end text line enables system handle documents complex layouts tables marginalia multi columns layouts shown system trained scratch manageably training samples complex homogeneous collection remarkably ground truth production quite cheap ground truth sample page annotated baselines done minutes per page therefore one expect adaptation collections covered neural network possible quite reasonable ground truthing effort applicability proposed method shown straight curved oriented text lines well combined scenario superiority proposed workflow classical simplified workflow shown statistically verified finally showed proposed method substantially outperforms previous state art nevertheless one see fig still errors made system missed baselines see fig bottom right segmentation errors see fig bottom left false positives see fig top left problems strongly degraded documents see fig top left errors seem follow certain deterministic principle surprising method based machine learning however plan test newly introduced concepts like capsules memory augmentation deeply supervised networks improve system performance acknowledgement nvidia corporation kindly donated titan gpu used research work partially funded european unions horizon research innovation programme grant agreement read recognition enrichment archival documents finally would like thank udo siewert valuable comments suggestions references references isaac clayphan haslhofer europeana 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multidimensional recurrent layers really necessary handwritten text recognition proceedings international conference document analysis recognition icdar efron better bootstrap confidence intervals journal american statistical association tukey quick compact two sample test duckworth specifications technometrics simistira bouillon seuret alberti ingold liwicki competition layout analysis challenging medieval manuscripts proceedings international conference document analysis recognition icdar figure results image bozen test set results trained training samples left right different data augmentation strategies top bottom shown figure results image bozen test set results two degraded images shown images arbitrarily curved rotated figure results images cbad test set images without layout information used figure results images cbad test set images without layout information used figure results images cbad test set images without layout information used
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stochastic recursive inclusions two timescales iterate dependent markov noise vinayaka yaji shalabh bhatnagar department computer science automation indian institute science bangalore vgyaji shalabh nov february abstract paper study asymptotic behavior stochastic approximation scheme two timescales drift functions presence markov noise shown recursion timescale tracks flow differential inclusion obtained averaging drift function recursion respect set measures take account averaging respect stationary distributions markov noise terms interdependence two recursions different timescales framework studied paper builds works ramaswamy allowing presence nonadditive markov noise application consider problem computing optimum constrained convex optimization problem objective function constraints averaged respect stationary distribution underlying markov chain proposed scheme neither requires differentiability objective function knowledge averaging measure introduction consider standard two timescale stochastic approximation scheme given denotes iteration index sequence random variables sequence random variables rdi lipschitz continuous function sequence rdi square integrable martingale difference sequence step size sequences sequences positive real numbers chosen satisfy addition monte carlo step size conditions condition ensures large number iterations time step recursion much smaller thus recursion appears static respect recursion using dynamical systems approach studied intuition shown hold precisely faster timescale recursion shown track ordinary differential equation given assuming every admits unique globally asymptotically stable equilibrium point say slower timescale recursion shown track given map assumed lipschitz continuous important application stochastic approximation scheme computation saddle point function given function respectively saddle point function inf inf sup sup prop know function admits saddle point every convex functions sub level sets functions inf compact sets years significant effort devoted developing algorithms compute points see references therein solutions proposed literature require computation partial derivatives function however practice closed form expressions partial derivatives often known expensive compute cases one often estimates partial derivatives using values objective function see one estimation method two timescale stochastic approximation scheme used compute saddle point noisy partial derivative values setting denote partial derivative operators respect respectively setting sequences denote partial derivative estimation errors map denotes correspondence minimum function vector field associated given shown conditions known envelope theorem mathematical economics see thus slower timescale maximizes function inf limit iterates recursion converge saddle point function cases function whose saddle point needs computed averaged respect certain probability measure example consider function compact metric space probability measure one wishes rto compute saddle point function every one access samples probability measure saddle point problem solved using recursion access samples available one uses markov chain monte carlo methods sample measure recursion markov noise component recursion takes form denote markov noise terms taking values appropriate state space recursion studied assumptions similar include lipschitz continuity maps often maps recursion lipschitz continuous map even single valued globally asymptotically stable equilibrium set motivates one study two timescale recursion drift functions recursion takes form maps quantities similar interpretation recursion recursion studied map allowed upper semicontinuous contributions paper comparisons state art paper study asymptotic behavior recursion given maps markov noise terms taking values compact metric spaces respectively show fast timescale recursion tracks flow differential inclusion given denotes set stationary distributions markov noise terms every integral denotes integral map respect measure assume every admits unique globally attracting set map also assumed upper semicontinuous slower timescale recursion show track flow given denotes map taking values space probability measures map defined captures equilibration fast timescale iterates averaging due markov noise terms comparison two timescale framework studied work allows drift functions map allowed upper semicontinuous much weaker requirement single valued lipschitz continuity imposed generalization case allows one analyze recursion drift functions single valued measurable since graph map embedded graph upper semicontinuous map refer reader several scenarios study stochastic approximation scheme maps becomes essential work generalizes two timescale framework studied allowing presence markov noise terms analysis paper extend straight forward manner requires results map approximation parametrization integration use probability measure valued functions however method analysis adopted paper adapted appropriately obtain convergence guarantees markov noise terms absent overview analysis organization paper known continuous convex compact maps taking values finite dimensional space admit continuous parametrization properties drift function ensure drift functions convex compact maps upper semicontinuous however maps admit continuous parametrization work around problem enlarging graph drift function since graph drift function embedded graph continuous convex compact map admit continuous parametrization thus sequence continuous maps obtained approximate drift function enables write inclusion form recursion additional parameter results needed accomplish stated section proceeding one needs identify mean fields recursion expected track end need results theory integration maps reviewed section measurablility integrability properties drift functions recursion investigated characterization integral continuous map terms parametrization established section compile definitions results theory differential inclusions needed later characterize asymptotic behavior recursion section state assumptions main result analysis single timescale stochastic recursive inclusions iterate dependent markov noise section define compile results needed space probability measure valued functions section state motivate assumptions recursion analyzed using results integration maps reviewed section mean fields defined main convergence result stated mean fields defined section possess properties ensure existence solutions associated differential inclusions properties established section section also shown appropriate modifications continuous maps approximate drift functions obtained section also approximate mean fields play important role analysis later analysis recursion consists two parts section recursion analyzed along faster timescale recursion viewed along faster timescale appears single timescale stochastic recursive inclusion iterate dependent markov noise section show recursion viewed along faster timescale satisfies assumptions associated single timescale recursion presented section applying main result single timescale analysis conclude faster timescale iterates converge section slower timescale recursion analyzed shown linearly interpolated sample path slower timescale iterates defined section tracks appropriate continuous functions tracking flow dynamical system known asymptotic pseudotrajectories see definition related results asymptotic pseudotrajectory argument paper presented section comprises following steps first step get rid additive noise terms involves defining appropriate piecewise constant vector field showing limit points shifted linearly interpolated trajectory slower timescale iterates coincide limit points solutions space continuous functions taking values simple argument gives set limit points shifted linearly interpolated trajectories slower timescale iterates second step show limit point obtained first step fact solution accomplished using probability measure valued functions reviewed section method also used analyzing stochastic approximation schemes recursion analysis references made explicit use lipschitz property underlying drift functions observe continuity sufficient carry analysis also analysis significantly differs equilibration faster timescale also accomplished using probability measures simplifies proof compared section limit sets slower timescale iterates characterized terms dynamics addition using convergence faster timescale iterates obtained section obtain main convergence result paper section application propose algorithm compute solution constrained convex optimization problem objective function constraints assumed convex affine respectively optimization problem obtained averaging quantities involved respect stationary distribution underlying markov chain problems arise optimal control controller must find optimum parameter changes state underlying system modeled markov chain cost function system constraints dependent state system controller seeks find optimum long run average cost function satisfying long run average constraints applications stationary distribution system states known one access sample path system state changes propose two timescale scheme performs primal ascent along faster timescale dual descent along slower timescale knowledge current state given iteration using theory presented paper shown limit set iterates proposed two timescale scheme contained set lagrangian saddle points underlying averaged constrained convex optimization problem algorithm assume differentiability objective function requires noisy estimate subgradient section conclude providing directions future research outline certain extensions believe analysis remains background section shall briefly review results needed theory maps differential inclusions present brief outline analysis single timescale version stochastic recursive inclusions markov noise define space probability measure valued functions metrizable topology needed later analysis two timescale recursion throughout paper denotes compact metric space metric denoted let denotes generic element upper semicontinuous maps approximation first shall recall notions upper semicontinuity lower semicontinuity continuity maps notions taken definition set valued map subsets upper semicontinuous every every exists depending denotes closed unit ball lower semicontinuous every every every sequence converging exists sequence converging continuous set valued maps taking compact set values mentioned notion equivalent standard notion see paper shall encounter set valued maps compact set valued hence chosen state definition upper semicontinuity maps studied later satisfy certain properties able approximate family continuous maps additional parameter properties natural extensions properties imposed maps studied case stochastic recursive inclusions markov noise choose call maps stochastic approximation maps sam definition sam stated definition sam map subsets stochastic approximation map every convex compact subset every every sequence say converging sequence converging exists every kzk sam appearing stochastic recursive inclusions condition stated replaced equivalent condition exists every kzk kxk kyk condition definition sam tells graph map defined closed hence said condition known closed graph property condition known boundedness condition makes sure size sets grow linearly distance origin condition differ conditions imposed easy show markov noise component absent condition imposed paper one consequence properties possessed sam one show map claim follows arguments similar cor stated lemma lemma map sam graph convex compact map embedded graph sequence decreasing continuous maps following statement made precise following lemma lemma continuous embedding map sam exists sequence valued maps every subsets continuous satisfies following every convex compact subset every iii exists every kzk map satisfies condition instead definition sam kzk kxk kyk furthermore every statement lemma found proof similar proof thm brief outline found appendix following useful observations proof lemma finite let every every exists depending every denotes closed unit ball continuous maps admit parametrization mean continuous map obtained represents map sense made precise lemma follows thm lemma parametrization let sam every map lemma every exists continuous map denotes closed unit ball every lemma iii every map satisfies condition instead definition sam kxk kyk throughout paper shall use denote closed unit ball dimension made clear context combining lemma lemma obtain approximation theorem stated theorem approximation sam exists sequence continuous functions every every convex compact subset lemma iii every map satisfies condition instead definition sam kxk kyk furthermore every measurable maps integration section shall review concepts measurability integration maps concepts needed define limiting differential inclusion recursion studied later paper expected track let denote measurable space subsets map every closed subset throughout subsection refers map defined definition measurable map map measurable every closed refer reader thm notions measurability relation definition definition measurable selection function measurable selection map measurable every map let denote set measurable selections next lemma summarizes standard results measurable maps measurable selections lemma measurable map castaing representation exists every denotes closure set refer reader thm thm proofs lemma respectively definition map let probability measure measurable map said exists definition aumann integral let probability measure integral map defined integrable next lemma states useful result properties integral map convex compact set valued lemma let probability measure frw map every convex compact convex closed subset proof lemma refer reader thm shall briefly investigate measurability properties sam first shall define slices sam shown lemma sam exists sequence continuous maps approximate every map parametrized maps lemma shall define similar slices well definition let subsets sam let lemma lemma respectively every define subsets every every every define subsets every iii every every define every every define subsets every every define every subsets every every every define every next two lemmas summarize properties slices inherit maps let denote borel sigma algebra associated metric space lemma let subsets sam let lemma respectively every let definition every lemma denote slices measurable map every convex compact subset exists every kzk satisfies condition instead condition definition sam kxk kyk every measurable map every convex compact subset exists every kzk satisfies condition instead condition definition sam kxk kyk iii probability measure every measurable selection hence every probability measure every measurable selection hence every continuous every part lemma proof lemma similar lemma shall provide brief outline fix order show measurable one needs establish closed using closed graph property one show closed subset hence bound claim convex compact every follows conditions definition sam respectively since measurable selections bounded probability measure arguments exactly claims associated slices approximating maps every finally part lemma follows properties maps stated lemma let probability measure denotes borel sigma algebra metric space metric max every fact product sigma algebra support measure denoted supp defined closed subset supp closed set supp probability measure set always exists unique see thm lemma let subsets sam satisfying condition instead condition definition sam let lemma lemma respectively every let denote slices definition every measurable map every convex compact subset every kzk kxk max kyk condition definition sam every measurable map every convex compact subset every kzk max kyk lemma iii iii every probability measure supp compact subset every measurable selection hence every every probability measure supp compact subset every measurable selection hence every continuous every kxk part lemma proof parts lemma similar corresponding lemma shall provide proof part iii proof part exactly proof fix iii consider part lemma kxk since supp compact subset exists every exists satisfying supp kxk hence supp supp supp kxk therefore every measurable selection hence lemma know map probability measure continuous parametrization every every similarly lemma know probability measure compact support continuous parametrization every every natural question ask relation integral map integral parametrization next lemma answers question stating lemma introduce following notation used throughout paper let denote space probability measures polish space prohorov topology also known topology convergence distribution see details probability measure let denote image measure projection similarly probability measure let belonging respectively denote image measure projections respectively lemma let subsets sam let lemma lemma respectively every every let every let denote slices definition every every probability measure suppose satisfies condition instead condition definition sam every every probability measure compact support remark support measure contained supp since exists unique measurable map supp therefore supp supp compact set support measure also compact lemma easy deduce measures compact support proof part lemma exactly lemma proof part similar minor technical modifications presented fix compact support consider exists proof let subsets every fact since every nonempty continuity closed every closed proof see appendix hence measurable since measurable lemma let let every let measure clearly therefore contained let cor exists unique measurable map since compact support see remark following lemma therefore compact support hence integrable lemma know every hence convex compact subset therefore every let every clearly measurable therefore gives contained differential inclusions limit sets section shall review results theory differential inclusions state definitions limit sets associated dynamical systems used later paper results section taken first shall define map whose associated differential inclusion known admit one solution every initial condition maps called marchaud maps definition map stated definition marchaud map subsets marchaud map every convex compact subset exists every kzk iii every every sequence converging every sequence converging let marchaud map associated map given since marchaud map known admits one solution every initial condition see sec solution initial condition mean function absolutely continuous shall recall notions flow invariant sets attracting sets attractors basin attraction internally chain transitive sets notions taken flow given map subsets every solution set let closed set invariant flow every exists solution every compact set attracting set flow exists open neighborhood say property every exists depending every stands compact set attractor flow attracting set invariant flow set basin attraction set denoted defined attractor whose basin attraction whole called global attractor given set exists chain exists integer solutions real numbers greater compact set said internally chain transitive every every every exists chain suppose invariant set flow restricted invariant set map subsets every solution every single timescale stochastic recursive inclusions iteratedependent markov noise section review results analysis single timescale stochastic recursive inclusions iterate dependent markov noise results presented found let probability space sequence random variables satisfying following assumptions hold map subsets compact metric space every convex compact subset exists every kzk iii every every valued sequence say converging sequence converging sequence measurable functions every every continuous sequence positive real numbers satisfying every sequence random variables min kxn detailed motivation assumptions found shall briefly explain consequences assumption ensures map sam assumption iteratedependent markov noise assumption consequence assumption every know markov chain defined transition kernel possesses weak feller property see addition since state space compact set stationary distributions markov chain whose transition probability given every let denote set stationary distributions markov chain whose transition kernel every also know every convex compact subset map closed graph see references therein assumption standard assumption assumption general additive noise assumption ensures contribution additive noise eventually negligible various noise models satisfying see assumption stability assumption iterate sequence map subsets serves vector filed differential inclusion iterates expected track defined every every denotes slice definition map appearing recursion map marchaud map see lemma associated given admits one solution every initial condition see sec let denote set solutions initial condition set possible solutions every subset set continuous functions set complete metric space metric defined min consequence lemma every closed compact subsets respectively let every define stochastic process continuous sample paths every every let main result analysis recursion follows theorem assumptions almost every family functions relatively compact every limit point solution formally lim iii limit set denoted defined compact internally chain transitive flow proof theorem see thm space probability measure valued functions section shall define space probability measure valued measurable functions shall introduce appropriate topology space show space compact metrizable spaces used theory optimal control diffusions see also analyzing stochastic approximation schemes see quantities defined section serve tools analyzing stochastic recursions later throughout section denote closed unit ball denotes closed ball radius centered origin every let denote set functions taking values space probability measures equipped prohorov topology measurable formally measurable similarly every denotes set functions taking values measurable formally measurable measurable every let denote coarsest topology renders continuous functions every every every similarly every let hdenote coarsest topology renders continuous functions every every every finally every let denote coarsest topology renders continuous functions every every every following well known metrization lemma topological spaces defined lemma metrization every topological space compact metrizable every topological space compact metrizable iii every topological space compact metrizable refer reader lemma proof metrization lemma next lemma provides continuous functions defined metric spaces used later proof lemma extention lemma defined metric spaces recall probability measure denotes image measure projection every similarly denotes image measure projection every easy see also image projection lemma every map every continuous map every continuous iii every every map every continuous map every continuous proof fix let sequence converging let denote projection map every clearly continuous continuous function continuous since compact metric space rthm get every definition every measure therefore every hence thm get gives continuity similar part lemma iii composition measurable functions measurable let sequence converging weh know every every every let denote projection map part lemma know thenh every every every arguments similar part lemma every every every therefore gives continuity similar part lemma recursion assumptions section shall formally define two timescale recursion well state motivate assumptions imposed let denote probability space sequence random variables sequence random variables satisfy every map subsets compact metric space metric every convex compact subset exists every kxk kyk iii every every sequence converging every sequence converging map subsets compact metric space metric every convex compact subset exists every kxk kyk iii every every sequence converging every sequence converging sequence random variables every every continuous sequence random variables every every continuous two sequences positive real numbers satisfying every every iii sequence random variables min sequence random variables min kxn kyn assumptions ensure sams assumptions markov noise assumptions every markov chain associated transition kernel given possesses weak feller property see addition since compact metric space markov chain associated transition kernel least one stationary distribution every stationary chain associated transition kernel every markov every let denote set stationary distributions markov chain associated transition kernel easily shown every convex compact subset graph map closed set closed subset proofs two statements similar similarly assumption every set stationary distributions denoted associated markov chain defined transition kernel convex compact subset map closed graph set defined analogous manner closed subset assumption standard two timescale step size assumption assumption iii tells eventually time step taken recursion smaller time step taken recursion hence recursion called slower timescale recursion recursion called faster timescale recursion assumptions conditions additive noise terms satisfy guarantee contribution additive noise terms eventually negligible various noise models additive noise assumptions satisfied refer reader assumption stability assumption ensures iterates remain within bounded set standard requirement study recursions highly nontrivial important future direction would provide sufficient conditions verification markov noise terms faster timescale limit average drift function stationary distributions given map appropriate map whose associated faster timescale recursion expected track given every every denotes slice map recursion consequence step size assumption respect faster timescale slower timescale recursion appears static one would expect family dis obtained fixing describe behavior faster timescale recursion proceed need ensure every solutions every initial condition next lemma states map marchaud map every ensures solutions lemma every map subsets marchaud map proof lemma given section next assumption ensure every global attractor one expects faster time scale iterates converge globally attracting set map every admits every subsets every kxk kyk every every sequence converging every converging respect slower timescale recursion faster time scale recursion appear equilibrated markov noise terms average drift function respect stationary distributions follows construct map slower timescale recursion expected track captures equilibration faster timescale averaging markov noise terms proceed recall denotes set probability measures prohorov topology denote images probability measure underr projections respectively similarly every define map subsets every supp every supp denotes support measure supp closed set supp every closed set supp natural question ask whether every properties map possesses relation stationary distributions markov noise terms lemma answers questions lemma map defined satisfies every convex compact subset every every sequence converging every sequence converging iii every denotes dirac measure proof fix consider product measure denotes dirac measure every otherwise stationary measure markov chain whose transition kernel given supp every since last equality follows fact therefore hence let consider measure clearly supp hence therefore supp every therefore gives convexity order show compact first show set closed set consider every converging clearly converges since every since supp every assumption compact subset thm lim therefore gives supp clearly converges since compact metric space thm know every let every easy see assumption continuous therefore since every every thus every therefore establishes hence closed establish compactness enough show set relatively compact measure support measure denoted supp contained compact set independent thus family measures tight prohorov theorem see thm set measures relatively compact therefore closed relatively compact hence compact let let denote closed unit ball assumption map therefore every exists depending every satisfying since compact compact since exists every kyn lim every since thm every since compact therefore supp let since every supp using property map fact get exists ery every every assumption map tildes continuous hence restriction compact set uniformly continuous compact set every conclude exists every every therefore every exists max every second term inequality goes zero use definition assumption thm therefore taking limit equation get every hence clearly therefore gives iii follows part lemma define map subsets every every denotes slice definition set valued map since every every supp compact lemma iii know slices every map well defined show later slower timescale iterates track given guaranteed solutions consequence lemma lemma map subsets marchaud map proof lemma given section remark order understand better consider cases map singlevalued case markov noise terms absent special cases also highlight fact results significant generalization results map since supp denotes dirac measure therefore measure since know every thus denotes set stationary measures markov chain transition kernel therefore every denotes slice definition map therefore nothing analogue slower timescale suppose markov noise terms absent analysis definition recursion see recursion rewritten form recursion markov noise terms taking values dummy state space transition laws every easy deduce stationary distribution maps every every form supp supp exactly slower timescale suppose following holds addition globally attracting set main result paper states almost every mean fields properties section prove every map map defined equations respectively marchaud maps recall assumptions maps sams setvalued know exist sequences continuous maps denoted lemma maps approximate respectively lemma appromating denoted throughout section continuous parametrization admit maps denote maps described similar definition maps define maps obtained averaging set valued maps every respect measures given maps definition let maps subsets subsets section every every define subsets denotes slice map every define subsets denotes slice map lemma prove every maps map marchaud maps every lemma every map subsets every convex compact subset lemma every kxk kyk every sequence converging every sequence converging every map marchaud map iii map marchaud map proof fix every lemma every hence every let lemma exist denotes closed unit ball clearly last inclusion follows fact convex subset lemma get therefore convex lemma every map bounded kxk kyk therefore every every thus every definition therefore every kxk kyk consequence arguments preceding paragraph order show compact enough show closed consider sequence converging definition lemma every exists since compact metric space coms pact hence exists subsequence converges clearly converges thm xnk since every fact closed get therefore thus gives closed sequence converglet sequence converging let ing lemma every exists since compact metric space compact metric space hence exists subsequence say converges clearly closed graph property map using continuity map easy show sup xnk ynk use thm inequality obh tain lemma gives follows part lemma iii proof similar part lemma minor modifications first modification use lemma instead lemma example order show closed fix sequence converging use lemma definition obtain denotes closed unit ball sequence every definition every supp hence supp compact subset prohorov theorem sequence relatively compact subset hence subsequence lemma compact hence every limit convergent point rest argument corresponding part lemma order show closed graph fix sequences converging converging use lemma obtain every every supp assumption set compact subset therefore exists large every supp prohorov theorem sequence measures tight convergent subsequence clearly lemma every limit point rest argument corresponding part lemma lemma know every every similarly next lemma states true well every lemma every every every iii every every every every proof proofs parts follow directly definition every every proof fact every part iii similar part present proof part proof part iii fact lemma fix definition every therefore let every exists since subset every supp hence quence probability measures tight prohorov theorem limit say let subsequence lemma know compact gives since every ery get every every every lemma know every exists denotes closed unit ball every every supp hence tight prohorov theorem convergent subsequence evk ery let denote limit point sequence since every thm every hence lemma therefore exr ists every hence proof part similar proof part present proof part proof part exactly lemma iii part lemma every fix let every exists let inf every every compact lemma compact convex inf inf inf inf inf last equality follows lemma lemma know every map measurable last equality follows every observation stated lemma every since supp every since compact exists every kxk lemma observation stated lemma every every kxk max bounded convergence theorem lim therefore lemma know closed subset hence arguments preceding paragraph every every thus every part lemma get every lemma map subsets defined eqn every convex compact subset exists every kxk kyk iii every every sequence converging every converging proof fix lemma iii every hence every lemma know convex compact subset lemma hence convex compact fix exists lemma know every kxk kyk therefore kxk kyk iii let sequence converging sequence converging lemma every every lemma every thus lemma lemma immediate consequence lemma similarly proof lemma follows fact marchaud maps see lemma iii approximate see lemma linear growth property map recursion analysis section present analysis recursion analysis comprises two parts first part deals analysis faster timescale recursion show faster timescale iterates converge almost surely second part deals slower timescale recursion analysis show slower timescale iterates track flow throughout section assume assumptions satisfied faster timescale recursion analysis every every two timescale recursion written every every recursion rewritten every every written form single timescale recursion every subsets every show quantities defined satisfy assumptions associated single timescale recursion section clearly assumption step size sequence satisfies assumption assumption markov noise terms satisfy assumption consequence stability assumption kzn hence assumption satisfied consider map defined clearly assumption every convex compact subset assumption every sup kxk kyk max kxk kyk every see thm assumption iii map closed graph hence map also closed graph therefore mapn satisfies assumption recall every every min let hold clear let fix sup sup sup assumption every exists kxn kyn since every every assumption kxn kyn assumption iii every exists every therefore every every thus every therefore every every sup sup sup taking limit equation using assumptions gives every lim sup therefore every every thus additive noise terms satisfy assumption therefore quantities recursion satisfy assumptions apply main result single timescale recursion see theorem iii conclude lemma assumptions almost every exists compact set depending recursion set internally chain transitive flow map associated clearly marchaud map use lemma solution every solution fix lemma holds let lemma let every assumption since internally chain transitive flow lemma know invariant let solution initial condition every solution assumption exists compact subset globally attracting set flow definition globally attracting set therefore since every get fact closed set invariant flow argument gives able show lemma obtain regard need impose following assumption compact set invariant flow open neighborhood exists open neighborhood subsets see section definition denotes flow restricted invariant set remark assumption weaker form assumption imposed implies assumption assumption basically lyapunov stability condition see defn flow restricted invariant set shall see application studied later assumption satisfied lemma assumptions almost every lemma therefore recursion proof present brief outline highlight assumption used let equation fix obtain lemma know internally chain transitive flow since also invariant assumption every thus every every solution every every open neighborhood exists lemma get every open neighborhood exists every every solution every fix assumption exists open neighborhood arguments previous paragraph find every every solution every exists therefore thus attracting set claim follows prop slower timescale recursion analysis present analysis slower timescale recursion present shall preliminaries denote maps define various quantities needed later throughout section let section shall allow assumptions satisfied slower timescale recursion analysis similar analysis single timescale inclusion minor modifications arising due presence faster timescale iterates throughout section denotes closed unit ball denotes closed ball radius centered origin preliminaries let every define every consider slower timescale recursion given every lemma every every therefore every following recursion follows lemma know every map admits continuous single valued parametrization next lemma allows write slower timescale inclusion terms parametrization result follows lemma lemma every every exists random variable every every define every denotes dirac measure every otherwise denotes dirac measure similarly denotes dirac measure lemma provides equicontinuity result used later lemma every every family maps equicontinuous proof fix lemma know map continuous hence map restricted compact set uniformly continuous therefore every exists every satisfying therefore every define every follows arguments sample path wise use smaller case symbols denote defined quantities along particular sample path example denote respectively fixed main pseudotrajectory every every every let denote solution every initial condition let assumptions first shall get rid additive noise terms regard prove lemma states every family functions limit points every proof lemma similar lemma given appendix lemma almost every every every lim sup lemma guarantees existence limit points proof similar lemma given appendix lemma almost every every family functions compact relatively consequence lemmas get almost every family functions relatively compact linearly interpolated trajectory slower timescale iterates uniformly continuous next proposition states every limit point solution proof along lines prop modifications arising due presence faster timescale iterates proposition almost every every limit point satisfies following every exists every every part proposition almost every iii absolutely continuous almost every proof let lemma holds proof thm clear fix let fix assumption exists kxn kyn therefore lemma get sequence convergent subsequence let limit point without loss generality assume definition every every max using defintion see recall get every every since lemma taking limit equation get every lim lim lim since choice topology bounded continuous form scalars bounded continuous functions respectively theorem functions uniformly approximate function thus convergence holds true real valued continuous functions implying thus dqk since converges uniformly function converges uniformly using lemma every exists depending every every dqk dqk dqk taking limit equation using get lim dqk every therefore every lim substituting limit equation get every fix let every since lemma get order prove almost every need show almost every supp every first present proof claim since lemma since proof lemma almost every exists subsequence subsequence natural numbers tnk fix holds definition every tnk tnk max tnk since using uniform continuity function converges uniformly compacts function therefore lemma definition part proposition get kxn hence every clearly compact exists large every hence pcp tnk thm every since every lim sup tnk lim sup therefore every gives hence supp since holds almost every almost every supp proof claimin similar proof prop provide brief outline sake completeness let convergence determining class appropriate affine transformation ensure every every define every every every square integrable martingale filtration theorem see appendix thm get martingale convergence almost every every converges let converges define arguments get therefore every every every choice fact step size sequence increasing definition get every every every lim max assumption every function continuous hence restriction function compact set uniformly continuous part proposition using uniform continuity fact follows definition uniform continuity get every every every lim know arguments similar lemma family functions equicontinuous therefore every every lim lim using equation get every every applying lebesgue differentiation theorem see get every every almost every since every also element supp therefore every every almost every since convergence determining class follows every almost every iii fix part proposition clearly absolutely continuous almost every part lemma know almost every hence lemma almost every since holds every get almost every last equality follows lemma continuous function said asymptotic pseudotrajectory flow denotes set solutions fix extend whole defining every assumption uniform continuity family functions relatively compact let limit point family functions proposition iii solution usually negative time argument omitted since follows positive time argument follows fix let proposition iii solution therefore absolutely continuous dydt since arbitrary solution therefore thm get following result theorem apt assumptions almost every linearly interpolated trajectory slower timescale recursion asymptotic pseudotrajectory characterization limit sets consequence theorem almost every limit sets slower timescale recursion defined characterized terms dynamics using lemma get main result paper stated theorem limit set assumptions almost every compact subset internally chain transitive flow assumption satisfied proof fix theorem know asymptotic pseudotrajectory flow claim follows thm part theorem know internally chain transitive flow since globally attracting set cor get therefore lemma get application constrained convex optimization section consideran application theory problem constrained convex optimization throughout section assume let objective function continuous every convex coercive exists kxk functions describing constraints given assume set non empty law markov noise terms given continuous let denote unique stationary distribution markov chain given transition kernel every let denote set subgradients convex function point formally easy show every convex compact map possesses closed graph property assume map satisfies linear growth property exists kxk every similarly define defined optimization problem wish solve given min subject standard approach solving optimization problem projected subgradient descent algorithm whose recursion given subgradient estimation error denotes projection operation onto affine subspace scheme implemented known case problems arising optimal control feasible set optimization problem given non empty since every since every coercive function coercive hence bounded therefore optimization problem least one solution let solution set optimization problem denoted origin every let denote closed ball radius centered pick compute max since functions coercive max exists max kxs every every brc every instead shall solve following optimization problem given kxk max kxk subject min determined constant associated linear growth property subgradient map arbitrary constant small value easy show least one solution set solutions denoted defined consider lagrangian associated optimization problem every max let obe defined every max kxk every max kxk transition law hence known propose following recursion performs primal descent along faster time scale minimization dual ascent slower timescale maximization recursion given step size sequences chosen satisfy assumption denotes subgradient estimation error assumed satisfy assumption example zero mean finite variance assumption satisfied generally satisfied martingale difference sequence satisfying assumption easy see maps satisfy assumptions respectively linear growth property map follows linear growth property prop get every every let every since convex coercive singleton since strictly convex exists set subgradients function max either max hence get kkyk thus max get every kyk set thm map clearly globally attracting flow since also single valued continuous hence map satisfies assumption iterates stable satisfied result section gives almost every exists compact set internally chain transitive flow arguments section let open neighborhood let thm map continuous hence open neighborhood easy show hence since compact exists consider solution starting satisfying every recall every solution hence descends along potential therefore solution remains within gives denotes flow restricted set thus assumption satisfied lemma get following result lemma faster timescale convergence almost every theorem gives iterates recursion track flow every equation since every singleton since unique stationary distribution markov chain given transition kernel get every therefore takes form order analyze asymptotic behavior need following version envelope theorem proof envelope theorem similar lemma envelope theorem let absolutely continuous function let defined every lebesgue measure every absolutely continuous exists every every exists function every inf absolutely continuous proof since absolutely continuous differentiable almost everywhere let set exists clearly lebesgue measure fix lipschitz continuous function since differentiable totally differentiable since linear total derivative given every since composition absolutely continuous function lipschitz continuous function absolutely thm every exists since absolutely continuous exists assumption every inf inf therefore every sup sup sup sup absolute continuity follows absolute continuity since absolutely continuous dvdq exists dvdq let exists therefore right hand derivative isfies thus considering repeating argument gives absolutely continuous since let defined inf concave function objective function dual optimization problem function strong duality theorem see prop dual optimization problem given least one solution let set solutions dual optimization problem denoted strong duality theorem also gives let solution initial condition absolutely continuous lemma since get hence every gives thus solution initial condition every therefore prop solution dual optimization problem hence also follows closed theorem summarize main convergence result associated recursion theorem convergence lagrangian saddle points solution initial condition bounded inf solution optimization problem iii iterates remain stable almost every satisfied almost every proof let solution initial condition assume since otherwise know every hence claim follows uniformly continuous since function uniformly continuous hence function uniformly continuous lemma claim equivalent claim suppose exists every exists uniform continuity exists every satisfying therefore obtain sequence every every lemma let get contradicts fact therefore definition let know hence feasible every claim follows prop iii let theorem holds theorem know exists non empty compact set internally chain transitive flow hence invariant let solution initial condition every since compact hence part theorem get sinced every get exists denotes set using easy show denotes flow restricted set see section therefore attracting set flow prop get therefore follows part iii theorem lemma conclusions directions future work presented detailed analysis two timescale stochastic recursive inclusion drift functions presence iterate dependent markov noise stationary distributions analysis section shows asymptotic behavior two timescale recursion faster timescale iterates recursion track flow fixed value slower timescale variable slower timescale iterates track flow assumptions two timescale recursion studied paper weaker current literature recursions behavior often required solve nested minimization problems arise machine learning optimization special case constrained convex optimization linear constraints considered application objective function assumed differentiable objective function constraints averaged respect stationary distribution underlying markov chain transition law hence stationary distribution known advance primal ascent algorithm recursion implemented knowledge sample paths underlying markov chain analysis presented paper guarantees convergence solution user specified choice outline important directions future work two timescale stochastic approximation schemes mean fields best knowledge sufficient conditions stability current literature believe extensions stability result single timescale stochastic approximation made case two timescale recursions another approach stability could along lines many applications iterates projected time step ensured remain within compact convex set projections often arise due inherent need application used ensure stability projected schemes tendency introduce spurious equilibrium points boundary feasible set complications arise due presence markov noise terms since projection map time differentiable directional derivatives known exist projected stochastic approximation schemes case without markov noise component analyzed serve basis analyzing general frameworks projection applications arising reinforcement learning noise terms markov lack markov property comes dependence control sequence controlled markov noise assumption two timescale stochastic approximation scheme analyzed lipschitz continuous drift functions extending analysis presented paper case drfit function controlled markov noise assumption straightforward requires major change overall flow analysis extension allows one analyse asymptotic behavior larger class reinforcement learning algorithms see several applications two timescale controlled stochastic approximation two timescale approximate drift problem also analyzed help results presented paper see definitions proof lemma fix prove claim along sequence claim lemma easily follows fix let min let exists definition solution since every lemma every thus sup since inequality independent therefore claim follows follows assumption proof lemma fix assumption know exists kxn hence let max every assume dqk kdq kxn kyn thus equicontinuous family claim follows theorem references borkar stochastic approximation two time scales systems control letters vol benaim dynamical system approach stochastic approximations siam journal control optimization vol bertsekas convex optimization theory athena scientific belmont ozdaglar subgradient methods problems journal optimization theory applications vol benzi golub liesen numerical solution saddle point problems acta numerica vol spall multivariate stochastic approximation using simultaneous perturbation gradient approximation ieee transactions automatic control vol milgrom segal envelope theorems arbitrary choice sets econometrica vol online available http karmakar bhatnagar two timescale stochastic approximation controlled markov noise temporal difference learning arxiv preprint ramaswamy bhatnagar stochastic recursive inclusion two timescales application lagrangian dual problem arxiv preprint borkar stochastic approximation dynamical systems viewpoint press cambridge university yaji bhatnagar stochastic recursive inclusions markov noise arxiv preprint hofbauer sorin stochastic approximations differential inclusions siam journal control optimization vol borkar stochastic approximation controlled markov noise systems control letters vol aubin cellina differential inclusions maps viability theory science business media vol springer ogura kreinovich limit theorems applications fuzzy random variables springer science business media vol parthasarathy probability measures metric spaces american mathematical vol borkar probability theory advanced course springer science business media meyn tweedie markov chains stochastic stability springer science business media borkar optimal control diffusion processes pitman research notes borkar controlled diffusion processes probability surveys citeseer kumaresan topology metric spaces alpha science int rudin principles mathematical analysis pure applied mathematics https ser international series online available borkar meyn ode method convergence stochastic approximation reinforcement learning siam journal control optimization vol ramaswamy bhatnagar generalization theorem stochastic recursive inclusions arxiv preprint andrieu moulines priouret stability stochastic approximation verifiable conditions siam journal control optimization vol nagurney zhang projected dynamical systems variational inequalities applications springer science business media vol perkins leslie asynchronous stochastic approximation differential inclusions stochastic systems vol
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apr jordan property lie groups automorphism groups complex spaces vladimir abstract prove connected real lie groups jordan implies algebraic groups necessarily affine fields characteristic zero transformation groups complex spaces riemannian manifods jordan introduction recall definition introduced def definition given group put sup min runs finite subgroups runs normal abelian subgroups called jordan group called jordan constant case also say enjoys jordan property informally jordan property means finite subgroups almost abelian sense extensions abelian groups groups taken finite list definition inspired classical theorem jordan claiming jgln holds every every field characteristic zero algebraically closed every fixed constant jgln independent denote simply computed particular examples jordan groups see variety means algebraic variety fixed algebraically closed field characteristic zero particular algebraic group defined either algebraic group topological group denotes identity component posed seven years ago sect see also sect following problem explored number researchers see recent brief survey references sect problem describe varieties group aut jordan work supported rsf grant vladimir popov present april still unknown whether varieties group aut note complex manifolds whose automorphism groups exist see remark hand many types varieties shown group aut jordan particular meng zhang recently proved following theorem thm every projective variety group aut jordan given variety denote aut identity component aut sense see also cor complete aut connected necessarily affine algebraic group jordan theorem cited implies claim every affine algebraic group jordan see thm key ingredient proof theorem given proof extension claim necessarily affine algebraic groups holds true latter proof rather involved present note obtain short proof general result extension immediately follows see theorem namely prove every connected real lie group jordan precise general statements formulated theorems corollary sections apply showing transformation groups complex spaces riemannian manifolds jordan see theorems question whether lie groups jordan posed vershik see thank grateful zarhin valuable comments lie groups explore jordan property real lie groups note groups type exist every discrete group real lie group nonjordan discrete groups see therefore jordan property expected constraint component group formulate restriction recall following definition introduced def definition given group put sup runs finite subgroups group called bounded particular every finite group bounded theorems corollary consider class finitedimensional real lie groups whose component group bounded jordan property lie groups complex spaces note every compact lie group belongs class finite theorem let real lie group whose component group bounded jordan proof lem thm may shall assume connected assumption implies existence compact lie subgroup every compact subgroup conjugate see chap thm iii particular every finite subgroup conjugate definition show jordan compact group admits faithful representation isomorphic subgroup glm see chap thm since latter group jordan jordan well see thm completes proof corollary every real lie group whose component group bounded set isomorphism classes finite simple subgroups finite dwell estimating jordan constants lie groups whose component group finite view proving class groups enjoys property stronger members jordan see corollary seeking goal seek improve estimates obtained lemma let simply connected simple affine algebraic group minimum rdim dimensions faithful linear algebraic representations given following table type rdim odd even remark proof lemma faithful representation dimension rdim explicitly specified type proof lemma lefschetz principle see may shall assume fix maximal torus let lie lie lie respectively system simple roots fundamental weights simple coroots lie respect fixed borel subalgebra lie containing lie number center finite subgroup fix subset lie whose image exponential map lie set nonidentity elements vladimir popov every dominant weight lie let irreducible representation lie highest weight dimension every specified ref table note weyl dimension formula implies every dim dim since simply connected differential linear algebraic representation since simple every finite set nonzero dominant weights ker hence faithful every well known dim minimum dimensions nonzero algebraic representations see type trivial hence case faithful therefore equality rdim dim proves claim lemma types type respectively since groups tautological faithful representation case holds well proves claim lemma types taken types apply set ref table used value lie lie consists one element type faithful therefore case holds proves claim lemma type type consists two elements since case faithful whence holds proves claim lemma type type consists one element imply faithful contains odd using infer faithful representation minimal dimension hence rdim dim proves claim lemma type type odd consists three elements infer faithful contains coprime either show jordan property lie groups complex spaces faithful representation minimal dimension hence rdim dim proving claim lemma type consists three elements type even hence faithful contains odd odd odd hand since case cyclic schur lemma implies difficult deduce faithful representation minimal dimension hence rdim dim dim completes proof lemma corollary every simply connected simple affine algebraic group rank admits faithful linear algebraic representation dimension proof clearly algebraic group admits faithful linear algebraic representation admits faithful linear algebraic representation bigger dimension view claim follows inequality rdim turn follows lemma indeed latter shows rdim type differs rdim respectively types theorem let real lie group whose component group bounded proof lem thm may shall assume connected particular use notation proof theorem since connected connected see chap thm hence see prop compact simply connected simple lie groups compact torus iii group epimorphism finite kernel thm iii infer jke every real form corresponding simply connected simple complex affine algebraic group rank latter equal corollary conclude admits embedding dim turn implies since dim admits embedding clearly admits embedding also gldim therefore view dim dim vladimir popov see show admits embedding definition direct product copies hence infer turn since view dim admits embedding gln whence jke putting together complete proof recall following definition family groups called uniformly jordan every group jordan integer every corollary fix integer let family connected real lie groups family uniformly jordan one take jln proof follows every corollary every integer set isomorphism classes finite simple groups embeddable connected real lie groups finite algebraic groups consider several applications theorems first apply algebraic groups answering question theorem every necessarily affine algebraic group algebraically closed field characteristic jordan moreover proof case finite lefschetz principle may shall assume structure real lie group whose identity component claim follows theorem statement next corollary one main results corollary fix integer let family necessarily affine connected algebraic groups algebraically closed field characteristic thm family uniformly jordan one take jan proof follows jordan property lie groups complex spaces automorphism groups complex spaces next application automorphism groups complex spaces let necessarily reduced complex space exists topology aut respect aut topological group see theorem every compact complex space group aut jordan proof compactness implies aut complex lie group claim follows theorem know whether statement theorem remains true aut replaced aut thm answer affirmative connected compact complex manifold theorem also affirmative projective variety hand recall cps connected smooth compact real manifolds whose diffeomorphism groups disproves ghys conjecture remark connected noncompact complex manifolds whose automorphism groups indeed countable group noncompact riemann surface aut isomorphic whence claim existence countable groups see sect actual fact using idea exploited earlier one prove said remark showing existence connected complex manifolds monstrous automorphism groups namely theorem simply connected noncompact complex manifold group aut contains isomorphic copy every finitely presentable particular every finite group every copy discrete transformation group acting freely proof follows see thm higman embedding theorem universal finitely presented group finitely presented group containing subgroup isomorphic copy every finitely presented group turn abckt cor finite presentability implies existence connected compact complex manifold whose fundamental group isomorphic simply connected consider universal cover noncompact complex manifold deck transformation isomorphic acts freely group subgroup aut hence one take remark theorem group aut every integer finite simple group order example vladimir popov automorphism groups hyperbolic complex manifolds next application complex manifolds hyperbolic sense kobayashi particular bounded domains theorem fix integer let family groups aut runs connected complex manifolds hyperbolic sense kobayashi complex dimension family uniformly jordan one take jhn iii every point aut aut jordan jaut proof let connected complex manifolds hyperbolic sense kobayashi complex dimension thms aut real lie group dimension whence theorems thm isotropy representation aut faithful image isomorphic subgroup unitary group whence iii remark group aut formulation theorem replaced aut indeed follows construction riemann surface remark hyperbolic sense kobayashi therefore connected hyperbolic complex manifolds group aut jordan however next theorem shows complex hyperbolic manifolds special type jordan property holds whole aut rather aut theorem every strongly pseudoconvex bounded domain smooth boundary group aut biholomorphic transformations jordan proof lie group aut compact claim follows theorem group aut theorem domain biholomorphic unit ball since aut see sect prop latter lie group connected see chap lem claim follows theorem corollary every strongly pseudoconvex bounded domain smooth boundary set isomorphism classes finite simple groups biholomorphic transformations finite isometry groups riemannian manifolds last application isometry groups iso riemannian manifolds topological groups respect topology jordan property lie groups complex spaces theorem fix integer let family groups iso runs connected riemannian manifolds family uniformly jordan one take jrn iii every point iso iso jordan manifold compact group iso jordan proof known see chap thms iso real lie group dimension group iso compact every group iso compact manifold compact claims follows combining facts theorems remark group aut formulation theorem replaced aut indeed follows construction riemann surface remark riemannian manifold aut iso therefore connected riemannian manifolds group iso jordan concluding remarks view computing jordan constants connected real lie groups reduced compact groups instance results may interpreted computing jordan constants unitary groups jun every leads following natural problem compute jordan constants simple compact connected real lie groups references akhiezer lie group actions complex analysis aspects mathematics vol vieweg braunschweig abckt burger corlette kotschick toledo fundamental groups compact manifolds mathematical surveys monographs vol american mathematical society providence bourbaki groupes lie chap groupes lie compacts masson paris collins jordan theorem complex linear groups group theory cps pyber diffeomorphism groups compact always jordan helgason differential geometry symmetric spaces academic press new york higman subgroups finitely presented groups proc soc london hochschild structure lie groups san francisco vladimir popov jordan sur les equations reine angew math kaup infinitesimale transformationsgruppen komplexer math ann kobayashi transformation groups differential geometry springer berlin kobayashi hyperbolic manifolds holomorphic mappings introduction world scientific new jersey meng zhang jordan property algebraic groups projective varieties appear amer available https onishchik vinberg lie groups algebraic groups springer series soviet mathematics berlin popov derksen invariants finite automorphism groups algebraic varieties affine algebraic geometry russell festschrift crm proceedings lecture notes vol american mathematical society providence popov jordan groups automorphism groups algebraic varieties automorphisms birational affine geometry levico terme italy october springer proceedings mathematics statistics vol springer heidelberg popov infinite dimensional algebraic transformation groups transform groups popov jordan groups talk petersburg division steklov institute mathematics russian academy sciences december videorecord available http phtml option popov finite subgroups diffeomorphism groups proc steklov inst math prokhorov shramov automorphism groups compact complex surfaces ramanujam note automorphism groups algebraic varieties math annalen rosay sur une boule parmi les domaines par son groupe dautomorphismes ann inst fourier grenoble rotman introduction theory groups graduate texts mathematics vol springer new york silverman arithmetic elliptic curves graduate texts mathematics vol springer dordrecht winkelmann realizing countable groups automorphism groups riemann surfaces documenta math wong characterization unit ball automorphism group invent math steklov mathematical institute russian academy sciences gubkina moscow russia address popovvl
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interdependency transmission distribution pricing sina parhizi amin khodaei department electrical computer engineering university denver denver colorado distribution markets among prospect considered future power systems would facilitate integration distributed energy resources ders microgrids via market mechanism enable monetize services provide paper follows ongoing work implementing distribution market operator dmo concept clearing settlement procedures focuses investigating pricing conducted dmo distribution locational marginal prices relationship transmission system locational marginal prices subject paper numerical simulations test distribution system exhibit benefits drawbacks proposed dmo pricing processes index distribution microgrids power system economics electricity markets nomenclature plmax pxmax pos elements incidence matrix load benefit customer payments dmo dmo payment iso dmo cost surplus superscript fixed loads index bid segments penalty factor power deviation index distribution system buses load demand power transfer main grid total assigned power main grid line power flow line power flow limit amount load awarded segment bid maximum capacity bid segment superscript responsive loads proactive customers index hours power transfer deviation transmission locational marginal price distribution locational marginal price variables used linearize absolute value neg variables used linearize absolute value introduction igh penetration proactive customers utilize variety distributed energy resources der electric vehicles flexible loads frequently mentioned characteristics future power grids growing trend advocated order handle challenges growing demand boost efficiency grid operation meeting standards put place various environmental regulations proactive customers distribution system completely relying utility grid provide demand many cases der deployment even actively participate electricity markets provide several valuable services order proactive customers able realize full capabilities monetize services provide play role system pricing clearing processes necessary modernize existing distribution system operation enable entities participate electricity market direct control dispatch proactive customers independent system operator iso would create several problems potential violation distribution companies responsibilities also creating complexities market optimization participation proactive customers grows hence several proposals offered trying define entity establishes electricity market facilitates market participation distribution level distributed system platform provider dspp introduced new york via reforming energy vision program one early efforts direction dspp proposed operate accord electric utility charge market operations would coordinate iso customers market participants according proposal distribution system platform dsp set functions provided utilities enable broad market participation similar effort distribution system operator dso introduced california entity charge providing reliable distribution services customers ensuring predictability iso wouldbe coordinating physical transactions interface necessarily financial aspects transactions propose still among responsibilities iso role dso introduced similar iso distribution system dso would responsible reliable operation distribution system also providing demand response dso would transactions wholesale market substation level one side proactive customers side depending extent iso responsibility dispatching resources distribution system would different levels dso autonomy operating distribution system degree iso control dso act aggregator deliver iso dispatch commands operate retail market distribution level something two extremes prospective roles dsos listed including managing multidirectional flows organizing auctions offering incentives order minimize operation costs dso control strategies coordinate several microgrids within distribution system via distribution network operator dno proposed dno trade energy microgrids transmission system problem formulated optimization microgrids optimize energy cost lower level dno ensures operational constraints higher level optimization asserted utilities transition distribution system operation responsibilities independent distribution system operator idso owning distribution assets argued operation distribution system utilities poses conflict interest utilities tend expand assets would reduce greater market participation proactive customers proposed idso would charge system reliability provide market mechanisms optimally schedule distribution level resources would make system management easier utilities determine value services provided proactive customers objectively proposals however conceptual lack detailed rigorous analysis distribution market operations necessary clearly define roles different parties involved distribution system undergoing evolution investigate detailed market processes among concerns needs addressed structure distribution market operation way settled procedures distribution market clearing settlement established study points necessity dsos capable providing settlements iso resource distribution system studies furthermore present marginal pricing method dso considering congestion problem distribution system optimizing social welfare system high penetration electric vehicles distribution market operator dmo rather similar entity discussed authors dmo function focused work facilitate establishment market mechanisms distribution systems interface interacting iso proactive customers enable participation customers wholesale market dmo receives demand bids customers distribution system aggregates submits single aggregated bid iso market clearing iso dmo divides assigned power awarded participated customers dmo part electric utility company formed separate entity either case independent operator guarantee fairness market operation dmo manages financial transactions electric distribution company edc carries physical transactions implementation distribution market establishment dmo offers several advantages customers system whole distribution markets proactive customers demand set dmo known certainty basis would enable efficient control peak demand increase operational reliability improve efficiency proactive customers participate electricity market player exchange power utility grid customers dmo would facilitate market participation coordinate proactive customers interactions utility grid minimize associated operational risks uncertainties iii considerable reduction required communication system proactive customers iso need communicate dmos considering listed advantages many obtained deployments widespread distribution markets considered beneficial necessary components modern power grids help accommodate large penetration proactive customers terms distribution market clearing settlement models proposed investigated detail constant variable power clearing schemes studied paper focuses locational marginal prices transmission side dmo reflected distribution system locational marginal prices furthermore microgrids considered studies representative proactive customers microgrids models however simplified model type proactive customer prosumers responsive consumers rest paper organized follows formulation proposed dmo market clearing settlement demonstrated section iii numerical simulations presented section paper concluded section distribution market model dmo collects demand bids microgrids distribution system creates aggregated bid submits bid iso typical microgrid bid shown fig iso collects demand bids dmos well curtailment service provides generation bids gencos determines generation load schedule dmo notified iso clearing process decisions determines amount power generation transfer demands within distribution system settles prices costs among various participants market objective ensuring optimal operation fair settlement price max load fig demand bid curve customer bus bid iii distribution market clearing settlement dmo objective maximize distribution system social welfare load benefits minus generation cost cost assigned power main grid paper proposes add last term objective penalize violations assigned power max bmg mgt power assigned dmo iso determined via wholesale market clearing hence constant tlmp also determined iso penalty coefficient multiplied deviation ensure assigned power followed distribution network objective subject distribution network microgrids prevailing constraints pllt dmt pllt dmt mgt dmtf max mgt plmax pllt plmax neg pos pos neg neg pos nodal power balance ensured power injected bus connected lines equal total bus load power balance interface ensured power transferred main grid distributed among lines connected bus bus number load passive customers constant proactive customers variable defined associated segments scheduled load microgrids determined based scheduled power consumption bid segment segment limited associated maximum capacity line power flows limited line capacity limits added penalty objective function represented absolute value deviation makes problem nonlinear order linearize term ensure linear programming problem used neg pos two variables used model absolute value variable inside absolute value positive neg would equal zero value negative pos would equal zero guaranteed happen since problem formulated linear programming minimization solved simplex method bus calculated dual variable power balance equation bus byproduct proposed clearing problem relationship depends value penalty factor paper distribution market clearing conducted two ways based penalty factor clearing clearing discussed following clearing proposed formulation bus distribution point connection transmission network equal case dmo permitted import power utility grid power assigned iso would penalty results reflected dlmps within distribution system buses however determined based marginal cost dispatchable units possible distribution line congestions clearing amount increased dmo seeks minimize deviation scheduled power assigned power transfer set iso case dependency lowered large values penalty factor would violation power transfer schedule determined iso merely functions dispatchable units marginal price possible distribution line congestions market settlement using obtained either methods market settled payments customers payments utility determined payment customer calculated times associated load total customer payments summation payments dmt includes consumers microgrids payment utility calculated tlmp times assigned power iso since losses ignored proposed market clearing model sum distribution loads equal total power assigned iso dmt case case assumed distribution market clearing results variety determined results case shown fig illustrates effect different values marginal price bus distribution system scaling factor used change scaling factor increases power purchased higher rate resulting lower power transfer iso local generation results lower congestion prices busses tend equal higher values scaling factor lower values follow also impact grid prices respond price variations modifying power injections lower scaling factors congestion line results rise buses buses lines downstream feeder become constrained dmt considering payments dmo cost surplus calculated difference two calculated payments dmt obtained negative positive zero cost settlement one challenges facing dmos operate radial networks opposed wholesale power system operated isos guarantee fair market participation customers different locations across feeders issue topic future research authors numerical results ieee test system used investigate viability merits proposed processes fig depicts system microgrids located buses customer submits power demand bid maximum large capacity distribution lines considered however assumed lines smaller capacities subject potential congestions various cases studied considering various values parameters fig daily average lmp bus different values using scaling factor case case second term assumed negligible varied independent operation distribution market analyzed results case shown fig illustrates effect raising increases dmo seeks minimize deviation assigned power transfer prices however tend approach prices scheduled power used without option deviate approaches infinity functions marginal prices dispatchable units become independent would result settlement costs distribution system higher lower equal payments iso depending marginal costs fig ieee standard test system fig daily average function case case nonzero results case shown figs fig tlmp scaling factor varies kept corresponding difference customers payment dmo payment iso depicted fig deficit dmo means dmo tries minimize deviation power transfer respect scheduled power term exists objective seeks minimize power transfer even expense higher deviation reduce payments iso fig daily average function scaling factors fig payment received dmo minus payments iso conclusion one challenges operation distribution markets clearing settlement method use consequent impacts distribution market prices paper studied interdependencies distribution network upstream node shown adding penalty factor could calculated way prices follow associated determined independent shown price would significantly change market settlement decision exact value penalty coefficient however made dmo references parhizi lotfi khodaei bahramirad state art research microgrids review ieee access vol khodaei microgrid optimal scheduling islanding constraints ieee trans power vol may khodaei shahidehpour cooptimization generation transmission planning power systems ieee trans power vol bahramirad khodaei svachula aguero building resilient integrated grids one neighborhood ieee electrif vol mar khodaei provisional microgrids ieee trans smart grid vol khodaei microgrid optimal scheduling ieee trans smart grid vol jul khodaei bahramirad shahidehpour microgrid planning uncertainty ieee trans power vol ipakchi albuyeh grid future ieee power energy vol mar kassakian schmalensee desgroseilliers heidel afridi farid grochow hogan jacoby kirtley others future electric grid massachusetts institute technology cambridge zinaman miller arent power systems future nrel taft grid architecture pnnl online available http analysis grid kristov martini century electric distribution system operations caltech new york state department public service developing rev market new york dps staff straw proposal track one issues martini kristov distribution systems high distributed energy resources future planning market design operation oversight lawrence berkeley national laboratory martinot kristov erickson distribution system planning innovation distributed energy futures curr sustain energy reports vol apr martini smart framework make distribution grid open efficient greentech leadership group rahimi mokhtari iso dso imagining new construct independent system operator distribution network public util vol keay rhys robinson distributed generation implications utility industry distributed generation implications utility industry elsevier wang chen wang begovic chen coordinated energy management networked microgrids distribution systems ieee trans smart grid vol tong wellinghoff rooftop parity solar everyone including utilities public util vol new york state embarks bold new vision electr vol jul kristov hou distributed generation implications utility industry distributed generation implications utility industry elsevier huang oren liu distribution locational marginal pricing quadratic programming congestion management distribution networks ieee trans power vol jul oren distribution locational marginal pricing optimal electric vehicle charging management ieee trans power vol parhizi khodaei microgrid optimal scheduling ieee parhizi khodaei investigating necessity distribution markets accomodating high penetration microgrids ieee pes transmission distribution conference exposition parhizi khodaei bahramirad distribution market clearing settelement ieee pes general meeting kersting radial distribution test feeders ieee power engineering society winter meeting conference proceedings cat vol
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observation mechanism microscopic acoustic crackling noises ghaffari university texas arlington yates arlington rock mechanics laboratory school earth environmental sciences university portsmouth burnaby building portsmouth department civil engineering lassonde institute university toronto toronto college street canada characterizing fast evolution microstructure key understanding crackling phenomena deformation solid brittle materials example proposed using atomistic simulations crack propagation elastic materials formation nonlinear hyperelastic zone around moving crack tips controls crack velocity date progress understanding physics critical zone limited due lack data describing complex physical processes operate near microscopic crack tips show analyzing acoustic emissions rock deformation experiments signature nonlinear zone maps directly crackling noises particular use functional network method characterize weakening zone forms near moving crack tips determine scaling law formation defects traversal rate across critical point transition rapid weakening crack tip results agreement mechanism kzm evolution acoustic signals known crackling noise direct result failing atomic bonds material fracture signals properly interpreted may used better understand dynamics rupture progress vicinity crack tips broad range scales conditions crackling part stored energy near crack tip consumed breaking molecular atomic bonds resulting new crack surface key understanding crackling process lies characterizing structure region stresses amplitudes large due microscopic size high speeds encountered area crack tip direct measurements difficult analysis typically relies computational techniques large strains present near crack tips nonlinear elastic contributions must occur recent work suggests around moving crack tip governs rupture velocity specifically local hyperelastic zone around moving crack tip enhances energy flow stiffening systems reduces energy flow softening systems resulting respectively increases decreases fracture velocity relative linear elastic case furthermore investigations using slow cracks gels demonstrate link spatial energy flow around rupture tip curvature tip link thought responsible inaccuracies linear elastic analyses commonly used material science simulating crack tip processes applications ranging metal fatigue earthquake nucleation study use functional network method applied acoustic emission data recorded rock deformation rock friction experiments see methods section show moving contain signatures discover onset nonlinear stage prior global failure coincides nucleation kink instabilities use network strings visualize evolution topological defects first time show emitted crackling noises hold signature mechanism kzm provides estimation defect density function traversing rate key output kzm relates faster ramp rates higher defect density result spontaneous symmetry breaking process measure evolution correlation function system networks verify main prediction kzm namely scaling correlation length ramp rate moreover show correlation length near transition remain effectively frozen main underlying hypothesis behind mechanism addition using laboratory data sets illustrate transition core hypothesis plays major role approximating weakening rate rupture velocity analyse waveforms laboratory analogue seismograms due rock fracture earthquake rupture different simulated depth pressure conditions loading paths two rock types westerly granite basalt mount etna italy datasets see methods section supporting information apply tools theory complex networks analyze emitted noises microscopic cracks acoustic time series recorded sensor represented node results allow develop interpretation reordered multiple signals involving microsecond evolution different dynamic crack tip phases encoded network modularity refer see methods section evolution broken three distinct phases fig fig initial strengthening phase preceding critical point point weakening catastrophic failure begins fastslip weakening phase slow slip decelerating phase better understand transition occurs critical point use reciprocal modularity profiles closely resemble dynamic stress profiles commonly used characterize rock failure fig following discussion describe observation defect formation prior onset phase proceed define critical zone onset value time first impulse inverse mean betweenness centrality profile indicates mean value nodes show profiles acoustic events laboratory tests events profile characterized narrow impulse transition phase later show first impulse corresponds first nucleated defect transition second spike indicates defects formation inverse transition define duration temporal length impulse zone transition also corresponds time rmax maximum value rmax critical point failure occurs onsets duration defect formation nucleation time varies order study spatial variability impulse regime visualize spatial evolution degree ith node see supplementary material using polar coordinates nodes indicates position node fixed outer circumference cylindrical sample figure refer configurations kstrings normal vector node indicates local direction increasing time evaluate variation position node consider temporal evolution single event fig show onset impulse zone coincides folding normal vectors flipped onset regime form local domain refer kink fig therefore formation domains course transition results behaviour phase since trend match mean value nodes infer stress nucleation kinks prior onset fastweakening occurs corresponds global instability collapsing kstring occurs almost nodes words normal vectors strings point inward emphasized polyakov string theory crumpled stings analogous heisenberg paramagnet undulations destroy order surface normals defects local defects initially ordered structures removed global collapse local bending twisting around defects remove topological defects analysis deformation map ising chain node assign sign negative mapping defects represented indicating flipped normal vectors kink separating zones nodes analogy spins functions locator change one ground state another degenerate ground state study evolution cracking noise typical ising chain goes phase transition critical point defined coincides rmax order parameter transition vice versa occurs continuously furthermore calculated correlation function including nodes approaching critical point fit correlation function exp total number nodes distance correlation length correlation length length correlation function shorter distances correlation length power law function fitted fully ordered state triangular function fully coherent system given green line figure adding defects destroys long range order among nodes interestingly approaching correlation length becomes essentially frozen shown crackling event means system approaches critical point finite rate point time correlation length follow diverging equilibrium value transition occurs much smaller system size correlation length sets mean final domains formation domains well observation frozen correlation length critical point acoustic networks indicate signature spontaneous symmetry breaking process mechanism kzm see methods section core idea behind mechanism near transition freezing correlation length unavoidable based theory resulting density defects left behind continuous transitions dependent rate critical point traversed rate system adjust relaxation time mechanism reflected density defects time scales test kzm density defect prediction carefully measured number flipped nodes vicinity critical point correlation length frozen fig key output analysis number defects flipped nodes final state observe larger local ramp rate faster fig fit frozen correlation length ramp scaling rate measure ramp rate measure slope prior impulse zone fully coherent system fig yield ramp rate analogous local loading rate prior nucleation kinks exponent obtained fitting laboratory data agreement within experimental error theoretical value mean field model parameters spatial dynamical critical exponents see methods section furthermore evaluate transition phase approach critical point right words heal reverse failure process approaching left right time reversal healing scenario critical point results slightly different characteristics defects density rate linear strengthening rate ramp rate recorded events higher rate linear weakening rate next estimate linear weakening rate using kzm scaling law defects obtain supplementary information determined microscopic details system time characteristics ramps phases respectively therefore linear weakening rate given faster local inversely scales weakening rate important result since shown weakening rate correlated global rupture velocity cracks verify prediction measured rate weakening rmin profiles max shown measurements qualitatively confirms aforementioned prediction shown first time via laboratory data microcrack evolution duration microseconds derived purely multiple aes event data mapped acoustic excitations crackling events complex networks chains using novel tools elucidated transition showed nature transition governed defect mediated transition process particular close critical point failure correlation length frozen density nucleated defects scales local stress ramp rate implying mechanism employed studying acoustic crackling noises determine kzm exponent estimation defect density made extensive measurements many reordered crackling noises experiments addition found dynamic profiles involve theory transitions occur interpretation could estimate linear rate weakening phase parameter closely global rupture velocity focus study characterize events definite continuous transition nevertheless could recognize events abrupt change order parameter characteristic first order transitions study needed explore frequency laboratory earthquakes boundary environmental conditions favor dominant firstorder crackling noises cracking noises nodes sets analogy spins would interesting study effect flipping nodes concept spinwaves crystal vibrations phonons interaction course transition furthermore future work focused studying thermal activation mechanism nucleated kinks prior time effects cleavage different crystals polarization patterns methods laboratory procedures use four sets recorded acoustic emissions labeled westerly granite basalt rock samples analyzed events precursor rupture fronts recorded acoustic waveforms evolution frictional westerly granite samples interfaces dry conditions smooth naturally rough surfaces respectively evaluated events different stages position limited particular stage tests experiment cylindrical sample westerly granite confining pressure approximately order faster events basalt samples described global loading rate experiments reordered amplified events using sensor networks short discrete events long timescale recorders records resolution recorded interval waveform networks acoustic emission waveforms concept behind studying single acoustic excitation event determine onset critical phase defects precipitate matrix elastic material excitation defects vicinity moving crack tip dislocations must reflected recorded acoustic event study acoustic event use functional network theory analyze multiple recorded waveforms full details employed algorithm found section summary algorithm uses metric establish functional networks nodes sensors per time step network assigned per network set properties nodes degree modularity index centrality measures extracted see definition temporal evolution monitored time interval forms carries generic universal time scales distinguishing details microsecond timescale following found three generic classes corresponding following phases nucleation main deformation phase signature initial strengthening slow slip stage distinguish role phase define parameter initial rest value also use parameter indicating mean qnorm norm local energy flow given network betweenness centrality characterizes importance node using number shortest paths nodes pass node mechanism idea behind kzm compare relaxation time healing time system equilibrium timescale change control parameter assuming linear change control parameter vicinity critical point ramp time relaxation healing time consider equilibrium static condition determines reaction time order parameter spatial dynamical critical exponents characteristic timescale system adiabatically follow change imposed local stress ramp relaxation time characterized outside interval centered around transition point time system cease maintain imposed change time reaching critical point correlation length system effectively frozen correlation length given topological defects formed density one defect fragment per domain estimate resulting density topological defects given frozen phase one define effective control parameter plotting events plane hold obtain estimate measure frozen correlation length given event rate transition procedure leads estimate match approximation scaling coefficients symmetry breaking test scaling exponent analyzed events definitive continuous transition order parameter measure frozen correlation length chains functional acoustic networks supplementary information supplementary information accompanies paper acknowledgements would like acknowledge thank thompson mine design engineering kingston canada providing part employed data set work first author would like acknowledge campo encouragements points results author contributions authors contributed analysis writing manuscript competing interests statement authors declare competing financial interests references sethna dahmen myers crackling noise nature buehler abraham gao hyperelasticity governs dynamic fracture critical length scale nature schubnel nielsen bhat madariaga supershear ruptures experiments crustal rocks science ghaffari young networks evolution precursor rupture fronts laboratory earthquakes scientific reports livne bouchbinder svetlizky fineberg fields fast cracks science ghaffari thompson young complex networks waveforms acoustic emissions laboratory earthquakes nonlinear processes geophysics ghaffari nasseri young pfaulting rocks stress field scientific reports kibble topology cosmic domains strings phys math zurek cosmological experiments superfluid helium nature del campo zurek universality phase transition dynamics topological defects symmetry breaking international journal modern physics kibble dynamics lab universe phys today september rubinstein fineberg evolution frictional strength nature polyakov fine structure strings nuclear physics kantor nelson phase transitions flexible polymeric surfaces physical review mermin topological theory defects ordered media rev mod phys anderson basic notions condensed matter physics pub advanced book program dziarmaga zurek zwolak quantum superpositions topological defects nature physics chaikin lubensky principles condensed matter physics cambridge univ press feynman statistical mechanics set lectures advanced book classics thompson young lockner premonitory acoustic emissions natural westerly granite geophys res thompson young lockner fracture westerly granite feedback constant strain rate loading nucleation propagation transition unstable fracture propagation rock damage fluid transport benson vinciguerra meredith young evolution volcano seismicity laboratory study earth planetary science letters newman networks introduction oxford university press figures figure representing dynamic crackling noises three main stages typical acoustic crackling noises shown normalized phases correspond strengthening weakening decelerating stages respectively typical profile generated dataset schematic representation impulse zone transition phase five recorded events dynamic strain gauges measurements centimeter scales event stresses dynamically drop mpa experiment figure two typical acoustic emission events cracking granite samples shown scaled recorded acoustic waveforms corresponding normalized different events tests signature inverse mean betweeness centrality shows divergence parameter vicinity nucleation zone based resolution measurements total precipitate time varies figure nucleation kinks formation domains mean number edges versus time event transition nucleation zone imprinted diverging inverse mean betweeness centrality schematic representation sensors location red filled circles radius ring proportional node degree shown accumulated patterns nodes degree polar system time interval panel transition linear stage regime indicated onset local defects black arrows inducing formation domains example reported three arrows normal strings crumpled strings destroy order string normal shown examples figure continuous phase transition results cracking noise mapping order parameter versus time goes zero approaches critical point failure point function close red see nearly overlapping patterns indicating frozen zone correlation length function versus normalized distance fully ordered state triangular function fully coherent given green line approaching fit correlation function red line determine correlation length event frozen correlation length blue points experimental measures correlation function normalization nodes nodes approaching point correlation length becomes frozen shown event correlation length roughly constant see supplementary figures examples figure description structural phase transition acoustic excitations formation domains occurs failure regime characterized undulation along schematic transition approaching left right time reversal critical point results slightly different characteristics defects particular rate linear strengthening rate approaching left higher rate linear weakening rate approaching right according kzm slower ramp rate marks longer time smaller defect density approaching critical point yields longer kzm time figure dependency frozen correlation length kink density ramping rate events faster transition critical points induce higher defect density shorter correlation length show typical rupture fronts size network nodes obtained dashed line agreement model prediction also see effect local rates regime events faster weakening rate scale slower ramp rate illustrated via plot normalized rate weakening observed function events
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quadratic minimum spanning tree problem variations ante ruonan abraham mar march abstract quadratic minimum spanning tree problem variations quadratic bottleneck spanning tree problem minimum spanning tree problem conflict pair constraints bottleneck spanning tree problem conflict pair constraints useful modeling various real life applications problems known paper investigate problems obtain additional insights structure problems identify possible demarcation easy hard special cases new polynomially solvable cases identified well instances simple graphs byproduct recursive formula counting number spanning trees characterization matroids context quadratic objective function keywords quadratic spanning tree complexity tree enumeration sparse graphs row graded matrix matroids introduction let undirected graph costs given edge pair edges respectively quadratic minimum spanning tree problem qmst formulated follows minimize subject family spanning trees associated cost matrix entry qmst viewed generalization many well known optimization problems travelling salesman problem quadratic assignment problem maximum clique problem used modeling various real life application areas telecommunication transportation irrigation energy distribution problem introduced assad along special case quadratic minimum spanning tree problem aqmst adjacent strong qmst aqmst proved along ideas solving problem using exact heuristic algorithms broad applications base inherent complexity qmst makes interesting topic research works qmst acustic department mathematics simon fraser university surrey surrey british columbia canada department mathematical sciences university ren road suzhou jiangsu china apunnen department mathematics simon fraser university surrey surrey british columbia canada focussed heuristic algorithms punnen provided characterization qmst instances solved minimum spanning tree problem exact algorithm aqmst qmst studied pereira gendreau cunha special case qmst one quadratic term studied buchheim klein fischer fischer many problems studied literature closely related qmst terms formulation applications list variations qmst investigate paper minimum spanning tree problem conflict pairs mstc given graph edge costs set mstc find spanning tree tree cost minimized edge pair one included tree feasibility problem mstc denoted fstc problem finding feasible solution mstc regardless costs given fstc instance construct qmst graph otherwise fstc instance feasible qmst optimal objective function value therefore fstc reduces qmst quadratic bottleneck spanning tree problem qbst replacing objective function qmst max obtain qbst problem introduced shown even bipartite graph values fqbst feasibility version qbst described given value exist spanning tree max fqbst equivalent fstc develops heuristic algorithms qbst using mstc heuristics subroutines bottleneck spanning tree problem conflict pairs bstc similar relation qbst qmst bstc defined substituting objective function mstc objective function furthermore define counterparts problems aqbst adjacent graph mstac fstac bstac edges conflict pairs restricted adjacent even though problems proved general exploring nicely solvable special cases provide additional insights structure problems opportunities developing effective heuristics primary research question focus paper extend qmst variations would retain status become amenable polynomial time solvability consider restricting structure graph cost matrix identify possible demarkation easy hard instances rest paper organized follows section introduces sparse graphs investigating include fans ladders wheels generalizations accordions recursive formula derived count number spanning trees generalizes well known sparring tree counting formulas fans ladders section study complexity qmst variations sparse graphs shown problems except case aqbst mstac bstac aqbst cases provide time algorithms problems general graph specially structured cost matrices discussed section particular show permuted doubly graded matrix qmst qbst solvable polynomial time case optimal solution value attains lower bound result extended case matroid bases provides new characterization matroids use notations denote respectively node edge sets graph quadratic costs sometimes denoted simplicity number spanning trees section define classes graphs called study qmst variations graphs also study number spanning trees graphs number feasible solutions corresponding qmst figure fan wheel given path fan obtained introducing new node edges add one edge resulting graph called wheel denoted deleting edges results examples fan wheel presented figure let two paths add edges resulting graph called ladder see figure figure ladder let define class graphs generalizes fans ladders given integers graph obtained recursively fusing together cnk along edge resulting graph two consecutive cycles edge common precisely graph constructed using following rules initialize graph embed plane designate edges free edges define follows choose free edge introduce say cik using edge new nodes get planar embedding resulting graph designate edge incident new node free edge every obtained graph set denoted figure presents examples let define subclass call defined way accordions except construction scheme designate edge free end points new nodes introduced note construction edge designated free least one end points new node denote set figure examples note figure also figure easy verify fan unique formulas counting number spanning trees already known generalize results deriving formula counting number spanning trees let number spanning trees graph following property holds lemma graph obtained deleting obtained coalescing endpoints proof straightforward due fact total number spanning trees number spanning trees contain plus number spanning trees without recursive formula number spanning trees given following theorem theorem every number spanning trees denote number every integers proof let generated cik cnk similarly let cork responding generated respectively edge called free edge likewise edge called free edge let graph obtained contracting free edge graph obtained contracting free edge free edge thus lemma note spanning tree either contains free edges contain exactly one free edge spanning tree contains free edges contain edge common cnk graph obtained deleting contracting path free edges isomorphic since contains free edges recall arbitrary depend respectively note one isomorphic hence fixed every number spanning trees recursion follows every fixed number spanning trees hence recursive formula given theorem gives implicit formula number spanning trees general case gives solving recursion obtain follows formulas consistent known spanning tree enumeration formulas fans ladders moreover generalize furthermore theorem used deduce explicit formulas fixed since every element contains exponential number spanning trees solving qmst variations class graphs complete enumeration would computationally expensive complexity qmst variations sparse graphs investigate complexity qmst variations sparse graphs discussed section intractability results recall section fstac feasibility version adjacent minimum spanning tree problem conflict pair constraints theorem fstac proof reduce problem fstac fan star let instance given conjunctive normal form instance construct graph node set edge set eij vij vij shown figure conflict set defined eij ekl xij xkl negations note edges conflict pair adjacent solution fstac let xij xij true eij false otherwise least one must true since contains least one moreover eij ekl eij ekl one xij xkl true negations hence true assignment problem conversely suppose true assignment problem eij xij true acyclic subgraph one edge conflict pair spans solution fstac otherwise add necessary edges form spanning tree gives solution fstac result follows figure since fstac feasibility version mstac mstac also subgraph fan wheel hence mstac reduced mstac fans wheels assigning large costs additional edges proves mstac fans wheels mstac special case mstc aqmst mstc special case qmst hence problems fans wheels furthermore theorem easily follows bottleneck versions problems fans wheels observations summarized following corollary corollary mstac mstc aqmst qmst bstac bstc aqbst qbst fans wheels next identify intractability results problems ladders theorem fstc ladders proof reduce problem fstc ladders let instance given conjunctive normal form construct ladder shown figure let eij ekl xij xkl negations note conflict edges necessarily adjacent exists solution fstc instance true assignment figure ladder constructed instance detailed proof similar one given theorem hence omitted fstc propagated mstc qmst bstc qbst summarized following corollary corollary mstc qmst bstc qbst ladders polynomially solvable special cases let examine complexity qmst variations covered section show remaining problems easy proposing linear time algorithm solve aqmst ladders let first introduce notations recall graph sequence consecutive cik intersect intersection edge corresponding two vertices let label edges cik eik edge also eik edge also adjacent note furthermore let denote vertices incident eik see figure illustration eik figure given denote determined first next define spanning trees minimum aqmst cost satisfy additional properties particular let spanning tree minimum aqmst cost contains eik contain analogously let minimum spanning tree contains eik contain let minimum spanning tree contains contain eik let minimum spanning tree contains eik note possible configurations eik covered similarly define minimum cost spanning forests made exactly two trees one tree contains tree contains particular let minimum cost forest contains contain eik let minimum cost forest contains contain eik let minimum cost forest contains contain eik note one tree forests exactly single vertices next show known tji tji fji fji calculate time two edge disjoint sets define graph spanned edges easy verify following recursive relations hold min cik min cik min cik eik min min cik min cik cik eik min cik eik min min min cik eik min cik eik since adjacencies edges cik graphs known minimization functions calculated easily function values calculated time remaining values constant time provided known easily calculated tji fji costs incrementally calculated along costs value increases optimal solution aqmst obtained call algorithm algorithm summarized algorithm algorithm input graph costs calculate determine tji fji using end minimum cost tree output theorem algorithm solves aqmst time proof correctness algorithm follows exploration possible cases resulting recursion relations iterations line algorithm iteration takes time calculate trees corresponding costs discussed hence overall complexity mstac easily reduced aqmst calculating maximum rather summation algorithm adapted solve bottleneck versions problems without describing detailed steps following corollary holds corollary mstac aqbst bstac solved time determined complexities problem variations graph classes investigated results summarized table represents means polynomially solvable fan wheel ladder mstac mstc aqmst qmst bstac bstc aqbst table polynomial solvability qmst variations qbst qmst row graded cost matrix section shown qmst variations mostly even restricted simple classes graphs wide variety hard optimization problems solved efficiently therefore shift focus special graphs specially structured cost matrices consider minimum spanning tree problem mst mst minimize subject family spanning trees let optimal objective function value mst consider minimum spanning tree problem mst minimize subject let optimal objective function value mst shown lower bound optimal objective function value qmst call natural lower bound qmst could find spanning tree objective function value surely optimal solution qmst matrix said row graded furthermore called doubly graded row graded given matrix permutation define matrix entry say permuted row graded permuted doubly graded exist permutation row graded doubly graded respectively note permuted row graded permuted doubly graded matrices recognizable polynomial time let graph vertices edge set given permutation spanning tree called spanning tree set lexicographically smallest among spanning trees recall mst solved greedy algorithm therefore spanning tree optimal permuting cost vector makes nondecreasing lemma let cost matrix qmst graph permuted row graded row graded spanning tree minimum spanning tree edge costs optimal solution qmst proof since row graded spanning tree optimal solution mst corresponding optimal objective function value also optimal mst edge costs optimal mst thus optimality qmst proved using lemma show qmst permuted doubly graded matrices polynomially solvable theorem cost matrix qmst graph permuted doubly graded doubly graded spanning tree optimal solution proof let since row graded optimal mst corresponding optimal objective function value also isp row graded thus minimum spanning tree edge costs lemma optimal qmst optimal objective function value rest section extend results general structure called matroid bases give new characterization matroids terms quadratic objective function best knowledge characterization matroids known uses optimization problem quadratic objective function let ground set family subsets call bases constant let call structure independence system structure base system independence system called matroid exists instance edge set graph collection spanning trees collection acyclic subgraphs called graphic matroid given base system weight quadratic minimum weight base problem qmwb formulated follows minimize subject associated cost matrix define minimum weight base problem follows mwb minimize subject let optimal objective function value mwb similar case qmst optimal objective function value problem mwb minimize subject called natural lower bound qmwb cost matrix given permutation say base base set lexicographically smallest among sets theorem following statements equivalent matroid let cost matrix qmwb base system permuted row graded row graded base minimum weight base costs optimal solution qmwb cost matrix iii cost matrix qmwb problem base system permuted doubly graded doubly graded base optimal solution moreover natural lower bound optimal objective function value proof using fact minimum weight matroid found greedy algorithm statements implies implies iii proved similarly lemma theorem hence proofs omitted show iii implies assume matroid aim show case iii true hence assume let permutation note base since let cost matrix entries otherwise clearly doubly graded hence permuted doubly graded see figure let consider figure objective function values qmwb cost matrix hence base optimal solution qmwb cost matrix implies iii true similarly prove implies showing matroid true complete proof theorem assume consider permutation cost matrix figure row graded base minimum weight base costs optimal solution corresponding qmwb end section noting theorem theorem hold true also qbst quadratic bottleneck base problem respectively sum objective value function replaced maximum proofs work since greedy algorithm obtains optimal solution also linear bottleneck objective functions base system matroid acknowledgments work supported nserc discovery grant nserc discovery accelerator supplement awarded abraham punnen references ahuja ergun orlin punnen survey neighborhood search techniques discrete applied mathematics assad quadratic minimum spanning tree problem naval research logistics buchheim klein combinatorial optimization one quadratic term spanning trees forests discrete applied mathematics cordone passeri solving quadratic minimum spanning tree problem applied mathematics computation punnen characterization linearizable instances quadratic minimum spanning tree problem darmann pferschy schauer minimal spanning trees conflict graphs optimization online http darmann pferschy schauer woeginger paths trees matchings disjunctive constraints discrete applied mathematics fischer fischer complete description spanning tree problem one linearised quadratic term operations research letters hao search approach quadratic minimum spanning tree problem engineering applications artificial intelligence gao fuzzy quadratic minimum spanning tree problem applied mathematics computation gupta punnen optimization problems operations research letters hilton spanning trees fibonacci lucas numbers fibonacci quarterly lozanoa glover javier tabu search strategic oscillation quadratic minimum spanning tree iie transactions maia goldbarg goldbarg biobjective adjacent quadratic spanning tree problem electronic notes discrete mathematics maia goldbarg goldbarg evolutionary algorithms adjacent quadratic spanning tree international journal innovative computing applications punnen local search intensified variable neighborhood search generalized assignment problem discrete optimization punnen quadratic minimum spanning tree problem lower bounding procedure efficient search algorithm computers operations research palubeckis rubliauskas metaheuristic approaches quadratic minimum spanning tree problem information technology control pereira gendreau cunha stronger lower bounds quadratic minimum spanning tree problem adjacency costs electronic notes discrete mathematics pereira gendreau cunha algorithms adjacent quadratic minimum spanning tree problem networks pereira gendreau cunha lower bounds exact algorithms quadratic minimum spanning tree problem computers operations research punnen zhang quadratic bottleneck problems naval research logistics number spanning trees finite graphs pro matematiky soak corne ahn new evolutionary algorithm spanning tree based communication network design ieice transaction communication soak corne ahn representation problems ieee transactions evolutionary computation zhang punnen quadratic bottleneck knapsack problems journal heuristics zhang kabadi punnen minimum spanning tree problem conflict constraints variations discrete optimization zhou gen effective genetic algorithm approach quadratic minimum spanning tree problem computers operations research
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towards automated deduction blackmail case analysis forensic lucid serguei mokhov joey paquet mourad debbabi faculty engineering computer science concordia university canada mokhov paquet debbabi may abstract focuses refinement application intensional logic cyberforensic analysis benefits compared automata approach work extends use scientific intensional programming paradigm onto modeling implementation cyberforensics investigation process backtrace event reconstruction modeling evidence multidimensional hierarchical contexts proving disproving claims intensional manner evaluation practical improvement finite state automata fsa approach seen related works base implementation language model use approach new dialect lucid programming language call forensic lucid paper focus defining hierarchical contexts based intensional logic evaluation cyberforensic expressions keywords intensional logic intensional programming cyberforensics forensic lucid lucid finitestate automata introduction problem statement first formal approach event reconstruction cyberforensic analysis appeared two papers gladyshev relies automata fsa transformation operation model evidence witnesses stories told witnesses possible evaluation one examples papers present proposed technique blackmail investigation aim case model implement using new approach promises friendly usable actual investigator work serve basis development area proposed solution intend show intensional approach dialect problem asset field cyberforensics promising practical usable fsa since lucid originally designed used prove correctness programming languages based temporal logic functional languages implementation backtracking proving disproving evidential statements claims investigation process evaluation expression either evaluates true false given facts formally specified context also attempt retain generality approach building problemspecific fsa fsa approach suffer state explosion problem logic perspective shown one model computations basic unit finite state machines logic armed contexts values model adopted implementation languages limits scope evaluation given set dimensions come intensional logic corresponding programming artifact essence model forensic computation unit intensional logic propose ways implement practice within intensional programming platform automated deduction blackmail case analysis forensic lucid mokhov see lot potential work successful beneficial cyberforensics well intensional programming communities based parameters terms defined papers various pieces evidence witnesses telling stories incident goal put together make description incident precise possible show certain claim may true investigator show explanations evidence agrees claim disprove claim investigator show explanation evidence agree claim authors fsa approach implementation algorithms cmu common lisp target improve lucid dialect call forensic lucid work focus specification hierarchical context expressions operators modeling examples lips unlike lucid entirely lacks contexts build logic syntax semantics thereby making implementation cases clumsy inefficient highly sequential system discussed offers distributed evaluation lucid programs efficient way general lisp compiler environment lucid overview lucid dataflow intensional functional programming language fact family languages built upon intensional logic turn understood multidimensional generalization temporal logic involving context parallel computation model program written lucid dialect expression may subexpressions need evaluated certain context given set dimensions dimi expression varies corresponding set indexes tags defined placeholders dimension context represented set dimi tagi mappings variable lucid called often stream evaluated defined context may also evolve using context operators generic version lucid general intensional programming language gipl defines two basic operators navigate switch query contexts gipl generic programming language intensional languages defined means two intensional operators proven intensional programming languages lucid family translated gipl general intensional programming system gipsy gipsy platform implemented primarily java investigate properties lucid family languages beyond executes lucid programs following demanddriven distributed architecture designed modular collection frameworks components related development compilation execution lucid programs separated allowing easy extension addition replacement components proposed testing investigation platform forensic lucid language forensic lucid overview section summarizes concepts considerations design forensic lucid language studied another related works end goal define forensic lucid language constructs concisely express cyberforensic evidence context second lucx integrated programming environment implemented general intensional programming compiler implemented general eduction engine implemented automated deduction blackmail case analysis forensic lucid mokhov evaluations initial state case towards actually observed final state fsm implementing system backtraces intermediate results provide corresponding event reconstruction path exists discuss work result expression basic form either true false guilty guilty given evidential evaluation context per explanation backtrace multiple backtraces correspond explanation evidence lack thereof properties define forensic lucid model evidential statements expressions representing evidence observations context execution trace running forensic lucid program designed expose possibility proposed claim events lead conclusion forensic lucid aggregates features multiple lucid dialects mentioned earlier needed tasks along extensions addition context calculus lucx stands lucid enriched context promotes contexts values operators simple contexts context sets union intersection etc used manipulate complex hierarchical context spaces forensic lucid additionally forensic lucid inherits many properties lucx objective lucid jooip intensional programming language comprising dialects former context calculus latter arrays structural representation data modeling case data structures events observations groupings related data eliminate aspects work well others conserve space instead focus context hierarchies syntax semantics hierarchical contexts also following example marfl using dot operator overloading accept different types left right arguments one basic requirements final target definition syntax operational semantics forensic lucid compatible basic lucx gipl necessary compiler run time system within implementing system called general intensional programming system gipsy translation rules equivalent provided implementing language compiler within gipsy environment general eduction engine gee execute minimal changes gee implementation context need provide ability encode stories told evidence witnesses constitute context evaluation return value evaluation would collection backtraces contain paths truth given trace contains truths values explanation story path trace enough supporting evidence entire claim true context task simplicity prototype language expressed integers strings attribute meaning description contexts finite navigated directions index potentially allowing negative tags tag sets dimensions concurrently contexts finite set symbolic labels values internally enumerated symbolic approach naturally appropriate humans machinery lucx implementation gipsy define streams observations context simple context context set fact forensic lucid defining dimensions dimensions one evidential statement finite unordered set observation sequences observation sequence finite ordered set observations observation eyewitness particular property along duration observation fsa observations tuples min opt generic form observations form specifically property exploded lucx context set atomic simple context context switching different observations automated deduction blackmail case analysis forensic lucid mokhov done naturally lucid context switching operator consider conceptual expression storyboard listing anything represents story context evaluation foo evaluated multiple contexts stories producing collection final results true false story well collection traces foo observed event observed event listing intensional storyboard expression notation may confusing respect notation dimension tag lucid specifically lucx fact simple syntactical extension allow groups contexts syntactical sugar later translated baseline context constructs tentative notation implies notion similar notion context set except syntactical sugar mentioned earlier allow syntactical grouping properties observations observation sequences evidential statements context sets generic observation sequence expanded context stream using min opt values translate index values thus obs expands property labels finite stream five indexed elements aaabb thus forensic lucid fragment listing would return third aaabb context stream observation portion therefore possible evaluations check properties shown figure higher min observation equivalent end end listing observation sequence duration property values anything context calculus allows observation sequence finite ordered context tag set allows integral duration given tag property may seem like allow duplicate tag values unsound classical lucid semantics however find way around little text implicit tag index semantics arrays computations part either gipl lucx however arrays provided jlucid objective lucid need notion arrays evaluate multiple computations context array computations conceptually equivalent running lucid program context array element separate instance evaluation engine automated deduction blackmail case analysis forensic lucid mokhov results expressions gathered one ordered storage within originating program arrays forensic lucid needed represent set results explanations evidential statements well denote properties observations explore notion arrays forensic lucid much greater detail near future work fsa approach computations correspond state event enable transition forensic lucid theoretically lucid expression observed property context index get position figure handling duration observed property context figure illustrating possibility query indices raw property persists produces finite stream valid indices used subsequent expressions alternatively supplying index get corresponding raw property index latter feature still investigation whether safe expose forensic lucid programmers make implicit times implementation level needed remedy problem duplicate tags previously mentioned observations form context allow durations means multiple duplicate dimension tags implied subdimension indexes allowed semantics traditional lucid approaches allow duplicate dimension tags noted however combination tag index stream still unique folded traditional lucid semantics transition function transition function described length derived works determines context evaluation changes computation general issue exists address transition function usually fsa approach transition function labeled graph first prototype follow graph model forensic lucid equivalent general lucid already basic operators navigate switch one context another represent basic transition functions intensional operators iseod first next fby wvr upon asa well inverse however specific problem modeled requires specific transition function plain intensional operators case transition function forensic lucid function matching state transition modeled sequence intensional operators question arises explicitly model transition function backtrace new language possible approach use predefined macros lucid syntax fact forensic operators functions rely traditional inverse lucid operators well context switching operators achieve defined automated deduction blackmail case analysis forensic lucid mokhov something similar transitions implementation level gee actually execution within gipsy fact intensional operators lucid represent basic building blocks operational semantics previously mentioned operational semantics forensic lucid large part viewed composition semantic rules gipl objective lucid lucx along new operators definitions explanation rules notation given great detail cited works trimmed extended abstract due shortage space objective lucid semantic rules affected refined semantic rules jooip also omit objective lucid jooip semantic rules due space limitation defer another publication new rules operational semantics forensic lucid cover newly defined operators primarily including reverse logical stream operators well operators refining semantics context set operators lucx box range also part work use notation referenced languages maintain consistency defining rules initial blackmail case modeling figure cluster data blackmail fragments case description section managing director company blackmailed contacted police handed evidence form floppy disk contained letter number allegations threats demands message known come friend police officers went interview found holiday abroad seized computer interviewed soon returned country admitted wrote letter denied making threats demands explained holiday access computer thus possible added threats demands letter discredit one blackmail fragments found slack space another letter unconnected incident police interviewed person letter addressed confirmed received letter day gone abroad holiday concluded must added threats demands letter going holiday could involved figure initial view incident diagram illustrating cluster data blackmail unconnected letters modeling investigation blackmail example functionality last cluster file used determine sequence events hence disprove alibi thus scope model restricted functionality last cluster unrelated file last cluster model store data objects three possible lengths lengt zero length means cluster unallocated length means cluster contains object size automated deduction blackmail case analysis forensic lucid mokhov simplified model cluster possible data lengths possible data values left part right part unrelated part threats slack data left part data right part observed final state lengt unrelated left part right part lengt left part right part threats slack figure simplified cluster model unrelated letter tip length means cluster contains object size data block threats figure therefore simplified model investigation events state last cluster changed three types events ordinary writes cluster rit direct writes file cluster allocated bypassing direct rit deletion file sets length file zero rit direct rit del formalization evidence final state observed investigators let inal denote observation state entire sequence observations inal inal observation sequence osunrelated says unrelated letter created time past received person addressed osunrelated ounrelated ounrelated denotes observation unrelated letter tip written cluster evidential statement esblackmail inal osunrelated finding explanation theory theory encoded using proposed notation oblackmail denotes observation unrelated letter written cluster time cluster contain blackmail fragment oblackmail denotes observation right part model contains blackmail fragment explanations two logically possible explanations represented state machine see corresponding state diagram blackmail case figure automated deduction blackmail case analysis forensic lucid mokhov figure blackmail case state machine first explanation finding unrelated letter written earlier adding threats last cluster letter editing suitable text editor vim restoring unrelated letter original content editing understand sequence events observe certain text editors vim configured edit text mode operation modified file written back disk blocks allocated original file result user forge file slack space appending desired slack space content end file saving reverting file back original content saving second explanation threats added slack space unrelated letter writing directly last cluster using example disk editor blackmail case example initial implementation steps listing conclusion proposed practical approach cyberforensics field also used normal investigation process involving crimes necessarily associated information technology combined expert system implemented clips also used training new staff investigation techniques focus hierarchical contexts values brings understanding process investigators cybercrime case management tools future work forensic lucid compiler environment prove equivalence fsa approach graphical based graph tool simplify forensic lucid programming investigators automated deduction blackmail case analysis forensic lucid mokhov mra mra sequence mra blackmail observation sequence observation possible dimensions box box events unordered symbolic observation map human unordered unrelated data part observation unordered end end end listing blackmail case modeling forensic lucid acknowledgments research work funded faculty engineering computer science concordia university montreal canada references pavel gladyshev ahmed patel finite state machine approach digital event reconstruction digital investigation journal pavel gladyshev finite state machine analysis blackmail investigation international journal digital evidence edward ashcroft william wadge lucid formal system writing proving programs siam automated deduction blackmail case analysis forensic lucid mokhov edward ashcroft william wadge erratum lucid formal system writing proving programs siam lalement computation logic prentice hall hoare series editor english translation french john plaice joey paquet hua gipsy platform investigation intensional programming languages proceedings international conference programming languages compilers plc pages las vegas usa june csrea press joey paquet serguei mokhov xin tong design implementation context calculus gipsy environment proceedings annual ieee international computer software applications conference compsac pages turku finland july ieee computer society joey paquet architecture distributed eductive execution hybrid intensional programs proceedings ieee workshop software engineering context aware systems secasa ieee computer society appear kaiyu wan vasu alagar joey paquet lucx lucid enriched context proceedings international conference programming languages compilers plc pages las vegas usa june csrea press xin tong design implementation context calculus gipsy master thesis department computer science software engineering concordia university montreal canada april gipsy research development group general intensional programming system gipsy project department computer science software engineering concordia university montreal canada http last viewed april serguei mokhov towards hybrid intensional programming jlucid objective lucid general imperative compiler framework gipsy master thesis department computer science software engineering concordia university montreal canada october isbn william wadge edward ashcroft lucid dataflow programming language academic press london edward ashcroft anthony faustini raganswamy jagannathan william wadge multidimensional declarative programming oxford university press london edward ashcroft william wadge lucid nonprocedural language iteration communication acm july kaiyu wan lucx lucid enriched context phd thesis department computer science software engineering concordia university montreal canada joey paquet scientific intensional programming phd thesis department computer science laval university canada joey paquet peter kropf gipsy architecture proceedings distributed computing web quebec city canada developing distributed component framework processing intensional programming languages phd thesis department computer science software engineering concordia university montreal canada march joey paquet peter grogono hua towards framework general intensional programming compiler gipsy proceedings annual acm conference programming systems languages applications oopsla vancouver canada october acm emil vassev joey paquet generic framework migrating demands gipsy execution engine proceedings international conference programming languages compilers plc pages las vegas usa june csrea press hua joey paquet intensional programming gipsy preliminary investigations proceedings international conference programming languages compilers plc pages las vegas usa june csrea press serguei mokhov joey paquet formally specifying proving operational aspects forensic lucid isabelle technical report mohamed department electrical computer engineering concordia university august theorem proving higher order logics emerging automated deduction blackmail case analysis forensic lucid mokhov trends proceedings serguei mokhov joey paquet mourad debbabi formally specifying operational semantics language constructs forensic lucid oliver sandra frings detlef jens nedon dirk schadt editors proceedings incident management forensics imf pages mannheim germany september serguei mokhov towards syntax semantics hierarchical contexts multimedia processing applications using marfl proceedings annual ieee international computer software applications conference compsac pages turku finland july ieee computer society xin tong joey paquet serguei mokhov context calculus gipsy unpublished serguei mokhov joey paquet mourad debbabi designing language intensional cyberforensic analysis unpublished aihua joey paquet serguei mokhov intensional programming intensional classes using java lucid submitted publication pppj bram moolenaar contributors vim editor improved online http gary riley clips tool building expert systems online http last viewed december lei tao warehouse garbage collection gipsy environment master thesis department computer science software engineering concordia university montreal canada
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logical methods computer science vol submitted published acyclic solos differential interaction nets thomas ehrhard olivier laurent preuves programmes cnrs paris france address abstract present restriction solos calculus stable reduction expressive enough contain encoding consequence shown equalizing names already equal required encoding particular induced solo diagrams bear acyclicity property induces faithful encoding differential interaction nets gives new proof differential interaction nets expressive enough contain encoding worked case finitary replication free systems without sum match mismatch introduction question extending correspondence intuitionistic logic concurrency theory open problem developed previous paper translation differential interaction nets shown differential linear logic logical system whose sequent calculus obtained first author precise analysis denotational models linear logic based vector spaces reasonable candidate correspondence concurrent computation since differential interaction nets appear expressive enough represent key concurrency primitives provided works trying relate variants linear logic process algebras work graphical syntaxes encoding see example however even try build bridges linear logic require hoc graphical constructions without logical foundations goal build differential interaction nets coming logical justification differential linear logic acm subject classification key words phrases solos calculus prefix typing differential interaction nets work partially funded french anr projet blanc curry howard pour concurrence choco logical methods computer science ehrhard laurent creative commons ehrhard laurent let tell bit genesis translation discovered notion communication areas particular differential interaction nets able represent communication primitives however due asynchronous flavour differential interaction nets immediate additional constructions prefixing could easily encoded led look solos calculus solos calculus allowed prove action prefixes thus sequentiality constraints encoded calculus without prefixes results kind also appear going much asynchronous done encoding solos calculus point view solos calculus seen low level concurrent language even basic main goal present paper stress close relation appears differential interaction nets graphical syntax differential linear logic solo diagrams graphical syntax solos calculus following ingredients possible give simple representation solo diagrams differential interaction nets based implementation nodes communication areas static correspondence interesting dynamic counterpart reduction semantics troubles come picture reduction semantics almost match mismatch solos calculus communication areas respect identification occurrence name another occurrence name reduction let call phenomenon selfidentification note identification never occurs since name passing handled substitution bound name another name fusion calculus solos calculus defines communication unification names two occurrences name could identified reduction identified considered dummy operation unification setting since already situation different differential interaction nets since communication areas keep track identification explicit link connecting communication area associated reduction able simplify link order give back communication area one gets complicated structure directly working thus name passing substitution thus without way direct interpretation whole finitary differential interaction nets possible approach propose different look restriction solos calculus avoiding static correspondence differential interaction nets also works dynamic level acyclic solos calculus requires restrict source calculus allows deal natural translation acyclic solos calculus obtained precise analysis translation solos calculus main idea able isolate inside unification mechanism symmetric operation one unifies name name flow one side example happens substitution mechanism reduction one consider flow going binding occurrence controlling occurrences first ingredient definition acyclic solos calculus simple typing system allowing label object occurrences names solos term acyclic solos differential interaction nets two protocols send receive kind uniform way main consequence typing system break symmetry two occurrences names unified one one however impose important structural behavioural constraints protocols given second part definition calculus acyclicity conditions justification conditions mimic structure input prefix inside solos calculus exactly one binding occurrence name solos term occurrences thus object occurrences acyclic solos coming sequentially binding one informally together leads name passing unification close substitution mechanism understand unification substitution flow roccurrence occurrences name formally prove never occurs acyclic solos calculus also show translation finitary solos calculus following range contained acyclic solos calculus proves restriction solos calculus still reasonable expressive power moreover gives alternative proof respect work presented existence translation finitary differential interaction nets first part paper devoted introduction required calculi solos calculus section solo diagrams section differential interaction nets section almost section elaborate material presented define simple static translation solo diagrams graphical syntax solos calculus differential interaction nets discuss problems arising dynamic level give sufficient condition solo diagrams translation differential interaction nets bisimulation occurrence name never unified another occurrence name main technical contribution paper comes last part section since property redexes solos term course preserved reduction solos find clever property define acyclic solos calculus typing system assigning protocols acyclicity conditions prove restriction well behaved respect reduction solos calculus show translation always acyclic solos term showing expressiveness system finally prove sufficient condition introduced section fulfilled solo diagrams corresponding acyclic solos terms showing obtain bisimulation respect differential interaction nets solos calculus section recall definition solos calculus going use also recall translation solos given two basic components communication output input prefixes occurrences bound communication occur case obtained substituting moreover prefix constructions induce sequential behaviour continuation resp ehrhard laurent interact agents output resp input triggered fusion calculus provides generalization communication modelized unification makes sending receiving perfectly symmetric thus distinction purely formal interaction constrained occur two dual entities symmetry broken one translates fusion terms unification restrained image translation becomes substitution properties binding restriction notice sequentiality aspect prefixing remains based unification mechanism provided fusion calculus solos calculus free explicit sequentiality constructions fusion prefixes restricted dummy continuations inactive process shown dyadic solos expressive whole fusion calculus comments chosen calculi since goal focus prefixing sequentiality deal calculi without replications recursive definitions without without sums see conclusion additional comments want spend time deal arbitrary arities calculi consider consider monadic three reasons makes presentation simpler lead loss expressiveness finally polyadic case already considered consequence see translation section led consider triadic solos calculus names arity exactly triadic solo diagrams multiedges arity exactly general case arbitrary arities could easily obtained introducing appropriate sortings various calculi terms monadic finitary given bind occurrences structural congruence least congruence containing reduction semantics given acyclic solos differential interaction nets solos calculus introduced goal solos calculus prove expressiveness calculus without prefix construction terms triadic solos calculus given binds occurrences structural congruence least congruence containing equivalence allows present terms solos calculus canonical forms either process bunch scope constructions followed solos parallel reduction semantics solos calculus given stands general unifier exactly names modified satisfy particular equivalence class names induced unification one name free alternative equivalent definition reduction semantics given together additional explanations example uxy zww zuy possible reduction uxy zww induces identifications thus two equivalence classes first one free thus possibility map second one elements bound choose one example choices would lead structurally congruent results consider unifier containing identity names obtain reduct uuy solos authors give different translations fusion calculus solos going focus one one introduce matching translation canonical embedding fusion calculus obtain translation solos present ehrhard laurent translated zzv uwy xvv uwy shown encoding adequate respect weak barbed congruence implies translated solos term structurally congruent uwy xwy yvv uwy zvv zwy applying reduction obtain particular following reducts structural congruence zvv zwy xwy yvv xwy yvv solo diagrams make relation differential interaction nets simpler graphical syntax associated solos calculus solo diagrams triadic solo diagram given finite set nodes finite multiset ternary multiedges directed edges list three nodes source one node target node tagged either free bound multiedge tagged either input output node must source target multiedge reduction reduction solo diagrams given choosing two multiedges opposite polarities one input one output target call dual multiedges respective sources acyclic solos differential interaction nets reduces figure reduction solo diagrams identifying two nodes two nodes two nodes reduction allowed occur identify two free nodes moreover free node identified bound node obtained node free two bound nodes identified obtained node bound chosen multiedges erased nodes anymore source target multiedge also removed applied free bound nodes reduction step number multiedges decreases two number nodes decreases remains graphical representation draw free nodes white dots bound nodes black dots output edges outgoing arrow input edges ingoing arrow example given figure relation solos term solos calculus easily translated solo diagram free names translated free nodes bound names bound nodes solos multiedges term translated empty graph edge solo translated graph free nodes corresponding elements one input multiedge source target node corresponding name node corresponding name solo translated way output multiedge parallel composition obtained graph union union sets nodes nodes corresponding name identified sum multisets multiedges restriction corresponds turning free node corresponding name bound node shown two solos terms structurally congruent corresponding solo diagrams isomorphic moreover reduction solo diagrams reflects faithfully reduction solos calculus two solo diagrams figure correspond uxy zww zuy respectively uuy examples used section labeled transition system follow methodology using labels distinguish actions process term syntax labels put prefixes solos graphical syntax solo diagrams put labels multiedges since multiedges correspond solos ehrhard laurent fix countable set labels used labeled transition systems solo diagram labeled multiedges equipped different labels belonging definition transition system objects labeled transition system labeled solo diagrams transitions labeled pairs distinct elements let two labeled solo diagrams obtained applying reduction step input multiedge labeled output multiedge labeled labels remaining multiedges must corresponding ones quite different usually done defining transition systems process algebra standard approach aims analyzing possible interactions process environment bisimulation used show two processes external behaviour goal use bisimulation compare internal behaviours solo diagrams differential interaction nets want illustrate differential interaction nets sufficiently expressive simulating concurrency mobility requires transition systems use bisimulation carry precise informations reductions processes particular taking part interactions solo diagrams identifications order compare solo diagrams differential interaction nets useful decompose reduction solo diagrams introducing notion solo diagrams identifications definition solo diagram identifications solo diagrams identifications solo diagrams equipped finite set undirected edges usual binary edges multiedges connects nodes solo diagram edges called identification edges decompose reduction solo diagrams presented using identification edges reduction step solo diagram defined follows choose two dual multiedges opposite polarities target respective sources build solo diagram identifications obtained erasing introducing three identification edges contract graph repeatedly choosing identification edge identifying two one nodes connects least one bound reduction succeeds valid reduction solo diagrams reach solo diagram remaining identification edge means particular step introduce identification edges two free names graphical representation draw identification edges dashed edges refine reduction example figure means solo diagrams identifications results step one application step figure acyclic solos differential interaction nets figure reduction step solo diagrams identifications differential interaction nets general formalism interaction nets recall general graphical syntax interaction nets introduced see also details statics interaction nets first define general typed syntax interaction nets assume given set symbols arity integer typing rule associated symbol typing rule list types arity associated symbol types formulae system linear logic particular type also type interaction net made cells cell associated exactly one symbol therefore arity typing rule cell one principal port auxiliary ports interaction net also finite set free ports ports free ports ports associated cells pairwise distinct set wires given wiring set pairwise disjoint sets ports cardinality ordinary wires union wires must equal set ports interaction net words port interaction net free associated cell connected exactly one port free associated cell wire wire connects exactly two ports ports shared free ports interaction net associated cell oriented wire interaction net ordered pair wire interaction net type associated oriented wire way associated associated last typing rules cells must respected sense cell arity whose ports typing rule denoting ports interaction net uniquely defined fact sets wires oriented wires types respectively free ports interaction net constitute interface free port associated type unique oriented wire whose endpoint type interface interaction net typical example typed interaction net precise one specify number loops net play role sequel ehrhard laurent cells symbols respective types interface cells represented triangles principal port located one angles ports opposite edge often draw black dot locate auxiliary port number dynamics interaction nets lafont traditional interaction nets reduction rule associates pair cells connected principal ports interaction net interface net moreover one rule given pair cells application reduction rule substitutes inside interaction net pair cells connected principal ports associated reduct ideas strongly related key steps cut elimination logics two rules interact principal formulas extensions interaction nets since interaction nets strong confluence properties extensions required take concurrency account possible way extending interaction nets concurrent behaviours modify statics introducing cells many principal ports breaks simple logical interpretations contradicting usual property logical rule sequent calculus introduces one principal formula differential interaction nets stick traditional statics syntax act dynamics side possible extension dynamics towards behaviours allow choice different possible reduction rules given pair cells make choice proposal lafont one many possible rules one rule consider one reduction rule pair cells extend interaction nets formal sums interaction nets allows represent one reduction finite set choices reduct interaction two cells proper sum interaction nets presentation cells define differential interaction nets interaction nets defined first present specific cells untyped seen typed equipped recursive equation following idea danos regnier defined notion untyped based single type symbol type outputs subject following recursive equation nets typed using typing system set tensor connective used premises dually par therefore types actually need typing nets present setting ten symbols par arity bottom arity tensor arity one arity dereliction arity weakening arity contraction arity codereliction arity coweakening arity cocontraction arity present various cell symbols typing rules pictorial way multiplicative cells par tensor cells well nullary versions bottom one follows acyclic solos differential interaction nets exponential cells typed according strictly polarized discipline first cells called dereliction weakening contraction bang cells called codereliction coweakening cocontraction differential interaction nets define differential interaction nets cells simple differential interaction net typed interaction net uses multiplicative exponential cells introduced differential interaction net finite formal sum simple differential interaction nets interface interface considered interface particular case differential interaction net empty sum differential interaction net given together interface net interface labeled differential interaction nets introduce labeled differential interaction nets differential interaction nets particular cells equipped labels labeled transition system differential interaction nets defined using labels section let countable set labels obtained adding distinguished element understood absence label introduced section labeled simple differential interaction net simple differential interaction net dereliction codereliction cells equipped labels belonging moreover two labels appearing labeled simple differential interaction net either pictures labels dereliction codereliction cells indicated label different label dereliction cell drawn without label differential interaction nets consider paper labeled sequel since confusion kinds interaction nets possible shall use net labeled differential interaction net simple net labeled simple differential interaction net reduction rules denote collection simple nets ranged letters without subscripts superscripts collection nets finite sums simple nets interface including empty sum ranged letters without subscripts superscripts consider subset identified sum made exactly one copy reduction rule subset consisting pairs simple net made two cells connected principal ports net interface actually reduction rules transform simple nets ones set finite infinite set easily extended relation arbitrary simple nets nets arbitrary nets nets ehrhard laurent note subnet simple net simple net resulting replacement relation defined simple last relation nets nets transitive closure reflexive defining reduction give reduction rules differential interaction nets multiplicative reduction first two rules concern interaction two multiplicative cells arity stands empty simple net simple net containing cell port thus wire confused net empty sum simple net next two rules concern interaction binary nullary multiplicative cell communication reduction let following reductions labels disappear reduction step reduction let following reductions according definition reduction arbitrary simple nets given application one last two steps simple net erases whole simple net gives net everywhere else paper means net appropriate interface two steps make labels disappear acyclic solos differential interaction nets structural reduction use symmetric transitive closure remark one check provided reduction rules redexes compatible typing system simple net made two cells connected principal ports reduction rule whose left member rule unique choice set labels choice influence right member rule confluence theorem let let union reduction relations relation confluent proof essentially trivial since rewriting relation critical pair see given consider particular following set deterministic denote symmetric transitive closure relation observe nets simple also simple reduction rules defined depend set labels dependence clearly monotone sense relation becomes larger set labels increases reduction nets normalizing mainly last structural reduction rule additional comments found transition system simple nets let distinct elements call redex communication redex whose codereliction cell labeled whose dereliction cell labeled definition transition system define labeled transition system whose objects simple nets transitions labeled pairs distinct elements let simple nets following holds simple net contains redex becomes one reduces redex remark steps allowed reduction involve codereliction dereliction labeled respectively communication steps involve derelictions coderelictions solos calculus solos communicate one step parallel composition single step becomes ehrhard laurent sequence many elementary steps restriction allows avoid considering steps nothing communication interested net contains possible communications corresponds branches choices paper compare transition systems consider strong bisimulations given following definition definition bisimulation given two labeled transition systems set labels relation bisimulation exists exists useful particular case lemma relation bisimulation toolbox process calculi interpretation introduce families simple nets built using previously introduced basic cells used basic modules interpreting processes nets communication areas considered compound cells reduction behave way cells interaction nets one could directly defined compound cells primitive constructions representation processes however prefer stress possibility implementing compound objects basic ones coming logically founded theory differential linear logic order emphasize possible application linear logic tools image encoding communication areas considered cells setting since would require one principal port interaction nets many principal ports studied particular representing concurrent behaviours theory quite different one principal port case compound cells generalized contraction cocontraction generalized contraction cell simple net one free ports connected auxiliary port cells called principal port free ports finitely many connected principal ports cells generalized cell called auxiliary ports generalized contraction cells inductively defined wire type generalized contraction cell select port type principal port generalized cell weakening cell gives generalized contraction cell connecting wire principal port free port obtained net principal port generalized cell given two generalized contraction cells connecting two auxiliary ports additional contraction cell principal ports generalized cells connecting principal port new contraction cell free port one obtains generalized contraction cell free port connected principal port added contraction cell principal port generalized cell acyclic solos differential interaction nets figure compound cell figure input output compound cells figure area order generalized cocontraction cells defined dually use graphical notations generalized contraction cells ordinary contraction cells superscript symbols avoid confusions observe infinitely many generalized contraction cells given arity even one could define choice one canonical contraction cell arity pluging two canonical contraction cells together would always give canonical one define choice cells let integer define compound cell figure decorated label dereliction cell mentioned section put decoration label number tensor cells compound cell equal particular derelictiontensor cell contains exactly two cells one cell dereliction cell define dually compound cell prefix cells define compound cells play main role interpretation solos thanks defined cells oriented wires nets shall define type therefore adopt following graphical convention wires orientation corresponding type let integer input cell output cell defined figure pairs auxiliary ports label prefix cells one carried outermost compound cell compound cells required unlabeled labeled ehrhard laurent figure communication areas order figure aggregation communication areas let define family nets free ports called communication areas order shall draw using rectangles beveled angles figure shows picture communication area order communication area order made pairs generalized cocontraction contraction cells wire auxiliary port auxiliary port wire auxiliary port auxiliary port note principal ports call pair associated ports communication area communication area order empty net communication areas order structures shown figure useful reductions one nice properties communication areas one connects two areas pair wires one gets another communication area lemma aggregation communication areas let communication area order communication area order let pair associated ports pair associated ports let simple net communication obtained connecting area order see figure proof prove particular case communication areas built contraction cells generalized ones connecting applying two structural reduction steps obtain communication area order see figure let simple net port say forwarded free port one two shapes given figure allows describe interaction derelictions coderelictions communication areas derelictions coderelictions meet connected common communication area lemma communication forwarding derelictions coderelictions communication areas let integer let let consider simple net acyclic solos differential interaction nets figure aggregation communication areas order figure port forwarding figure dereliction codereliction communicating communication area obtained connecting principal ports codereliction labeled weakening pair associated ports communication area order connecting principal ports dereliction labeled coweakening another pair associated ports communication area applied leads sum simple nets one contains codereliction dereliction connected principal ports together communication area order summands principal ports codereliction dereliction forwarded see figure proof consider particular case communication area built contraction cells generalized ones reduction pictured figure first apply possible structural reduction steps focus codereliction labeled reduction associated redex gives sum two simple nets one principal port codereliction forwarded apply simple net reduction step involving dereliction labeled finally obtain sum two simple nets first one principal ports dereliction codereliction forwarded second one connected remaining part net communication area order generalization communication areas order easy obtain case decomposing communication area two communication areas one order one order lemma theorem help conclude consider interaction two prefix cells ehrhard laurent figure forwarding dereliction codereliction figure prefix reduction lemma reduction prefixes let connect output prefix labeled input prefix labeled principal ports obtain simple net reduces net reduces simple wires figure proof key point check simple net obtained connecting compound cell cell principal ports reduces min wires weakening cells coweakening cells solo diagrams differential interaction nets translation relying toolbox described define translation labeled solo diagrams identifications labeled simple differential interaction nets node appears times source multiedge times target multiedge times member identification edge translated communication area order example node target one input multiedge one output multiedge thus source one multiedge thus member one identification edge thus input multiedge sources target translated input cell principal port auxiliary ports pair connected pair associated ports communication area corresponding acyclic solos differential interaction nets figure translation second solo diagram figure connected port weakening cell connected pair associated ports communication area corresponding label prefix cell label multiedge output multiedge sources target translated output cell principal port auxiliary ports pair connected pair associated ports communication area corresponding connected port coweakening cell connected pair associated ports communication area corresponding label prefix cell label multiedge identification edge connecting translated pair wires connecting pair associated ports communication area corresponding pair associated ports communication area corresponding since communication areas order uniquely defined generalized contraction cocontraction cells given arity unique either translation solo diagrams differential interaction nets fact relation denote solo diagram simple differential interaction net remark let solo diagram without identification edges differential interaction net redex except redexes involving weakening coweakening cells indeed redexes given labeled codereliction cell whose label coweakening cell facing weakening cell contraction cell labeled dereliction cell dually dereliction cell weakening cell facing coweakening differential interaction net associated second solo diagram identifications figure given figure ehrhard laurent figure proof lemma bisimulation goal establish bisimulation labeled transition systems section section unfortunately translation provide bisimulation going show problems restrict get bisimulation mismatch crucial point comes step reduction solo diagrams contraction identification edges solo diagrams corresponds aggregation communication areas given following lemma lemma solo diagrams identifications obtained contracting identification edge step connecting nodes obtained aggregating communication areas corresponding proof see figure obtained replacing two distinct nodes unique node communication area corresponding communication area corresponding since connected identification edge connected pair wires aggregating see lemma obtain communication area exactly corresponds hypothesis crucial since two communication areas connected together pair wires reduce one communication area lemma except thus able encode reduction solos calculus ensure never contract identification edge connecting node typical example problem given xyz abc shown figure solo diagrams loop created first step eventually disappears remains differential interaction nets consequence transitions last differential interaction net transition differential interaction net translation solo diagram restriction since goal make bisimulation result true going constrain syntax solo diagrams way hypotheses lemma always valid want restrict reduction step case edge connecting node equivalent asking gid containing identification edges acyclic definition call acyclic solos differential interaction nets figure reductions solo diagrams differential interaction nets corresponding term xyz abc ehrhard laurent acyclic reduction step reduction step associated two dual multiedges gid acyclic another problem comes constraint freeness names contraction identification edges solo diagrams translation differential interaction nets forgetting fact nodes free others bound consequence reduction might happen differential interaction net side without possible solo diagram side example given term xyz avoid situation introduce notion acyclic redex definition acyclic redex solo diagram pair dual multiedges acyclic redex redex meaning induced freeness conditions satisfied induced reduction step acyclic reduction step definition sac labeled transition system biggest labeled transition system object sac object transition transition objects transitions lost object sac transitions starting belong pairs dual multiedges objects sac acyclic redexes means object belongs sac soon none paths starting allows reach object redex decidable property since finitely many paths definition acyclic solo diagram solo diagram object sac called acyclic solo diagram proposition sac simple nets proof hypothesis communication area two dual prefixes labeled connected apply forwarding derelictions coderelictions communication area lemma gives sum simple nets containing simple net redex summands forwarded following figure let simple net obtained reducing redex simple net obtained applying prefix reduction starting reduction see lemma easy see multiedges labels lemma step acyclic contraction identification edge corresponds aggregation two associated communication areas thus lemma diving let solo diagram without identification containing redex codereliction labeled dereliction labeled connected communication area redex generated forwarding lemma proof according remark redexes involving codereliction dereliction weakening coweakening acyclic solos differential interaction nets figure simulation solos reduction step confluence theorem assume reductions involving codereliction coming involving involving weakeaning coweakening cells arrive look port connected principal port codereliction auxiliary port prefix cell reduction reduction change situation prefix cell labeled changing situation means erasing auxiliary port generalized cocontraction cell wire reduction principal port prefix cell labeled change situation principal port prefix cell labeled immediately result communication area order part connecting simple wire principal port generalized contraction cell arity reductions involving eventually erase never reach redex contradicts hypotheses principal port generalized contraction cell arity order reach redex reductions involving must start destroying generalized contraction order eventually reach redex must simple net corresponding case case consider last two configurations cases followed see figure see figure note infinitely many consecutive cases since induced reduction makes size net decrease cases reach case connected communication area result cases give forwarding cases reach case turn attention reduction steps involving look port connected principal port dereliction ehrhard laurent figure proof lemma sequence figure proof lemma sequence auxiliary port prefix cell reduction reduction change situation prefix cell labeled changing situation means erasing auxiliary port generalized contraction cell wire reduction could reductions involving coweakening would eventually erase reach redex contradicts hypotheses principal port prefix cell labeled reach redex contradicts hypotheses principal port generalized cocontraction cell arity reductions involving eventually erase never reach redex contradicts hypotheses principal port generalized cocontraction cell arity order reach redex reductions involving must start destroying generalized cocontraction order reach redex must escape four cases means case reach simplest case facing note infinitely many consecutive cases since size net decreases conclude order reach redex must sequence configurations see figure see figure face sequence must connected communication area reduction sequence corresponds forwarding proposition sac solo diagram simple net sac proof notations coincide figure definition exist simple net containing redex net obtained reducing redex lemma contains two dual multiedges connected node labeled let simple net obtained applying prefix reduction starting reduction see lemma first show lemma way redex generated acyclic solos differential interaction nets forwarding codereliction dereliction communication area order connected gives prefix redex communication area order lemma firing prefix redex generate pairs wires connecting communication areas exactly correspond identification edges let solo diagram without identifications obtained firing redex possible since acyclic redex iteratively apply lemma starting reach possible since sac let simple net obtained way corresponding get bisimulation result define composition exists simple net theorem bisimulation bisimulation sac proof sac proposition simple nets lemma exists simple net consequence thus conversely lemma exists simple net proposition solo diagram simple net sac consequence thus rest paper devoted definition solos calculus whose translation solo diagrams lives inside sac acyclic solos calculus following approach previous section analyzing translation solos calculus define solos calculus called acyclic solos calculus key informal properties calculus expressiveness contains image translation solos sections stability respect reduction solos since stable reduction propositions acyclicity solo diagram associated term acyclic solos calculus acyclic solo diagram thus bisimulation differential interaction nets holds section definition acyclic solos calculus given two steps first means typing system assigning polarities occurrences names second structural constraints typed terms intuitively restrict terms structure ehrhard laurent reduction occurring roots trees forest fact deal general structures forests abstraction forest structure induced sequentiality translation solos polarities side represent asymmetry solos setting types consider system two types use denoting either typed term term scopes decorated types typing judgment shape contains typing declarations associating types names typed term typing rules zyy either intuitive way understanding types given two mutually recursive definitions generally starting mutually recursive definitions finite set types one could derive typing systems kind would also satisfy subject reduction property example main purpose typing system able associate communication protocol object occurrence communication protocols send receive add communication protocols decoration definitions object occurrence typed term context decorated communication protocol input solo whose subject type decoration given according components type output solo given dual way formally directly enrich typing derivations way assign communication protocols object occurrences names typed term examples given image translation section acyclic solos differential interaction nets intuition behind comes difference mechanism solos calculus substitution entails flow information given interaction output prefix input prefix receiving names substituted sending names unification fusion solos establishes perfect symmetry agents trying communicate communication protocols used trace flow information coming translation solos even communication symmetric inside solos calculus reasonable term decorated communication protocols understand unification substituted consequence constraints given definition preservation reduction first check elementary properties typing system respect reduction lemma free names free names appear lemma weakening lemma substitution proof three lemmas proved simple inductions typing derivations proposition subject reduction reduces proof simply consider case definition given section interesting one assume entails thus lemma thus finally conversely thus lemma interesting case contains exactly names hypothesis deduce lemma typability translation prove typability translation see section definition translation free names finite set define environment environment proposition typability free names finite set ehrhard laurent proof first show process typable translations prefixes typable see figure cases immediate finally get typability sum induced communication protocols acyclicity relying decoration communication protocols induced typing system able define restriction solos calculus given typed term first define properties relations names solos occurring solo write subj subject root object occurrence object occurrence communication protocol subj subj subj one output input denote transitive closure reflexive transitive closure definition acyclic solos calculus acyclic solos terms typed terms satisfy following five properties name one implies roots implies implies input free name roccurrence acyclic solos differential interaction nets figure typing derivations translations prefixes ehrhard laurent acyclic solos calculus given restricting terms acyclic solos terms structural congruence reduction relation induced solos calculus make remarks state immediate consequences definition let consider following definition communication protocols occurrences names occurrence object occurrence input prefix communication protocol otherwise assuming names appearing made different possible remark name one free name entails reduction occurs roots case induces forest ordering means bigger respect forest ordering says name solo input place breaking symmetry solos calculus output solos input solos allows rule processes like would lead twice identification almost identifying would break acyclicity associated solo diagram ensures identification edges always contractible freeness problem preservation reduction want prove reducts acyclic solos term reduction solos calculus acyclic solos terms proposition consider reduction interaction solos definition reduction see section substitution solo shape solo say residue solos residue solos unique residue section order simplify notations reduces use name solo unique residue exists make difference relations different processes use notation relation process following three lemmas hold assuming conditions lemma typed term solos calculus satisfying reduces substituted reduction proof let solos destroyed reduction name involved reduction taking part unification occurring solo different either thus one root contradicting lemma typed term solos calculus satisfying reduces solo whose subject modified reduction subject subject residue residue root root acyclic solos differential interaction nets proof let subject subject appears either root thus lemma symmetrically since belongs thus root contradicting root root since subject modified means also substituted thus lemma lemma typed term solos calculus satisfying reduces reduction introduces object occurrence name proof reduction put must occur assume occurs lemma since would entail root contradicting turn preservation result proposition acyclicity preservation acyclic solos term reduces acyclic solos term proof first prove preservation conditions independently conditions preservation condition given lemma move condition note one root none since subj subj assume subject neither modified reduction since subject modified lemma contradicts condition one subjects modified root lemma concerning condition assume lemma lemma path relation broken reduction entails solo path got subject modified lemma impossible since roots assume hold condition preserved output solo contains name lemmas also case condition preserved lemma shows acyclic solos calculus well behaved respect reduction solos calculus acyclicity translation order prove expressiveness acyclic solos calculus check contains image translation ehrhard laurent relation following translated arrow represents relation theorem acyclicity acyclic solos term proof proposition typable let mention additional facts easily checked induction definition free names free names free names iii subject thus subject root forest one reasons name acyclic solos points conditions easily verified first check five conditions induction using facts iii entail free name easy induction fact shows entails subj subj conclude fact iii induction using facts iii immediate induction consequence facts iii easy check case back acyclic solo diagrams last property want show solo diagram associated term acyclic solos calculus definition acyclic solo diagram definition let solo diagram associated term acyclic solos calculus show pair dual edges acyclic redex lemma enough show belongs sac theorem lemma let solo diagram associated acyclic solos term let two dual multiedges pair multiedges defines acyclic redex definition want show graph gid identification edges introduced section acyclic generate freeness problem contraction identification edges acyclic solos differential interaction nets construction edges gid connecting two nodes correspond occurrences names term going unified definition acyclic solos calculus one occurrences one roccurrence consider directed graph obtained orienting edges gid towards first prove following lemma directed graphs lemma let finite directed graph underlying graph connected root node without incoming edge node one incoming edge acyclic proof function maps edge target injective function hypotheses image set nodes roots consequence number edges thus less equal number vertices thus minus one since connected conclude acyclic prove lemma proof prove connected components gid equipped orientation satisfy hypotheses lemma thus gid acyclic let assume input multiedge redex since solos solos term correspondence multiedges corresponding solo diagram use notations introduced beginning section multiedges well solos following properties occurrences another moreover thus root contradicting occurrence neither finally iii neither otherwise apply consider connected component gid underlying graph appropriate finite connected according condition node one incoming edge show root name connected component first show connected name without start corresponding occurrence name done also corresponding occurrence thus without iii since connected component contains either done iii finally property roots free nodes according labeling nodes property preserved contraction identification edges thanks moreover property entails one extremity identification edge free thus freeness problem arises contraction identification edges ehrhard laurent theorem acyclic diagrams acyclic terms solo diagram associated acyclic solos term definition acyclic solo diagram definition proof prove range translation acyclic solos calculus solo diagrams satisfies conditions definition lemma know translation contains acyclic redexes translation acyclic solos term reduces exists term translates reduces thus acyclic solos term proposition conclusion spirit correspondence logical device concurrent one shown stress strong connection differential interaction nets solos calculus links allow one share methods two worlds possibility giving concurrent interpretations procedure logical foundations process calculi applying tools concurrency theory behavioural equivalences formal proofs conversely denotational semantics linear logic concurrent languages let discuss constraints restrictions face finitary calculi considered finitary calculi without replications recursive definitions nevertheless exponential boxes natural device linear logic compatible differential interaction nets representing replicable processes shown represent boxes restricted expressive form replication restriction related difficulties controling reduction auxiliary ports exponential boxes exploiting kind restriction language solos possible represent restricted form replication translation logical correctness linear logic provide graphical syntax representing proofs sequent calculus however requires impose correctness criterion order characterize exactly could sequentialized sequent calculus proof correctness criterion differential interaction nets relating sequent calculus differential linear logic however differential interaction nets obtained translation satisfy correctness criterion general comes particular unconstrained use communication areas logical tools care logical correctness directly applied setting example relational model denotational model differential interaction nets even satisfy correctness contrary many important results logic hold logically correct case would important understand translation interacts correctness expressive would concurrent calculus whose translation differential interaction nets satisfies correctness criterion particular kind communications could represented communication areas logically correct setting acyclic solos differential interaction nets acyclicity trying build strong bridge differential interaction nets solo diagrams able define suitable restriction solos calculus used intermediary step differential interaction nets led new proof bisimulation result presented technical choices design acyclic solos calculus completely determined mentioned goal nevertheless would interesting study acyclic solos computational behaviour many ways similar happens name passing unification style setting example would interesting see contain specific communication primitives somehow behaviour acyclic solos term always mimics behaviour another approach would extend translation relation take cycles figure account requires understand precise impact cycles behaviour differential interaction nets way seems possible extend bisimulation result whole solos calculus following link provided present work mentioned points addressed strengthen idea underlying correspondence however possibilities already open relational semantics geometry interaction processes acknowledgement would like thank cosimo laneve helpful discussions solos also thank anonymous referees useful comments references samson abramsky computational interpretations linear logic theoretical computer science vladimir alexiev interaction nets thesis university alberta emmanuel beffara maurel concurrent nets study prefixing process calculi theoretical computer science may boudol asynchrony research report inria gianluigi bellin philip scott linear logic theoretical computer science thomas ehrhard sequence spaces linear logic mathematical structures computer science thomas ehrhard olivier laurent interpreting finitary differential interaction nets information computation june thomas ehrhard laurent regnier differential interaction nets theoretical computer science claudia faggian maurel ludics nets game model concurrent interaction proceedings annual ieee symposium logic computer science pages ieee computer society girard parallel syntax aldo ursini paolo agliano editors logic algebra volume lecture notes pure applied mathematics pages new york marcel dekker kohei honda olivier laurent exact correspondence typed polarised theoretical computer science may ehrhard laurent kohei honda mario tokoro object calculus asynchronous communication proceedings ecoop volume lecture notes computer science pages springerverlag kohei honda nobuko yoshida combinatory representation mobile processes popl proceedings acm symposium principles programming languages pages new york usa acm ole jensen robin milner bigraphs mobile processes revised technical report cambridge university computer laboratory lionel khalil des interaction avec amb agent mccarthy applications doctorat ecole normale paris june yves lafont proof nets interaction nets girard yves lafont laurent regnier editors advances linear logic volume london mathematical society lecture note series pages cambridge university press cosimo laneve joachim parrow victor solo diagrams proceedings conference theoretical aspects computer science tacs number lecture notes computer science pages cosimo laneve victor solos concert mathematical structures computer science damiano mazza multiport interaction nets concurrency proceedings concur number lecture notes computer science pages damiano mazza interaction nets semantics concurrent extensions doctorat degli studi roma tre robin milner graphical form esop proceedings european symposium programming pages london robin milner joachim parrow david walker calculus mobile processes information computation david park concurrency automata infinite sequences peter deussen editor giconference theoretical computer science volume lecture notes computer science pages march joachim parrow victor fusion calculus expressiveness symmetry mobile processes proceedings thirteenth annual symposium logic computer science pages indianapolis june ieee ieee computer society press laurent regnier doctorat paris vii work licensed creative commons license view copy license visit http send letter creative commons second suite san francisco usa eisenacher strasse berlin germany
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quantum algorithms graph connectivity formula evaluation stacey shelby qusoft cwi amsterdam netherlands middlebury college middlebury usa aug august give new upper bound quantum query complexity deciding certain classes planar graphs show bound sometimes exponentially better previous results show boolean formula evaluation reduces deciding connectivity class graphs applying algorithm boolean formula evaluation problems match bound quantum query complexity evaluating formulas variables give quadratic classical query complexity certain class promise boolean formulas show approach yield superpolynomial separations results indicate approach may right way looking quantum algorithms formula evaluation introduction deciding whether two points connected network problem significant practical importance work argue problem also important quantum algorithmic primitive heiligman mhalla designed quantum algorithm deciding stconnectivity requires queries adjacency matrix graph vertex set belovs reichardt later discovered especially elegant quantum algorithm problem requires logarithmic space belovs reichardt algorithm improves query complexity algorithm connecting path promised short exists belovs reichardt algorithm already adapted used subroutine deciding graph problems detecting certain subgraphs deciding whether graph forest deciding whether graph bipartite work modify span program algorithm used inheriting space time efficiency restrict deciding class planar graphs effective resistances set graphs question planar duals small find quantum algorithm requires far fewer queries suggested analysis fact obtain polynomial constant improvement query complexity classes graphs stacey jeffery jeffery completed parts work institute quantum information matter iqim caltech shelby kimmel completed parts work joint center quantum information computer science quics university maryland accepted quantum click title verify addition improving understanding quantum query complexity stconnectivity problems show boolean formula evaluation reduces extremely naturally problems kind improved analysis holds therefore finding good algorithms lead good algorithms boolean formula evaluation one might expect reduction would produce good algorithms find reduction gives optimal performance certain classes boolean formulas boolean formula evaluation fundamental class problems implications algorithms complexity theory quantum evaluating formulas like spurred interest better understanding performance quantum algorithms boolean formulas research culminated development span program algorithms optimal quantum query complexity problem using span program algorithms shown queries sufficient formula inputs classically query complexity evaluating query complexity evaluating arbitrary formulas simple bounds quantum query complexity total formula evaluation problems promise versions still fully understood kimmel showed certain promise version called trees quantum query complexity zhan kimmel hassidim showed classical query complexity log logkn giving superpolynomial quantum range values general treatment promises inputs give superpolynomial query found since analysis shows graphs small effective resistance decided efficiently turn means boolean formula evaluation problems promise inputs correspond low resistance graphs also evaluated efficiently result gives new insight structure quantum promise boolean formulas contributions summarize main results paper follows improved quantum query algorithm deciding input subgraph graph additional planar analysis involves effective resistance original graph planar dual find families graphs analysis gives exponential polynomial improvements respectively previous quantum analysis algorithm boolean formula evaluation via reduction using reduction provide simple proof fact boolean formulas input variables evaluated using queries show quadratic superpolynomial using reduction certain classes promise boolean formula evaluation problems open problems would like better bounds classical query complexity evaluating problems would provide new approach finding separations classical quantum query complexity additionally reduction accepted quantum click title verify boolean formula evaluation could helpful design new classical algorithms formulas another open problem concerns span programs general view span programs solving problems could useful understanding span programs since analysis span programs straightforward see appendix section appendix important class span programs arising learning graph framework provides means designing quantum algorithms much simpler intuitive designing general span program limitation framework respect whereas learning graph algorithm designed detect framework capable giving optimal quantum query algorithms decision problem would likely treat symmetrically analysis equal footing duality problems could give insights extend learning graph framework powerful framework without losing intuition relative simplicity organization section provides background information section describe improved analysis span program algorithm subgraphs graphs planar section show every formula evaluation problem equivalent problem section apply results promise able prove significant separation using approach also section use ideas create improved algorithm playing game associated preliminaries graph theory undirected weighted multigraph let denote vertices edges respectively work consider undirected multigraphs henceforth often refer graphs refer edge multigraph specify endpoints well label edge written although label assumed uniquely specify edge include endpoints convenience let denote set directed edges planar graph implicit planar embedding let denote faces call infinite face planar graph external face graph connected vertices imagine fluid flowing traveling graph along edges finally exits fluid spread along number possible linear combination called precisely definition unit let undirected weighted graph connected unit function accepted quantum click title verify definition unit flow energy given unit graph unit flow energy definition effective resistance let graph connected effective resistance runs unit connected intuitively characterizes connected vertices shorter paths connecting smaller effective resistance effective resistance many applications random walk equal commute time expected time random walker starting takes reach return models electrical network edge unit resistor potential difference applied corresponds resistance network determines ratio current voltage circuit see extend connections considering weighted edges network consists graph combined positive weight function definition effective resistance weights let network effective resistance runs unit random walk network models reversible markov chain walker vertex traverses edge probability proportional commute time models electrical network edge represents resistor resistance corresponds resistance network single edge weight resistance calculating effective resistance use rule edges series path generally graphs connected series resistances add edges parallel generally graphs connected parallel follow rule conductances parallel add conductance graph given one resistance conductance edge equal weight edge precisely easy verify following claim let two networks nodes create new graph identifying nodes nodes connecting graphs parallel define however create new graph identifying node node relabeling node connecting graphs series define accepted quantum click title verify bit foreshadowing let take values representing false representing true clearly computes function since also computes function use definition given graph connected function whenever words defines subset edge one endpoint one endpoint witness different components path exists finally consider dual graphs definition dual graph let planar graph implicit embedding dual graph defined follows every face vertex two vertices adjacent corresponding faces share edge call edge two vertices dual edge convention always label faces either side edge span programs quantum query algorithms span programs first introduced study quantum algorithms reichardt since proven immensely important designing quantum algorithms query model definition span program span program made inner product spaces vector space iii target vector linear operator every string associate subspace operator orthogonal projector onto definition positive negative witness let span program let string call positive witness define positive witness size min exists positive witness otherwise let denote set linear maps call linear map negative witness define negative witness size min exists negative witness otherwise finite say positive wrt finite say negative let denote set positive inputs set negative inputs way span program defines partition function say decides use design quantum query algorithm decides given access input via queries form accepted quantum click title verify theorem fix let span program decides let bounded error quantum algorithm decides quantum query complexity boolean formulas boolean formula expressed rooted tree leaves uniquely labeled variables internal nodes labeled gates set specifically node degree must labeled whereas higher degree nodes labeled gate defined number children depth boolean formula largest distance root leaf define formula also called monotone formula boolean formula every internal node labeled restricting formulas lose much generality since formula equivalent formula distance one leaf gates affect query complexity formula moreover although consider formulas techniques applied general formulas single variable may label multiple leaves since equivalent larger formula promise input hereafter refer formula mean formula slight abuse notation times denote boolean variable times denote bit instantiating variable instantiation variables labeling leaves formula value input defined follows depth depth greater express recursively terms subformulas root labeled root labeled former case define latter case define family formulas variables gives rise evaluation problem input string output say mean composed evaluates important formula evaluation problem evaluation full binary tree arbitrary depth every internal node two children every leaf node distance root internal node labeled even distance leaves odd distance leaves use nandd denote depth nandd sometimes defined boolean formula composed depth instead think formula alternating even two characterizations identical instance nandd binary string example formula depth denotes identity function nandd instance associated game rooted binary tree represents nandd leaves take values figure game starts root node call current node round game long current node leaf current node even respectively odd distance leaves player resp player chooses one current node children become current node current node leaf leaf value player wins leaf value player wins sequence accepted quantum click title verify moves two players determines path root leaf simple inductive argument shows exists strategy player always win matter strategy employs exists strategy player always win say input value value improved analysis algorithm section give improved bound runtime quantum algorithm subgraphs planar let problem parameterized family multigraphs takes input string input defines subgraph including edge exists path connecting write quantum algorithm accesses input via queries standard quantum oracle defined authors present quantum query algorithm complete graph easily extended multigraph generalize algorithm depend weight function similar construction also implicit call following span program span span span choice weight function span program decides soon see choice may impact complexity resulting algorithm using authors ref show query complexity evaluating max connected analysis case complete graph easily seen apply general multigraphs fact straightforward show bound improved max connected max connected min connected otherwise particular complete graph vertex set promise exists length gives bound accepted quantum click title verify worst case analysis improve previous quantum algorithm gives bound paper consider particular multigraphs planar even additional added equivalently exists planar embedding face graph figure case figure external face given graph define three related graphs denote first define graph extra edge labeled connecting denote planar dual every planar dual one edge crossing edge original graph exists edge dual also labeled denote two vertices endpoints finally denote graph except edge removed figure example derive planar graph face obtained adding edge planar dual diagram labeled gray graph black obtained removing edge note dual edges inherit labels case primal edge construction always number edges defines subgraph including edge let subgraph include edge path must cut note looking figure one convince oneself defines simply define vertices path vertices path let weight function define weight function every path either hence finite path case finite state main lemma lemma let planar multigraph also planar let weight function let using lemma theorem immediately following theorem let planar multigraph also planar bounded error quantum query complexity evaluating min max max minimization positive functions accepted quantum click title verify might difficult general find optimal edge weighting choice least give upper bound query complexity however see sometimes structure graph allow efficiently find good weight functions proof lemma appendix positive witness result follows generalizing proof weighted multigraphs idea witnesses connected linear combination paths effective resistance characterizes size smallest possible positive witness linear combination similarly negative witness turns linear combination argued every corresponds using correspondence cuts paths negative witness linear combination allows show correspondence negative witnesses minimal connecting appendix show quantum walk step network implemented time efficiently algorithm query efficient also time efficient let show following theorem let defined let upper bound time complexity implementing property planar time complexity deciding min max connected max connected appendix also show space complexity implementing algorithm referred theorem space complexity time max log log comparison previous quantum algorithm planar algorithm always matches improves algorithm see compare eqs choose value edges first terms bounds need analyze second term however using duality paths cuts shortest path length obtain inequality create value one edges shortest path zero edges flow unit flow energy equal shortest path however true effective resistance smaller minimum energy possible present two simple examples algorithm analysis better first example highlight change complexity gives advantage graphs second example show accepted quantum click title verify able choose weight function give advantage graphs let length vertices arranged line vertex connected neighbors left right single edge vertices either end line figure let string length hamming weight string figure example graph analysis better analysis even edges promise always contains least edges connected choosing value edges max connected case unit flow value edge flow energy however max connected edges least edges thus define unit flow value parallel edges giving energy hand max fact since counts minimum number edges across cut always least exists whereas small choosing example applying eqs analysis gives query complexity analysis gives query complexity section show bound tight consider graph figure consists edges line connecting vertices assign weights edges path edges max occurs one present case edges contribute effective resistance final edge contributes also max accepted quantum click title verify figure example graph analysis quadratically better analysis taking advantage weight function values edge edge shown boldface maximum occurs one path path edges edge weight contributes flow energy however max cut across max occurs one present thus analysis gives query complexity analysis gives query complexity section give example analysis provides exponential improvement analysis formulas section present useful relationship formula evaluation problems problems certain graphs mentioned section simplicity restrict analysis formulas algorithm extends simply formulas case primarily concerned query complexity input formula given via standard quantum oracle defined given formula variables recursively construct planar multigraph two distinguished vertices labeled respectively every edge uniquely labeled variable single variable single edge vertices labeled edge label otherwise suppose graph obtained graphs identifying vertex labeled vertex labeled labeling vertex labeled vertex labeled connect graphs series figure formal definition see appendix possibility case construct starting identifying vertices labeled labeling accepted quantum click title verify resulting vertex identifying vertices labeled labeling resulting vertex connect parallel see figure note graphs constructed way exactly set graphs two terminals see def equivalent graphs without minor figure let obtain connecting series connecting parallel note planar furthermore always face thus define section show following lemma let formula variables every exists path furthermore every exists path give formal proof lemma appendix intuition subformulas evaluates true subformulas evaluates true likewise two vertices connected multiple subgraphs parallel vertices connected path subgraphs subformulas evaluates true every subformula evaluates true likewise two vertices connected multiple subgraphs series vertices connected path every subgraph thus show induction connected see connected use similar argument make use fact lemma implies solve formula evaluation problem solving associated problem input subgraph construction always planar graph external face moreover apply theorem obtain following theorem family formulas bounded error quantum query complexity input promised come set min max max minimization positive functions proof lemma query complexity query complexity since planar face apply theorem immediately implies result accepted quantum click title verify comparison existing boolean formula algorithms reichardt proved quantum query complexity evaluating formula variables corollary algorithm recovers result theorem let formula variables exists choice quantum algorithm obtained span program computes bounded error queries need following claim prove appendix claim formed composing series formed composing parallel intuition behind claim following although defined via dual constructed sequence series parallel compositions also built sequence series parallel compositions formula variables define formula variables replacing morgan law entrywise negation simple inductive proof shows see lemma appendix proof theorem make use following fact network positive real number min min proceed proof formula repeated applications morgan law push leaf since learned one query restrict attention formulas variable easy see taking value single edge prove induction true choice completing proof complexity algorithm obtained suppose formulas variables variables let denote bits induction hypothesis weight function using construction formed composing series thus every edge corresponds edge create weight function edge originating graph new weight function combining old weight functions scaling factors using lemma claim pgp accepted quantum click title verify thus recall weight function define weight function edge claim formed composing parallel lemma claim whenever set exactly set continuing thus combining desired proof case similar immediate corollary theorem following corollary deciding subgraphs graphs edges accomplished using queries chosen two terminal nodes many results field characterizing classical complexity seems difficult quantum complexity however show lower bound classical query complexity class boolean formulas terms effective resistance corresponding graphs achieving quadratic query complexity consider formulas restricted domains let let analyze formulas input every gate formula comes case denote case denote promises domains make easier evaluate functions example evaluates promised one input value least using sabotage complexity bound classical query complexity following theorem whose somewhat long technical proof found appendix accepted quantum click title verify theorem let randomized query complexity evaluating quantum query complexity evaluating note theorem compose formulas promises input implicitly assume promise input composed formula precisely defined theorem proven showing max max using sabotage complexity show lower bound randomized query complexity gives quadratic separation randomized quantum query complexities class formulas details see appendix using composition lower bound promise boolean functions lower bound grover search multiple marked items quantum query complexity theorem tight additionally light reduction boolean formula evaluation see example figure section equivalent problem query bound example also tight based theorem one might guess evaluating formulas using stconnectivity reduction one obtain quadratic classical randomized query complexity however fact possible obtain superpolynomial certain promise problems using approach discuss section results query separations section prove two query separations stronger previous results query separations rely formula promise inputs restriction promise defined shortly let fkd set inputs nandd satisfy condition two results following theorem using approach formula evaluation theorem one solve evalnandd input promised flog queries log log classical algorithm requires queries different choice example demonstrates dramatic improvement algorithm give analysis case exponential precisely polynomial constant improvement theorem consider problem analysis gives bound quantum queries decide problem number edges gnandd analysis shows problem decided quantum queries define mean zhan find relationship difficulty playing game associated accepted quantum click title verify witness size particular span program nandd find trees fewer faults critical decisions player playing associated game easier evaluate quantum computer show algorithm least well algorithm zhan evaluating trees see relate fault complexity effective resistances gnandd consider nandd instance recall relationship nandtree instance game described section call sequence nodes player player choose course game path path root leaf node call set paths player wins player never makes move would allow opponent win path resp encounters nodes roots resp subtrees never passes node player resp player could make decision move resp subtree whether node tree root subtree determined evaluating subformula corresponding subtree see figure example let set nodes along path player turn thus resp contains nodes even resp odd distance leaves zhan call node fault one child root tree child root tree node constitutes critical decision point let denote number faults define fault complexity input min max otherwise set trees fkd instances log trees winning player encounter fault nodes path leaf kimmel shows exists span program evaluating whose witness size instance fault complexity player first turn player first turn player second turn second turn figure depiction game let input shown figure instance consists paths shown using solid lines fault nodes double circles path encounters two faults nodes player makes decisions therefore first show relationship effective resistance gnandd actually used refined definition accepted quantum click title verify lemma even gnandd odd gnandd proof lemma found appendix immediate corollary lemma theorem following corollary span program nandd decides evalnandd restricted domain queries particular decides trees domain fkd queries proof theorem theorem immediate consequence corollary set log along fact classical query complexity evaluating formulas dlog log use corollary along following claim prove theorem claim let two graphs nodes let suppose create new graph identifying nodes nodes connecting graphs parallel create new graph identifying node node relabeling node connecting graphs series min proof theorem using corollary analysis shows decided queries apply compare analysis must characterize quantity max gnandd connected since already max connected prove every nandd gnandd thus promise input long exists nandd nandd gnandd intuitively every subgraph gnandd cuts across edges gnandd proof induction base even case single edge connecting input nandd case cut across unique edge induction step treat even odd separately suppose odd nandd thus gnandd involves composing two graphs series since assuming connected gnandd least one must connected without loss generality suppose connected induction thus using claim gnandd accepted quantum click title verify suppose even nandd thus gnandd involves composing two graphs parallel since assuming connected gnandd must connected induction since even thus using claim gnandd therefore using analysis inputs gives query complexity number input variables comparing analysis gives query complexity see polynomial constant improvement winning game section describe quantum algorithm used help player make decisions playing game particular consider number queries needed player make decisions throughout course game order win probability section focus trees case trees similar first describe naive strategy uses quantum algorithm decides tree winnable bounded error log queries player must decide move node evaluates subtree rooted amplifying success probability using log repetitions moves one evaluates since player decisions make strategy succeeds bounded error since evaluating depth costs quantum queries total query complexity log log log log log strategy use fact subtrees may easier win others example one choice leads subtree leaves labeled whereas subtree leaves labeled player needs distinguish two disparate cases generally one subtrees might small positive witness size winnable whereas large positive witness size winnable strategy move subtree whose formula corresponds graph smaller effective resistance unless two subtrees close effective resistance case matter one choose depth game instance show gnandd small player plays randomly strategy better naive strategy average estimate effective resistance subtrees current node using witness size estimation algorithm particular appendix prove lemma est algorithm let formula constant quantum query complexity estimating resp relative accuracy accepted quantum click title verify resp let est algorithm lemma nandd estimating effective resistance two subtrees care subtrees smaller effective resistance want wait iterations est terminate let qbe polynomial function est always terminates queries define subroutine select takes two instances outputs bit select works follows runs est est parallel one programs say est outputs estimate terminates program steps algorithm running terminated time outputs programs terminated outputs bit appendix prove following lemma lemma let instances nandd least one let wmin gnandd gnandd min wmin queries outputs select terminates gnandd gnandd bounded error using lemma prove following inductive proof appendix theorem let input nandd every node player makes decision let player use select algorithm following way let two children inputs respective subtrees given respectively player moves outcome occurs majority times select run log times player decision nodes chooses left right equal player probability win game probability least use gnandd queries average average taken randomness player choices acknowledgments funded department defense acknowledges funding provided institute quantum information matter nsf physics frontiers center nfs grant support gordon betty moore foundation nwo wise fellowship references aaronson sculpting quantum speedups proceedings conference computational complexity ccc pages germany schloss fuer informatik isbn doi url http aldous fill reversible markov chains random walks graphs unfinished monograph recompiled available http belovs span programs functions proceedings symposium theory computing stoc pages doi accepted quantum click title verify belovs reichardt span programs quantum algorithms stconnectivity claw detection proceedings european symposium algorithms esa pages doi kothari randomized query complexity sabotaged composed functions proceedings international colloquium automata languages programming icalp volume pages doi boyer brassard tapp tight bounds quantum searching fortschritte der physik issn doi sici url http sici brassard mosca tapp quantum amplitude amplification estimation contemporary mathematics cade montanaro belovs time space efficient quantum algorithms detecting cycles testing bipartiteness chandra raghavan ruzzo smolensky tiwari electrical resistance graph captures commute cover times computational complexity doi dirac property graphs remarks critical graphs journal london mathematical society doi doyle snell random walks electrical networks volume carus mathematical monographs mathematical association america duffin topology networks journal mathematical analysis applications issn doi http url http heiligman mhalla quantum query complexity graph problems siam journal computing doi farhi goldstone gutmann quantum algorithm hamiltonian nand tree theory computing doi arxiv grover fast quantum mechanical algorithm database search proceedings annual acm symposium theory computing stoc pages acm doi heiman wigderson randomized deterministic decision tree complexity boolean functions computational complexity doi ito jeffery approximate span programs proceedings international colloquium automata languages programming icalp pages doi karchmer wigderson span programs proceedings ieee annual conference structure complexity theory pages doi kimmel quantum adversary upper bound chicago journal theoretical computer science doi accepted quantum click title verify lee mittal reichardt szegedy quantum query complexity state conversion proceedings annual ieee symposium foundations computer science focs pages doi reichardt span programs quantum query complexity general adversary bound nearly tight every boolean function proceedings ieee symposium foundations computer science focs pages doi arxiv reichardt span programs quantum query algorithms electronic colloquium computational complexity eccc reichardt reflections quantum query algorithms proceedings annual symposium discrete algorithms soda pages siam reichardt quantum algorithm evaluating formulas theory computing doi saks wigderson probabilistic boolean decision trees complexity evaluating game trees proceedings annual symposium foundations computer science focs pages ieee doi valdes tarjan lawler recognition series parallel digraphs proceedings annual acm symposium theory computing stoc pages acm doi url http zhan kimmel hassidim quantum boolean evaluation trees hidden structure proceedings innovations theoretical computer science conference itcs pages new york usa acm isbn doi url http analysis span program section analyze complexity algorithms proving lemma first stated section relates witness sizes span program effective resistance graphs related need concept circulation like flow source sink definition circulation circulation graph function following easily verified observations useful several remaining proofs section claim let unit multigraph consider corresponding vector written linear combination accepted quantum click title verify vectors corresponding cycles paths let circulation written linear combination cycles furthermore written linear combination cycles cycle goes around face next claim shows direct correspondence positive witnesses claim fix span program call positive witness note necessarily positive witness particular input unit positivepwitness furthermore positive witness unit proof proof straightforward calculation let unit used provepthe second half claim let define immediately see furthermore thus thus unit next claim shows direct correspondence negative witnesses accepted quantum click title verify claim planar graph fix span program call linear function negative witness unit furthermore every negative witness figure duality cycle star proof consider edges directed edges assign edge directions dual orienting dual edge radians primal edge note without loss generality negative witness assume invariant affine transformations first show negative witness defined unit begin define agrees everywhere defined addition clearly next every corresponds face edges coming dual edges going clockwise around face see figure directed edges going clockwise around thus circulation since remove flow edge recovers get unit next show unit exists negative witness define circulation obtained defining define express linear combination cycles around faces clockwise cycle around clockwise cycle around face counterclockwise cycle around correspondance vertices faces define claim let edge edge part clockwise cycle around one face call accepted quantum click title verify counter clockwise cycle around one face call since two faces containing edge must since thus particular means negative witness prove main result section lemma lemma let planar multigraph also planar let weight function let proof connected exists unit unit supported let flow claim positive witness since supported positive witness thus hand let optimal positive witness claim unit since unit thus last inequality thus prove let connected fix optimal negative witness claim linear function defined unit since negative witness also accepted quantum click title verify since exactly supported unit thus direction let minimal energy claim negative witness since supported edges exactly edges negative witness thus completing proof time space analysis span program algorithm section give upper bound time complexity terms time complexity implementing step quantum walk analysis follows relatively straightforwardly section include completeness end section show space complexity algorithm max log log first describe algorithm derived span program following conventions throughout section let denote orthogonal projector onto inner product space span program corresponding algorithm performs phase estimation unitary applied initial state denotes orthogonal projector onto denotes orthogonal projector onto kernel denotes decide function domain sufficient perform phase estimation precision case span program simple exercise see implemented quantum operations including queries reflection independent requires queries implement however could still require number gates grows quickly size show implementing reduced implementing quantum walk task could quite easy depending structure example case complete graph vertices done log gates accepted quantum click title verify multigraph weight function define quantum walk step unitary acts follows theorem let defined let upper bound time complexity implementing property planar time complexity deciding min max connected max connected theorem follows lemma stated lemma deals construction algorithm initial state lemma let defined let upper bound time complexity implementing implemented time complexity proof analysis follows see also let define spaces follows span span define isometries whose respectively ihu note following hzu hzv thus calculate accepted quantum click title verify note rows multiples rows row row thus ker ker define claim maps ker eigenspace ker see note ker ker ker thus next suppose ker ker since isometry ker rowmz thus thus implement remains argue implemented time complexity first show implement isometry rather unitary acts first use first qubit perform map conditioned value first register copy either second register get thus implement time takes write vertex log using ability implement implement reflection acts identity computational basis states form reflects computational basis states without form next implement unitary acts first use quantum walk step implemented time perform conditioned bit first register swap third fourth registers get total cost implementing log thus implement quantum gates lemma let defined let upper bound complexity implementing initial state algorithm approximated time proof without loss generality assume includes edge simply include subgraph furthermore set positive specified later effect edges note ker accepted quantum click title verify ker thus constant precision phase estimation maps using quantum amplitude amplification amplify amplitude part arbitrarily close using number calls proportional fact straightforward show rowa vector smallest norm satisfies using fact along claim definition let effective resistance without edge think two graphs parallel using claim setting thus using calls approximate initial state finally note space required algorithm addition whatever auxiliary space need implement max log log act hilbert space dimension less principle implemented max log log qubits however implementation may also make use number auxiliary qubits use unitaries perform phase estimation precision min max max thus need log qubits store output phase estimation putting everything together gives claimed space complexity formula evaluation section prove correspondence evaluating formula solving graph first give formal definition definition formula define letting define accepted quantum click title verify define order prove lemma first prove lemma lemma formula define formula obtained replacing denotes bitwise complement furthermore isomorphism maps edge labeled label edge labeled proof first part proof induction suppose depth variable suppose induction hypothesis second last equality morgan law case similar figure shown black dual shown grey thick lines represent graphs edges dual edges dotted edge dual prove furthermore dual edges label induction depth result follows immediately formula case edge labeled edge labeled inductive step show dual therefore suffices exhibit bijection faces separated edge label first consider case consists graphs chained together series figure additional edge faces exactly interior faces well two faces either side denote adding label internal face accepted quantum click title verify since external face since also use labels anticipation isometry induction hypothesis exists bijection faces separated edge label define induction hypothesis see edge separated edge labeled edge one dual edge induction hypothesis see figure edge edge exactly faces either side completing proof nearly identical proof shows shown isomorphism note subgraph includes edges hand graph includes edges taking bitwise negation find lemma allows prove claim claim formed composing series formed composing parallel proof lemma using definition composed series using isomorphism lemma composed series proof similar prove lemma relates existence path value function lemma let formula variables every exists path furthermore every exists path proof prove statement induction depth depth edge edge thus connected case evaluates connected case evaluates consists connected series moreover consists connected series thus consists followed thus connected connected happens hand claim consists connected parallel one equivalent one thus lemma connected accepted quantum click title verify morgan law true case similar classical lower bound class promise boolean formulas section consider query complexity classical algorithms formulas proving theorem use recent tool kothari show classical randomized query complexity function denoted satisfies randomized sabotage complexity defined presently furthermore prove composed function let fsab dsab dsab consistent say consistent fsab fsab finally randomized sabotage complexity given fsab randomized query complexity classical query complexity definitions see first bound sabotage complexity lemma sab consistent proof must furthermore number must least thus sabotaged problem reduces finding least one marked item promised least marked items randomized query complexity task theorem sab sab proof similar next corollary follows immediately lemma composition property sabotage complexity corollary let understand query complexity symmetric composed formulas look compares quantum query complexity evaluating functions prove following lemma lemma let let domain accepted max quantum click title verify max proof proof follows induction number compositions first suppose consists edges connected parallel consists edges connected series input zeros input therefore notice using claim however domain orni inputs thus max max similar analysis holds base case inductive step let either let domain let denote bits input copy suppose first formed taking graphs connecting series way input functions satisfies claim max max hand formed taking graphs connecting parallel using claim input function thus larger values come cases large max occurs promise domain must least subformulas evaluate subformulas want input maximizes effective resistance subformula therefore max therefore using inductive assumption max max max max inductive step similar corollary lemma give theorem accepted quantum click title verify proofs relationship faults effective resistance section prove lemma lemma even gnandd odd gnandd proof give proof case similar first gnandd connected gnandd lemma occurs means exactly holds thus suppose case rest proof induction need look odd even cases case input case using since decision nodes player since single edge let input let first bits last bits first consider odd using definition section fact odd root node see gnandd consists connected series claim gnandd root fault decision node player know tree choice player makes would allow win game therefore subtrees connected root node must using max combining eqs inductive assumption even depth trees odd gnandd consider case even root decision node player consequently root node claim gnandd suppose root fault without loss generality let assume subtree input becomes gnandd using inductive assumption odd depth trees fact root fault gnandd accepted quantum click title verify root fault finite using inductive assumption max max gnandd similar analysis completes proof estimating effective resistances section prove lemma bounds query complexity estimating effective resistance graph corresponding boolean formula ito jeffery describe quantum query algorithm estimate positive negative witness size span program given access describe use algorithm estimate effective resistance graphs ref define approximate positive negative witness sizes similar positive negative witness sizes conditions relaxed definition approximate positive witness span program define positive error min say approximate positive witness define approximate positive witness size min case approximate positive witness positive witness negative inputs positive error larger define similar notion approximate negative witnesses definition approximate negative witness span program define negative error min called approximate negative witness define approximate negative witness size min case approximate negative witness negative witness positive inputs negative error larger ito jeffery give following result accepted quantum click title verify theorem fix let span program define quantum algorithm estimates relative error uses exists queries similarly let span program exists define quantum algorithm estimates accuracy uses queries apply theorem span program defined throughout section always set weight function take value one edges graph case simplify notation denote span program apply theorem need bounds prove lemma formula largest number nodes path root leaf let formula maximum applying lemma theorem main result section first stated section lemma est algorithm let formula constant quantum query complexity estimating resp relative accuracy resp proof lemma theorem since lemma estimate quantity using number queries depends estimate lemma queries similarly estimate queries prove lemma use following observation gives upper bound length longest terms bound tight general claim let formula constant longest path connecting length proof prove statement induction thus easy see two vertices number edges connecting every length suppose first suppose since consists connected series consists followed etc longest length accepted quantum click title verify maxi consists connected parallel must include exactly one edge adjacent one edge adjacent however path includes edge must must one edge adjacent one edge adjacent path never thus must contained completely one longest path thus longest induction maxi prove lemma suppose proof lemma begin prove upper bound optimal approximate positive witness claim approximate positive witness since unit value edges unit flow since approximate positive witness minimal error minimizes since optimal minimizes approximate positive witnesses define know also maps also positive witness last inequality uses also approximate positive witness similarly optimal claim consider decomposition paths cycles cycles disjoint paths easy see case unit edge weights let since common edges hci also hci error also minimal error furthermore equality cycles decomposition optimality approximate accepted quantum click title verify positive witness conclude since max max since longest length length path thus approximate negative witness next prove bound function minimized claim since function defined unit witness complexity create usual way argument similar previous argument optimal approximate negative witness upper bounded twice length longest lemma claim upper bounded thus winning analyze algorithm winning game associated proving lemma theorem lemma let instances nandd least one let wmin min gnandd gnandd wmin queries outputs select terminates gnandd gnandd bounded error proof since least one using description select section least one programs terminate suppose without loss generality est first terminate outputting two possibilities est terminate steps case gnandd gnandd select outputs est outputs steps passed select outputs prove first case contradiction suppose gnandd gnandd est must terminate gnandd gnandd steps select run est relative accuracy gnandd gnandd accepted quantum click title verify gnandd plugging est must terminate steps contradiction thus gnandd gnandd qso outputting correct furthermore since terminate gnandd steps since gnandd gnandd running time consider second case programs output estimates gnandd gnandd suppose gnandd gnandd gnandd gnandd thus gnandd gnandd required furthermore running time algorithm bounded running est second terminate know est running time gnandd steps sumption est terminated less gnandd steps total running time theorem let input nandd every node player makes decision let player use select algorithm following way let two children inputs respective subtrees given respectively player moves outcome occurs majority times select run log times player decision nodes chooses left right equal player probability win game probability least use gnandd queries average average taken randomness player choices proof first note player must make choices course game amplify player probability success repeating select decision node log times taking majority probability player chooses wrong direction node ensure probability choosing wrong direction course algorithm analyze error free case let polynomial function select inputs terminates min gnandd gnandd queries prove trees odd depth expected number queries player course game gnandd even depth trees gnandd thus proving main result prove result induction depth tree depth zero trees decisions result holds inductive case treat odd even depth cases separately first consider instance nandd odd thus nandd accepted quantum click title verify root odd distance leaves root decision node player tree matter choice player makes end subtree depth inductive assumption holds trees expected number queries player must make win subtree input averaged player choices assuming player chooses left right equal probability thus expected number queries player must make player choices throughout game gnandd claim jensen gnandd gnandd proving case odd consider instance nandd even thus nandd root even distance leaves root decision node player player runs select returns lemma requires min queries making choice move subtree input inductive assumption expected number queries player need make throughout rest game averaged player choices two cases consider combining total number queries averaged accepted quantum click title verify player choices gnandd gnandd used gnandd claim fact bound value gnandd proves even induction step case case case using fact gnandd gnandd gnandd inequality follows thus average total number queries gnandd gnandd gnandd proves induction step case accepted quantum click title verify
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oct layered heaps beating standard fibonacci heaps practice huggins march abstract consider classic problem designing heaps standard binary heaps run faster practice fibonacci heaps worse time guarantees present new type heap runs faster practice standard binary fibonacci heaps asymptotic insert times arbitrarily better log namely log arbitrary positive integer heap defined recursively maximum run time speed occurs recursion depth used heap heaps layered heaps define heaps arbitrary integer heap standard binary heap stored array heap used log inductive hypothesis insert operations children heaps log log furthermore height ary heap also log insert operations take log time want functions take standard log time may need children heap operation takes log time total log times height layered heap operations running times explained following pseudocode insert heap swapping start placing element end array position following set log swap values else insert children layered heap contains position array recursive break set pop heap set log store remove root element heap put last item heap root swap downwards top children heap top children heap greater element recursively balance ary heap time log break element greater top current children heap return popped top heap seen running time insert log log running time pop log log popular competing heaps fibonacci heap presented amortized constant insert time standard log time amortized running times later improved strict running time bounds per operation later publication however practice constants associated various fibonacci heaps large outperform standard binary tree thus due simplicity faster running time binary heaps traditionally taught used running time comparisons simulate situations asymptotically faster insertions heaps may better traditional heaps simulation elements would added elements would popped ith insert inserted value heap repeated running times recorded function size heap binary heaps faster fibonacci heaps practical data sizes practice furthermore analysis recursively defined layered heaps made clear constants become large overcome practice unless thus compared heap traditional binary heap results shown figure number elements heap heap grows results computed extrapolated cover feasible data sizes figure shows despite log time heaps pop performed layered heap good cache performance processing children heap elements usually fit cache one memory access children heap thus running time practice inserts layered heap look like inflated log memory use identical heaps discussion although layered heaps interesting theoretical point view arbitrary giving asymptotic insert running time arbitrarily closer closer constant practice layered heap fastest practice run times faster binary heap reasonable data sizes fact heaps easy enough describe implement analyze directly opposed using probably taught data structures courses standard binary heaps presented bibliography fredman michael lawrence tarjan robert fibonacci heaps uses improved network optimization algorithms journal association computing machinery
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oct structure sally module integrally closed ideals kazuho ozeki maria evelina rossi shiro goto occasion seventieth birthday abstract first two hilbert coefficients primary ideal play important role commutative algebra algebraic geometry paper give complete algebraic structure sally module integrally closed ideals local ring satisfying equality minimal reduction denote first two hilbert coefficients respectively multiplicity chern number almost extremal value respect classical inequalities holds complete description homological numerical invariants associated graded ring examples given introduction notation throughout paper let denote local ring maximal ideal positive dimension let ideal simplicity assume residue class field infinite let denote length integers equality holds true integers called hilbert coefficients respect polynomial known polynomial denoted hpi encodes asymptotic information coming hilbert function defined generating function numerical function power series hsi series called hilbert series well known series rational even exists polynomial integers coefficients hsi key words phrases local ring associated graded ring hilbert function hilbert coefficient mathematics subject classification notice hilbert coefficients computed follows denotes derivative evaluated choose parameter ideal forms reduction let denote respectively rees algebras let following vasconcelos consider sally module respect notion filtration sally module introduced vaz pinto follows denote graded graded whose grading given definition set let graded following natural exact sequences graded every notice finitely generated graded since extension graded ring set shall explore structure assume integrally closed inequality holds true equality holds case associated graded ring behavior function known see corollary thus integrally closed ideal enjoys nice properties seems natural ask happens integrally closed ideal satisfies equality problem trivial even consider notice holds true see instance depend minimal reduction let polynomial ring indeterminates field main result paper stated follows theorem assume integrally closed following conditions equivalent rankb graded linearly independent linear forms polynomial ring case following assertions hold true depth deptht depth iii suppose hpi suppose hpi suppose hpi hilbert series hsi given hsi let briefly explain paper organized shall prove theorem section section introduce auxiliary results structure stated general setting hope information successfully applied give new insights problems related structure sally module section introduce consequences theorem particular shall explore integrally closed ideals section construct class local rings satisfying condition theorem preliminary steps purpose section summarize results structure graded need throughout paper remark section ideal necessarily integrally closed lemma following assertions hold true integers hence dimt homogeneous components graded given proof let notice graded graded finitely generated therefore since assertion follows definition module assertions readily follow assertion following result need holds true condition automatically satisfied case integrally closed see proposition suppose asst dimt let minimal reduction proof proposition need following lemmata lemma suppose therefore nonzero divisor proof show holds true proceed induction may assume assertion holds true suppose take hypothesis induction write since notice since holds true hence get therefore assume assertion holds true take hypothesis induction write hypothesis induction therefore get thus required lemma suppose proof show take write denote images respectively polynomial ring ring thus holds true lemma suppose following sequences graded exact proof let consider homomorphism graded denotes image show ker take ker write xtn xtn lemma forms nonzero divisor notice polynomial ring ring therefore hence ker thus get first required exact sequence let consider homomorphism graded denotes image need show ker take ker write xtn write lemma hence therefore hence ker consequently get second required exact sequence following lemma crucial fact proof proposition lemma assume asst proof take asst assume htt height one prime ideal consider following exact sequences qtp ttp follow exact sequences lemma since depthtp notice polynomial ring indeterminates ring depthtp depthtp exact sequence notice polynomial rings indeterminates ring hence depthtp ttp exact sequence annt therefore depthtp lemma however impossible thus required let give proof proposition proof proposition take asst lemma suppose htt height one prime ideal look following exact sequences follows canonical exact sequences notice depthtp nonzero divisor thanks depth lemma exact sequence get depthtp depthtp depthtp since depthtp depthtp exact sequence therefore lemma impossible thus required following techniques due vaz pinto section let graded exist canonical exact sequence graded definition set notice forms graded polynomial ring indeterminates ring let denotes epimorphism graded denote images following lemma lemma suppose map isomorphism graded proof show ker assume ker let least integer ker notice ker take ker set may write denote images respectively may assume may write lemma therefore hence ker contradiction thus ker consequently map isomorphism thanks lemma prove following result proposition suppose proof exact sequence isomorphisms graded see lemma therefore gno proposition following result specifies gno proposition using proposition proof takes advantage techniques proposition suppose let combining lemma proposition obtain following result proven elias valla theorem case corollary suppose equality holds true case ring let introduce relationship depth module associated graded ring lemma suppose let deptht depthg particular depthg proof notice lemma dimensional therefore deptht deptht exact sequence depthg deptht deptht depthg deptht gno proposition assertions follow proof theorem purpose section prove theorem throughout section let integrally closed ideal theorem suppose integrally closed following conditions equivalent rankb exists graded ideal graded prove theorem need following bound lemma theorem corollary corollary suppose let integrally closed ideal proof theorem let see lemma asst proposition since asst rankb clear assertion equivalent saying obvious asst torsion free graded hence graded suppose dim rankb exists graded ideal graded since every height one prime polynomial ring principal may choose htb since enough show applying exact sequence graded lower terms htb therefore thanks proposition lower terms therefore lemma thus graded direct consequence theorem following result holds true proposition assume integrally closed suppose let depth deptht depth suppose hpi suppose hpi suppose hpi hilbert series hsi given hsi proof since lemma therefore thanks theorem graded ideal generated linear forms hence get let consider exact sequence graded resp resp therefore assertions follow proposition hsc hsb exact sequence denotes hilbert series graded modules also hss hsl hsc exact sequence isomorphisms graded lemma hsi hss proposition get required result prove deptht exact sequence depthg depthg lemma let consider case need show depthg assume depthg gno proposition taking local cohomology functors respect graded maximal ideal exact sequence graded get monomonophism graded graded notice hand hdm lemma however impossible therefore depth prove theorem assume assertion theorem isomorphism graded graded ideal theorem able show lemma ideal generated linearly independent linear forms recall lemma therefore implication theorem follows also notice last assertions theorem follow proposition thus theorem proven modulo following theorem theorem assume integrally closed suppose proof proceed induction suppose result follows proposition since assume assertion holds true since residue class field infinite without loss generality may assume superficial element integrally closed page proposition set since inductive assumption says holds true depthg thanks sally technique lemma divisor assume depthg proposition since homomorphic image let take isomorphism graded graded ideal theorem since denote linearly independent linear forms enlarge basis lemma assume since ideal principal need following claim proof claim assume denotes graded maximal ideal therefore nakayama lemma impossible thus show assume let write denotes image notice forms let choose elements let respectively images let consider relations notice lemma write forms regular sequence since lemma exist elements equality hence argument works see therefore hence exist elements equality since regular sequence write equality since denotes integral closure ideal hence therefore impossible thus therefore completes proof theorem theorem well consequences purpose section present consequences theorem let begin following exactly case theorem theorem assume integrally closed following conditions equivalent graded case following assertions follow depthg iii hilbert series hsi given hsi proof last assertions see theorem since lemma epimorphism graded must isomorphism dimt proposition let denote closure ideal largest ideal let note following remark remark assume integrally closed holds true denotes integral closure ideal thus following result correspond case theorem section give example maximal ideal satisfy assertion theorem theorem suppose assume integrally closed following conditions equivalent graded qien case associated graded ring buchsbaum ring depth buchsbaum invariant proof set ien let ien let denote ith hilbert coefficients filtration follows theorem theorem therefore graded since depthg theorem apply local cohomology functors respect graded maximal ideal exact sequences graded get derived monomorphisms deptht deptht recall ring dim deptht lemma furthermore since since integrally closed isomorphisms graded hence hence since remark therefore qien theorem qien remark ring theorem since furthermore ring proposition assume one equivalent conditions proof implication let integer ien ien ien therefore ien set look exact sequence graded since ien ring let unique graded maximal ideal applying local cohomology functors exact sequence yields since thus buchsbaum ring buchsbaum invariant hence graded ring completes proof theorem rest section explore relationship inequality northcott structure graded module integrally closed ideal well known inequality holds true equality holds theorem case associated graded ring suppose integrally closed thanks corollary associated graded ring cohenmacaulay thus integrally closed ideal seems satisfactory understood section briefly study integrally closed ideals let begin following theorem assume integrally closed suppose following assertions hold true graded depth hilbert series hsi given hsi proof follows corollary therefore hence let since thanks theorem graded assertions follow depth theorem since ring otherwise depthg completes proof theorem notice following result also follows theorem corollary assume integrally closed suppose depth graded ring cohenmacaulay closing section briefly study integrally closed ideal suppose corollary depthg equality holds true associated graded ring corollary thus need consider following theorem suppose assume integrally closed let following assertions hold true either graded exists exact sequence graded suppose depthg suppose depthg hilbert series hsi given hsi proof since also see proof proposition thanks theorem graded denote linearly independent linear forms thus assertions follow theorem remark measures far multiplicity minimal value see corollary depth still open problem whether depthg assuming theorem confirms conjectured bound corollary assume integrally closed suppose depth proof corollary depthg suppose depthg theorem completes proof corollary example goal section construct example local ring maximal ideal satisfying equality theorem class examples exhibit includes interesting example given wang see example theorem let integers exists local ring dim minimal reduction construct necessary examples may assume fact suppose assume already chosen certain local ring dim minimal reduction let let formal power series ring ring set local ring dim maximal ideal ideal reduction forms super regular sequence respect recall polynomial ring initial forms thus observation allows concentrate attention case let integers let formal power series ring indeterminates infinite field let set denote images respectively since dim let maximal ideal set therefore minimal reduction system parameters interested hilbert coefficients maximal ideal well structure associated graded ring module theorem following assertions hold true local ring dim graded therefore buchsbaum ring depthg hilbert series hsm given hsm notice wang example quoted corresponds particular case let divide proof theorem two steps let begin following proposition let set proof let algebraically independent let get isomorphism substituting corresponds prime ideal algebras prime ideal get thus thanks associative formula multiplicity hand therefore thus local ring let set let denote ith hilbert coefficients filtration lemma following assertions hold true qkn therefore proof since yvi therefore since assertion remark routine check qkn thus theorem also proposition prove theorem proof theorem since lemma therefore lemma ring buchsbaum ring depthg theorem completes proof theorem references elias valla rigid hilbert functions pure appl algebra gno goto nishida ozeki sally modules rank one michigan math guerrieri rossi hilbert coefficients hilbert filtrations algebra huneke hilbert functions symbolic powers michigan math huckaba marley hilbert coefficients depth associated graded rings london math soc itoh hilbert coefficients integrally closed ideals algebra itoh integral closures ideals generated regular sequence algebra northcott note coefficients abstract hilbert function london math soc ratliff rush two notes reductions ideals indiana univ math rossi valla conjecture sally comm alg rossi valla local rings embedding dimension proc london math soc rossi valla hilbert functions filtered modules umi lecture notes springer sally hilbert coefficients reduction number alg geo sing sally ideals whose hilbert function hilbert polynomial agree algebra vasconcelos hilbert functions analytic spread koszul homology contemporary mathematics vol vaz pinto hilbert functions sally modules algebra wang links symbolic powers prime ideals math department mathematical science faculty science yamaguchi university yoshida yamaguchi japan address ozeki dipartimento matematica universita genova via dodecaneso genova italy address rossim
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bernoulli embeddings graphs vinith sumit netflix los gatos usa ibm india research laboratory new delhi india vmisra sumitbhatia mar march abstract semantic hashing salakhutdinov accelerate information retrieval binary valued embeddings significantly reduce latency retrieval graphical data introduce simple effective model learning binary vectors nodes graph imagining embeddings independent coin flips varying bias continuous optimization techniques applied approximate expected loss embeddings optimized fashion consistently outperform quantization spectral graph embeddings various learned embeddings ranking tasks variety datasets introduction consider users perhaps research intelligence recruiting community seek explore graphical data perhaps knowledge graphs social networks graph small reasonable users directly explore data examining nodes traversing edges larger graphs graphs noisy edges rapidly becomes necessary algorithmically aid users problems arise setting essentially information retrieval recommendation graphical data well studied hasan blanco identifying important edges predicting links exist like responsiveness retrieval systems critical gray driven numerous system designs hardware hong software low alternative seek algorithmic solutions latency challenge models perform link prediction node retrieval evaluated across two axes relevance retrieved nodes speed retrieval gold standard relevance typically set trained models rely observable features quantify connectivity two nodes models often quite slow evaluate due complexity features question extreme embeddings accelerate retrieval graphical data much manner semantic hashing salakhutdinov assist efficient retrieval text image data roughly speaking binary representation node allows one search similar nodes constant time directly binary embedding space much faster alternatives table challenge efficient binary representations difficult learn reasonable metric accuracy finding optimal binary representations one solution lean large body work around learning continuous embeddings graphs utilize modern quantization techniques binarize continuous representations catch approach continuous embeddings optimized future binarization mind hurts relevance retrieved nodes sec part work conducted authors ibm almaden research center technique preprocess query observable features real embeddings binary embeddings sparse similarities real embeddings quantized binary embeddings dense similarities slow slow fast table complexity five different node retrieval approaches ranked highest lowest accuracy table nodes edges latent dimensions primary contribution thus method learning embeddings explicitly optimized binarization use link retrieval mind concretely manner similar mikolov likelihood edge two nodes modeled function hamming distance bit embeddings rather directly optimizing bit embeddings eij task task weiss instead imagined drawn matrix independent bernoulli random variables eij parametrized independent probabilities success pij minimizing expected loss product distribution embeddings applying efficient approximations hamming distance sec continuous optimization techniques applied convenience refer bit embeddings learned manner bernoulli embeddings comparisons performed five different graphical datasets described section bernoulli embeddings found achieve significantly higher mean average precision variety alternative binary embedding options including various quantizations deepwalk vectors perozzi fiedler embeddings several embeddings introduce table also found hold reranking scenario binary hashes used preprocessing step accelerate computationally intensive algorithms node retrieval performed using binary embeddings orders magnitude faster alternatives especially larger datasets table related work approaches node retrieval roughly categorized table evaluated terms relevance speed bernoulli embeddings occupy unexplored valuable niche spectrum binary embeddings retrieval appropriate large graphs thousand nodes learned directly adjacency matrix higher relevance retrieved nodes following describe categories represented table observable features methods link prediction node retrieval graphs typically rely observable neighborhood features dong however computing similarity using tools significant computational cost low second order neighborhood features jaccard index hasan adamicadar score adamic either implicitly explicitly involve operations paths neighborhoods generally requires either one joins graph table multiplications graph adjacency matrix even efficient indexing operation complexity number edges graph higher order path features katz metric rooted pagerank hasan path ranking algorithm dong involve even longer paths even ations link prediction often harnesses dozens features parallel cukierski offline precomputation dense node similarity matrix dramatically help latency quadratic complexity number nodes leaves option smaller graphs embeddings unquantized embeddings use lsh node retrieval typically involves search nearest euclidean neighbors query embeddings appeared prominently context text reasons unrelated retrieval perozzi perozzi apply machinery mikolov mikolov sentences generated random walks graph yang yang illustrate graph embedding powerful tool context semisupervised learning world knowledge graphs particularly welcoming embeddings starting work hinton continuing models bordes bordes sutskever sutskever socher socher others additionally graph laplacian eigenvectors find prominent use spectral clustering interpreted latent embedding fiedler embedding analogous lsa knowledge graphs naturally represented tensors adjacency matrices analogously suggest use tensor factorizations approximate factorizations nickel discrete embeddings hinton salakhudtinov salakhutdinov introduce semantic hashing solution similar problem different domain instead relying indexed vectors retrieval relevant documents discrete embedding learned every document corpus query time user rapidly retrieve shortlist relevant documents simply scanning query neighborhood discrete embedding space embedding sufficiently compact neighborhood nonempty scan fast embedding compact retrieval yields list may reranked using computationally demanding algorithms results fueled development variety similar techniques seeking learn compact binary encodings given dataset quantized embeddings popular approach taken weiss weiss gong lazebnik gong others assume data consists short real vectors apply algorithms graphical data one must first learn embedding nodes graph compare baselines sec binary embeddings similarity matrix approach also known supervised hashing liu kulis norouzi matrix pairwise similarities data points supplied objective preserve similarities discrete embedding space surface appears similar graphical setting interest however sparsity assumption placed similarity matrix proposed solutions typically variations coordinate descent fall victim complexity application limited graphs thousand nodes note approach specifically avoids issue exploiting sparse structure graphs liu liu also assume one supplied matrix similarities data rather directly attempting replicate similarities embedded space perform constrained optimization forces embedded bits uncorrelated graph setting acts approximation sign bits fiedler embedding appears amongst empirical baselines architecture bernoulli embeddings consider generic graph consisting nodes adjacency matrix goal formulated learning matrix probabilities pij node embeddings sampled matrix independent bernoulli random variables eij pij convenience one may reparameterize embeddings eij pij matrix iid random thresholds distributed uniformly interval model objective recall use case short binary codes retrieve shortlist nodes similar query node one merely look entries indexed nearby locations embedding space seek model hamming distance embeddings monotonically reflects likelihood connection nodes embeddings natural simple choice used instance mikolov treat conditional link probability nodes cosine distance embeddings exp softmaxj eit exp translate setting binary vectors natural substitute hamming distance cosine distance distance limited taking values set transformation required empirically find complex transformations unnecessary purpose learning good simple linear scaling suffices softmaxj adh eit represents hamming distance potentially negative scaling parameter given model seek maximize expected log likelihood observed graph edges log softmaxj aeit expression unfortunately introduces two obstacles softmax requires summation candidate nodes expectation functional presents difficulties optimization first addressed means noise contrastive estimation gutmann detailed sec second addressed via several approximation techniques detailed sec noise contrastive estimation nce sidestep softmax summation follow steps mnih kavukcuoglu mnih employ nce gutmann whose minimum coincides specifically softmax normalization replaced learnable parameter one instead optimizes model specifically complex choice transformation improve optimization objective find consistent significant improvement test set metrics reported sec essentially beyond point parametrization mapping appears improve probabilities produced forward pass network noticeably improve returns backwards pass effectiveness distinguishing true data point randomly generated noise kij objective given eadh kij log log kij kij kij negative sample drawn conditional noise distribution gutmann gutmann argue one choose noise distribution resemble data distribution closely possible experiment distributions ranging powers unigram distributions node neighborhood accordance locally closed world assumption dong dong mixtures thereof curricula transition easily identified noise complex noise models empirically find none techniques outperform uniform noise distribution either significantly consistently approximation objective objective function presents challenge optimization expectation difficult evaluate analytically argument expectation discrete reparameterization trick kingma help introduce two continuous approximations discrete random variable maneuver around difficulty first according model embedding matrix eij pij normalized hamming distance two embeddings mean independent identically distributed bernoullis eil ejl eil ejl kolmogorov strong law sen quantity converges almost surely expectation therefore sufficiently large approximated lmean log kij eadh log kij amenable optimization approximation accurate larger dimensionality recall goal learn short binary codes sharper approximation possible smaller means central limit theorem follows pik pjk pik pjk applying reparametrization trick kingma objective takes form lclt log kij log kij zero mean unit variance gaussian observe argument expectation differentiable respect parameters case common approach kingma optimizing loss lclt use monte carlo integration approximate gradient noise yields unbiased estimate gradient variance converges however monte carlo integration generally appropriate approximating higher dimensional integrals single dimensional integral numerical quadrature deterministic therefore biased significantly faster error convergence midpoint rule small accuracy improved performing quadrature respect gaussian cdf letting denote computation figure compares mean approximation quadrature normal approximation range embedding dimensionalities consider smaller embedding dimensionalities greater accuracy quadrature approximation leads lower test set log loss figure comparison bernoulli models optimized lmean lclr samples dataset left error absolute approximate loss true loss right true loss optimization discretization training set loss given lclt approximated minimized stochastic gradient descent diagonalized adagrad update rule duchi generate discrete embedding bernoulli matrix entry rounded accordance maximum likelihood experiments datasets results evaluated five datasets five percent edges dataset held test set remainder used training entities directed edges knowledge graph extracted wikipedia corpus using statistical relation extraction software castelli edges filtered one supporting location text entity relation types ignored despite filtration possesses large number spurious edges entities noisy dataset representative automatically constructed knowledge graphs commonly encountered enterprise settings bhatia wordnet entities directed edges comparatively manually constructed graph consisting relations wordnet dataset ignore edge type attribute slashdot entities directed edges flickr entities undirected edges blogcatalog entities undirected edges standard social graph datasets consisting links users respective websites leskovec tang baselines comparisons little work specifically obtaining embeddings graphs compare bernoulli embeddings quantizations three classes embeddings fiedler embeddings computed using unnormalized graph laplacian symmetric graph laplacian graph laplacian deepwalk embeddings computed using model perozzi perozzi distance embeddings distemb obtained modifying bernoulli embedding objective predict link probabilities hamming cosine distance embedded vectors note hamming distance case equivalent using bernoulli probabilities embeddings cosine distance variety interpreted deepwalk modified nce window size three different quantizations embeddings computed lsh selected due explicit goal representing cosine similarity used deepwalk variety distemb spectral hashing chosen similarity objective fiedler embeddings iterative quantization another embedding found outperform several datasets gong consider well observable predictor additionally use perform logistic regression several observable neighborhood features form indicates neighborhood specifically compute number common neighbors score variations transformations log score despite far find predictor significantly improve performance reranking results produced binary embeddings embeddings trained scale graphs consider sparsely populated latter useful via densely populated former useful reranking furthermore find quantizations real embeddings perform best dimensional embeddings quantized bit vectors sought evaluation metrics methods considered likely excel metric optimized bernoulli embeddings expected log loss deepwalk context prediction etc interest however lies node http retrieval specifically ranked list nodes returned algorithm mean average precision map commonly used relatively neutral criterion appropriate task subtlety however lies choice set evaluate map document retrieval algorithms commonly evaluate precision recall documents training set salakhutdinov lead overfitting algorithms typically make use training set documents labeled algorithms however explicitly learn labeled similarity information represented edge list adjacency matrix alternatively stated rather extrapolating similarities input text goal generalize predict additional similarities already observed measure generalization via test set precision query node retrieved node judged relevant edge query appears test set observe much smaller typically reported map edges appearing training set judged specifically small test sets value rarely expected exceed inverse average degree furthermore reporting observed results scores corresponding distemb deepwalk fiedler embeddings best observed test set score amongst variations quantizer type lsh itq graph laplacian type unnormalized symmetric random walk distance embedding hamming optimizations almost certainly represent test set overfitting present challenging baseline bernoulli embeddings bernoulli deepwalk lsh precision precision fiedler observable bernoulli distemb distemb fiedler deepwalk recall recall figure left ranking right reranking mean binary embeddings flickr test set averaged random queries real embeddings quantized bits dimensions highest test set map parameterizations shown empirical results case directly retrieving results binary embeddings bernoulli embeddings found significantly outperform various alternative binary embeddings fig table also interesting compare unquantized real embeddings last four rows table despite informational disadvantage bernoulli embeddings competitive datasets predictor achieves significantly higher map latent embedding models real binary line expectations one fact expect even better results link predictor harness predictive power without computational expense computing observable features compelling option cos similarity consistently outperformed dataset dimensionality bernoulli reranked binary bernoulli observables distemb ranked real fiedler deepwalk ranked binary wordnet slashdot flickr blogcatalog table test set map dimensions distemb deepwalk fiedler quantizations present maximum map across laplacian varieties choice distemb distance function choice quantizer bold indicates category leaders nodes retrieved binary embeddings salakhutdinov observe two computational bottlenecks play searching embedded space populate list entities computing observable features entities constraints operations locations embedded space may searched nodes subject illustrated fig documented second four rows table bernoulli embeddings found outperform alternatives finally note bernoulli objective function map evaluation center around prediction unknown links generalization metrics also translates qualitatively meaningful ranking known links table illustrates dataset notice observed quality retrieved entities roughly reflects map scores respective query new delhi roger federer bruce wayne rank distemb ministry education indonesian poet gael monfils third round second round dick grayson damian wayne gotham bernoulli delhi mumbai british india grand slam final maria sharapova rafael nadal joker batman arkham asylum bernoulli reranked delhi india indian rafael nadal french open novak djokovic batman robin joker observable delhi india alumni rafael nadal french open wimbledon batman robin joker table retrieved nodes excluding query node dataset several queries efficiency results training embedding dataset largest problem consider takes roughly per epoch intel ram validation loss typically minimized epochs table reports average time taken different methods retrieve node similar query node considering binary embedding space retrieval reduces look hashtable embeddings table note retrieval speed independent embedding dimensionality dataset size method time binary embeddings hash based retrieval ranking using observables using observables binarybruteforce nodes binarybruteforce nodes binarybruteforce nodes binarybruteforce nodes realbruteforce nodes realbruteforce nodes realbruteforce nodes realbruteforce nodes oom table time taken milliseconds different methods retrieving similar nodes given query node times reported averaged runs ran system running ubuntu ram core ghz intel xeon processor oom indicates memory extreme observables model dataset entities takes scales linearly number nodes dataset finally goldilocks solution takes embedding used obtain shortlist reranked using observables model hash based lookup way exploit binary embeddings one replaces brute force nearest neighbor search amongst real valued embeddings brute force search amongst binary embeddings memory speed advantages remain despite runtime binary embeddings take times less space real embeddings hamming distance computed much faster euclidean distance table illustrates synthetically generated embeddings graphs million nodes conclusion introduce problem learning embeddings graphs graphical analog semantic hashing sidestep difficulties optimizing discrete graph embedding problem reformulated learning continuous parameters distribution discrete embedding sampled bernoulli embeddings correspond simplest distribution find computable appropriate approximations sampling techniques variety datasets addition memory time efficient bernoulli embeddings demonstrate significantly better precision recall alternative hashed real embeddings performance gap continues hold retrieval binary embeddings followed reranking powerful cumbersome link predictor latter case precision recall rival exceed link predictor performing retrieval time less independent data set size references adamic lada adamic eytan adar friends neighbors web social networks bhatia sumit bhatia alok goel elizabeth bowen anshu jain separating wheat chaff relationship ranking algorithm semantic web eswc satellite events pages blanco roi blanco berkant barla cambazoglu peter mika nicolas torzec entity recommendations web search semantic pages springer bordes antoine bordes jason weston ronan collobert yoshua bengio learning structured embeddings knowledge bases conference artificial intelligence castelli vittorio castelli hema raghavan radu florian han xiaoqiang luo salim roukos distilling exploring nuggets corpus proceedings international acm sigir conference research development information retrieval pages acm moses charikar similarity estimation techniques rounding algorithms proceedings annual acm symposium theory computing pages acm cukierski william cukierski benjamin hamner yang features supervised link prediction neural networks ijcnn international joint conference pages ieee dong xin dong evgeniy gabrilovich geremy heitz wilko horn lao kevin murphy thomas strohmann shaohua sun wei zhang knowledge vault approach probabilistic knowledge fusion proceedings acm sigkdd international conference knowledge discovery data mining pages acm duchi john duchi elad hazan yoram singer adaptive subgradient methods online learning stochastic optimization journal machine learning research gong yunchao gong svetlana lazebnik iterative quantization procrustean approach learning binary codes computer vision pattern recognition cvpr ieee conference pages ieee gray wayne gray deborah 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updating silent speech challenge benchmark deep learning yan licheng liu hongcui wang zhilei liu zhibin niu bruce denby tianjin university tianjin china abstract silent speech challenge benchmark updated new results obtained deep learning strategy using input features decoding strategy original article word error rate obtained compared published value additional results comparing new features original features reduced dimensionality well decoding scenarios two different language models also presented silent speech challenge archive updated contain original new features addition original raw data index speech interface multimodal speech recognition deep learning language model introduction speech interfaces challenges silent speech interface ssi defined device enabling speech processing absence exploitable audio signal example speech recognition obtained exclusively video images mouth electromyographic sensors ema glued tongue classic applications targeted ssis include persons lost ability vocalize illness accident yet retain ability articulate speech communication environments silence either necessary desired responding cellphone meetings public places without disturbing others avoiding interference call centers conferences classrooms private communications police military business personnel ssi concept first identified outgrowth speech production research tandem proliferation use cellular telephones special issue speech communication ssis based seven different sensor types presented mhz range medical ultrasound video imaging tongue lips surface electromyography semg sensors applied face neck electromagnetic articulography ema sensors attached tongue lips jaw vibration sensors placed head neck murmur microphones nam placed neck eeg electrodes cortical implants ssi figure overview ssi showing sensors automatic speech recognition asr followed speech synthesis retained phonetic text digital representation depending application technology ssis initially stood somewhat apart main body speech processing standard techniques intrinsically associated audio signal nevertheless novelty ssi concept exciting range applications perhaps aided accrued interest speech processing gradually allowing ssi technology join speech processing main stream activity ssi research remained strong since publication received best paper award recent survey literature reveals dozens publications ssi systems using original seven technologies mentioned also two additional ones namely low frequency ultrasound micropower radar despite activity ssis today remain part specialized laboratory instruments performance automatic speech recognition asr system often characterized word error rate wer expressed percentage total number words appearing corpus date ssi asr system able achieve wer parity acoustic asr indeed number practical issues make ssi asr systems considerably involved implement acoustic counterparts sensor handling acoustic asr may amount routine microphone protocol ssis sensors often rather specialized expensive require physical contact minimum careful placement respect speech biosignalproducing organs introduces problems invasiveness sensor placement bringing added complexity ssi experiments interference ssi principle silent certain ssi modalities vibration sensors radar low frequency ultrasound example actually associated signals propagate beyond area utilization ssi possibility interference interception may limit adoption modalities outside laboratory feature extraction easily calculated mel frequency cepstral coefficients mfcc acoustic asr features choice decades feature selection specialized sensors ssis remains open question particularly since many ssi modalities ultrasound imaging eeg example much higher intrinsic dimensionality simple acoustic signal furthermore identification stable phonetic signatures acoustic data today mature field existence salient landmarks speech biosignals arising imaging modalities electromyography example less evident medical operating mhz frequency range propagate outside body well established documented technique speech production speech pathology research whose first use context ssis discussed also relatively modality requiring transducer placed speaker chin coupled small video camera front mouth capture lip movement sensors easily accommodated lightweight acquisition helmet thus minimizing sensor placement issues tongue imaging added lip video thus many ways attractive modality building practical ssi silent speech challenge benchmark lip video ssi trained timit corpus achieved aid language model single speaker wer correct word rate independent test corpus representing promising early ssi result benchmark continuous speech recognition task subsequently raw image data original tongue ultrasound lip videos made available online silent speech challenge ssc archive purpose archive provide stable data set newly developed feature extraction speech recognition techniques applied ssc data serve basis experiments reported article although wer ssi trained timit corpus promising must remembered standard acoustic asr obtain similar superior scores training full acoustic timit corpus much challenging task advances thus still necessary order truly put silent speech recognition ssr par acoustic asr past several years improvements acoustic speech recognition using deep neural networkhidden markov model systems rather traditional gaussian mixture modelhmm become common approach deep learning strategy used improve estimation emission probabilities hmm used speech decoding natural ask extent approach improve ssr performance well despite ssi implementation challenges outlined earlier applications deep learning techniques ssr indeed begun appear example tests reported phonetic feature discrimination emgbased ssi without small speech corpus deep learning ema based ssi explored giving ssr phone error rates per lower wer corpus used training testing development specific bigram applied ssc benchmark data albeit wer study comparing efficacy different feature extraction methods present article reports first application approach ssc recognition benchmark using input features decoding strategy reported thus allowing direct comparison performances ssr results obtained significantly improved compared archive giving best scenario wer correct word recognition rate nearly threefold improvement benchmark value contrast furthermore used also employed developed completely independent speech corpus adition results second less included present article finally tests reduced dimensionality feature vectors well completely new input features created raw ssc archive data also reported new features added ssc archive future use researchers remainder article details ssc data acquisition system description available archive data first summarized section section describes feature extraction strategy developed present study full details based speech recognition procedure appear section results summarized section conclusions perspectives future work outlined final section ssc data acquisition archive resources ssc data acquisition system consisted acquisition helmet holding element mhz probe tongue imaging black white video camera capture lips pixel tongue images pixel lip images created system acquired synchronized manner frames per second fps using ultraspeech multisensory acquisition system ssc training corpus consists lip video data single native english speaker pronouncing utterances lists sentences timit corpus punctuation manner speech recorded silently without vocalization therefore audio track test set comprised one hundred short sentences selected corpus read speaker data available web address indicated archive initially contained raw ultrasound lip images training test sets original features used well feature vectors new features created present article see section appended speech recognition challenge data carried standard scheme made use also included archive details appear section feature extraction introduction mentioned earlier speech recognition sensor data faces problem discovering effective feature recognition strategy lip video ssis although attractive many ways share drawback based images intrinsic input dimensionality may order million pixels means feature extraction thus critical following discussion centered tongue features lip features much easier handle overall coherence extracted way tongue features contour extraction approach tongue contour extraction tempting choice reducing dimensionality retains visual interpretability features ultrasound imaging tongue boundary upper surface tongue produces bright continuous contour referred scan sagittal contour image processing tools automatically extracting characterizing contour make ultrasound imaging powerful tool study speech production unfortunately despite extensive literature techniques extracting tongue contours ultrasound data see references therein tongue contour tracking remains extremely challenging task high level speckle noise ultrasound images multiplicative noise arising coherent nature ultrasound wave coupled variations acoustic contact transducer speaker skin blocking ultrasound wave hyoid bone jaw poor reflectivity muscle fibers certain orientations tongue lack complete echogenic tissue path parts tongue surface particular tongue tip often result sagittal contours incomplete contain significant artifacts even totally absent even imperfect contours remove tedium valuable qualitative studies difficult integrate information coherent way labeled training datasets intended machine learning tasks speech recognition consequence ssis tended use projective feature extraction techniques rather contour finding work performed thus far principal component analysis pca discrete cosine transform dct methods choice pca dct approaches pca used ssis eigentongues approach wherein ultrasound image represented linear combination set orthogonal determined training set representative images first eigentongues found sufficient represent discriminative power contained ultrasound images dct widely used lossy image compression based notion information image concentrated lower spatial frequencies note dct direct multiplicative transform related fourier transform make use training set technique calculating dct coefficients presented next section article ssc archive based found dct features provided substantially better recognition scores well faster execution times eigentongue approach result leads important consequence ssc benchmark refers recognition scores obtained using dct features note addition original eigentongue features longer available consequently quantitative comparison approach analysis used original benchmark major impetus article must make use identical dct features baseline result ssc archive dct features constructed following way first fixed regions interest roi tongue lip images resized pixels resizing necessary order keep number dct coefficients tractable image matrix size dct computed cos cos dimensionality reduction achieved retaining lowest frequency dct components feature size selected based performance comparisons acoustic speech recognition usual concatenate first derivative mfcc feature vector vector procedure also carried dct features archive thus creating feature vector tongue lip frame new features created deep auto encoder although dct features provided promising recognition results lip video ssis necessary make certain compromises extracting notably resizing original images calculating retaining small fixed number dct coefficients computational tractability issues prevent present removing first restriction presence raw tongue lip data ssc archive allows consider taking closer look second one first interesting examine appearance tongue lip images reconstructed using dct coefficients example result frames given figure figure original lip top row tongue third row images compared reconstructions second fourth rows respectively using dct coefficients although lip reconstructions sufficiently clear distinguish overall degree mouth opening acoustically pertinent quantity visual fidelity tongue images rather poor information tongue images necessary distinguishing different acoustic configurations evidently coded dct way retain high level visual fidelity tantalizing ask however whether one might better creating original images present archive new feature representation reduces dimensionality explicitly preserving visual fidelity rather relying somewhat arbitrarily placed cut spatial frequency space done dct deep auto encoder dae used explore possibility deep neural network used reducing dimensionality learning representation input data contains encoder decoder symmetrically arranged code layer shown figure action encoder defined activation function sigmoid function weight matrix bias decoder output defined dimension weight matrix equal autoencoder trained minimizing image reconstruction error computed log log training complete code layer may regarded compressed representation input suitable use feature vector details dae training procedure found figure architecture dae obtain new features rois selected resized computational tractability purposes via interpolation lip tongue pixel arrays form inputs dae tests various architectures network chosen number inputs lips tongue desired dimensionality created feature vector intermediate figures number neurons layer features calculated encoder symmetric decoder networks trained lists images selected random ssc timit training corpus reconstructed images bottom row panel tongue lips compared original images top row panel figure show results using dae features respectively figure shows remarkable visual fidelity obtained using dae features contrast images reconstructed using dct features shown previously barely recognizable even dimensional features although one may ask extent dae solution similar pca case seen later ssr results allows obtain nonetheless intriguing figure original top row panel reconstructed bottom row panel images tongue lips using two dimensionalities dae features approaches past years approaches pattern recognition speech signal image processing based convolutional neural networks cnn proven effective cnn multilayer neural network consisting multiple shared weights overlapping receptive fields alternated pooling layers reduce dimensionality retaining subset afferent inputs use shared weights across different instances identical greatly reduces number weights learned thus allowing training cnn remain relatively tractable cnns thought able learn hierarchy features progressively higher order information pass input output network recently cnn begun make entry field ssi actually lipreading application cnn trained transform video frames large video database directly synthesized speech using video sound track create type training labels cnn trained recognize phonetic targets tongue video lip images single speaker database using phonetically labeled sound track ground truth speech recognition task cnn used recognize tongue gestural targets tongue images isolated phoneme nonsense word recognition task latter reference extensive use made data augmentation increase size training set often concern using cnn require large training sets effective due large number weights must learned conceivably cnn technique could applied raw images ssc archive try improve dct dae features tested thus far archive contains sound track however cnn feasible cnn training take place conjointly hmm probabilities study possibility appear upcoming article speech recognition system overview kaldi deep learning toolkit used build ssr system whose overall architecture illustrated figure features extracted archive data first normalized zero mean unit variance mean variance normalization mvn figure features mvn monophone dnn triphone triphone figure overall ssr training procedure ssc benchmark htk used perform speech recognition using standard architecture order ensure meaningful comparison possible benchmark result without actually htk recognition kaldi performed first using gmmhmm procedures used asr adapted standard recipes acoustic speech recognition deep learning described kaldi acoustic model training stage name acoustic model retained even though input feature data used monophone model first trained using combined tongue lip feature vectors type dct dae dimension alignment monophone used training stage features also included subsequent phase model created using alignment result applying linear discriminant analysis lda maximum likelihood linear transformation mllt methods replace features appearing produce new feature vector dimension monophone acoustic models trained consecutively time using previous model alignment training alignment used deep belief network dbn implemented illustrated figure using dimensional feature vectors inputs restricted boltzmann machines rbm cascaded means weight vector figure top layer dbn softmax output layer transition probabilities hmm trained previous phase system parameters summarized table including total numbers gaussians tied state regression tree search space beam weight acwt parameters dnn training operates two phases phase rbms trained using contrastive divergence algorithm hidden layers rbms made units learning rate except first learning rate second phase training set used train dnn optimizing remaining training used test weights learned phase used initialize dnn model dnn architecture implemented cuda gpu machine figure dnn structure used ssr table ssr system parameters monophone regression tree leaves regression tree leaves number hidden layers units per hidden layer number hidden layers units per hidden layer beam acwt dnn pretrain dnn training model lexicon issues used decoding stage ssc benchmark derived fixed subset wall street journal wsj text corpus also adopted tests obtaining realistic wer scores small corpora however problematic using closed vocabulary case tends towards underestimation attainable wer hand unbiased lexicon derived exclusively small training set might contain words present test set thus leading overly pessimistic wer estimate help address issues second estimate achievable wer data also made using another less namely contains newswire text wsj san meteor associated press along spontaneous dictation journalists hypothetical news articles results appear next section results analysis table shows comparison kaldi results corpus ssc benchmark using dct input features decoding strategy formula used wer number insertion error number deletions number substitutions total number words test set although test htk repeated fact quite similar results obtained using kaldi column provides reassurance figures obtained using kaldi reasonable table shows strategy reduced wer almost factor compared benchmark table comparison original htk result using dct features error htk kaldi kaldi ssc benchmark hmm hmm insertion deletion substitution number correct rate wer words correct words perform tests proposed section procedure repeated using alternate shown table feature vectors types dct dae one notes first dct dae features give similar performances barring monophone case return point discussion tables nonetheless although wer performance obtained less specific somewhat worse expected still significantly better ssc benchmark types features table comparing results different feature vectors types wer dct dae monophone dnn monophone dnn explore different types input features dct dae feature vectors dimension visual modality also tested results given table wsj table csr overall higher scores obtained task specific expected tables also show similar results obtained two types features dct slightly better dimensionality input vectors however dct features longer salient dae retains effectiveness thus although dae completely successful simultaneously optimizing saliency low dimensionality results furnishes intriguing suggest may possible better sophisticated approach table recognition results wsj wer dct dimension monophone dnn wer dae dimension monophone dnn table recognition results csr wer dct dimension monophone dnn wer dae dimension monophone dnn conclusions perspectives confrontation ssc recognition benchmark ssr techniques using kaldi deep learning package led improvement wer almost factor favorable scenario thus helping establish highly attractive ssi modality tests performed using original wsj less csr give wer values data using significantly improved compared benchmark tests kaldi also used test architecture order demonstrate compatibility methods used benchmark new features derived raw benchmark data using dae give results slightly worse obtained original dct features retaining effectiveness even low dimensionality new original features appended ssc benchmark data results promising ssr still remains somewhat less accurate acoustic speech recognition work necessary future ssc benchmark interesting experiment feature extraction strategies example convolutional neural networks cnn might allow step information may lost skipped ssi generally interesting accumulate much larger possible multispeaker data sets mentioned problems associated small speech data sets may avoided acknowledgements research supported national nature science foundation china foundation china ministry education references denby schultz honda hueber brumberg silent speech interfaces speech communication hueber benaroya chollet denby dreyfus stone development silent speech interface driven ultrasound optical images tongue lips speech communication volume issue april schultz wand modeling coarticulation continuous speech recognition speech communication volume issue april brumberg kennedy guenther interfaces speech communication speech communication volume issue april hirahara otani shimizu toda nakamura nakajima shikano enhancement system utilizing 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relaying system optics channels aug jaedon park member ieee chae senior member ieee giwan yoon member ieee paper analyze performance twoway subcarrier sim relaying system optics fso communication channels analysis takes consideration attenuations due atmospheric turbulence geometric spread pointing errors time derive generalized infinite power series expressions tight upper lower bounds overall outage probability average probability errors system study finds subcarrier intensitymodulated relaying system using binary phase shift keying bpsk modulation could used practical applications case weak turbulence regime required snr obtain average bit error probability also noted pointing errors clearly degrade performance subcarrier relaying system relay atmospheric turbulence average probability error optics overall outage probability pointing errors subcarrier intensity modulation twoway relaying system upper lower bounds introduction optics fso systems communication techniques free radio frequency spectrum regulations attracted enormous amounts scholarly attention although intensity modulation direct detection system using keying ook widely used due simplicity system appropriate applications relaying systems indeed system requires adaptive decision threshold practice difficult implement circumvent implementation difficulty consider paper subcarrier intensity modulation sim scheme scheme requires adaptive decision threshold ameliorates irradiance fluctuation therefore subcarrier intensity modulation scheme suitable relaying systems fso channels research part supported agency defense development add basic science research program national research foundation korea nrf funded ministry education grant park agency defense development add daejeon korea jaedon yoon school electrical engineering korea advanced institute science technology kaist daejeon korea gwyoon chae school integrated technology yonsei university korea cbchae fso systems highly affected attenuations caused atmospheric turbulence geometric spread pointing errors result variations refractive index fading also known scintillation causes irradiance fluctuations received signals intensity apart scintillation effects pointing errors due building sway also cause significant performance degradations fso systems addition performance degradation caused atmospheric turbulence pointing errors fso communication systems suffer significant degradation environments overcoming problems left relaying technologies relaying systems classified either relays relaying systems also called regenerative systems decode received signal fully reencode retransmitting another hop relaying systems also called nonregenerative systems amplify received signal forward another hop less complexity relaying systems since relaying systems decode received signal need less power lower system complexity relaying systems recently enormous amount research interest devoted relaying system techniques conventional applications interest largely due technique even efficient signaling scheme two nodes communicate two phases via relay resulting improvement spectral efficiency upadhyay studied performance opportunistic relaying system analog network coding fading channels guo analyzed overall outage probability well symbol error probability relaying system exponential distribution jang investigated performance multiuser relay channel han analyzed tight upper lower bounds average sum rate relaying system alamouti orthogonal space time block code ostbc considering gamma distributions yang analyzed performance relaying system fading channels ikki analyzed performance relaying presence cochannel interferences rayleigh fading channels case general relaying system relaying system attractive practice twoway relaying system due simple processing relay terminal also similarly conventional relaying systems relaying systems able lower effectively system power consumption complexity relaying systems case relaying systems fso applications tang proposed cooperation relay scheme optical relay networks investigated optimal bit decision algorithm receiving node fading distributions puri investigated performance fso system considering protocol distribution distribution also proposed analyzed relay selection protocols fso relays assuming distribution pointing errors authors later analyzed relay selection derived outage probability average error probability expressions studied fso relay systems using difficult gain immediate insight analyzed performances fso systems since error probabilities fso channels usually expressed complex meijer due modified bessel function second kind moreover performance analysis large challenge especially case relaying systems fading distribution complex form meijer novel approach simplify mathematical expressions introducing generalized infinite power series representation modified bessel function second kind representation express error probabilities fso systems power series expansions composed elementary gamma functions offers readers mathematical insight fso systems paper analyze first time performance relaying system using subcarrier intensity modulation scheme fading environments simple generalized infinite power series expression analyze fso communication channels considering attenuations due atmospheric turbulence geometric spread pointing errors main contributions paper follows overall outage probability analysis relaying system fso channels derive upper lower bounds overall outage probability relaying system fading channels considering attenuations due atmospheric turbulence geometric spread pointing errors average probability error analysis relaying system fso channels based derived results overall outage probability upper lower bounds derive average probability errors subcarrier relaying system fading channels considering attenuations due atmospheric turbulence geometric spread pointing errors rest paper organized follows section discuss system channel model relay photo detector photo detector amplifier bias obpf obpf laser driver terminal laser driver bias terminal obpf obpf laser driver photo detector photo detector bpsk demodulator bpsk demodulator output data output data bpsk modulator input data fig bias bpsk modulator input data sim relaying system using bpsk modulation subcarrier relaying system fading distributions section iii derive generalized infinite power series expressions upper lower bounds overall outage probability system consideration section derive average probability errors corresponding upper lower bounds overall outage probability given section iii sections present numerical results conclusion ystem channel model system model consider relaying system described fig two source terminals communicate along relay terminal using optical subcarrier intensity modulation scheme binary phase shift keying bpsk modulation simplicity throughout paper assume bpsk modulation modulation schemes however also applicable channels assumed stationary independent identically distributed intensity fading statistics channel state information csi also assumed available receiver terminal source data modulated onto subcarrier signal using bpsk modulation signal added bias signal drive laser diode positive values relay received optical signal radiated terminal passes optical band pass filter obpf reject background radiation noise finally photodetector generates photocurrent proportional similarly photodetector generates photocurrent proportional subcarrier signal park relaying system radiated source terminal two signals added photocurrent given photodetector responsiveness irradiance terminal relay irradiance terminal relay constant fultill additive white gaussian noise awgn mainly due thermal background noise channel state considered product two random factors iia iip iia attenuation due atmospheric turbulence modeled fading distribution iip attenuation due geometric spread pointing errors noted channel model random variable described detail following section components received photocurrent filtered photocurrent given relay amplifies electrical signal gain given meet average transmit power constraints amplified electrical signal added bias signal drive laser diode positive values finally relay retransmits optical signal terminals represent subcarrier signal powers relay terminal respectively terminal passing optical band pass filter received photocurrent given ybra awgn substituting filtering component subtracting parts received photocurrent rewritten ybra substituting without loss generality assume received snr terminal expressed snr absence turbulence pointing errors similarly received snr terminal expressed turbulence misalignment fading model assumed random variable iia follows distribution using generalized power series representation method probability density function pdf cumulative distribution function cdf random variable iia expressed fiia iia lim iia iia sin atmospheric turbulence parameters exp fiia iia lim iia exp optical wave number wavelength diameter receiver collecting lens aperture link distance meters index refraction structure see details pdf iip considering circular detection aperture radius gausssian beam given fiip iip iip wzeq ratio equivalent beam radius receiver pointing error displacement standard deviation jitter receiver parameter relation wzeq calculated using erf exp erf error function beam waist radius calculated distance combined pdf iia iip given fii fii fiia iia diia fii conditional probability given iia state expressed fiip fii iia iia iia iia iia fig rectangular integral region substituting rewritten fii lim iia iia diia fig integral region oabc two random variables assumed independent necessarily identically distributed gammagamma fading hence rectangular integral region rewritten calculating definite integration derived fii lim lim easily confirmed goes nonpointing errors case converge respectively iii overall outage probability analysis analyze overall outage probability first study integral rectangular region described fig lemma integral rectangular region rectangular integral region expressed proof random variable define new random variable integral rectangular region composed shown fig written since cdf given integrating pdf cdf obtained fii fxi fii fii substituting obtained subcarrier relaying system overall outage probability defined pout min threshold snr let outage probability rewritten pout upper probability difficult derive described fig integral region calculation oabc analyze upper lower bounds instead exact form park relaying system upper bound overall outage probability theorem upper bound overall outage probability given pout pout proof integral region oabc lower bounded integral region obtained lemma follows thus integral region oabc lowerbounded substituting obtained also cdf min approximated outage probability upper bound variable change thus integral region oabc upperbounded substituting obtained also cdf min approximated outage probability lower bound variable change average robability rror nalysis section average probability error derived subcarrier relaying system let denote conditional error probability awgn channel average probability error expressed lower bound overall outage probability theorem lower bound overall outage probability given conditional error probability given pout pout proof integral region oabc upper bounded oaebf integral region oaebf expressed oaebf integral regions obtained lemma follows bpsk modulation substituting rewritten cdf random variable variable change average probability error given theorem substitute obtain average probability errors corresponding upper lower bounds overall outage probability lim lim lim lim overall outage probability str low str mod low mod weak low weak sim fig overall outage probability sim relaying system set given lim lim lim lim proof see appendix umerical results section present numerical results derived overall outage probability upper lower bounds average probability errors corresponding bounds overall outage probability subcarrier intensitymodulated relaying system considered using bpsk modulation fso channel fso channel modeled fading distribution atmospheric turbulence parameters strong turbulence regime parameters moderate turbulence regime weak turbulence regime pointing errors normalized jitter normalized beamwidth considered numerical evaluations parameter used infinite power series truncated figs show overall outage probability performances subcarrier relaying system respect snr absence turbulence pointing errors threshold snrs respectively consider pointing errors set according fig given threshold snr obtain overall outage probability required snr values upper bounded strong moderate weak turbulence regimes also seen figure obtain overall outage probability required snr values lower bounded strong moderate weak turbulence regimes according fig given threshold snr obtain overall outage probability required snr values upper bounded strong moderate weak turbulence regimes also seen figure obtain overall outage probability required snr values lower bounded strong moderate weak turbulence regimes park relaying system overall outage probability overall outage probability str low str mod low mod weak low weak sim str low str mod low mod weak low weak sim fig overall outage probability sim relaying system set fig overall outage probability sim relaying system pointing errors set black color red color blue color overall outage probability overall outage probability str low str mod low mod weak low weak sim fig overall outage probability sim relaying system set according fig given threshold snr obtain overall outage probability required snr values upper bounded strong moderate weak turbulence regimes also seen figure obtain overall outage probability required snr values lower bounded strong moderate weak turbulence regimes clearly seen figs derived upper lower bounds gaps tight various atmospheric turbulence regimes snr scenarios also present simulation results exact outage probability evaluate tightness upper lower bounds analyzed outage probability given fig overall outage probability different normalized beamwidth normalized jitter upper bound strong regime figs present overall outage probability performances subcarrier relaying system considering pointing error effects fig shows overall outage probability performances various values normalized beamwidth threshold snr set pointing errors normalized jitter set fig depicts upper bound performance overall outage probability strong regime varying normalized beamwidths normalized jitters time threshold snr set snr set clearly seen figures better overall outage probability performance achieved using narrow beamwidth small jitter average probability error overall outage probability str low str mod low mod weak low weak sim fig convergence characteristics overall outage probability respect without pointing errors upper bound moderate regime fig average probability error relaying system sim bpsk pointing errors set black color red color blue color average probability error average probability error str low str mod low mod weak low weak sim fig average probability error relaying system sim bpsk set fig average probability error different normalized beamwidth normalized jitter upper bound strong regime fig illustrates convergence performance derived overall outage probability respect varying number power series threshold snr set upper bound outage probability considered moderate regime observed figure outage probability converges high snr fig shows average probability error performances subcarrier relaying system respect snr absence turbulence pointing errors figure simplicity sake consider pointing errors described section since average probability errors derived based cdf functions outage probability upper lower bounds performances also show upper lower bounds seen figure obtain average probability error bpsk subcarrier intensity modulation scheme required snr values bounded strong moderate weak turbulence regimes also clearly seen figure derived upper lower bounds gaps various atmospheric turbulence regimes snr scenarios tight similar results outage probabilities noted subcarrier relaying system using bpsk modulation could used practical applications case weak turbulence regime required snr obtain average park relaying system average probability error snr obtain average probability error also noted performance subcarrier relaying system clearly degraded pointing errors believe proposed system could combined technologies leave future work future work includes ppendix fig convergence characteristics average probability error respect without pointing errors upper bound moderate regime symbol error probability figs present average probability error performances subcarrier relaying system considering pointing error effects fig shows average probability error performances various values normalized beamwidth pointing errors normalized jitter set fig depicts upper bound performance average probability error strong regime varying normalized beamwidths normalized jitters time snr set clearly seen figures better average probability error performance achieved using narrow beamwidth small jitter fig shows convergence performance derived average probability error respect varying number power series pointing error considered upper bound average probability error considered moderate regime observed figure series term enough make average probability error converge high snr onclusion paper derived upper lower bounds overall outage probability average error probability subcarrier relaying system fso channels considering attenuations caused atmospheric turbulence geometric spread pointing errors time derived performances based generalized infinite power series expressions model gammagamma fading distribution fso channel according analysis results overall outage probability average probability error gaps upper lower bounds noted twoway subcarrier relaying system using bpsk modulation could used practical applications case weak turbulence regime required substitute average probability errors corresponding upper lower bounds overall outage probability expressed respectively using average probability errors corresponding upper lower bounds overall outage probability expressed substitute given substitute derived respectively lim lim apply gamma function finally obtained given substitute product cdfs expressed lim lim lim lim lim lim lim lim product cdfs expressed lim apply multiplication power series lim park relaying system substitute product cdfs derived sandalidis tsiftsis karagiannidis uysal ber performance fso links strong atmospheric turbulence channels pointing errors ieee comm vol sandalidis tsiftsis karagiannidis optical wireless communications heterodyne detection turbulence channels pointing errors lightwave vol lim lim 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vol partial selection protocols relayed fso networks lightwave vol puri chatzidiamantis garg aggarwal karagiannidis relay selection multiple relayed fso networks ieee wireless comm vol puri garg aggarwal asymptotic analysis twr assisted fso links partial selection ieee comm vol ghassemlooy aslam physical layer network coding relay free space optical communication link internet technologies applications ita ieee bayaki schober mallik performance analysis mimo optical systems fading ieee trans vol simon alouini digital communication fading channels john wiley sons gradshteyn ryzhik table integrals series products jeffrey zwillinger eds academic press chung sim kim kim chae prototyping full duplex radios ieee comm mag vol lim chae caire performance analysis massive mimo users ieee trans wireless vol sim park chae heath compressed channel feedback correlated massive mimo systems jour comm networks vol yilmaz koo park park han chae frequency assignment problem net filter discrimination constraints appear jour comm networks place photo jaedon park received degree electronics engineering hanyang university seoul korea degrees school electrical engineering korea advanced institute science technology kaist daejeon korea respectively currently senior researcher agency defense development add daejeon korea research interests mimo systems relay systems fso systems chae underwood distinguished professor school integrated technology college engineering yonsei university korea member techplace nical staff research scientist bell laboratories photo murray hill usa joining bell laboratories school engineering applied sciences harvard university cambridge usa postdoctoral research fellow received degree electrical computer engineering university texas austin usa member wireless networking communications group wncg prior joining research engineer telecommunications center samsung electronics suwon korea worked samsung participated ieee mobile wimax standardization made several contributions filed number related patents current research interests include capacity analysis interference management wireless mobile networks nano molecular communications editor ieee omm ieee ireless omm etters ieee rans ireless omm ieee rans olecular iological ulti scale omm ieee rans mart rid omm ets guest editor ieee reas omm special issue molecular biological ieee senior member chae outstanding teaching award yonsei university underwood distinguished professor award yonam research award yonam foundation best young professor award college engineering yonsei university ieee infocom best demo award joint award young engineer year haedong young scholar award ieee signal processing magazine best paper award ieee comsoc outstanding young researcher award ieee vts dan noble fellowship award gold prize humantech paper contests scholarship also received korea government fellowship kosef studies park relaying system giwan yoon received degree seoul national university snu seoul korea degree korea advanced institute science technology kaist seoul korea place degree university photo texas austin usa employed engineer group seoul korea worked development bipolar transistors employed senior engineer digital equipment corporation dec usa developed oxynitride gate dielectric cmos devices faculty member information communications university daejeon korea developed devices wireless communications since kaist currently professor school electrical engineering teaching research activities areas nano devices integrated systems energy generation harvesting devices sensing devices healthcare iot sensor networks applications yoon member ieee
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complexity analysis presence control operators functions oct ugo dal lago giulio pellitta abstract polarized version girard scedrov scott bounded linear logic introduced normalization properties studied following laurent logic naturally gives rise type system whose derivations reveal bounds time complexity underlying term first example type system guaranteeing time complexity bounds typable programs introduction among properties programs bounds amount resources like computation time space programs need executed particularly significant problem deriving bounds indeed crucial systems undecidable whenever programming languages considered units measurement become concrete close physical ones problem becomes even complicated architecturedependent typical example one wcet techniques adopted systems need deal many machine instructions program corresponds also much time instruction costs executed possibly complex architectures including caches pipelining etc task becoming even harder current trend towards multicore architectures different approach consists analysing abstract complexity programs example one take number instructions executed program measure execution time course less informative metric however becomes accurate actual time taken instruction kept low one advantage analysis independence specific hardware platform executing program hand latter needs analysed variety complexity analysis techniques employed context abstract interpretation type systems program logics interactive theorem proving properties programs written functional languages various reasons wellsuited verified way type systems includes safety properties welltyped programs wrong complex ones including resource bounds paper delineate methodology complexity analysis programs control operators latter constructs available concrete functional programming languages including scheme ocaml allow control flow ways technique introduce takes form type system groote derived girard scedrov scott bounded linear logic bll following prove sound typable programs indeed reduced number steps lesser equal polynomial bound read underlying type derivation similar result given cost model one induced abstract machine authors knowledge first example complexity analysis methodology coping well functions also control operators rest section explain crucial role linear logic work meantime delineating main features linear logic complexity analysis linear logic one successful tools characterizing complexity classes setting correspondence subsystems indeed shown correspond polynomial time computable functions logarithmic space computable functions many introduced fragments turned type systems relatively complete intensional sense reason success lies way linear logic decomposes intuitionistic implication linear implication low complexity exponential modality marks formulas structural rules applied gives proper status proof duplication without performed linear number steps tuning rules governing exponential modality one define logical systems performed within appropriate resource bounds usually coupled encoding functions complexity class system hand makes system characterization rules governing exponential modality constrained least two different ways one hand one rules governing dereliction digging dropped restricted happens example light linear logic soft linear logic logic refined enriched control number times structural rules applied words rules still refined form happens bounded linear logic similarly one could control modal impredicativity system levels first approach corresponds cutting space proofs axe many proofs among many corresponding efficient algorithms part system require one forbidden logical principles second approach milder terms class good programs left behind strong evidence approach one obtain quite expressive logical system much known whether approach scales languages functions also continuations control operators present understanding impact features complexity programs interesting research topic however received little attention past linear logic control operators hand twenty years passed since classical logic shown amenable paradigm interestingly enough classical axioms pierce law law excluded middle seen type control operators like scheme callcc meantime various facets new form correspondence investigated detail many extensions classical logic naturally provides type discipline introduced moreover decomposition provided linear logic known scale classical logic actually linear logic known admit involutive notion negation inception satisfying embedding classical logic linear logic however requires restricting latter way polarities way one left logical system desirable dynamical properties paper define bllp polarized version bounded linear logic kind enrichment resource polynomials provide bll shown cope well polarization following close relationship polarized linear logic bllp gives rise type system proofs typable shown reducible normal forms number steps bounded polynomial weight result former translates similar result latter since reduction step corresponds one reduction steps proofs analysis extended reduction abstract machine bounded polarized linear logic sequent calculus section define bllp sequent calculus although section familiarity bounded polarized linear logic would certainly help polynomials formulas resource monomial finite product binomial coefficients form distinct variables integers resource polynomial finite sum resource monomials given resource polynomials write denote resource polynomial also resource polynomials closed addition multiplication bounded sums composition polarized formula formula either positive negative generated following grammar ranges countable sets atoms throughout paper formulas also terms contexts etc considered modulo formulas either positive negative ranged metavariables like formulas like sometime denoted polarized setting contraction performed negative formula consequence need notion formula namely labelling formula respect occurrences bound metavariables labellings positive respectively negative formulas respectively labelled formulas sometimes denoted metavariables polarity essential negation usual classical linear system applied possibly labelled formula morgan resource variable appear need mention writing becomes similarly space formulas space labelled formulas seen partial orders stipulating two labelled formulas compared iff exactly skeleton polynomials occurring compared formally iff iff iff iff iff iff iff iff keep mind linear logic contains subset formulas isomorphic polarized classical logic resp thought roughly resp sense think labelled formulas formulas hiding implicit exponential modality sense polynomials occurring next atoms whynot operator positive position occurring next bang operator negative position cases defined natural way iff finally iff dually iff lemma iff moreover iff proof iff proved induction structure consider second part statement suppose positive call respectively case negative similar certain operators resource polynomials lifted formulas example want able sum labelled formulas provided proper form assuming course free either construction generalized bounded sums suppose labelled formula form freepin free labelled formula see details constructions defined abstraction formula arity simply formula resource variables meant bound result substituting second order abstraction term arity free occurrences propositional variable arity defined formally induction structure interesting clauses following two sequents rules easiest way present bllp give sequent calculus actually proofs structurally identical proofs laurent llp course llp proofs legal bllp proofs giving rise exponential decorated according principles bounded linear logic sequent expression form multiset labelled formulas one among positive contains labellings negative formulas indicate metavariables like operator extended one formulas write expressions like amounts summing polynomials occurring occurring similarly bounded sums rules sequent calculus bllp figure please observe relation implicitly applied formulas polynomials whenever possible way smaller formulas always derived see section llp structural rules act negative formula exponential ones since formulas occurring sequents labelled however still keep track many times formulas used spirit bll cut figure bllp sequent calculus rules byproduct taking sequents multisets labeled formulas multiplicative rules need deal labels example consider rule resource polynomial labelling conclusion anything smaller equal polynomials labeling two premises sequent calculus introduced could extended additive logical connectives sake simplicity however kept language formulas simple already mentioned bllp proofs seen obtained decorating proofs laurent llp resource polynomials given proof llp proof obtained erasing resource polynomials occurring two bllp proofs write iff iff two decorations llp proof even structural rules applied negative formulas certain proofs copied erased along process soon realize box proof ends occurrence rule systems boxes copied erased process applied proofs inductively defined follows either last rule proof proof obtained two applying rule said closed contain axiom box auxiliary doors formula context rules malleability main reason strong intensional expressive power bll malleability conclusion proof modified many different ways without altering structure malleability crucial make system expressive also prove bllp enjoys section give four different ways modifying sequent way preserve derivability two anyway expected also hold bll two make sense polarized setting first taking smaller formulas general preserves derivability lemma subtyping proof simple induction crucial cases last rule used axiom know follows thus take know follows thus take suppose last rule used necessarily hence induction hypothesis thatp consequence simply defined derive thesis transitivity last rule used induction hypothesis immediately yields thesis concludes proof substituting resource variables polynomials preserves typability lemma substitution let proof moreover proof easy induction structure lemma proof already mentioned one key differences ordinary linear logic polarized version latter arbitrary proofs potentially duplicated erased along process former special ones namely boxes consequence fundamentally different nature structural rules two systems since bllp refinement llp means phenomenon expected beware bounded setting contraction symmetric two copies proof duplicating identical need prove proofs indeed split bllp preliminary following technical lemma lemma shifting sums formula proof fact exists implies exist consequence call last formula consequence exists equal concludes proof lemma splitting exist moreover proof induction last rule used axiom form know observe form following derivations building second one made use particular lemma last rule used write consequence thus apply induction hypothesis easily reaching thesis last rule used promotion following shape simply observe conclusion lemma allowed form consequence observe conclusions appropriate relation lemma concludes proof observe every proof split parametric version splitting also necessary lemma parametric splitting exists splitting allows cope duplication parametric splitting implies arbitrary proof modified lifted box one auxiliary doors please observe continues upper bound even natural number substituted free variables easy consequence lemma following useful dealing cuts involving rule lemma proof hypothesis consequence concludes proof cut elimination section show procedure bllp defined start showing logical cuts reduced cut logical two immediate subproofs end rule introducing formula involved cut describe logical cuts reduced critical cases figure needs explained reducing multiplicative logical cuts extensively use subtyping lemma dereliction reduction step obtained lemma conclusion lemma consequence follows proof contraction reduction step suppose apply lemma lemma obtain multiplicatives cut cut cut dereliction cut cut contraction cut cut cut digging cut cut figure logical steps digging lemma find instances cut rule logical said commutative induce relation proofs example proof cut equivalent cut way define equivalence relation space proofs general cuts proof logical cut turned logical one lemma let proof containing occurrence rule cut two proofs effectively obtained proof lemma goes follows given instance cut rule cut consider path sequence formula occurrences starting going upward inside path starting going upward inside paths end either rule instance rule introducing main connective game play show two paths always shortened way commutations thus exposing underlying logical cut lemma implicitly defining procedure given instance cut rule turn logical cut procedure lemma fire way implicitly defining another reduction relation next question following procedure going terminate every proof strongly weakly normalizing many steps take turn form actually produces reduction sequences long length anyway strongly normalizing relatively easy way prove goes follows bllp proof corresponds llp sequent calculus proof latter corresponds polarized proof net moreover implies canonical relation polarized finally identical whenever since known strongly normalizing admit infinite reduction sequences proposition relation strongly normalizing mean performed reasonably bounded time already bll take hyperexponential time whole elementary linear logic embedded soundness get soundness result somehow need restrict underlying reduction relation following one could indeed define subset imposing dereliction contraction box steps involved closed moreover could stipulate reduction external take place inside boxes closed external reduction however enough simulate able reduce scope make much sense anyway forced consider extension closed reduction fact new notion reduction still guarantees polynomial bounds technically remarkable strengthening respect bll soundness theorem quite natural notion downward path proofs occurrence negative formula proceed downward either find main premise cut rule conclusion first case occurrence said active second said passive proofs endowed new notion reduction dereliction contraction box digging cuts fired negative formula occurrences rightmost argument passive literature sometimes called special cut moreover reduction needs external usual notion reduction see enough mimic head reduction denoted next step consists associating weight form resource polynomial every proof similarly happens bll proof conclusion consists resource polynomial disjoints sets resource variables corresponding formula cause ambiguity set resource variables corresponding formula denoted similarly multiset formulas weight simply however variables substituted proof defined induction structure following rules figure please notice cut figure proofs negative formula conclusion associated fresh variables accounting application rule applied cut variables set allows discriminate case rules produce time complexity along case ultimately leads lemma main idea behind lemma even logical cut perform going dangerous contraction involved closed residual negative rules null weight passive conclude theorem polystep soundness every proof sense weight proof resource polynomial easily computed rules figure anyway inductively defined also upper bound number logical steps separating normal form please observe continues upper bound even natural number substituted free variables easy consequence lemma talking polynomial bounds bll consequence also bllp one write programs way size input reflected resource variable occurring type please refer type system describe version introduced groote terms follows figure weak head notions reduction range two infinite disjoint sets variables called respectively contrast originally formulated parigot restricted terms form notions reduction reduction rules consider following ones usual fired following weak reduction denoted reduction simply take place scope binders head reduction denoted generalization concept pure details figure please notice head reduction redexes indeed fired even lie scope however involved redex harmless restriction corresponds taking outermost reduction order needed technical reasons become apparent soon type system following laurent types negative formulas used types however particular legal type form use following abbreviation case particular abbreviated typing formulas negative formulas either form typing formulas modal formula one form typing formula please observe constructions section including labellings sums etc easily apply typing formulas finally use following abbreviation labeled modal formulas typing judgement statement form context assigning labelled modal formulas typing formula context assigning labelled typing formulas way typing judgments defined allows see bllp sequents way various concepts section lifted sequents judgments remarkably includes subtyping relation typing rules figure typing rule applications particular seen overly complicated fact premises except first two allow necessary degree malleability contexts without even subject reduction would danger alternatively one could consider explicit subtyping rule price loss syntax directness indeed malleability results section transferred defined type assignment system var abs app figure additive type assignment rules subject reduction polystep soundness aim section show head reduction preserves types corollary number reduction steps normal form bounded polynomial along lines theorem actually latter easily follow former subject reduction formulated sense proved precise correspondence type derivations proofs mind order facilitate task subject reduction proved modified system called bllpmult proved equivalent fundamental difference two systems lies structural rules contraction weakening reflected type system already noticed additive flavour since structural rules implicitly applied binary typing rules particular makes system syntax directed type derivations compact problem approach correspondence type derivations proofs weak directly lifted dynamic level one step could correspond possibly many steps bllpmult contrary structural rules explicit turns useful technical tool prove properties bllpmult typing judgments precisely ones changes typing rules figure whenever derivability one system needs distinguished derivability put system name subscript position surprisingly two bllpmult type exactly class terms lemma iff proof implication follows weakening contraction lemmas easy prove implication direct since additive var app multiplicatively derivable given bllpmult type derivation one define bllp proof following rules figure work induction structure way one gets guiding principles also prove underlying transformation process nothing lemma moreover proof obtained cutting guaranteed var abs app figure multiplicative type assignment rules figure mapping multiplicative derivations bllp proofs app abs var proof usual induction structure need careful generalize statement case simultaneous substitution many variables needed lemma moreover proof obtained tensoring axiom cutting result guaranteed theorem subject reduction let suppose moreover proof induction structure interesting cases application reduction takes place inside follows app app exists induction hypothesis omit trivial cases looks follows abs app lemma ensures required type derivation actually exists looks follows app lemma ensures exists looks follows since know derivation obtained applying subtyping lemma derivation concludes proof observe performing head reduction corresponds instead permissive following easy corollary theorem theorem theorem polystep soundness terms let let var abs app abs figure type derivation control operators section show powerful enough type natural encoding two popular control operators namely scheme callcc felleisen control operators change evaluation context expression simulated operators respectively save restore stack arguments passed subterms idea way starting point extension krivine machine groote see section extension groote calculus named satisfies separation theorem fails parigot calculus hence untyped setting original parigot strictly less expressive groote calculus callcc encoding callcc could operational behavior would expect callcc first satisfy following property see indeed second step replaces since hypothesis important observe second step replaces variable term free hence weak reduction gets stuck actually notion weak reduction even restrictive one proposed groote head reduction contrary somehow liberal moreover also straightforward check reduction callcc simulated head reduction typable answer positive derivation typing instance pierce law figure obvious derivation felleisen canonical way encode felleisen term behavior something like fresh variable indeed var abs app abs figure type derivation type derivation figure derivation worth noting weak reduction strong enough properly simulating operational behavior possible type parigot unless open term used alternatively free continuation constant must used obtaining yet another calculus one reasons picked version proposed groote calculi see discussion felleisen abstract machines theorem main result paper far tells number steps performed terms typable bounded weight underlying type derivation one may wonder however whether taking number reduction steps measure term complexity sensible substitutions involve arguments possibly much bigger original term recent work accattoli first author however shows case endowed head reduction unitary cost model polynomially invariant respect turing machines conjecture invariance results extended section show polystep sound another cost model namely one induced groote abstract machine done following similar proof pcf typed linear dependent types krivine abstract machine natural extension configurations built around environments closures stacks defined mutually recursively follows environments partial functions makes correspond closures correspond stacks metavariables environments etc closures pairs whose first component whose second component environment metavariables closure etc stacks finite sequences closures metavariables stacks etc configurations pairs whose first component closure whose second component stack indicated etc reduction rules configurations figure sound complete respect head reduction however reduction take place scope scope actually turned type system configurations closely follow laurent next step assign weight every type derivation authors aware work krivine machine derived semantically rather syntactically independently groote paper also extension machine allows reduce even paper essential purposes since abstract machine groote enough work control operators still even though important differences respect setting calculus considered untyped variant parigot might worthwhile investigate future figure transitions similarly done type derivations terms idea prove weight typable configurations decreases every transition step lemma allows generalize polystep soundness theorem polystep soundness let let please observe theorem holds particular initial configuration typable term hht conclusions paper presented evidence enrichment intuitionistic linear logic provided bounded linear logic robust enough lifted polarized linear logic paves way towards type system one hand guarantees typable terms reduced normal forms number reduction steps read type derivation allows naturally type useful control operators many questions purposely left open particular language programs pure whereas structure types minimal allowing form polymorphism expect endowing bllp second order quantification constants recursion particularly problematic although laborious extensions already considered similar settings absence control actually particularly interesting direction would turn type system ong stewart way extending linear dependent paradigm language control course outside scope paper whose purpose delineate basic ingredients logic underlying type system stressed introduction convinced work first one giving time complexity analysis methodology programming language functions one could course object complexity analysis could performed translating equivalent terms way suitable however would force programmer whomever complexity analysis deal programs structurally different original one course translations could introduce inefficiencies maybe harmless purely qualitative viewpoint could make difference complexity analysis tatsuta investigated maximum length language without rta references beniamino accattoli ugo dal lago invariance unitary cost model head reduction rta volume lipics pages zena ariola hugo herbelin minimal classical logic control operators icalp volume lncs pages springer patrick baillot paolo coppola ugo dal lago light logics optimal reduction completeness complexity information computation patrick baillot damiano mazza linear logic levels bounded time complexity theoretical computer science patrick baillot kazushige terui light types polynomial time computation lambdacalculus information computation volume pages curien hugo herbelin duality computation icfp pages acm ugo dal lago marco gaboardi linear dependent types relative completeness logical methods computer science ugo dal lago martin hofmann bounded linear logic revisited tlca volume lncs pages springer david walter theorem journal symbolic logic pages jacobus bakker arie bruin jeffrey zucker mathematical theory program correctness international series computer science prentice hall philippe groote caap volume lncs pages springer philippe groote relation syntactic theory sequential control logic programming automated reasoning pages springer philippe groote environment machine mathematical structures computer science matthias felleisen expressive power programming languages esop volume lncs pages springer marco gaboardi simona ronchi della rocca soft type assignment system lambdacalculus csl volume lncs pages springer girard linear logic theoretical computer science girard new constructive logic classical logic mathematical structures computer science girard light linear logic information computation girard andre scedrov phil scott bounded linear logic modular approach computability theoretical computer science timothy griffin notion control popl pages acm press sumit gulwani speed symbolic complexity bound analysis cav volume lncs pages springer steffen jost kevin hammond loidl martin hofmann static determination quantitative resource usage programs popl madrid spain acm press yves lafont soft linear logic polynomial time theoretical computer science olivier laurent polarisation logique doctorat aixmarseille march olivier laurent krivine abstract machine overview unpublished note september olivier laurent polarized theoretical computer science luke ong charles stewart foundation functional computation control popl pages acm press michel parigot algorithmic interpretation classical natural deduction lpar volume lncs pages springer alexis saurin separation streams lics pages ieee ulrich stratified bounded affine logic logarithmic space lics pages thomas streicher bernhard reus classical logic continuation semantics abstract machines journal functional programming reinhard wilhelm jakob engblom andreas ermedahl niklas holsti stephan thesing david whalley guillem bernat christian ferdinand reinhold heckmann tulika mitra frank mueller isabelle puaut peter puschner jan staschulat per worst case execution time problem overview methods survey tools acm transactions embedded computing systems
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discontinuous galerkin time domain framework periodic structures subject oblique excitation feb nicholas miller andrew baczewski john albrecht balasubramaniam shanker nodal discontinuous galerkin method derived analysis scattering doubly periodic structures oblique interrogation field transformations employed elaborate formalism free issues causality common applying spatial periodic boundary conditions simultaneously incident fields arbitrary angles incidence upwind numerical flux derived transformed variables retains form original maxwell problem domains without explicitly imposed periodicity conjunction amenability framework meshes provides natural means accurately solving first order maxwell equations number periodic systems engineering interest results presented substantiate accuracy utility method index structures discontinuous galerkin methods time domain analysis ntroduction periodic structures play significant role electromagnetics optics generating unique spectral responses readily engineered applications periodicity include frequency selective surfaces fss electromagnetic band gap ebg structures biomimetic structures metamaterials etc computational analysis fields increasingly intricate periodic unit cells plays significant role design optimization frequency domain integral equation finite element discontinuous galerkin methods successfully applied variety periodic electromagnetic systems timedomain methods studying periodic systems include finite difference time domain fdtd methods remain relatively unexplored analysis periodic structures provides number advantages characterization broadband response structure single simulation treatment nonlinearities integral differential formulations maxwell problem attendant disadvantages well integral formulations discretization yields dense linear system fast efficient methods applied problems stable formulations tdies remain research problem much recent progress recent work also presented obtaining transient response using entire domain laguerre polynomials results system wherein time variable completely avoided alternatively differential formulations problem yield sparse linear systems stability better understood proper imposition boundary conditions bcs becomes challenging particular asymptotic boundary condition fields receding infinity must enforced approximately absorbing boundary condition abc perfectly matched layer pml periodic bcs perimeter unit cell trivial enforce systems excited normal incidence issues associated causality oblique incidence set field transformations mitigate causality issues introduced fdtd later adapted fetd framework sequence papers frequency domain boundary condition exploited wherein fields unit cell boundaries related one another phase shift depends exciting wave vector lattice vectors frequency domain maxwell equations posed terms set transformed variables phase shift built inverse transform applied return equations time domain additional terms appear maxwell equations transformed variables work apply field transformations time domain discontinuous galerkin framework conservation form maxwell equations first time time domain analysis periodic structures methods received relatively little attention exceptions unique contributions paper extensions time domain framework permit analysis doubly periodic structures oblique incidence first field transformations used remove causality issues reviewed demonstrate form upwind flux utilized discretizing transformed maxwell equations invariant whether one utilizing original transformed fields issues addressing use meshes across periodic boundaries discussed relevant implementation details provided finally results presented validate accuracy utility method number doubly periodic test cases athematical ormulation fig illustration doubly periodic structure periods ellipses indicate structure periodic consider domain depicted fig doubly periodic distribution isotropic lossless dielectric pec scatterers reside periodicity system described defined subscript defined basis vectors lattice vectors orthogonal work extensions basis vectors simply realized incident system planewave wavevector sin cos excitation sin sin cos incident wavevector decomposed within orthogonal span respectively evident eqns using auxiliary field components tantamount zero phase propagation boundaries delay boundaries unit cell time domain analog bloch functions typical frequency analysis applying field transformations first order time domain maxwell equations yields equations discretize within framework iii iscontinuous alerkin ethod discretization fig illustration single unit cell doubly periodic structure periods fields obey following boundary conditions spatial translation lattice vector direct implementation periodic boundary conditions requires knowledge future values fields one periodic boundary order update fields periodic boundary context time integration scheme fields updated time based upon sequence previous values possible without extrapolation alternatively transformed fields identified periodic boundary conditions remain causal done introduce delayed auxiliary variables allow seamless extension previous formulations write eqns conservation form matrix field defined flux matrix represents ith cartesian unit vector isotropic permittivity isotropic permeability identity matrix solving system equations requires discretizing domain using tetrahedra domains denoted boundaries equipped outward pointing normal vector unknowns expanded set globally discontinuous nodal polynonp use nodal basis mials functions defined following standard practice strong form problem obtained zzz shown trivially transformed fields obey called numerical flux rewrite problem eqn defined function nodal values element boundaries replacing flux matrix matrix xnp identity matrix mass matrix stiffness matrix face matrix defined zzz mij evident system two distinct characteristic values implies three rankinehugoniot jump conditions needed relate fields across discontinuities using convention integrating single element reducing integration limits faces elements arrive jump conditions equivalent transformed equations sij since equations hold time periodic numerical flux may written fij periodic numerical flux heart choice nodal values formulations hesthaven warburton proven upwind flux stable convergent maxwell equations maxwell equations upwind flux takes form eqn jump nodal field values element boundaries boundary conditions table boundary condition jumps jump defined terms nodal field values element boundaries impedance twice average impedance shared boundaries derive periodic numerical flux note using conservation form maxwell equations pec abc abc inc sinc applying boundary conditions periodic system equations requires constraining jumps across face present list common jumps first presented denotes total fields scattered fields respectively addition angle incidence jumps planewave abc allows periodic numerical flux satisfy condition transformed fields polarization respectively impedance medium must also consider boundary conditions interfaces unit cells implement eqns map must created periodic planes unit cell natural first choice creating maps create meshed unit cell whichhh periodic planes conformal set jumps alternatively significantly easier generate meshed unit cell without meticulous constraints periodic planes nodes periodic plane align information regarding triangles generated interface first decomposed list four different types fragments fragments polygon clipping algorithm employed generate data fragments defined facilitate definition quadrature rules numerically integrating surface terms esults demonstrate validity computational framework discuss several scattering results cases fourth order integration used time step size determined minimum edge length polynomial order reflection transmission data presented structure obtained eqn fig power reflected planewave obliquely incident nonmagnetic lossless dielectric slab top power reflection broadband frequency range polarization top left polarization top right bottom minimum edge length polynomial order error convergence polarization given fundamental coefficient zrt dxdy fourier transform planewave excitahere tion reflected transmitted field denoted calculated magnitude fourier transform fig reflection coefficient planewave normally incident periodically arranged pec minkowski fractals unit cell dimensions fractal dimensions fractal shown abc surfaces placed away pec fractal electric field coefficient integrated zrt plane located either scattering structure reflection transmission respectively first result scattering plane wave normally incident minkowski fractal fss result validates implementation normal incidence serves check treatment periodic boundary conditions independent oblique incidence framework fig displays illustration fractal dimensions unit cell dimensions air fig left illustration pec rods oriented unit cell dimensions radius rods right power reflected normally top obliquely bottom incident planewave electric field cases fig left illustration nonmagnetic lossless dielectric slab periodically arranged pec strips located center slab slab thickness pec strips shown illustration right power reflected normally top obliquely bottom incident planewave nonmagnetic lossless dielectric slab periodically arranged pec strips residing center slab thickness electric field cases box placed pec fractal heights numerical results displayed fig reference data minkowski fractal drawn next structure simple dielectric slab thickness relative permittivity slab lossless nonmagnetic unit cell dimensions chosen arbitrarily height air box slab chosen fig displays power reflected slab angle incidence structure show excellent agreement theoretical numerical power reflection coefficient across frequency range demonstrate higher order accuracy computational framework fig displays average absolute error numerically theoretically calculated reflection frequency band next structure consists two infinite pec rods oriented unit cell dimensions displayed fig respectively length structure chosen reduce number unknowns infinite fig left illustration single unit cell periodically arranged dielectric slabs outlined black slab heights widths chosen based ratio respectively right reflected power obliquely incident planewave electric field air boxes rods centers rods centers rods placed apart radius rods fig displays numerical results periodic method compared numerical results periodic method framework demonstrates excellent results compared framework effect planewave abc past next higher order floquet mode also captured next structure array pec strips embedded dielectric slab dielectric slab lossless nonmagnetic dimensions shown fig air box placed dielectric slab height reference data agrees well numerical results code shown fig see effect planewave abc much like framework last validation structure consists dielectric slabs alternating dielectric constants dielectric slabs lossless nonmagnetic unit cell displayed fig slab heights width slabs set based ratio slab width set air box placed set slabs arbitrarily chosen height relative permittivity slab results structure shown fig reference data drawn results show good agreement reference data shown several cases validate dgtd framework final topic work addressing stability explicit time integrator respect planewave angle incidence speed floquet modes proportional therefore cfl bound sufficient higher angles incidence simplest solution problem scale cfl condition vcf fig displays smallest stable time step scale respect angle incidence planewave passing freespace unit cell dimensions freespace mesh smallest edge length polynomial order parameters held constant angle incidence unit cell mesh conformal respect periodic boundaries simple result provides empirical evidence explicit time integration scheme conditionally stable even near grazing angles incidence satisfying cfl condition near grazing angles fig angular dependence time step scale vcf angles less required unity scaling stability however requires scales two orders magnitude thus increases number time steps accordingly onclusion uture ork paper presented threedimensional time domain discontinuous galerkin method analyzing interaction obliquely incident planewaves doubly periodic structures employed field transformation provide formulation free causality issues periodic boundary conditions time field transformations applied first order maxwell equations numerical flux derived using equivalent set transformed equations computational framework validated using existing results literature particular examples elaborated paper employed planewave abc currently developing exact time domain floquet radiation boundary condition future applications include optimization photonic band gap structures complex frequency selective surfaces acknowledgment work supported national science foundation grant authors would like thank general electric support acknowledge computing support hpc center michigan state university east lansing eferences munk frequency selective surfaces theory design john wiley sons yang electromagnetic band gap structures antenna engineering munk metamaterials critique alternatives john wiley sons capolino theory phenomena metamaterials volume crc press baczewski dault shanker accelerated cartesian expansions rapid solution periodic multiscale problems ieee trans antennas baczewski miller shanker rapid analysis scattering periodic dielectric structures using accelerated cartesian expansions josa april lucas fontana hybrid finite element method unified radiation scattering analysis general infinite periodic arrays ieee trans antennas sotirelis albrecht numerical simulation photonic crystal defect modes using unstructured grids wannier functions phys rev august chun accurate methods solving maxwell equations applications may petersson jin analysis periodic structures via formulation floquet abc ieee trans antennas march petersson jin formulation periodic structures ieee trans antennas january chen capolino shanker michielssen floquet analysis transient scattering doubly periodic discretely planar perfectly conducting structures radio august harms mittra implementation periodic boundary condition algorithm fss structures ieee trans antennas dault nair shanker log method evaluating convolutions time domain periodic green function international conference electromagnetics advanced applications pages ieee september pray nair shanker stability properties time domain electric field integral equation using separable approximation convolution retarded potential ieee trans antennas august jung mei sarkar transient wave propagation general dispersive media using laguerre functions mod methodology progress electromagnetics research jin theory computation electromagnetic fields john wiley sons hoboken usa november veysoglu shin kong timedomain analysis wave scattering periodic surfaces oblique incidence case journal electromagnetic waves applications january sirenko bagci sirenko accurate characterization diffraction gratings using time domain discontinuous galerkin method exact absorbing boundary conditions ieee conference hesthaven warburton nodal methods unstructured grids computat september niegemann stannigel busch methods analysis systems photonics nanostructures fundamentals applications february busch niegemann discontinuous galerkin methods nanophotonics laser photonics reviews november leveque finite volume methods hyperbolic problems cambridge university press mohammadian shankar hall computation electromagnetic scattering radiation using finitevolume discretization procedure computer physics communications november vatti generic solution polygon clipping communications acm july carpenter kennedy kutta schemes bertoni cheo frequency selective reflection transmission periodic dielectric layer
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rocket robust confidence intervals via kendall tau transelliptical graphical models sep rina foygel barber mladen kolar september abstract understanding complex relationships random variables fundamental importance statistics numerous applications biological social sciences undirected graphical models often used represent dependencies random variables edge two random variables drawn conditionally dependent given measured variables large body literature exists methods estimate structure undirected graphical model however little known distributional properties estimators beyond gaussian setting paper focus inference edge parameters transelliptical model generalizes gaussian nonparanormal graphical models propose rocket novel procedure estimating parameters latent inverse covariance matrix establish asymptotic normality rocket ultra setting mild assumptions without relying oracle model selection results rocket requires number samples known necessary obtaining consistent estimator element precision matrix gaussian model hence optimal estimator much larger family distributions result hinges tight control sparse spectral norm kendall tau estimator correlation matrix independent interest empirically rocket outperforms nonparanormal gaussian models terms achieving accurate inference simulated data also compare three methods real data daily stock returns find rocket estimator method whose behavior across subsamples agrees distribution predicted theory introduction probabilistic graphical models lauritzen widely used explore complex system aid scientific discovery areas ranging biology neuroscience financial modeling social media analysis undirected graphical model consists graph set vertices set edges random vector markov respect particular conditionally independent given remaining variables txc puzta buu one central questions statistics estimation undirected graph given independent realizations well quantifying uncertainty estimator paper focus asymptotic inference elements latent inverse covariance matrix semiparametric elliptical copula model embrechts also known transelliptical model liu let independent copies random vector follows transelliptical distribution correlation matrix nonnegative random variable univariate strictly increasing functions recall follows transelliptical distribution marginal transformation pxp follows centered elliptically contoured distribution covariance matrix fang let inverse covariance matrix also known precision matrix gaussian model nonzero elements correspond pairs variables conditionally dependent form edge graph elliptical model nonzero elements correspond variables conditionally correlated general possible pxa pxb conditionally uncorrelated conditionally independent model construct estimator fixed element precision matrix asymptotically normal furthermore construct confidence interval unknown parameter valid robust model selection mistakes finally construct uniformly valid hypothesis test presence edge graphical model main theoretical result establishes given initial estimates regression coefficients pfa pxa pxb pfj pxj one obtain asymptotically normal estimator initial estimators need converge sufficiently fast rate see section particular note require strict sparsity regressions allow error rate achievable known methods nonconvex lasso loh wainwright see section achieve rate estimator requires scaling sample size gaussian case sample size scaling minimax optimal ren given accurate initial estimates order construct asymptotically normal estimator prove key result vector signpxi subgaussian scale condition number dependence dimension coming problem initially posed han liu subgaussianity proved special cases result allows construct asymptotically normal estimator combining initial regression coefficient estimates kendall tau rank correlation matrix particular subgaussianity result allows establish new concentration result operator norm kendall tau correlation matrix holds exponentially high probability result allows uniformly control deviations quadratic forms involving kendall tau correlation matrix approximately sparse vectors results independent interest could used extend recent results mitra zhang wegkamp zhao han liu elliptical copula setting furthermore subgaussianity signpxi turn leads bound error kendall tau estimate sparse spectral norm allows study properties penalized rank regression base confidence intervals hypothesis tests asymptotically normal estimator element see section point results hold milder conditions required ren treats special case gaussian graphical models notably give estimator elements precision matrix without requiring strong parametric assumptions relationship literature work contributes several areas first contribute growing literature graphical model selection high dimensions extensive literature gaussian graphical model assumed case edge set graph encoded elements precision matrix meinshausen yuan lin rothman friedman aspremont fan lam fan yuan cai liu wang zhao liu learning structure ising model based penalized studied tibshirani ravikumar xue recently yang studied estimation graphical models assumption nodes conditional distribution belongs exponential family distribution see also guo guo lee hastie cheng yang yang studied mixed graphical models nodes conditional distributions necessarily family instance may nodes well nodes parametric gaussian assumption relaxed liu graph estimation studied gaussian copula model recently liu xue zou liu show graph recovered gaussian elliptical semiparametric model class conditions sample size number nodes maximum node degree graph estimation done gaussian assumption paper construct novel estimator element precision matrix without requiring oracle model selection properties second contribute literature inference recently much interest performing valid statistical inference setting zhang zhang belloni belloni van geer javanmard montanari javanmard montanari farrell developed methods construction confidence intervals low dimensional parameters linear generalized linear models well hypothesis tests methods construct honest uniformly valid confidence intervals hypothesis test based estimator first stage similar results obtained context least absolute deviation quantile regression belloni lockhart study significance input variables enter model along lasso path lee taylor perform postselection inference conditional selected model liu ren chen construct estimators elements precision matrix gaussian assumption extend results perform valid inference semiparametric ellitical copula models recent independent work propose procedure inference nonparanormal model provide detailed comparison section section notation let rns denote set let denote indicator function vector let supppaq support set let defined iprns usual extensions maxiprns matrix sets write ast denote submatrix obtained extracting appropriate rows columns sets replaced single indices example asj vector square matrix rns may write denote square submatrix matrix use notation vecpaq denote vector formed stacking columns denote frobenius norm operator norm spectral norm largest singular value norms applied entrywise maxij write cpaq denote condition number ratio largest smallest singular values two matrices denotes kronecker product bqik aij bkl two matrices size denotes hadamard product entrywise product bqij aij bij kronecker products hadamard products defined also vectors treating vector matrix one column throughout denotes cumulative distribution function standard normal distribution ptn preliminaries method introducing method begin preliminary definitions properties transelliptical distribution related models gaussian nonparanormal graphical models suppose follows multivariate normal distribution gaussian graphical model represents structure covariance matrix graph edge nodes indicates precision inverse covariance matrix model generalized allowing arbitrary marginal transformations variables liu study resulting distribution nonparanormal model also known gaussian copula write marginally transformed vector pxp follows centered multivariate normal distribution pxp sparse structure underlying graphical model representing sparsity pattern recovered using similar methods gaussian case note gaussian model special case nonparanormal model setting identity function linear functions would like nonzero mean elliptical transelliptical graphical models elliptical model generalization gaussian graphical model allows dependence variables random vector follows elliptical distribution mean vector covariance matrix random variable radius denoted write aaj cholesky decomposition unit vector drawn uniformly random independently radius note level sets distribution given ellipses centered shape determined gaussian model special case elliptical model taking transelliptical model also known elliptical copula combines elliptical distribution marginal transformations much nonparanormal distribution applies marginal transformations multivariate gaussian random vector write denote marginally transformed vector pxp follows centered elliptical distribution specifically pxp marginal transformation functions assumed strictly increasing note gaussian nonparanormal elliptical models special cases model pearson rho kendall tau point assume distribution correlation matrix diagonal elements equal one case gaussian distribution entries pearson correlation coefficients pair variables case also write erxa setting estimate sample covariance nonparanormal setting longer case equal correlation corrpxa due marginal transformations however estimate performing marginal empirical transformations standard normal distribution taking empirical transformations estimated via empirical covariances similarly elliptical model rescaling also erxa therefore estimate via empirical covariance transelliptical distribution contrast longer possible taking scaling simplicity generalize calculations erfa pxa qfb pxb therefore estimate marginal transformations estimate using empirical covariance transformed data however unlike nonparanormal model estimating straightforward reason elliptical distribution marginal distributions known unless distribution radius known therefore marginally estimate know marginal distribution transformation marginal distribution pxa contrast nonparanormal model pxa marginally normal alternative liu use kendall rank correlation coefficient kendall tau population level given pxa signpxa signpxb copy unlike pearson rho kendall tau coefficient invariant marginal transformations since strictly increasing functions see signpfa pxa signpfb pxb signpxa signpxb sample level kendall tau estimated taking comparing pair distinct observations signpxia signpxib follows elliptical distribution theorem lindskog gives following relationship kendall tau pearson rho coefficients given covariance matrix sin rps since kendall tau invariant marginal transformations identity holds transelliptical family well reason liu estimate covariance matrix sin necessarily positive semidefinite note however spearman rho like kendall tau also invariant marginal transformations liu comment equivalence spearman rho values analogous kendall tau holds uniformly across entire elliptical transelliptical family therefore type estimator could carried kendall tau denotes estimate given matrix remainder paper kendall tau coefficients denoted entries tab denotes empirical estimate entries comparing models tail dependence clear compared gaussian graphical model nonparanormal model allows data may extremely marginal distributions subtle consideration question tail dependence two variables particular nonparanormal model allow tail dependence two variables stronger gaussian distribution specifically consider pairwise dependence given pxa corr marginal distribution taking measure correlation extreme right tail extreme right tail course also consider left tail distribution note marginal transformations variable affect measure since quantiles take transformations account particular nonparanormal distribution tail correlations pxa multivariate gaussian distribution contrast elliptical transelliptical model exhibit much higher tail correlations since real data often exhibits heavy tail dependence variables flexible transelliptical model may better fit many applications demonstrate behavior simple example figure take corresponding multivariate degrees freedom note equivalent taking note relevant quantiles tail correlation equal kendall tau coefficient value figure shows tail correlation decreases towards zero normal distribution grows low values therefore shift nonparanormal transelliptical model important since allows model variables high tail dependence high dependence extreme events tail dependence gaussian quantile figure tail dependence normal elliptical distributions data generated figure displays estimated empirically sample size rocket asymptotically normal estimator suppose data points drawn transelliptical distribution covariance matrix would like perform inference particular entry precision matrix specifically interested producing confidence interval prespecified node pair move towards constructing confidence interval introduce definitions calculations first let puzta observe matrix inversion calculate follows define nonparanormal graphical model setting regression coefficients pxa pxb regressed tfj pxj linear model setting idea used sun zhang belloni therefore rewrite follows somewhat redundant formulation allow favorable cancellation error terms later abuse notation index entries indices denote lying rta rather next define oracle estimator defined plugging true values empirical estimate given later theorem show due standard results theory known would achieved oracle estimator asymptotically normal weakly converges normal random variable goal inference model centered variance scales calculate variance later course practice know true values must instead use available estimators denoted discuss obtain preliminary estimates later given estimators regression vectors define estimator follows since interested rather matrix final step define estimator function order make inference approximate distribution first treat distribution corresponding entry oracle estimator let rpn vectors entries observe sin linear approximation write sin taking cos cos signpxi signpxi study variability error consider kernel signpx gpx signpx cos see later understanding behavior kernel allow characterize empirical estimator ultimately distribution oracle estimator course gpx depends unknown quantities namely replace estimates empirical version kernel define random kernel gqpx signpx qvqj cos signpx vqa vqb vqi define note defined gqpxi meanpq sqab qpxi see later sqab meanpq estimates variance expression arises naturally theory main result theorem prove follows distribution approximately standard normal therefore approximate interval given appropriate quantile normal distribution notation fixed random quantities point much possible throughout main body paper quantities depend data depend initial estimates quantities depend data denoted check accent example quantities neither denoted hat accent example depend hat check population quantities random two important exceptions course data oracle estimator depend main results section give theoretical result showing confidence interval constructed asymptotically correct coverage probability long reasonably accurate estimators asymptotic result considers problem whose dimension grows sample size also allow sparsity level true inverse covariance matrix rpn use denote approximate bound sparsity column details given begin stating several assumptions distribution data initial estimators constants appearing assumptions interpreted values depend dimensions problem assumption data points rpn draws transelliptical distribuiid tion fpn fpn strictly monotone functions etc depend sample size since dimension problem grows abuse notation write etc dependence implicit random variable covariance matrix rpn positive definite bounded condition number ccov constant ccov assumption columns true inverse covariance denoted approximately csparse constant csparse assumption constant cest probability least preliminary estimate vector satisfies logppn logppn cest cest assumption define kernel hpx signpx signpx rpn let pxq rhpx define total variance varphpx conditional pxqq iid fpn constant ckernel ckernel assumption assumes smallest largest eigenvalues correlation matrix bounded away zero infinity respectively assumption commonly assumed literature learning structure probabilistic graphical models ravikumar liu assumption require precision matrix exactly sparse commonly assumed literature exact graph recovery see example ravikumar requires rows norm grow fast note vector ccov could set csparse ccov assumption condition assumes existence initial estimators converge fast enough rate next section see assumption together stronger version assumption sufficient assumption satisfied specific estimator efficient compute finally assumption imposed assumption depends allow estimation asymptotic variance correlation matrix without reference distribution radius assumption depends therefore derived consequence choice state main result theorem assumptions exists constant cconverge depending ccov csparse cest ckernel dimensions problem ppn cconverge sup tpr sab use positive semidefinite ordering matrices note second part inequality always true law total variance note result holds uniformly large class data generating processes satisfying assumptions relatively weak assumptions compared much sparse estimation inference literature emphasize result holds without requiring exact model selection oracle properties hold restrictive sequences data generating processes example require condition lower bound true edges incoherence conditions van geer may implausible practice instead requiring perfect model selection require estimation consistency given assumption weaker assumptions would sufficient guarantee model selection consistency immediate corollary see confidence interval constructed asymptotically correct corollary assumptions notation theorem interval constructed fails cover true parameter probability higher result holds uniformly large class data generating distributions theorem striking shows form asymptotically normal transelliptical distribution family sample complexity ppn sample size requirement shown optimal obtaining asymptotically normal estimator element precision matrix multivariate normal data ren precisely bprpn maxaprpn constants theorem ren proves inf inf sup logppn therefore estimator rate optimal terms sample size scaling infimum measurable function data also consider taken estimator related optimality question whether confidence interval produce optimal lowest possible width given desired coverage level gaussian setting ren method produces interval asymptotically minimal length given sample size due fact variance estimator matches fisher information rocket method enjoy theoretical property empirically observe confidence intervals slightly wider produced ren method gaussian data point also worth mentioning result study inference gaussian copula graphical models base inference procedure decorrelating pseudo score function parameter interest showing normally distributed main result stated theorem requires sample size satisfy logppn logppn maxaprpn bprpn potentially large immediately clear result achieves much better scaling sample size work constants instead denoted constant use different notation distinguish used plays different role denote necessarily constant initial estimators validity inference method relies part accuracy initial estimators assumed satisfy error bounds high probability stated assumption high probability logppn logppn cest cest cest constant prove required error rates obtained additional sparsity assumption lasso estimators argmin penalty parameter chosen appropriately fact optimization necessarily positive semidefinite problems may convex turn proving local minima satisfy required error rates assumption proceed use theoretical results loh wainwright gives theory local minimizers nonconvex regularized objective functions particular local minimizers two optimization problems satisfy requirements assumption therefore need able run optimization algorithms find local minima specialize main result setting theorem adapted loh wainwright theorem consider suppose satisfies restricted strong convexity conditions logppq logppq max logppq local minimum objective function set holds apply loh wainwright results theorem problem estimating setting assume exact sparsity likely similar results would hold approximate sparsity use exact sparsity fit assumptions existing theorem corollary suppose assumption holds assume additionally columns true inverse covariance exist constants csample classo depending ccov csample logppn probability objective function least local minimizer set ccov satisfies choose classo logpp holds estimating using corollary see local minimizer satisfies assumption cest classo remark practice constant classo suggested theory general unknown choosing small multiple logpp generally performs instance logppn use simulations choice constant ensures penalty term dominates variance elements objective function derivative true solution elements prove corollary follows loh wainwright result theorem sufficient check restricted strong convexity condition holds high probability compute necessary values parameters theorem matrix proof technical relies novel results concentration kendall tau correlation matrix details given appendix provided sufficient condition local minimizer satisfy assumption however many estimators used initial estimators example one could use dantzig selector tao potential benefits dantzig selector optimizab tion program twofold first optimization program convex even positive second one need know upper bound norm using techniques similar used prove corollary also prove assumption holds dantzig selector used initial estimator large problems however dantzig selector type methods computationally much slower lasso type methods empirical results implement lasso rather dantzig selector since study graphs many nodes practice found simulations using lasso model selection refitting without penalty leads better empirical performance specifically first fit lasso argmin precisely find local minimum nonconvex optimization problem ball large radius practice every iteration stay inside ball therefore long see convergence iterative algorithm solving nonconvex lasso concern theoretical boundedness constraint extract combined support two solutions supppq supppq refit coefficients using following work belloni chernozhukov sun zhang shown refitted estimators also satisfy assumption practice refitting improves accuracy preliminary estimators reducing shrinkage bias finally remark would like perform inference potential edges require many initial estimators computed course quite computationally demanding however ren propose simple modification significantly reduces comqaall putation time node first regress variables call solution solution already optimal regressing node next nodes rpn szta case nodes due sparsity modification actual number regressions required far node forms edges nodes qaall require pkn many regressions total form initial estimators main technical tools section outline proof theorem section state key technical result establishes property vector following transelliptical distribution section also illustrate application technical result establishing bound section sketch proof main result proof theorem two key steps first step prove distribution oracle estimator asymptotically normal sab explicit form sab given proof theorem sab asymptotic variance next step prove difference estimator oracle estimator converges zero fast rate variance estimator sab converges sab fast rate asymptotically normal estimator detailed combining steps prove proofs step found appendix outline main results step step establishes type bound centered normalized oracle estimator linear function kendall tau approximate oracle estimator statistic data prove variance linear approximation bounded away zero apply existing results convergence following result proved appendix theorem suppose assumptions hold exist constants cnormal cvariance depending ccov csparse ckernel logppn cnormal sup tpr sab defined proof satisfies sab cvariance step contains main challenge problem since requires strong results concentrap covariance matrix main ingredient tion properties kendall tau estimator step new result proving signs vector signpxi subgaussian observations results discussed secb around given section using tion application concentration tools able prove following theorem proved appendix theorem suppose assumptions hold exists constant coracle depending ccov csparse cest logppn probability least coracle logppn note additional condition logpp assumed hold main result theorem since inequality hold claim theorem trivial logppn sab coracle random vectors recall definition subgaussian random vector definition random vector fixed vector holds graphical models data points come subgaussian distribution sample covariance matrix pxi xqpxi xqj known concentrate near population covariance measured different norms example elementwise convergence sample covariance population covariance convergence sufficient establish rates convergence graphical lasso clime graphical dantzig selector estimating sparse inverse covariance ravikumar cai ayuan similar results beaobtained also transelliptical family since logppq hence logppq shown liu liu however order construct asymptotically normal estimators elements precision matrix stronger results needed convergence sample covariance population covariance ren particular result convergence spectral norm uniformly sparse submatrices required one relate convergence elementwise norm sparse spectral norm convergence however would lead suboptimal sample size one way obtain tight bound sparse spectral norm convergence utilizing subgaussianity data points exactly proceed establish recall kendall tau estimator covariance sin signpxi signpxi therefore crucial determine whether vector signpxi subgaussian scale depend heavily ambient dimension using past results elliptical distributions reduce simpler case using arguments lindskog proved appendix iid lemma let suppose positive definite probability signpx equal distribution signpzq previous work shown gaussian random vector signpzq subgaussian variance proxy depends special cases covariance identity equicorrelation matrix han liu however result general covariance structures previously unknown following lemma resolve question proving gaussian vectors recall condition number lemma let signpzq rsignpzqs lemma primary tool main results key ingredient sqab sab lemma proof theorem bounds errors proved appendix also use result establishing results following section note signpx obviously subgaussian distribution sum subgaussian random variables since bounded however scale could grow linearly deterministic probabilistic bounds results given section lemma instrumental obtaining probabilistic bounds crucial establishing theorem corollary let set vectors unit ball abusing notation let denote sparse spectral norm matrices maxu vpsk following lemma provides bound error kendall tau sparse sectral norm proof given appendix lemma suppose satisfy probability least holds following deterministic bound sparse spectral norm next relate proven appendix error covariance estimator lemma following bound holds deterministically result pena theorem bounds high probability details bound given appendix combining bound lemmas immediately obtain following corollary corollary take holds probability least following bound log finally use result based work sun zhang order extend sparse spectral norm bound bound holding approximately sparse vectors lemma based proposition sun zhang fixed matrix vectors results lemma corollary compared theorem mitra zhang proves essentially result kendall tau estimate nonparanormal gaussian copula model technique extend immediately transelliptical model extend result transelliptical model special case provides alternative proof result gaussian copula model note result depend condition number covariance matrix maximum eigenvalue however context graphical models commonly assumed smallest eigenvalue constant furthermore results lemma corollary also compared theorem han liu give similar bounds spectral assumption norm sparse submatrices distribution rigorously establish bounds covariance matrices without explicitly making assumption simulation studies section illustrate finite sample properties rocket described section simulated data real data experiment additional simulations presented appendix use rocket construct confidence intervals edge parameters report empirical coverage probabilities well length constructed intervals comparison also construct confidence intervals using procedure ren based pearson correlation matrix nonparanormal estimator correlation matrix npn proposed liu pseudo score procedure first two methods use plugin estimate correlation matrix together estimate recall liu estimate correlation matrix based marginal transformation observed data let fba pxq fea pxq fba pxq fba pxq fba pxq fba pxq txia empirical cdf logpnq fea pxia feb pxib corr correlation matrix estimated asymptotic variance estimators based pearson nonparanormal correlation matrix estimate obtained sab kendall tau estimator covariance matrix initial estimator precision matrix suitable conditions asymptotically normal asymptotic variance consistently estimated corollary however find empirically suggest using clime estimator cai construct method performs better using estimate row similar sun zhang simulations set tuning parameter logppn suggested constant large enough penalty dominates variance element score computations carried matlab simulation generate data model follows degrees freedom inverse covariance matrix encodes grid node connected four nearest neighbors nonzero elements equal diagonal element equal let set pdiag pdiag additional simulations appendix show experiment chain graph structure take grid size take sample size figure shows plots based independent realizations test statistic error four methods together reference line showing quantiles standard normal qab distribution figure observe quantiles test statistic error based rocket closest quantiles standard normal random variable quantify results table reports empirical coverage width confidence intervals based table observe coverage confidence intervals based rocket pseudo score closest nominal coverage three node pairs displayed figure table namely correspond true edge nonedge nearby nodes therefore easy mistake edge distant nodes respectively quantiles rocket pearson standard normal quantiles quantiles pseudo score standard normal quantiles standard normal quantiles standard normal quantiles standard normal quantiles quantiles nonparanormal standard normal quantiles standard normal quantiles standard normal quantiles standard normal quantiles standard normal quantiles standard normal quantiles standard normal quantiles figure simulation transelliptical data plot corresponds grid graph structure row corresponds edge row close row far rocket pearson npn pseudo score table simulation transelliptical data percent empirical coverage average length confidence intervals based independent simulation runs results surprising since neither pearson nonparanormal correlation matrix consistently estimate true contrast rocket pseudo score method able construct test statistic qab asymptotically distributed normal random variable asymptotic distribution provides good approximation finite sample distribution simulation illustrate performance rocket data generated normal nonparanormal distribution consider corresponding grid simulation generate samples fep fej pxq pxq signpxq pxq pxq pxq exppxq table summarizes results simulation observe data multivariate normal methods perform well rocket pseudo score slightly wider intervals similar coverage data generated nonparanormal distribution using pearson correlation results confidence intervals nominal coverage due bias setting nonparanormal estimator rocket pseudo score still correct nominal coverage note however kendall tau equal zero pearson correlation transf gaussian gaussian rocket pearson npn pseudo score table simulation gaussian nonparanormal data percent empirical coverage average length confidence intervals based independent simulation runs transform gaussian rocket pearson nonparanormal pseudo score marginal transform figure simulation power plots simulated data generated gaussian distribution multivariate distribution also equal zero coverage pearson improves see example coverage simulation simulation illustrate power test based statistic qab reject null hypothesis samples generated distribution covariance matrix form entries zero note implies multivariate normal also consider marginal transformation described simulation figure plots empirical power curves based independent simulation runs different settings data follow normal distribution methods similar power distributions tests based pearson nonparanormal correlation correct coverage shown illustrative purpose discussion proposed novel procedure rocket inference elements latent inverse correlation matrix elliptical copula models paper established surprising result states rocket produces asymptotically normal estimator element inverse correlation matrix elliptical copula model sample complexity required obtain asymptotically normal estimator element precision matrix multivariate normal distribution furthermore sample complexity optimal ren result surprising family elliptical copula models much larger family multivariate normal distributions example contains distributions heavy tail dependence discussed section rocket achieves optimal requirement sample size without knowledge marginal transformation result also significant practical importance since normal distribution convenient mathematical approximation data generating process recommend using rocket whenever making inference inverse correlation matrix instead methods heavily rely normality simulation studies even data generated normal distribution rocket lose power compared procedures specifically developed inference normality main technical tool developed paper establishes sign normal random vector taken elementwise random variable parameter depending condition number covariance matrix dimension based result able establish tight tail bound deviation sparse eigenvalues kendall tau matrix result independent interest would allow improve number recent results sparse principal component analysis factor models estimation structured covariance matrices mitra zhang han liu fan sharpest result nonparametric estimation correlation matrices spectral norm gaussian copula model established mitra zhang results establish similar result family elliptical copula models provide alternative proof gaussian copula model acknowledgments work partially supported ibm corporation faculty research fund university chicago booth school business alfred sloan fellowship work completed part resources provided university chicago research computing center references richard baraniuk mark davenport michael wakin simple proof restricted isometry property random matrices constructive approximation jan issn doi url http alexandre belloni victor chernozhukov least squares model selection sparse models bernoulli may issn doi url http alexandre belloni victor chernozhukov christian hansen inference treatment effects selection amongst controls rev econ nov issn doi url http 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dimensional inverse covariance matrix estimation via linear programming mach learn yuan lin model selection estimation gaussian graphical model biometrika zhang stephanie zhang confidence intervals low dimensional parameters high dimensional linear models stat soc jul issn doi url http tuo zhao han liu calibrated precision matrix estimation high dimensional elliptical distributions ieee trans inf theory pages issn doi url http tuo zhao han liu kathryn roeder john lafferty larry wasserman huge highdimensional undirected graph estimation url http huge package version real data experiment section evaluate performance rocket method real data set compare gaussian graphical model based approach ren using pearson correlation nonparanormal estimator proposed liu details methods given section use stock price closing data obtained via package huge zhao gathered publicly available data yahoo data consists daily closing prices companies days transform data consider closing price stock day xij log closing price stock day practice dependence across time data set treat row independent perform two experiments data set experiment test whether empirical results agree asymptotic normality predicted theory three splittting data disjoint subsamples comparing estimates across subsamples experiment use full sample size compare estimates confidence intervals produced three methods experiment checking asymptotic normality real data example available ground truth compare know true distribution data compare estimates exact true precision matrix however still check whether estimators produced methods exhibit asymptotic normality claimed theory splitting data many subsamples considering empirical distribution estimators across subsamples construct subsampled data randomly select disjoint sets size denoted due small sample size restrict attention companies categories materials consumer staples consist companies respectively total companies define data set identifies stocks interest sqp using next pair stocks subsample compute rocket true distribution data follows transelliptical model precision matrix sqp approximately standard main result theorem implies normal since sqab concentrates near sab see theorem zab sab plq particular implies sample variance vector pzab zab expectation approximately figure show histogram sample variances samplevarpzab across pairs variables compare pearson nonparanormal methods repeat procedure estimators estimated variances produced two methods well also displayed figure see rocket produces mean sample variance near two methods give mean sample variances pearson nonparanormal substantially higher theoretical value indicates normal approximation distribution estimator may approximately valid rocket http rocket nonparanormal pearson sqab pair variables figure sample variances rescaled estimator using subsampled stock data sample variances approximately according theory see section correct scale scale predicted theory two methods data set plq vector pzab zab addition sample variance near also exhibit tails according theory check calculate proportion values vector lying near mean approximately according theory using standard normal quantiles results rocket coverage pearson coverage nonparanormal coverage see rocket method achieves appropriate coverage experiment estimating graph second experiment use full sample size estimate sparse graph stocks selected experiment using three methods method first produce approximate testing presence edge pair variables recall according main result theorem pair variables edge sab approximately distributed sqab variable using calculate pab figure show resulting graphs edge drawn whenever passes threshold pab whenever pab number edges selected method shown figures overall see rocket selects roughly number edges pearson method less nonparanormal method data set since pearson nonparanormal methods exhibit approximately normal behavior across subsamples experiment interpreted power comparison methods additional edges selected nonparanormal method instance may reliable since calculation based approximating distribution estimator using theoretical scaling appear hold method additional simulations simulation chain graph repeat simulation paper chain graph rocket edges pearson edges materials consumer staples nonparanormal edges pearson edges rocket edges nonparanormal edges figure estimated graph stock data using rocket pearson nonparanormal methods see section edge displayed pair variables pab top row pab bottom row graphs drawn using igraph package csardi nepusz core team rocket pearson npn pseudo score table simulation transelliptical data percent empirical coverage average length confidence intervals based independent simulation runs corresponds chain graph structure structure instead grid graph structure inverse covariance matrix chain structure set take sample size figures show plots based independent realizations test statistic error four methods together reference line showing quantiles standard normal distribution first row two figures illustrates actual performance methods second row illustrates performance oracle procedure need solve variable selection problem instead knows sparsity pattern two figures observe quantiles test statistic error based rocket pseudo score estimators closest quantiles standard normal random variable quantify results table reports empirical coverage confidence intervals based table observe coverage confidence intervals based rocket pseudo score closest nominal coverage simulation additional simulation evaluate robustness procedures three rocket pearson estimated neighborhood nonparanormal pseudo score standard normal quantiles standard normal quantiles standard normal quantiles standard normal quantiles oracle standard normal quantiles standard normal quantiles estimated neighborhood rocket pearson standard normal quantiles figure simulation transelliptical data plot edge corresponds chain graph structure standard normal quantiles qab true nonparanormal pseudo score standard normal quantiles standard normal quantiles standard normal quantiles standard normal quantiles oracle standard normal quantiles standard normal quantiles standard normal quantiles standard normal quantiles figure simulation transelliptical data plot close true edge corresponds chain graph structure gaussian rocket pearson npn pseudo score table simulation gaussian transelliptical data row corruption mechanism percent empirical coverage average length confidence intervals based independent simulation runs corresponds grid graph structure denotes fraction corrupted rows data contamination mechanisms random row contamination deterministic row contamination element contamination mechanism let contamination level set precision matrix chosen grid structure used earlier simulations row contamination element contamination consider two settings uncorrupted samples drawn either row contamination mechanism corrupts tnru rows data matrix random row contamination mechanism corrupted row filled entries drawn distribution contaminated rows heavy tailed come elliptical copula family uncorrupted data deterministic row contamination mechanism corrupted row equal vector numbers occur alternating way element contamination mechanism selects tnpru elements substitutes value drawn either equal probability results simulation summarized tables deterministic row corruption mechanism malicious hurts procedures simulation observe procedures using kendall tau correlation matrix tend robust furthermore coverage elements tends severely affected corruption coverage gaussian rocket pearson npn pseudo score table simulation gaussian transelliptical data deterministic row corruption mechanism percent empirical coverage average length confidence intervals based independent simulation runs corresponds grid graph structure denotes fraction corrupted rows gaussian rocket pearson npn pseudo score table simulation gaussian transelliptical data element corruption mechanism percent empirical coverage average length confidence intervals based independent simulation runs corresponds grid graph structure denotes fraction corrupted elements gaussian vectors property section prove lemma shows gaussian vector satisfies property centered sign vector signpzq rsignpzqs subgaussian paper apply lemma case rsignpzqs signpzq therefore subgaussian lemma let signpzq rsignpzqs proof lemma without loss generality rescale note mean therefore rescaled well write aaj matrix iid write fixed vector signpzq signpzq evi pay last step holds conditional terms pay depends therefore terms conditionally independent next observe pay sign pay pay pay pay define evi pay evi pay pay evi pay evi evi pay inequality proved applying hoeffding lemma see example massart lemma bounded random variable pay pay since conditioning treated random combining calculations far signpzq applies function elementwise vector next show ayq lipschitz note since density standard normal distribution bounded uniformly ayq ayqi ayqi last step true therefore ayq apply standard concentration results lipschitz functions gaussian massart proposition qsq therefore signpzq finally pay pay pay rsignpzi words rsignpzqs combining everything proved signpzq ersignpzqs desired proof main result preliminaries first compute bounds use many times proofs first note ccov assumption next pby matrix blockwise inversionq jpi ccov jpi jpi since ccov therefore ccov csparse applying assumption proof theorem asymptotic normality oracle estimator theorem suppose assumptions hold exist constants cnormal cvariance depending ccov csparse ckernel logppn sup cnormal tpr sab defined proof satisfies sab cvariance approximated linear function proof theorem first show error kendall tau estimator define vectors rpn entries error given definition sin sin take taylor expansion see next since cos ptp sin tpq ptp ptp next rewrite linear term signpxi cos signpxi cos order respect data note erls cos cos ertps cos ptp define kernel gpx cos let pxq rgpx iid let pxqq callaert janssen section npl erlsq sup custat tpr universal constant custat next bound ratio appendix following lemma proved lemma suppose assumptions hold let gpx pxq defined proof theorem pxqq variance cmoment cvariance cmoment constants depending ccov ckernel particular lemma implies sab cvariance define next linear term analysed provides approximation erlsq sin tpq ptp ptp sin tpq ptp ptp csparse last inequality holds finally next lemma proved pena lemma pena theorem probability least log applying lemma least logppn probability erlsq gives summarize computations far asymptotic normality result linear term erlsq gives bound asymptotically normal use following lemma proved prove therefore appendix lemma let random variables sup tpr variable converges standard normal distribution rate sup tpr apply lemma sab sup custat tpr lemma furthermore csparse cvariance logppn sab cmoment cvariance logppn lemmas noting cnormal defining csparse custat cvariance proved desired result sab cmoment cvariance proof theorem gap estimator oracle estimator estimation variance theorem suppose assumptions hold exists constant coracle depending ccov csparse cest logppn probability least coracle logppn logppn sab coracle first part theorem bounds distance estimator oracle estimator established using bounds section details given appendix second part theorem bounds error estimating variance sab treated appendix bounds derive bound difference use bounds covariance error bounds give deterministic empirical estimator oracle estimator given initial assumptions hold following lemma proved appendix lemma assumptions hold probability least logpp logpp csubmatrix csubmatrix constant depending ccov cest csparse point combine corollary lemma obtain probabilistic bound theorem looking first corollary setting see assumption assumption logppn stated theorem conditions corollary must hold probability least log logppn ccov note additional condition logpp assumed hold main result theorem since inequality hold claim theorem trivial choose universal constant guarantees last inequality holds using assumptions logppn combining result lemma obtain logppn csubmatrix ccov taking coracle csubmatrix ccov proved first bound theorem holds probability least variance estimate second part theorem bounding error variance estimate sqab state bound lemma defer proof appendix since need develop additional technical results treating bound lemma assumptions definitions theorem probability least logppn event bounds assumption hold logppn sab coracle combining lemma work using assumption proved bounds stated theorem hold probability least desired proof theorem main result prove main result theorem proof theorem recall goal prove qab converges distribution recalling using formula matrix inverse separate random variable several terms sqab detpab sqab sqab sab sab sab sab sab sqab sab show converges standard normal distribution apply lemma qab stated appendix order apply lemma obtain desired result assemble following pieces first variable sab satisfies suptpr cnormal logpp shown theorem second define variables sab sab set coracle coracle coracle logppn cvariance coracle cvariance qab sab logppn show theorem probability least holds variable trivial consequence bound sab theorem combined lower bound sab cvariance theorem coracle logppn turn bound prove bound observe theorem stated probability also variance last step follows theorem ccov therefore sab sab ccov cvariance last step follows theorem along fact ccov combining everything cvariance cvariance coracle coracle logppn coracle cvariance ccov logppn logppn main result theorem holds trivially assuming logppn proved desired bound given convergence results apply lemma obtain following result sup tpr sab logppn coracle coracle coracle cvariance logpp coracle logppn cvariance cnormal logpp oracle ccvariance logpp oracle ccvariance result theorem holds trivially assuming case ppn sup cconverge tpr sab cconverge coracle coracle coracle cnormal cvariance cvariance accuracy initial lasso estimator corollary suppose assumption holds assume additionally columns true inverse covariance exist constants csample classo depending ccov csample logppn probability objective function least local minimizer set ccov satisfies choose classo logpp result holds estimating proof corollary define apply theorem sparse recovery problem order need check conditions hold feasible condition conditions satisfied result theorem applied setting feasibility define ccov proved ccov furthermore true therefore ccov condition restricted strong convexity need check restricted strong show corollary implies convexity conditions hold matrix exists constant crsc depending ccov logppn probability least rpn crsc logppn apply lemma obtain see holds set crsc logpp see probability applying corollary value least long set constant crsc large enough event holds cov crsc logppn therefore probability least restricted strong convexity condition holds crsc prove probability least ccov logppn logppn cfeasible csample condition penalty parameter constant cfeasible depending ccov long set csample ccov given true require condition holds logppq max define classo ccov max cfeasible ccov csample plugging bound see lower bound satisfied classo check upper bound need logppn classo logppn assuming ccov logppn follows directly therefore satisfied probability least condition sample size satisfy plugging definitions see sufficient require crsc logppn conclusion combining work see conditions feasibility satisfied probability least long csample logppn csample max ccov ccov crsc therefore applying theorem high probability events hold local minimizer lpxq set csparse holds arguments results hold estimating proving since consider term rpn fixed vector calculated taylor expansion max max cos ptp max cos ptp ptp ptp sin next bound term final separately beginning expression second term know ccov ccov bound use calculation furthermore lemma probability least logppn using therefore second term bounded logpp logpp cov ccov csample use assumption csample logppn next turn first term order bound term begin stating two lemmas proved appendix lemma exist vectors sequence cos lemma fixed cov ccov exp exp lemma write cos ptp cos par ptp qpbr cos note ccov ccov exp cos ptp exp par ptp qpbr exp par ptp qpbr pby jensen inequalityq ccov ccov pby lemma exp ccov ccov exp observe set logppn satisfies long set ccov csample ccov due assumption csample logppn see logppn cos logppn logppn logppn pcosp logppn logppn logppn pcosp ccov logppn ccov logppn exp exp logppn ccov set cfeasible ccov therefore logpp max cos ptp cfeasible combining returning logpp logpp cov cfeasible csample probability least proves proofs lemmas proof normal convergence lemma lemma let random variables sup tpr variable converges standard normal distribution rate sup tpr proof lemma first define truncated versions signpbq signpcq probability note next fix suppose since density pby applying lemma stated belowq prove reverse bound suppose therefore since density pby applying lemma stated belowq therefore identical arguments prove lemma proof last step holds sign vector transelliptical distribution iid lemma let suppose positive definite probability signpx equal distribution signpzq proof lemma first since strictly monotone see signpx distribution regardless choice therefore suffices consider case iid identity function elliptical distribution case lindskog lemma distribution random variable obeys ptq characteristic functions iid respectively note two independent copies therefore since probability proves probability next take uniformly distributed unit sphere distribution using fact probability see signpx equal distribution sign signpzq desired proof lemma lemma following bound holds deterministically proof lemma taylor theorem cos sin entries tab tab tab taking cos sin first bound matrix term note sin last step holds since function lies furthermore definition next bound matrix term lemma express cos convex combination cos satisfy satisfy furthermore note due bound cos sup ptp psk maxu vps proves lemma using definition proof lemma lemma suppose satisfy probability least holds proof lemma lemma straightforward combination lemma stated appendix together following result lemma adapted lemma theorem baraniuk let random matrix satisfying exp exp unit vectors constants satisfying probability least holds unit vectors combined lemma applied place cov lemma immediately yield bound sup probability least long next take ptp ptp ptp proves lemma desired next turn proof lemma proof lemma adapted lemma theorem baraniuk first fix rps let following arguments baraniuk lemma take set unit vectors sup sup rpu unit uprs furthermore fixed exp exp last step set similarly exp therefore exp sup rpu finally taking exp unit uprs sup choices see sup exp upsk proof corollary corollary take holds probability least following bound log proof corollary proof straightforward combination lemmas lemma log last step holds applying lemma lemma proof lemma lemma based proposition sun zhang fixed matrix vectors sup psk proof lemma first introduce two vector norms used proposition sun zhang sup dual norm note sup zpsk norm definition definition dual norm sup sup wpsk wpsk sup wpsk sup zpsk combining everything sup zpsk finally proposition sun zhang applied notation proves last inequality follows trivially norm definition proves lemma proof lemma lemma assumptions hold probability least logpp logpp csubmatrix csubmatrix constant depending ccov cest csparse proof lemma choose bound dqth entry error first probability least bounds write assumption hold assume event occurs point qcj qcj bound terms bound first term side ccov pcest logppn step holds must last step holds assumptions finally bound second term side qed basis vectors corresponding nodes vector vector ccov csparse logppn assumption pccov csparse logppn trivially desired result applying lemma noting lemma follows trivially bounds setting csubmatrix appropriately proof lemma lemma suppose assumptions hold let gpx pxq defined proof theorem pxqq variance cmoment cvariance cmoment constants depending ccov ckernel proof lemma first pxq signpx cos signpx signpx signpx vec cos signpx signpx vec cos pxqj vec cos pxq defined assumption variance therefore pxqq vec cos vec cos ckernel vec cos vec cos pby assumption ckernel var vec cos hpx signpx ckernel var signpx cos ckernel var signpzq cos signpzq ckernel csigns cos step take apply lemma see signpx distribution signpzq last step apply following lemma proved appendix lemma take positive definite distinct matrix mja let exists constant csigns depending var signpzqj signpzq csigns mab applying lemma cos yields lower bound since mja finally cos tab pccov cos last step holds cos tab sin tab pccov summarize ckernel csigns cvariance ccov next give upper bound pxqq vec cos vec cos vec cos vec cos signpx pas lower bound earlierq var signpx cos var gpx finally compute upper bound lemma exists decomposition cos note ccov similarly ccov ccov cos signpx signpx signpx pby jensen inequalityq signpx signpx max signpx max max ccov max ccov ccov cmoment max last inequality holds signpx ccov lemmas proofs lemmas initial estimators lemma exist vectors sequence cos proof lemma use matrix max norm defined matrix min max max apiq bpjq denote ith row jth row respectively matrix max norm satisfies several key properties use srebro shraibman first max wii second finally convexhull abj puv unit vectors matrices recall matrix nuclear norm sum singular values matrix cos wegkamp zhao show cos matrix entries given elementwise powers applying since correlation matrix max max last identity comes wegkamp zhao finally cos convexhull abj cos expressed convex combination stated lemma lemma fixed exp ptp exp ccov proof lemma start simple observation vqj ptp qpu vqj ptp qpu gives via exp ptp exp qpu qpu exp qpu exp qpu note therefore sufficient show unit vector cov exp ptp exp ccov prove using chernoff bounding technique end denote group permutations rns let xpiq denote row fixed define sign observe iprn fixed identically distributed sign using lemma lemma fixed unit vector sign xpiq xpi ccov random variable sign sign applying lemma stated ccov exp exp exp referring back ccov exp ccov exp twt iprn pby jensen inequalityq iprn exp since fixed equal distribution continuing step exp exp exp ccov ccov exp lemma suppose rexp ptzqs ect exp tpz erz exp remark well known square subgaussian random variable satisfies subgaussian tails near mean see example lemmas vershynin obtain small explicit constants proof first part proof follows arguments vershynin lemma first bound integers exp exp pby stirling approximation long also use fact simplify next trivially erz exp tpz erz erz erz exp pby place exp lower bounds variance signs gaussian lemma take positive definite distinct matrix mja let exists constant csigns depending var signpzqj signpzq csigns mab proof lemma let denote set rpsztau law total variance var signpzqj signpzq var signpzqj signpzq let denote jth row ath entry removed written column vector recalling mja var signpzqj signpzq var signpza signpzj var signpza var signpza var last step holds since distribution conditional given continuing step var signpzqj signpzq give lower bound expectation quantity first consider term note variance bounded next note lemma signpzq signpzqj recall sin furthermore wegkamp zhao section gpkq nonnegative scalars hadamard product wegkamp zhao section show also therefore applying lemma stated min set see probability least therefore combining everything var signpzq signpzq noting mab proves desired result define csigns lemma suppose random variable erw erw eretw proof lemma erw erw erw erw erpew since erew erpe ptw ptw ptw ptw rearranging terms proved lemma bounding error estimating variance lemma lemma assumptions definitions theorem probability least logppn event bounds assumption hold sab coracle logppn proof lemma recall proof theorem defined gpx signpx cos signpx pxq rgpx recall proof lemma given appendix vec cos pxqq vec cos pxqq pxq signpx signpx rpn note estimate variance define vectors entries vqa vqb vqi define pxi pxi abusing notation write pxi define writing define also hpxi signpxi signpxi vec qvqj cos vec cos vec qvqj cos vec cos qvqj cos vec following lemma proved appendix carries elementary calculations norms vectors lemma define proof lemma assume logppn bounds assumption hold constants depend ccov csparse cest probability least following bounds hold well logppn logppn logppn reshapes vector matrix define matrix norm jth column continue bounding error estimating bound term separately first term apply following lemma proved appendix lemma assumptions notation lemmas universal constant cstudentized logppn cstudentized second term since triangle inequality norm must satisfy appears twice expression bound difference term applying triangle inequality several times xqj qpq xqj xqj xqj xqj xqj logppn logppn xqj next state two lemmas proved appendix lemma probability least logppn lemma let defined assumption every rpn defined statement lemma point assume bounds derived lemmas hold lemmas shown true probability least event bounds assumption hold lemmas xqj ccov logppn applying bound along high probability events lemmas return obtain logpp logppn logppn logppn ccov logppn logppn logppn ccov last step define ccov use assumption logppn next returning logppn logppn logppn cmoment cstudentized last step applies high probability event uses fact cmoment lemma defining cmoment cstudentized using assumption logppn logppn finally returning applying lemma see logpp logppn cstudentized next logpp studentized cvariance denominator apply lemma finally since know sab sqab defining logpp studentized sab cvariance coracle see cstudentized cvariance sab coracle logppn calculations variance estimate lemma proof lemma assume logppn lemma define bounds assumption hold constants depend ccov csparse cest probability least following bounds hold well logppn logppn logppn reshapes vector matrix define matrix norm jth column proof lemma calculate cos csparse last inequality apply next uvqj cos cos cos applying assumption fact bounds assumption hold csparse cest logppn furthermore applying lemma probability least logppn csparse cest finally since logppn last step holds assumption theorem logppn csparse cest logppn csparse cest next logppn logppn uvq cos logppn uqv cos upq vqj cos logppn logppn logppn logppn logppn csparse cest cest logppn csparse cest logppn step applies assuming bounds assumption hold last step use assumption logppq uvq cos calculate norm finally noting vec matrix uvq cos uvq cos qvq qvqj next applying bounds assumption hold logpp logppn logppn ccov cest cest cest ccov logppn define ccov est cest use assumption logppn lemma assumptions notation lemmas universal constant cstudentized logppn cstudentized proof lemma definition pxqq pxq signpx signpx rpn therefore since fixed pxqqx varpxj pxqq pxqq recall pxq rgpx define kernel signpx hpx gpx signpx cos define gpx qgpx gpx xqgpx gpx xqgpx note order sup sup last step applies lemma sup sup cmoment use lemma last bound next gpx qgpx gpx qgpx gpx gpx therefore pxqq pxqs gpx obtain next examining definition npn gpxi gpxi therefore using fact always gpxi ergpx using bernstein inequality peel theorem therefore probability least logppn use assumption logppn using bernstein inequality using fact always probability least gpxi ergpx logppn use assumption logppn therefore gpx ergpx gpx ergpx logppn gpxi ergpx set cmoment use logppn apply lemma bound rgpx combining everything proves probability least logppn logppn setting cstudentized using fact logppn logppn cstudentized lemma probability least logppn proof lemma definitions see pxqq pxqj pxqsj pxi pxi pxi pxi first bound pxi pxi pxqj pxq pxi pxi pxi pxi pxi pxi pxqj pxi pxi pxi pxi handle two terms separately first bound convenience define pxi pxi pxi pxi since positive semidefinite matrices ones diagonal bqfjk kprpn fjk rpn vector pfjk pfjk zeros elsewhere next bqfjk afjk bfjk fjk afjk fjk bfjk pxi fjk pxi fjk pxi pxi qqj fjk first inequality follows fact second inequality follows triangle inequality next observe pxi fjk psignpxi signpxi fjk conditioning mean variables taking values since furthermore conditioning erp pxi pxi therefore applying hoeffding lemma see example lemma massart exp pxi pxi qqj fjk exp applying lemma stated pxi pxi fjk taking union bound returning pxi pxi pxi pxi next turn second term since always see rpn pxi pxi mean terms taking values applying hoeffding inequality pxi pxi erhpxqhpxqj setting taking union bound see pxi pxi erhpxqhpxq returning probability least pxi pxi pxq proves bound next complete proof bound pxqs pxqs pxi pxi pxqs pxqs pxi pxi pxqs pxi pxqs pxqs pxi pxqs therefore since pxqs pxqs pxqs sign writing denote jth basis vector rpn exp pxi pxqs exp pxi pxqs exp first inequality follows convexity second applies hoeffding lemma pxi pxqs therefore taking union bound therefore combining probability least logppn last step uses fact lemma let defined assumption every rpn defined statement lemma proof lemma since statement deterministic treat matpzq rpn fixed vecpm vecpm vecpm var vecpm pxq var vecpm erhpx var ervecpm hpx pby law total varianceq var vecpm hpx pvecpm hpx pvecpm signpx signpx signpx signpx signpx signpxj pwhere jth column mjj rsignpx signpx mjj mkj mkj rsignpx signpx finally wegkamp zhao theorem lemma let fixed vector let random vectors necessarily independent pzi rzi erexpttv pzi rzi squs probability least pzi erzi proof lemma assumption exp pzi erzi exp vershynin lemma tracking constants carefully lemma exp pzi erzi convexity pzi erzi exp exp pzi erzi therefore pzi erzi exp pzi erzi exp setting proved desired result
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model demand response aggregator competition selling stored energy regulated unregulated power markets mahdi motalleb reza abstract work concerned application principles model competition demand response aggregators selling excess energy stored electrochemical storage devices directly aggregators power market market framework presented alternative traditional market structure may better suited developing demand response technologies addition increasing penetration independent renewable energy generation devices demand power generated utility combustion fuel could replaced lowering emission pollutants energy used charge batteries produced sustainably traded smaller scales four variants game considered unregulated competition stackelberg regulations transaction price size without scheduling nash equilibrium derived game variant order serve decision making criteria determines optimal bidding strategy aggregator sell market model applied case study involving completion selling two aggregators bidding strategy dependent parameters inherent aggregator energy storage hardware strategy selected aggregator vary variations game conditions considered demand response scheduling offers greater payoff aggregators implement compared addition transaction price volume regulations market affect participants optimal bidding strategies nash equilibrium lowers payoffs aggregators participating market relative unregulated competition index terms demand response aggregator dra game theory game stackelberg game demand response scheduling nash equilibrium introduction development implementation smart grid technologies offers substantial advantages traditional vertically integrated electric utilities parties stand benefit proliferation technologies endusers electrical power primarily wish minimize cost pay utility energy require power devices opposed utility company wishes maximize net profit concerned load scheduling must remain able provide adequate supply paper market framework developed based concepts demand response aggregators dras compete sell energy stored consumer battery systems model determines optimal bidding decision dra maximize pay conditions incomplete information game game theory used power system markets interpret participant behavior deregulated environments allocate costs among pool participants two different hybrid algorithms presented generation expansion planning gep problem electric market algorithms divided two programing levels master slave static computational game theoretic model also developed investigate impacts competition wholesale price electricity mahdi motalleb reza ghorbani renewable energy design laboratory redlab department mechanical engineering university hawai manoa honolulu hawai phone fax motalleb rezag demand electricity profits firms levels various polluting emissions medium run electricity market simulator mrems based game theory presented simulator incorporated two different games one unit commitment thermal units one strategic bidding hourly market clearing common electricity bidding mechanisms electricity auction markets analyzed using signaling game theory also swarm platform used develop simulation model based multiple agents role sustainable energy volatility investigated context market participant competitive expansion planning problem incomplete information method generation company genco perceives strategies market participants applied make decisions strategic generation capacity expansion formulation presented investigate learning speed traders strategic collaboration dynamic electricity markets shown learning speed traders decreases large fluctuations power exchange market authors expanded application equilibrium problem equilibrium constraints epec model typically used games spatial market model prevalent market setting investigated international market metallurgical coal whose market characteristics provide arguments wide variety alternate market structures framework proposed employing swarm optimization pso hybrid simulation approach provide suitable platform sustainable gep bayesian game model analyzing transfer electric power presented explain transfer within market containing players controlling unreliable marginal generators dynamic model developed analyze impacts natural gas market reformation promoting natural electricity ngfe generation hourly realtime pricing rtp applied natural gas electricity markets innovative game theoretic framework proposed retail electricity market energy internet high penetration distributed residential electricity suppliers energy cells envisioned energy internet proposes large number distributed renewable energy generation energy storage systems connected grid interfacing devices framework economic operations future residential distribution systems presented cases involving extensive participation distributed electricity prosumers defined novel roles utilities customers game theoretic algorithms used determine retail electricity market clearing price consider group coalition scenarios multiple electricity prosumers game theory useful mathematical tool handle problems related demand side management dsm several demand response programs proposed differing objectives determining optimal hourly incentive offered customers sign load curtailment scheduling load usage creating several possible tariffs consumers net demand remains threshold adjusting demand meet supply well smoothing aggregated load system evaluating impact response capability consumers promoting distributed penetration game theory used present optimization model minimize load curtailment needed restore equilibrium operating point system fault condition loss generation optimal pricing evolutionary perspective proposed urban gas markets power structure demand response program employed simulate user demand response processes principle product game theory basically game theory formal study competitive conditions choices potentially affect interests players approach liquefied natural gas lng projects presented approach based consensus algorithm addressing consensus output common value using cost functions within framework based game theory another work process conceptual planning project evaluation oil gas industry presented set strategic decisions generated binary genetic algorithm processes industrial environmental concerns evaluated game theoretic approach industry environment considered two players conflicting interest find optimal strategies governing energy policy matrix model proposed assess economic impact make operational decisions constrained restructured electricity markets game theoretic modeling approach performed develop financial transmission rights bidding strategies power suppliers assuming adequately forecast locational marginal prices lmps game theoretic model considered multiple participants well network contingencies evolutionary game approach proposed analyze bidding strategies electricity markets demand research work presented characterized impact plans maintenance decisions gencos applying cournot model used strategic generation dispatch generating units electricity markets order characterize process nash equilibrium used set strategies nash equilibrium player better unilaterally changing strategy authors suggested use theory evolutionary games concept near nash equilibrium simulate electricity market presence two producers novel approach presented generation maintenance scheduling gms problem electricity markets coordination procedure independent service operator iso modeled gencos gms process modeled dynamic game genco optimal strategy profile determined nash equilibrium game conventional model dynamic cournot electricity market game gencos assumed hold uniform accurate belief concerning market dynamics replaced realistic model subjective demand errors coining new equilibrium concept termed subjective equilibrium system performance equilibrium output profit customer surplus analyzed results suggested system equilibriums strongly influenced gencos knowledge market demand game theory different types games utilized analysis different types problems different types games categorized number players involved symmetry game whether cooperation among players allowed literature power markets different game models used include cooperative stackelberg forchheimer one leader bertrand games players leaders beside application power markets game theory used diverse related fields analysis electric vehicle charging station construction charging method plugged hybrid evs analysis power grid vulnerability performance evaluation thermal power plants analysis effects higher domestic gas prices russia european gas market interactive energy management networked active distribution systems present work demand response market framework developed based considerations market commodity energy stored consumers batteries end users electricity charge batteries electricity cheap sell back grid consumers electricity expensive four basic concepts game theory including players nodes moves payoffs proposed market framework players dras node belongs one dra possible moves game player options chosen given dra player strategy determine action player take stage game proposed market structure strategies different biddings dras payoff player dra difference monetary gain upon selling stored energy cost storing energy comprehensive quadratic cost function proposed study discharging energy stored batteries also assumed players privy information player moves therefore problem cast real incomplete information game two different types game considered stackelberg game dras compete sell energies stored batteries without limitation constraints transaction size price stackelberg game utility game leader controls market restricting game two different programs considered demand scheduling water heaters thermostatic devices dynamic programing used scheduling section explains principles game theory application energy markets section cost function payoff function developed order utilize proposed method section describes proposed market framework including alternate game types schedules considered presented market framework applied case study dissection results provided section finally conclusions possible future works presented section principles definitions game theory power market first let briefly present fundamental principles definitions concepts need analyze power market following subsections cover proposed function use model aggregator cost bidding process dras payoff function dras game theory power market game theory beliefs formulated risky alternatives order maximize expected revenue payoff function aggregators real competition player dra remains unaware detailed data players instance data strategies payoffs probability theory gives expected value payoff given probability events bay law used revise beliefs given new information objective analysis find nash equilibrium best strategy set possibilities sense incentive exist alter equilibrium strategy despite strategies chosen players fig illustrates power market aggregators acting buyers others acting sellers submit bids center iso center sets market spot price aggregators fig general structure power market including buyer dras seller dras iso dras participate market sell available stored energy consumed powering local loads power transaction game aggregators transactions modeled game strategies maximize payoffs two types games considered research stackelberg game takes place player dra interested maximize payoff cooperation aggregators coordinate strategies game aggregators choose strategy aggregators try identify best response strategy stackelberg game one leader utility limit market price power transferred ensure policies respected market aggregator cost function subsection shows dra cost function presented quadratic function since energy resources available sale given dra aggregation stored energy residential batteries aggregator cost function battery cost function cost discharging battery battery function chose variant widely used logarithmic barrier function used penalty function interior point methods hbat log bat cost battery house function stored energy utility give higher prices ahi pricing coefficient determined total load kwh parameter used give cost values close values given quadratic one also serves maximum typical value relation presented cost function quadratic one understood taylor expansion since cibat log taylor series expansion ahi ahi updates load data resolve every minutes measurement devices assume load constant minutes updates quadratic cost function battery rewritten function power bat ahi order obtain coefficients ahi ahi two factors considered electricity price grid charging battery capital maintenance costs battery thus cost battery house charging power amount cost house pays utility sell electricity charge battery amount power contribution capital maintenance costs battery house chbat chgrid chci grid power coefficients ahi obtained using since individual house cost functions quadratic form aggregation houses also produce quadratic cost functions dra cost function aggregator selling stored power coefficients number houses aggregator bidding process dras section explains dras bid price selling stored energy batteries based cost function defined section marginal cost aggregator dpi since marginal cost linear function stored power batteries assume aggregators bids also linear functions stored power bid price per unit power discharging power level marginal cost electricity slope bid curve market coordinator iso receives sale bids dras given matches lowest bid buyer aggregators fig aggregator intends increase battery discharging beyond spot market price greater net power interchanged positive negative aggregator selling power buying power aggregators pmax maximum stored power aggregated storages dra pmax minimum local load dra fed local batteries established one governing policies aggregator max slope buy sell pmax fig aggregator biding curve aggregator payoff function mathematical formulation dra payoff function proposed framework described given spot market price ith dras payoff difference discharging cost batteries aggregator power transactions net power transacted let denote power transaction associated pair aggregator payoffs participants characterized region payoff point without trading payoff trade would take place nash bargaining problem two aggregators denoted aggregators agree upon solution represents acceptable deal participants involved problem formulated follows given initial solution find different solution satisfies conditions point point nash equilibrium solution bargaining game definition payoff function dra objective function maximize pij tij rijk pij tij pgi tij pgi payoff dra seller transmitting power tij aggregator receiving power tij aggregator contract difference tij tij due transmission losses negligible transmission losses tij tij parameter transaction price pgi amount local loads aggregator fed local storages power transaction first term gross revenue aggregator due transaction second bracketed term change aggregator batteries discharging cost owing transaction case two aggregators shown aggregator seller buyer objective function maximize rseller rbuyer rseller rbuyer tranction price rseller rbuyer aggregators payoffs aggregator constrained storage capacity payoffs constrained rseller rbuyer indicating negative payoffs excluded transactions payoff aggregator zero fig system including two dras seller buyer real competition aggregators participants information participants cost function coefficients availability batteries biding prices following section shows proposed market framework competition dras sell energy stored batteries incomplete information game proposed market framework dras proposed methodology explained section first subsection includes details incomplete information game make market competition among dras proposed market framework based game theory presented second subsection mathematical probabilistic details developed market framework including two different game types two demand scheduling programs provided second subsection fig shows visual schematic proposed market framework dra manages number buildings control signals sent wireless network house contains various devices possibly including water heaters whs batteries air conditioners acs electric vehicles evs dras decide bid buying selling power market scheduler outputs signals deciding setting function satisfy conditions game social welfare games scheduling games implemented given inputs hot water usage data games possibly electricity price signals utility games scheduling dras communicate wireless network servers communicate market directly arrange transactions stackelberg games servers also mitigate size price transactions accordance utility policy constraints transmitted wireless network fig general schematic proposed market framework two types games two scheduling program market competition dras incomplete information section market model proposed model competition among aggregators electricity market participants incomplete information based aggregator payoff function storages discharging costs power transactions spot market price hence individual aggregator payoff function bids offered aggregators participant estimates participants bids order maximize payoff aggregator complete information concerning payoff lacks information critical predicting others actions precisely competition known incomplete game aggregator unknown characteristics modeled classification participant types dra type contains information concerning payoff function aggregators pay functions grid electricity prices availability charged batteries etc aggregator type corresponds battery discharging cost structure coefficients aggregator would full knowledge costs estimate remaining aggregator costs participant adjusts slope bidding curve fig order maximize payoff choice corresponds bidding strategy bid low bid high bid marginal cost game main goal game determine optimal process used determine value choose order maximize payoff despite unknown parameters participants dras use bayesian approach deal incomplete information approach probability distribution represents unknown parameters expected value payoffs aggregators maximized estimating probability distribution aggregator uses information common aggregators consider scenario described dra perspective competing dra selling stored energy batteries aggregator depicted fig assume participants types drawn random hypothetical populations containing types respectively index possible types dra respectively instance model uncertainty discharging cost would assume possible types corresponding cost generally aggregator knows know hence know opponent type know discharging costs dra estimates probability distributions random variable type competitors based freely available published information electricity prices demand curves water usage water heaters availability storages aggregators parameters fig grid including three dras sellers buyer market framework purposes goal game calculate expected payoffs different strategies aggregators mentioned scenario find nash equilibrium choose strategy maximizes payoff epam epam expected payoff function aggregator type type opponent aggregator conditional probability dra type dra type prob prob shows conditional probability function basic probability distribution corresponds probability type type dra strategy bidding determined sam vector strategies type example vector strategies may bid high bid marginal cost bid low conditional payoff dra depends strategies dra dra type sbn aggregator needs know opponent type maximize since information available aggregator maximizes expected value payoff epam similarly aggregator expected payoff function epbn prob game considered previous converts game imperfect information complete game number different types aggregators respectively case aggregator knows others payoff functions basic probability distributions involved game know opponent type nash equilibrium solution consider two variants noncooperative stackelberg scenario involving three dras competing sell energy stored batteries game effect demand response scheduling whs similar thermostatic storages considered market model proposed scheduling algorithm shown fig depicts procedure used calculate requisite parameters used obtaining nash equilibrium fig steps calculating required parameters market following subsections explain determine parameters needed process game demand response scheduling program game without scheduling assuming dras know payoff function may determine nash equilibrium following procedure outlined fig define aggregators types first step set different possible discharging cost function coefficients defined aggregator number defined types aggregators respectively participant type determined largely electricity price availability batteries discharging electricity price significant batteries charged directly grid availability stored energy batteries may estimated considering use patterns high power consumption devices water heaters whs thermostatic devices example whs consuming power based water usage water temperature data energy stored batteries consumed locally power devices considering two factors may classify aggregator one four scenarios elecexp whoff elecexp whon elecch whoff elecch whon game without demand response scheduling depends entirely water usage completely independent price electricity since thermostat functions maintain social welfare demands customer without considering price two events independent probability events occurring product probabilities event aggregator scenarios elecexp elecexp elecexp elecexp elecch elecch elecch elecch probability event probability whs power state whoff whon obtained scheduler block described fig case scheduler determined typical water consumption profile objective function maintain social maintains temperatures range require customer ensure sufficient hot water available let probability distributions modeling uncertainties aggregator discharging cost probability aggregator type scenario probability aggregator type define basic probability distribution game probability would represent participants types respectively would depend electricity price local demand whs conditions expected probability type type defined participation coefficient scenario equal one participants selling energy market otherwise example electricity expensive water usage high energy stored electrochemical cells used preferably powering local loads price electricity cheap stored energy available sale loads may supplied directly grid lower cost whereas even price high absence local demand leaves energy available sale given scheme predicting original may treated imperfect information aggregator knows cost function coefficients may compute discharging costs without knowing opponent discharging cost step calculated step define aggregators strategies aggregator type corresponds set strategies defined bid slopes fig assume aggregator three strategies based slope bid curve gth aggregator fig set three different strategies strategies bidding less marginal cost biding marginal cost bidding marginal cost coefficient defined step sam sbn calculated step define aggregators conditional payoff step following system equations solved combination participant types strategies power remaining storages cells dra feeding local loads power sold aggregator power purchased aggregator transaction price determining conditional payoffs dra may calculated conditional payoff matrixes take form row corresponds strategy type column corresponds strategy type instance corresponds type payoff decides bid marginal cost situation type bids marginal defined similarly step define expected payoff matrices expected values payoffs epa epb calculated using final form epam matrix epam column epa corresponds presumed strategy participant opponent instance column type payoff uses strategy type strategy type notation applies epb step obtain nash equilibrium strategies utilizing epa epb nash equilibrium pairs obtained look collection strategies aggregator strategy would represented best response aggregator strategies nash equilibrium prediction game played aggregator optimal bid derived equilibrium point aggregators predict particular nash equilibrium occur incentive play differently game scheduling game scheduling water heaters considered steps game game without scheduling section single exception first step noncooperative game without scheduling whs keep water temperature given range satisfy social welfare constraints situation whs independent electricity price game scheduling objectives social welfare demand scheduling first step finding nash equilibrium game scheduling requires knowledge conditional probabilities since events independent define aggregators types section set possible discharging cost function coefficients defined dra acting seller similarly define four scenarios depending variables two variables electricity price whs condition game events dependent due scheduling two events dependent probability occurring defined using conditional probability elecexp elecexp elecexp elecexp elecexp elecexp elecch elecch elecch elecch elecch elecch second part equation conditional probability instance elecexp probability water heater given electricity expensive conditional probabilities calculated using scheduler mentioned fig considering two objective functions social welfare scheduling game batteries available provide power scheduling disabled loads thus dra stands make larger payoff game without scheduling stackelberg game without scheduling previous sections considering games assumed transaction price transaction power depend dras bid prices dras storage capabilities tend charge batteries energy less expensive hours order sell dras peak hours price high thereby maximizing gains per transaction enough dras phenomenon termed reverse peaks becomes apparent shift peak consumption accompanied high tendency aggregators sell stored energy back high price hours undesirable effect resolved allowing market controlling center utility center controlled utility play role market controller game leader allowed set bounds market transaction quantity price thereby ameliorating undesirable effects market associated reverse peaks competition including market controller modeled stackelberg game nash equilibrium exist stackelberg rational player chose particular predictable strategy offers incentive alter despite strategies chosen competitors finding nash equilibrium stackelberg game follows procedure outlined fig similar previous sections exception calculating conditional payoff step transaction powers transaction price controlled leader utility values pamax pbmax fall within range permitted controller transactions proceeds obtained values higher maximum allowable values maximum values used clear market applying utility policies stackelberg game prevents huge effects players aggregators market also prevents reverse peaks stackleberg game scheduling analysis stakleberg game scheduling follows reasoning similar one without scheduling section exception must assume electricity price water usage dependent events conditional probabilities used calculating probabilities intersection dependent events possible aggregator scenarios step analogous done section considering scheduling game case study discussion results illustration purposes consider system three dras labeled depicted fig load data three dras taken real grid model island maui states fig case study including agregators sellers buyer dra manages houses contain electrochemical storage cells dra manages house storage devices dra manages houses equipped storage devices houses whs assumed draw majority load nominal power residential storage device thus maximum generation capacity dra pagen pbgen pcgen case study assume nominal power thus total power demand pademand pagen similarly aggregators pbdemand pcdemand since marginal cost dra greater aggregators compete sell power assume dras discharge cost coefficients simplicity assumed sellers know discharging cost coefficient buyer fig shows normalized price electricity maui island function time course single day price updates resolve every fifteen minutes normalized average water consumption houses aggregators shown fig time intervals fig normalized electricity price one day case study resolution normalized water consumption time hour fig normalized average water consumption houses seller aggregators case study resolution results stackelberg games without scheduling presented following sections results game without scheduling purposes take prices half normalized price electricity expensive normalized price data fig elecexp elecch status whs may determined water consumption data fig order keep water temperatures range typical runs consumes power fifteen minute intervals day fig output status scheduler shows status seller whs game without scheduling time hour fig status whs game without scheduling seller aggregators probabilities whon follow immediately known ratio intervals using equations section probabilities four scenarios probability distributions dras mentioned scenarios expected probabilities conditional probability vectors dras calculated using assume dra choses one following strategies biding marginal cost marginal cost biding biding marginal cost therefore strategies finally equations step section expected pay matrixes dras chose bidding strategies finding nash equilibrium inspecting expected pay matrices find dra epa epa see bottom row matrix bid marginal cost type type results greatest expected payoff regardless competitors strategies say strategy dominates others dra also conclude rational player choose strategy bid marginal cost regardless type dra need consider best response strategy represents payoffs possible strategies competing player bids marginal cost examining last column epb done considering column tells type receive greatest payoff bidding marginal cost type receive greatest payoff bidding marginal cost player knows play way incentive change strategy strategy represented column column epa step section pair strategies column epb column epa nash equilibrium game nash equilibrium consistent prediction game played rational players participants predict particular nash equilibrium occur incentive play differently strategy pairs nash equilibrium participants maximum strategies maximize participants conditional payoffs participant could obtain least payoff equilibrium point may obtain depending opponent strategy section scheduling whs based one objective function social welfare situation based water consumption fig effected electricity price fig results game scheduling case scheduler used schedule whs aggregator based electricity price joint objectives social welfare keeping water temperature given range using fig scheduling using similar section elecexp elecch fig status depicts scheduling whs aggregator using inputs data shown fig highlighted areas show times electricity expensive prices higher half maximum price time hour fig status whs game scheduling aggregator case based fig equations section elecexp elecexp elecexp elecexp elecch elecch elecch elecch aggregator comparison fig fig shows scheduling curtail load whs much therefore new demand aggregator assumed aggregators schedule whs therefore demands aggregators demand demand section gen pcdemand load curtailment aggregator power stored batteries sell aggregator probability distributions strategies section following game process expected payoff matrices epa epb following reasoning used section finding nash equilibrium aggregator obtains higher learn rational participant would bid marginal cost type marginal cost type strategy represented column epa epa pair strategies column epb payoff bidding marginal cost inspecting last column column epa nash equilibrium game nash equilibrium strategies bid higher marginal cost regardless type bid marginal cost depending type type respectively stackelberg game without scheduling game limitation power transactions transaction prices selling stored energy aggregators transaction price transaction power obtained real markets exist controller limits transaction power transaction price independent controller empowered utility communicates utility control market case study noncooperative games section transaction prices varied assumed controller limits selling price stored power less time interval case stackelberg game without scheduling probabilities strategies section final expected pay matrixes epa limitations transaction power price expected payoff values decreased game comparison unscheduled game section nash equilibrium strategies still reasoning bid higher marginal cost regardless type bid marginal cost depending type type respectively pair strategies column epb column epa nash equilibrium game stackelberg game scheduling case aggregator schedules whs based electricity price fig water consumption fig limitations market due constraints imposed controller must considered situation closest reality comparison situations considered objective functions social welfare keeping water temperature houses range scheduling based electricity price water consumption also real market policies real limitations applied situation game probabilities strategies section final expected pay matrixes epa epb nash equilibrium strategy bid higher marginal cost regardless type bid marginal cost depending type type respectively pair strategies column epb column epa nash equilibrium game presence controller stackelberg games enforcing market restrictions game payoffs aggregator decreased section comparison also section comparison conclusions work applied principles elementary game theory model competition dras selling power stored electrochemical storage cells power market dras buyers situation presents competitors incomplete information game dra unable determine parameters associated cost operating player equipment order utilize better developed theory imperfect information approximated local power demands using known water usage data assuming major component local loads using public statistical data bayesian approach derive probability distributions dra types examined four types consider effects pure competition noncooperative without scheduling regulated competition stackelberg without scheduling also considered types game addition scheduling examine effect adding games games quite idealized would practical implement various reasons explained stackelberg games relatively realistic provide possible beneficial alternative currently popular market structures application model data compiled island maui served case study analyze results model likely conditions results showed bidding strategies dras dependent parameters associated hardware conditions games considered addition scheduling increased payoffs dras implemented compared payoffs decreased regulated stackelberg games compared purer competition scheduling conserves energy resources spent acquiring competition dras selling power independently generating company provides alternative market structure currently prominent competition selling power market decreases market price electricity allows smaller independent energy produces enter market thereby lowering demands placed generating company treatment power stored batteries bought utility power produced locally renewable energy technologies treatment would analogous increasing penetration renewable energy generating technologies photovoltaic systems great effect gaining independence combustible economy reducing pollution believe proliferation optimization smart grid technologies play vital role building sustainable power economy university hawaii renewable energy design lab redlab currently developing hardware systems capable implementing presented model work forthcoming exploring implications competition model alternative market structure state hawaii currently face difficulties integration larger numbers systems legacy grid much work required reach states renewable energy goals acknowledgements project sponsored national science foundation award number thanks john branigan helping edit paper bibliography shayanfar saliminia lahiji aghaei rabiee generation expansion planning pool market hybrid modified game theory improved genetic algorithm energy conversion management vol shayanfar saliminia lahiji rabiee aghaei generation expansion planning pool market hybrid modified game theory particle swarm optimization energy conversion management vol zeinalzadeh alptekinoglu arslan learning inventory competition world congress game theory 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efficient computation dct via summation parts mar abstract paper introduces new fast algorithm discrete cosine transform dct based formula proposed method converts dct matrix alternative transformation matrix decomposed sparse matrices low multiplicative complexity method capable scaled exact dct computation associated fast algorithm achieves theoretical minimal multiplicative complexity dct depending nature input signal simplifications introduced overall complexity proposed algorithm reduced several types input signal analyzed arbitrary null mean accumulated null signal proposed tool potential application harmonic detection image enhancement feature extraction input signal level discarded signal required integrated keywords dct fast algorithms image processing introduction discrete transforms play central role signal processing noteworthy methods include trigonometric discrete fourier transform dft discrete hartley transform dht discrete cosine transform dct discrete sine transform dst well haar transforms among methods dct applied several practical contexts noise reduction watermarking methods compression techniques harmonic detection cite fact processing signals modeled stationary type random process dct behaves asymptotic case optimal transform terms data decorrelation approximation holds true correlation coefficient related stochastic process tends unit case many real images moreover recent increase processing demand consumer electronics big data manipulation emphasizes necessity fast efficient dct computation consequence dct adopted several image video coding schemes jpeg hevc avs china aiming minimizing computational cost dct evaluation number fast algorithms dct proposed including chen dct algorithm lee method loeffler algorithm dct factorization arai dct multiplication operations required dct others transforms implemented via long sequences additions operation sign changes thus algorithms require multiplications often higher computational costs therefore methods developed order reduce overall number multiplications arai dct particularly useful furnishes scaled version dct spectrum applications harmonic detection image compression scaled dct often sufficient tool contexts relative value spectrum necessary therefore part cost computing dct avoided coelho department electrical computer engineering university calgary calgary canada cintra signal processing group departamento universidade federal pernambuco department electrical computer engineering university calgary calgary canada rjdsc dimitrov department electrical computer engineering university calgary calgary canada among fundamental mathematical tools separate technique counterpart method although applied several contexts computational physics approximate second derivatives approximations linear equation computational fluid dynamics rapid calculation slow converging series electromagnetic problems particularly overlooked signal processing community early attempts employ numerical analysis tool due collaborators context dft computation evaluation fourier coefficients errors calculations aim paper propose new fast algorithm dct computation based formula periodic signals introduced method sought achieve theoretical minimal multiplicative complexity exact dct computation moreover minimize computational costs proposed algorithm also sought provide scaled version dct spectrum proposed algorithm finds application important problems feature detection level may relevant also applied scenarios input signal natively accumulated integrated situation occurs face recognition problems usual algorithms require data integrated paper organized follows section furnish mathematical background technique dct considering matrix formalism detail proposed algorithm dct section section introduced method assessed terms computational complexity comparisons competing algorithms shown section brings final comments remarks mathematical background technique equivalent method let two signals prescribes denotes forward difference operator given expression simplified assumption following additional weak conditions admitting considered signals periodic period established condition restrictive indeed fourier analysis often assume input signals periodic particular dct obtained solution harmonic oscillation problem expression interpreted transformation let input signal transformed ker given discrete transformation kernel kth component therefore ker table common discrete transform kernels transform ker dft exp dht dct cos dst sin offset removal accumulation system formula figure block diagram proposed architecture transformed output signal table summarizes common transformation kernels therefore applying yields following expression components ker ker comparing notice original transform expression alternative form input data kernel function processed notice output accumulator system input signal whereas ker derives forward difference system input signal ker although forward difference system causal fact poses difficulty formalism ker random deterministic sequence whose values known priori moreover possesses null mean following expression holds true trigonometric transforms condition implies null value therefore simplified written ker summation ranges means transformation matrix linked dimension fact contrasts original transformation matrix size thus effected dimension reduction transform computation consequence computational cost associate algorithms expected reduced figure depicts overall diagram transform computation based formula assumed arbitrary signal notice power two removal block accumulation system multiplierless operations discrete cosine transform dct linear transformation maps signal another signal according following relationship cos expression given compact format means matrix representation indeed considering signals column vector format dct matrix whose given cos transformation matrix following cos particular definition dct adopted therefore component considered derive loeffler dct algorithm notice evaluated without multiplications heideman introduces mathematical analysis multiplicative complexity major discrete transforms result multiplicative complexity theory relevant current work following transform blocklength power two minimum multiplicative complexity dct general formula given theorem obtain dct computation via section apply technique propose alternative computation dct next analyze resulting expressions order derive fast algorithm means matrix factorization matrix formalism facilitate development sought dct fast algorithm formula matrix formalism first forward difference operator needs adapted manipulate matrices let square matrix matrix resulted cyclically applying forward difference operator row therefore considering formula shown written according vector represents dct spectrum accumulated input signal considering identities symmetry identities entries matrix given multiplicative form shown sin input signal possesses null mean therefore first row last column neglected follows remaining submatrix necessary computation particular matrix given sufficient computation dct level notice scaling matrix shows repeated multiplicands along rows thus following factorization examination matrix obtained diag expression tells matrix contributes scaling factors actual dct computation considering applications require scaled version harmonic detection color enhancement computational cost disregarded additionally context image compression diagonal matrices directly absorbed quantization matrix representing extra computation notice also scaling factor trivial multiplication implemented via simple operation therefore computational cost scaling matrix actually fast algorithm highly symmetrical matrix factornow aim provide sparse matrix factorization ization methods based butterfly structures directly applied therefore obtain following factorization matrix simple permutation matrix represents computational cost terms hardware translates simple wiring matrix additive matrix consisting usual butterfly stage present algorithms remaining matrix still contains mathematical redundancies due symmetrical nature considering matrix factorizations fast algorithm design described following expression obtained matrix contains rotation block decomposed thus figure sfg proposed algorithm dashed lines represents multiplication removal umulation system figure sfg stage consisting component removal coupled accumulation system diag means usual trigonometric manipulations notice required multiplicands satisfy finally complete sparse matrix factorization given signal flow graph sfg proposed algorithm shown figure computational complexity assessment comparison section assess computational complexity proposed algorithm measured arithmetic cost evaluate additive multiplicative count required introduced method results distinguish following scenarios input data arbitrary signal null mean signal iii accumulated signal null mean accumulated signal scenario general case thus less redundancy exploited aiming minimization computational cost scenario commonly found image compression applications scenario pertinent context feature detection instance level may relevant scenario iii represents case input signal natively accumulated integrated situation occurs face recognition problems usual algorithms require data integrated scenario combination last two cases notice scenario require offset removal block figure scenario iii require accumulation system needs removal block scenario requires neither offset removal block accumulation system table compares proposed method several prominent techniques dct evaluation discussed scenarios details arithmetic complexity assessment considered technique found scenario emphasized bold best results arai method proposed algorithm scaled dct methods show multiplicative complexity computation well parenthesis applicable furnish full multiplicative complexity scaled methods scaling factors considered multiplicative complexity loeffler lee chen dct algorithms limited case methods admit scaled computation proposed algorithm could outperform competing method scenarios iii although proposed method requires five multiplications compute scaled arai demands fewer scaling multiplications scenario ideally suitable proposed algorithm benefits level removal accumulated input signal however even case introduced algorithm could achieve theoretical minimum multiplicative complexity discussion scenario offset removal block requires seven additions compute mean value seven subtractions remove level input samples thus total additions necessary accumulation system requires total six additions thus removal block combined accumulation system requires total additions figure details inner structure abovementioned blocks multiplicative costs concentrated matrices account nontrivial multiplications minimum multiplicative complexity dct matrices contribute additions thus scenario proposed algorithm requires total additions demonstrate scenario proposed method outperform relevant method dct computation nevertheless even scenario proposed method attained theoretical minimal multiplicative complexity relevant figure merit comparison competing methods extra required computation consists additions scenario proposed algorithm demand removal block one instantiation accumulation system figure rightmost subblock therefore total additions savings attained compared scenario consequently considering scenario proposed algorithm requires additions usual methods presented literature adapted scenario however compared costs scenario methods could save three additions instead seven additions computation table comparison products additions dct algorithms algorithm scaled scenarios mult additions loeffler iii lee iii chen iii arai yes iii proposed yes iii figure sfg removal block finite difference operator accumulated input signal level particularly true loeffler lee algorithms demand number multiplications proposed method exact spectrum required additionally proposed algorithm furnish scaled version dct spectrum performance required arai algorithm scenario iii proposed method demand accumulation system still requires removal block case removal block must adapted process accumulated input signals figure shows modified removal block requires additions generate sought sequence contrast traditional methods dct computation faced accumulated signals input data difference system figure shows sfg difference system completely eliminated proposed algorithm scenario combination scenarios iii appropriate scenario proposed method case proposed algorithm require stage input data directly applicable thus proposed method limited operations described architecture shown figure scenario usual methods requires stage consisting difference system figure demands seven extra additions proposed algorithm extended different sequence sizes benefit encoding methods hevc extension derived assuming arbitrary length considering dct kernel table applying forward difference operator obtain ker sin sin given despite similarity dst kernel expression contains factor sin separated diagonal matrix thus design associate fast algorithm becomes equivalent problem factorizing matrix entries given sin factorization accomplished means techniques detailed approach described paper dct showcase proposed dct algorithm readily extended two dimensional case thus suitable image processing applications indeed dct kernel separability dct computed successive calls dct achieved procedure computation dct row given input image computation dct column image derived step resulting image dct original image therefore principle dct algorithm immediately extended case without additional modification regarding dct computation domain different image representation schemes employed order reduce overall execution time works consider different image representation based slice intensity representation isr although representation useful scenarios pattern recongnition attain usual image representation adopted signal processing community remarks conclusion paper proposed new method dct computation based formula matrix formalism furnished arithmetic complexity assessed proposed method attained theoretical multiplicative complexity lower bound computation exact dct detailed theoretical minimum consists multiplications per achieving minimum multiplications trivial task many dct algorithms capable loeffler dct popular method literature achieve minimum moreover introduced method could also compute scaled dct five multiplications matching arai dct thus proposed method simultaneously match loeffler arai dct terms multiplicative complexity could attain multiplicative complexity minimum could also outperform several competing methods three different scenarios also consider additive complexity secondary comparison metric fact outperformed several popular dct algorithms input signal assumed possess null mean integrated accumulated signal proposed design understood fundamental mathematical block considered software hardware realizations additionally field integer approximate transforms benefit proposed scheme indeed design extremely approximation methods relies exact algorithms therefore exact algebraically precise methods useful resource even approximation could enough particularly proposed method alternative arai algorithm selected scenarios suggested method suitable context feature detection level may relevant well situations traditional algorithms require data integrated face recognition problems future works aim applying formula different transforms various blocklengths acknowledgements work partially funded cnpq capes facepe references oppenheim schafer yoder padgett discrete time signal processing vol upper saddle river britanak yip rao discrete cosine sine transforms academic press ahmed rao orthogonal transforms 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international journal multimedia applications ijma february review face detection systems based artificial neural networks algorithms omaima assistant professor faculty sciences university jordan amman jordan abstract face detection one relevant applications image processing biometric systems artificial neural networks ann used field image processing pattern recognition lack literature surveys give overview studies researches related using ann face detection therefore research includes general review face detection studies systems based different ann approaches algorithms strengths limitations literature studies systems included also keywords face detection face recognition artificial neural networks introduction past years face recognition received significant attention regarded one successful applications field image analysis human faces represent complex multidimensional meaningful visual stimulant developing computational model face recognition difficult face detection regarded fundamental part face recognition systems according ability focus computational resources part image containing face process face detection images complex variability present across human faces pose expression position orientation skin color presence glasses facial hair differences camera gain lighting conditions image resolution analysis facial expression primarily research field psychologists past years time advances many domains face detection tracking recognition pattern recognition image processing contributed significantly research automatic facial expression recognition face detection performed recognition system done extract relevant information face facial expression analysis two classes techniques face representation relevant information extraction geometrical feature extraction relies parameters distinctive features eyes mouth nose time face represented array pixel intensity values suitably appearance based approaches texture array compared face template using suitable metric research compared performances representation techniques face recognition therefore according complexity face detection process many applications based human face detection developed recently surveillance systems digital monitoring intelligent robots notebook cameras digital cameras cell phones doi international journal multimedia applications ijma february applications play important role life nevertheless algorithms applications complicated hard meet requirements specific past decade many approaches improving performance face detection proposed time many literature studies focused survey face detection techniques artificial neural networks ann used largely recent years fields image processing compression recognition encryption pattern recognition many literature researches used different ann architecture models face detection recognition achieve better compression performance according compression ratio reconstructed image quality peak signal noise ratio psnr mean square error mse literature surveys give overview researches related face detection based ann therefore research includes survey literature studies related face detection systems approaches based ann rest paper organized follows section includes main steps face detection recognition section includes literature studies related face detection systems based ann section includes comparisons literature studies section includes recommendations finally section concludes work face detection recognition general face recognition system includes many steps face detection feature extraction face recognition shown face detection recognition includes many complementary parts part complement depending regular system part work individually face detection computer technology based learning algorithms allocate human faces digital images face detection takes sequences input locates face areas within images done separating face areas background regions facial feature extraction locates important feature eyes mouth nose positions within detected face feature extraction simplifies face region normalization detected face aligned coordinate framework reduce large variances introduced different face scales poses accurate locations feature points sampling shape facial features provide input parameters face identification face analysis task facial expression analysis face animation face synthesis simplified accurate localization facial features face identification generates final output complete system identity given face image based normalized face image facial feature locations derived previous stages feature vector generated given face compared database known faces close match found algorithm returns associated identity main problem face identification large differences face images person compared different persons therefore important choose suitable face classification technique provide good separate ability different persons face identification wide range applications offers way human identification face used important biometric security applications recently face recognition received wide interest number countries integrating facial information electronic passport biometrics fingerprints iris addition security law enforcement face recognition also applied entertainment consumer electronics means natural user interface recognizing existence user identity consumer devices offer customized services thereby creating international journal multimedia applications ijma february enhanced user experience achieve face recognition system processing stage system designed satisfy application requirements face recognition involves comparing image database stored faces identify individual input image related task face detection direct relevance face recognition images must analysed faces identified recognized detecting faces image help focus computational resources face recognition system optimizing systems speed performance figure framework system main steps face detection system shown face detection separate image windows two parts one containing faces one containing background process difficult commonalities exist faces vary terms age skin color facial expression also differing lighting conditions image qualities geometries face detector would able detect presence face set lighting conditions upon background figure general face detection system artificial neural networks face detection recent years different architectures models ann used face detection recognition ann used face detection recognition models simulate way neurons work human brain main reason role face recognition research includes summery review researches related face detection based ann international journal multimedia applications ijma february retinal connected neural network rcnn rowley baluja kanade presented face detection system based retinal connected neural network rcnn examine small windows image decide whether window contains face figure shows approach system arbitrates many networks improve performance one network used bootstrap algorithm training progresses training networks add false detections training set eliminates difficult task manually selecting training examples must chosen span entire space images first step adapted applied window image window passed neural network decides whether window contains face used three training sets images test seta collected cmu consists scanned photographs newspaper pictures images collected www pictures frontal views faces require ann examine pixel windows test setb consists images containing faces windows test setc similar test seta contains images complex backgrounds without faces measure false detection rate contains images faces window detection ratio approach equal faces two large test sets small number false positives figure rcnn face detection rotation invariant neural network rinn rowley baluja kanade presented neural face detection system unlike similar systems limited detecting upright frontal faces system detects faces degree rotation image plane figure shows rinn approach system employs multiple networks first router network processes input window determine orientation uses information prepare window one detector networks present training methods types networks also perform sensitivity analysis networks present empirical results large test set finally present preliminary results detecting faces rotated image plane profiles international journal multimedia applications ijma february figure rinn face detection principal component analysis ann pca ann jeffrey norris used using principal component analysis pca class specific linear projection detect recognized faces video stream figure shows pca ann face detection system sends commands automatic sliding door speech synthesizer touchscreen door control server matlab java used developing system system steps search face image select every region input image use intensity values pixels inputs ann feed values forward ann value region represents face repeat steps several times time resized version original input image search faces different scales figure pca ann face detection fast neural networks fnn hazem proposed fast neural networks fnn approach reduce computation time locating human faces image divided small sub images one tested separately using fast ann experimental results comparison conventional neural networks showed high speed achieved applying fnn international journal multimedia applications ijma february polynomial neural network pnn huang proposed face detection method using polynomial neural network pnn local regions multi scale sliding windows classified pnn two classes face locate human faces image pnn takes binomials projection local image onto feature subspace learned principal component analysis pca inputs investigated influence pca either face samples pooled face samples convolutional neural network cnn masakazu matsugu described algorithm robust facial expression recognition combined face detection using convolutional neural network cnn figure shows cnn approach problem subject independence translation rotation scale invariance facial expression recognition addressed study evolutionary optimization neural networks stefan used ann get decision whether image region represents human face described optimization network hybrid algorithm combining evolutionary computation learning evolved solutions perform considerably faster architecture without loss accuracy proposed hybrid algorithm tackles problem reducing number hidden neurons face detection network without loss detection accuracy speed classification whether image region corresponds face could improved approximately figure shows evolutionary optimization ann figure cnn face detection international journal multimedia applications ijma february figure hybrid algorithm visualization input field connectivity multilayer perceptron mlp according rowley work marian beszedes milos oravec presented neural network based face detection system detect faces unprocessed input image figure shows mlp approach used image processing techniques normalization rotation position light conditions improvement small windows extracted input image multilayer perceptron mlp used detect rotation input window also decide whether window contains face system based method distribute decision among multiple sub networks algorithm used train ann result system form set containing locations human faces set includes face randomly generated samples sets extended according boosting method face samples samples obtained false face detections testing ann trained basic sets images containing faces system succeeded percent consider number faces input picture amount classified correctly three faces total number found shilbayeh proposed face detector based multi layer perceptron mlp ann maximal rejection classifier mrc improve efficiency detection comparison traditional ann ann organized reject majority patterns image backgrounds improving detection efficiency reducing computation cost maintaining detection accuracy international journal multimedia applications ijma february figure mlp face localization system back propagation neural networks bpnn zoran samcovic used ann face detection video surveillance ann trained multilayer back propagation neural networks bpnn three face representations taken pixel partial profile eigenfaces representation three independent generated based three face representation figure shows bpnn approach aamer mohamed proposed face detection system based bpnn via gaussian mixture model segment image based skin color approach start skin non skin face candidates selection features extracted discrete cosine transform dct coefficients based dct feature coefficients color spaces bpnn used train classify faces bpnn used check image include face dct feature values faces represent data set face candidates obtained gaussian mixture model fed bpnn classify whether original image includes face gabor wavelet faces ann sahoolizadeh proposed hybrid approaches face recognition based combined gabor wavelet faces ann feature classifier gabor wavelets used represent face image representation face images using gabor wavelets effective facial action recognition face identification reduced dimensionality linear discriminate analysis sampled gabor wavelet faces increase discriminate ability nearest feature space extended various similarity measures shows good performance achieve recognition rate orl data set international journal multimedia applications ijma february figure bpnn architecture face detection avinash raina presented face detection approach gabor wavelets transform feed forward neural network finding feature points extracting feature vectors gabor filter used feature extraction face detection classifier ffnn take feature vectors input location feature points contains information face approach graph constructed general face idea instead fitting graph feature points obtained characteristics face automatically facial features allow make decision face parts facial features compared locally instead using general structure two measures used study false negative false positive two measures calculated using equation equation follows mohammad abadi proposed approach based ann gabor wavelets detect desirable number faces fixed photo gray background used correlation window face photo estimated areas candidate face presence used step algorithm referred areas around section extraction gabor wavelets characteristics neural network classifier resultant areas lead detection face locations photo examined result estimation efficiency method different tests method simulated matlab used face photos non face photos training phase every face photo mirror photo angle degrees positive negative directions photos one pixel shift every directions placed training set reducing network sensitivity face photos also mirror degrees transformation placed training data obtained right answer error limit false negative tested image positive also test image size positive international journal multimedia applications ijma february anissa bouzalmat presented bpnn face recognition bpnn input feature vector based fourier gabor filters used algorithm detecting face regions images using color skin presents overlooked different background accessory clothing introduced gabor filters orientations resolutions get maximum information extract maximum information varying resolution orientation done generate extract features vector whole face image bpnn applied perform recognition task solution implemented using java environment results indicate proposed method achieves good results figure shows bpnn gabor wavelet face detection figure description proposed solution architecture skin color bpnn construction robust face detection system regarded one practical applications vigorous development kalavdekar prakash described face detection system process images based neural network detect face images achieving high detection rates analyzing video sequence current challenge since faces constantly dynamic motion presenting many different possible rotational illumination conditions solutions task face detection presented detection performances many systems dependent upon environment suggested system includes skin color filter image filtering mlp detection figure shows skin color bpnn face detection face detection attempt locate regions contain face given still image image sequence two main solutions face detection approaches elmansori khairuddin presented face detection method combines two algorithms skin color based face detector bpnn skin color based face detector used modeling distribution skin color identify areas likely regions skin identify potential areas skin equalizing probability likelihood problem space linearly separable linear threshold function offered solution supported sparse feature mapping architecture bpnn used represent function using arbitrary decision surfaces utilizing nonlinear activation functions experiments showed methods show closer performances classification face space method achieved high detection rates acceptable number false negatives false positives system implemented using international journal multimedia applications ijma february figure skin color bpnn face detection cascaded neural network finally zuo proposed fast face detector based hierarchical cascade neural network ensembles enhance detection accuracy efficiency used number neural network classifiers form neural network ensemble classifier specialized sub region space classifiers complement perform detection task organized neural network ensembles pruning cascade reduce total computation cost face detection stage simpler efficient ensembles used earlier stages cascade able reject majority non face patterns image backgrounds improving overall detection efficiency maintaining detection accuracy results showed proposed neuralnetwork ensembles improve detection accuracy compared traditional ann approach reduced training detection cost achieving detection rate equal comparisons different ann approaches literature studies based ann building face detection systems described study one studies based special architecture ann face detection many studies based one architecture ann multilayer perceptron mlp backpropagation neural networks bpnn retinal connected neural network rcnn rotation invariant neural network rinn fast neural networks fnn convolutional neural network polynomial neural network pnn studies based ann combination techniques methods principal component analysis ann pca ann evolutionary optimization neural networks gabor wavelet faces ann finally skin color bpnn studies based ann one studies includes experiments based different database training testing images many studies take detection rate performance measure studies take error rate performance measure studies talk exactly used database hand literature studies strengths limitations exactly determine best topology used face detection system high performance table shows information topology database images performance face detection systems taken many literature studies international journal multimedia applications ijma february table shows literature studies used different data bases image training testing set studies included table use known database used camera image samples number samples different one study example research take image samples whereas studies used one image sets samples note table also studies adopted different image dimensions table performance measures used literature studies research topology retinal connected neural network pca ann pnn data base training testing performance three training sets images test seta scanned photographs test setb images contain faces test setc images faces images complex backgrounds without faces measure false detection detect faces set test images select pictures sung data set faces train ann random noise pictures negative examples remaining faces data set followed random noise pictures first set images downloaded several websites one face image second set images downloaded website cmu acceptable number false detections error training epochs examples mis classifications made error detection rate false cnn training cnn number facial fragment images used layer face layer respectively number images also used layer recognition rate still images subjects bpnn training set contains face images collected various face dbs samples also include scaled versions face factor gabor wavelet ann orl dataset frontal faces tightly cropped grey images individuals variations pose illumination etc detection rates measured separate test set faces detect faces set test images mlp mrc bpnn training set face images mit images scaled test set images mit patterns generated different locations scales real images taken different lighting conditions digital camera images web images several websites detection rate error rate detect faces set real images processing time image processing time image international journal multimedia applications ijma february finally note table factors used performance evaluations different one study many studies used detection rate others used false rate figure shows detection rate different ann approaches many studies note figure highest face detection rate obtained using cnn approach time bpnn approach adopted result obtaining good high detection rate figure detection rate different studies recommendations face detection system face detection first step face recognition systems localize extract face region image background literature studies related face detection systems based ann described earlier research summarized follows face detection techniques presented based image detection many literature researches give overview exactly used database system training testing many researches give sufficient information performance measures used face detection lack equations related performance measures studies face detection systems adopted ann combination approaches algorithms obtain better results detection improve performance face detection system may increase system complexity required memory time face detection lack using significant ann architectures map patternnet ann fast bpnn lack literature related face detection based combination ann genetic algorithm according points many recommendations must taken consideration suggest build strong face detection system try design real time face detection system based video taken real time camera give sufficient details exactly used database system training testing give sufficient details performance measures equations used face detection international journal multimedia applications ijma february ann adopted combination algorithms obtain better results face detection time must focus simplify combined algorithms steps reduce memory required processing time try use ann architectures map patternnet fitnet fast bpnn try use different optimization ann training algorithms trainlm trainbfg bayesian regularization trainbr traincgf algorithm gradient descent traingd gradient descent momentum traingdm obtain best results face detection system try use genetic algorithm optimization algorithm obtain best values ann algorithm parameters result optimal results conclusion paper includes summary review literature studies related face detection systems based anns different architecture approach programming language processor memory requirements database images performance measure face detection system used study study strengths limitations future work face detection system suggested based using pattern net back propagation neural network bpnn many hidden layers different network architectures parameters values bpnn patternnet adopted determine patternnet architecture result best performance values face detection system acknowledgements author would like thanks university jordan supporting research references zhao face recognition literature survey technical report university maryland october turk pentland eigenfaces recognition journal cognitive neuroscience phil brimblecombe face detection using neural networks meng electronic engineering school electronics physical sciences urn bouchra abboud facial expression recognition synthesis based appearance model signal processing image communication vol issue viola jones robust object detection technical report cambridge research laboratory usa february yan detecting faces images survey ieee transactions pattern analysis machine intelligence vol minyoung kim face tracking recognition visual constraints videos ieee conference computer vision pattern recognition june brunelli poggio face recognition features versus templates ieee transaction pattern analysis machine intelligence vol chen lin simple algorithm based minimum facial features annual conference ieee industrial electronics society iecon nov taipei taiwan sanjay singh robust skin color based face detection algorithm tamkang journal science engineering vol international journal multimedia applications ijma february abdenour hadid matti timo ahone discriminative feature space detecting recognizing face proceedings ieee computer society conference computer vision pattern recognition vol elise arnaud robust automatic face tracker dedicated broadcast videos ieee international conference image processing zhonglong zheng jie yang yitan zhu face detection recognition using colour sequential images journal research practice information technology vol may fei zuo embedded face recognition using cascaded structures thesis technische universiteit eindhoven china zuo automatic human face detection distributed video security system proceedings progress workshop embedded systems stan anil jai handbook face recognition springer science business media jang kim fast robust face detection using evolutionary pruning ieee transactions evolutionary computation bernd menser michael brunig face detection tracking video coding application conference record asilomar conference signals systems computers pacific grove usa pedro alexandre dias martin active appearance models facial expression recognition monocular head pose estimation master thesis dept electrical computer faculty sciences technology university coimbra hjelmas low face detection survey computer vision image understanding vol http yongzhong jingli zhou shengsheng survey face detection extraction recognition computing informatics vol zhao face recognition literature survey acm computing surveys december cha zhang zhengyou zhang survey recent advances face detection technical report microsoft research microsoft corporation one microsoft way redmond http abboud davoine dang facial expression recognition synthesis based appearance model signal processing image communication mohammad alia abdelfatah tamimi omaima integrated system monitoring recognizing students class session aircc international journal multimedia applications ijma december jain ross prabhakar introduction biometric recognition ieee transactions circuits systems video technology henry rowley baluja kanade neural face detection computer vision pattern recognition neural face detection carnegie mellon university phd thesis kahkay sung tomaso poggio example based learning view based human face detection massachusetts institute technology artificial intelligence laboratory center biological computational learning memo cbcl paper mit december henry rowley shumeet baluja takeo kanade rotation invariant neural face detection december jeffrey norris face detection recognition office environments thesis dept electrical eng master eng electrical massachusetts institute technology hazem face detection using neural networks image decomposition lecture notes computer science vol huang face detection cluttered images using polynomial neural network neurocomputing masakazu matsugu subject independent facial expression recognition robust face detection using convolutional neural network neural networks stefan christian uwe evolutionary optimization neural networks face detection proceedings european symposium artificial neural networks evere belgium publications marian beszedes milos oravec system localization human faces images using neural networks journal electrical engineering vol international journal multimedia applications ijma february shilbayeh face detection system based mlp neural network recent advances neural networks fuzzy systems evolutionary computing issn isbn zoran bojkovic andreja samcovic face detection approach neural network based method video surveillance seminar neural network applications electrical engineering neurel faculty electrical university belgrade serbia september aamer mohamed face detection based neural networks using robust skin color segmentation international systems signals devices ieee sahoolizadeh sarikhanimoghadam dehghani face detection using gabor wavelets neural networks world academy science engineering technology vol avinash kaushal raina face detection using neural network gabor wavelet transform international journal computer science technology ijcst vol september issn mohammad abadi face detection help gabor wavelets characteristics neural network classifier american journal scientific research issn http anissa bouzalmat face detection recognition using back propagation neural network fourier gabor filters signal image processing international journal sipij vol september doi kalavdekar prakash face detection using neural network international journal computer applications mansaf elmansori khairuddin omar enhanced face detection method using skin color neural network european journal scientific research issn http zuo fast face detection using cascade neural network ensembles eurasip journal advances signal processing volume article hindawi publishing corporation hudson hagan demuth neural network user guide mathworks apple hill drive natick author omaima received degree dept faculty computers mathematical university iraq received degree cis cis faculty aabfs jordan currently assistant professor cis faculty sciences alzaytoonah university jordan research interests include image compression recognition artificial neural networks genetic algorithms member international association engineers iaeng
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shape restricted density estimation apr baraud abstract purpose paper pursue study built observations defined baraud based model means estimator belongs true distribution observations also belongs risk squared hellinger loss bounded quantity viewed dimension function model often related metric dimension model defined minimax point view pessimistic typically bound accurate points model may pessimistic true distribution belongs specific part situation want investigate models like set decreasing densities exist specific points model shall call extremal risk substantially smaller typical risk moreover risk point model bounded sum risk bound extremal point plus square distance point implies true density close enough extremal point risk point may smaller minimax risk model actually remains true even true density belong model result based refined bounds suprema empirical processes established baraud introduction present paper pursues study introduced baraud versatile estimation strategy based models want explain specific property estimators shall call superminimaxity study motivated conference adityanand guntuboyina gave cambridge june talk gaussian regression shall deal density estimation given observations unknown density respect reference measure estimator measure performance using loss function hellinger distance shall focus properties lead superminimaxity first properties robustness exist various notions robustness robustness model contamination robustness possible outliers etc see huber illustrations cases problem formulated following way know performance estimator true density belongs model deteriorate actually form small arbitrary density proportion data actually corresponds sample density since density one check natural wonder happens risk estimator mixture form also date april generally belongs small hellinger ball around leads notion robustness respect hellinger deviations shall use illustrate problem contamination assume choose statistical model unknown density set uniform densities case mle maximum likelihood estimator uniform density largest observation risk bounded belongs denoting expectation true density true density belong unfortunately situation may become quite different mixture larger easy check probability order least larger mle uniform distribution terrible estimator although quite good since hellinger distance larger previous example shows mle definitely robust sense since may sensitive small deviations model contrary precise let consider model densities based risk function bounded robustness expressed following property proven baraud whatever density universal positive constant fondamental property following reasons quite close simple density estimated small risk bound essentially behave true density risk bound plus small additional term viewed squared bias intuitively based sample density based sample density remain close parametric example based uniform distributions everything happens considered data values ignored others consequently risk remains order even instead notion robustness quite flexible shows risk estimator deteriorate much small hellinger neighbourhood point model many models risk uniformly bounded sup corresponds minimax point view leads whatever density turns models exists subset risk bounds substantially smaller call superminimaxity although exists analogy denomination notion quite different one superefficiency described famous counterexample hodges theorem cam points superefficiency apart fact deals property estimation faster points however superefficiency asymptotic property point superminimaxity definitely nonasymptotic defined given value number observations detailed study superefficiency one could look paper brown low zhao moreover superminimaxity together robustness following consequence either close enough risk bound inf may substantially smaller typical risk bound leading superminimaxity actually combination robustness existence local risk bounds form lead phenomenon also described different framework chatterjee paper strongly influenced research direction showing risk particular points bounded quantity smaller order global minimax risk requires specific probabilistic tools established baraud tools allow bound expectation supremum empirical process neighbourhood element quantity smaller order one could get using global entropy class example van geer existence points model estimator superminimax already noticed grenander estimator density see grenander groeneboom interval known value shown grenander estimator piecewise constant density based intervals bounded positive universal constant therefore smaller order typical risk nonincreasing densities order shall see estimation problem perform similarly possible logarithmic factors superminimaxity property piecewise constant densities moreover need know robust respect hellinger distance case monotone densities far unique many examples families densities one find subset rates convergence faster rate typical point moreover happens set often possesses good approximation properties respect much larger space approximation properties combined robustness expressed allow derive minimax risk bounds large subsets sets possibly therefore neither possess finite metric dimension finite entropy view illustrating superminimaxity phenomenon shall consider present paper models densities defined shape constraints namely piecewise monotone piecewise convex concave densities large amount literature dealing density models shall content mention references refer reader bibliography therein monotone densities refer books groeneboom wellner van geer estimation convex density mention groeneboom refer papers doss wellner rufibach cule samworth estimation density regression setting let mention guntuboyina sen estimating convex regression function chatterjee isotonic regression recently bellec extended results chatterjee properties estimators convex polyhedral cones homoscedastic gaussian regression framework general closed convex subsets also derived results superminimaxity specific gaussian framework two papers results restricted convex models opposed convexity play special role presentation models shall use necessarily convex allows deal general shape constraints like piecewise monotonicity paper organised follows statistical setting main notations conventions well brief reminder density estimation framework found section introductory example model monotone densities section gives first flavour results establish along paper main result found section applications different density models piecewise constant piecewise monotone piecewise densities detailed section problem model selection addressed section section devoted proofs statistical setting let measurable set measure set probabilities absolutely continuous respect shall denote set functions subset consisting functions satisfying set probability densities respect element density denoted turn metric space via hellinger distance recall cam hellinger distance two elements given shall write observe random variables values distribution density although might uniquely defined density respect distribution observations shall refer density simplicity avoid trivialities shall always assume sequel log estimators shall consider based models defined follows definition density model model short subset exists countable subset dense respect hellinger distance shall say dense separable respect hellinger distance density model chosen corresponding probability model approximates true distribution respect hellinger distance model may may contain course model good distance large set inf aim paper study performance built definition properties described great details baraud give brief account context density estimation based variables one consider provides robust sense almost rate optimal estimator model densities cases know order avoid long developments restrict construction specific situations shall encounter namely observations let increasing function onto defined given model densities countable dense subset density defined following way densities set define measurable element closure set inf sup calculation involves ratio use convention constant chosen convenience calibration numerical constants original paper baraud replaced positive number clear construction given model unique however risk bounds derived baraud valid version notations conventions definitions set max log max min inf denotes cardinality finite set numerical constants may vary line line function denote respectively plimits whenever limits exist shall also use following conventions definition partition open interval size either finite set real numbers shall call endpoints partition numbers intervals partition open intervals partition also identified set intervals shall equally write set partitions endpoints intervals denoted length means monotone densities view illustrating main result paper presented section let consider example model consisting densities respect lebesgue measure arbitrary interval open left end vanish elsewhere case set densities form nonnegative may unbounded neighbourhood results get would similar set densities interval vanish elsewhere define set densities form note densities take value two unbounded extremal intervals partition instance corresponds family uniform densities intervals situation prove following result theorem defined section satisfies ces inf universal constant remark since side always bounded one useless consider values lead bound smaller one particular actually equivalent ces inf although shall repeat systematically remark hold subsequent results bound means risk function quite small neighbourhood specific densities belongs close enough density risk order logarithmic factors precisely sup becomes large remains fixed rate convergence towards element therefore almost parametric particular interest densities bounded supported compact interval numbers depending given introduce set densities form variation function defined following way definition let function defined interval positive length monotone interior variation given sup inf note set uniform densities intervals compact contains densities arbitrarily large functional remains invariant translation scaling implies also invariant translation scaling turns densities lying well approximated elements precisely following approximation result holds proposition using triangle inequality side bounded following way inf inf inf finally since arbitrary inf inf optimizing side respect using facts arbitrary log derive following corollary theorem corollary probabilities satisfies constant log ces inf log particular log risk bound estimator larger universal constant log smaller values bounded log logarithmic factors rate respect optimal since corresponds lower bound order minimax risk subset consisting densities supported bounded lower bound follows proof proposition result actually stated paper proof shows applies hellinger distance well property means although set compact support densities unknown minimax risk finite know estimator performance also robust respect hellinger deviations note corollary also used determine rate estimation decreasing densities possibly unbounded support maximum value provided assumption behaviour function goes infinity main result let start definitions definition class subsets said shatter finite subset class subsets equal class subsets equivalently class subsets dimension exists integer finite subset cardinality shattered smallest property definition let class functions set values shall say weak dimension smallest integer class subsets dimension larger may introduce main property used paper definition let class functions shall say element extremal extremal point degree class functions weak dimension proposition let class nonnegative functions element extremal degree larger dimension larger proof let bound dimension according value therefore dimension larger let assume set using lemma section suffices prove two dimensions larger equivalent showing thus therefore dimension larger let turn case means either conventions equivalent therefore dimension larger hence dimension concludes proof let state main result theorem let model set extremal points satisfies universal constant inf whatever true distribution consequently ces inf note boundedness implies values lead trivial bound infimum could reduced know extend factor necessary believe optimal although appears necessary situations shown example section applications throughout section lebesgue measure particular shall consider densities respect lebesgue measure start following useful lemma lemma class subsets element union intervals dimension proof let points easy check elements form disjointed intervals pick subset points piecewise constant densities let consider model section build belong form union disjointed intervals applying lemma proposition sets obtain elements extremal degrees larger therefore deduce theorem sup logarithmic factor corresponds parametric rate respect although partition defines arbitrary support unknown follows massart proposition lower bound minimax risk form shows power necessary suspect power three logarithm optimal piecewise monotone densities let see theorem applied simple situation piecewise monotone densities definition given partition function called piecewise monotone pieces based monotone open interval set functions denoted since density monotone set densities respect lebesgue measure belong clearly proposition element extremal degree larger proof let piecewise monotone density based partition therefore endpoints let piecewise constant density based partition endpoints let partition endpoints therefore intervals interval monotone constant implies belongs follows lemma sets unions intervals conclusion follows lemma proposition applied lemma whatever set written union intervals set well proof let partition open intervals associated endpoints partition either interval decomposition shows union disjointed intervals nevertheless bound refined follows situation need consider case belongs closure one intervals form set counts one interval situation adds extra interval occurs number points larger therefore union intervals proof application theorem elements extremal proposition leads following result corollary satisfies distributions inf ces inf universal constant note bound trivial using also leads trivial bound restrict since inf inf inf analysis necessary evaluate inf order shall use approximation result based following functional definition let partition based using convention define vij variation given functional defined inf infimum runs among partitions based denote subset densities note convention equal zero case summation restricted requires equal sense functional translation scale invariant means takes value whatever besides possesses following property lemma proof let based partitions satisfying viewed element based consequently suffices show fact suffices show simply divide interval length intervals respective lengths amounts show setting vjj follows inequality approximation elements elements controlled following way proposition let applying corollary leads following bound valid whatever distribution observations ces inf final optimization respect leads ces log log since result valid densities optimize respect finally leads theorem based model satisfies log ces inf log distributions particular sup log log want estimate bounded unimodal density support finite length may build case bounded since density performance unimodal density given log piecewise densities previous sections considered densities piecewise monotone constant implied properties follows proposition actually approximation properties matter derives fact hellinger distance square roots densities going sophisticated properties monotonicity state accounts slightly complicated structure section definition let partition intervals function piecewise based either convex concave open interval partition set functions varies denoted denote set functions form affine function sets sets densities belongs respectively recall either concave convex open interval continuous admits derivative monotone following result prove useful find extremal points lemma proof since exists either convex concave open interval derivative monotone sets two disjointed subintervals monotone proposition elements extremal degrees larger proof let consider exists partition endpoints either convex concave interval partition endpoints affine interval partition contains intervals intervals either convex concave hence function belongs subset lemma follows lemma union intervals since arbitrary conclude lemma dimension larger shows proposition elements extremal degrees larger conclusion follows application lemma section may apply theorem consists extremal points deduce following risk bound proposition corollary satisfies distributions inf ces inf universal constant control approximation term inf analogue one derived previous section inf based new functional definition let partition based either convex concave monotone derivative using convention define vij variation functional defined inf infimum runs among partitions based subset denote densities note finite density necessarily zero two extremal unbounded intervals partition therefore analogue lemma holds functional similar proof saying omit details proposition let arguing previous section derive corollary proposition concluding result satisfies theorem based model ces inf log log particular log log densities want investigate situation close previous one case densities line densities form exp open interval possibly infinite length concave function let denote set densities subset densities piecewise affine pieces instance exponential density belongs laplace density belongs also note exp holds square root exp proposition elements extremal degrees larger proof let consider exp exp set exp exp subset log log convention log set otherwise equal union log since interval concave piecewise affine pieces function log piecewise concave pieces hence belongs lemma also belongs follows lemma log log union intervals consequently set exp exp union intervals derive lemma exp larger conclusion follows proposition may apply theorem following risk bound use proposition derive corollary satisfies distributions inf ces inf universal constant particular means elements estimated parametric rate log factor case uniform densities exponential densities translates laplace density among many others remark simplicity restricted study densities could well handle case piecewise densities several pieces densities form exp concave functions extension would similar leads monotone piecewise monotone straightforward model selection results sections based use single model section section section implies risk bounds depend first case cases order get best possible value either unknown distribution may use selection procedure different ways shall explain using theorem section simplify presentation assume number observations even split sample two parts size also consider models simultaneously models fix weight follows exp exp may use models build based sample results family estimators sbj sbk risks estimators bounded according theorems second step consider preliminary estimators based sample set points may apply selection procedure described section via based second sample theorem paper applies parameters follows selection procedure results estimator satisfies ces min inf sbj inf sbk may take expectation respect get ces min inf sbj inf sbk applying theorems order bound sbj sbk respectively derive two following bounds hold simultaneously ces inf log log ces inf log log simple example procedure could applied larger family models preliminary estimators shall insisit important point may easily extend results got single model large families models get final bound corresponding best bound among models involved procedure alternative selection procedure leading result described baraud section also possible avoid splitting device using models simultaneously penalized indicated section baraud would get end type risk bounds simplicity shall insist approach proofs preliminaries sequel shall use following elementary properties lemma subsets dimension larger holds class defined let class functions set extremal point degree positive let class functions exists extremal degree larger let class subsets partition dimension larger vcp class dimension larger proof let set cardinality either shattered empty form proves first statement second statement follows fact third one argue follows could shatter points would exist points could shattered hence would contradictory fact dimension larger proof theorem let since assumed separable also separable may therefore choose countable dense subset let choose countable dense subset possibly changing may assume loss generality finally define estimator based following construction described section baraud well notations paper set note may empty start proof following lemma lemma weak class dimension larger proof since weak dimension larger map increasing follows baraud proposition weak dimension larger let proof theorem fix follows baraud proposition page place definition since countable bounded family also countable elements bounded besides lemma ensures weak vcmajor class dimension larger may therefore apply corollary baraud family get sup log log log log log particular hence log recall quantity defined section baraud sup implies either log since cases max deduce log positive numerical constant use theorem baraud wesrecall notation defined densities means since obtain probability least inf inf inf follows finally dense inf inf bracketed term side becomes inf inf inf conclusion follows proof theorem let based partition positive part consequently possibly empty therefore dimension larger lemma dimension larger proposition element extremal dimension larger finally theorem follows theorem proof proposition relies series approximation lemmas shall also prove useful sequel lemma let monotone function finite variation interval finite length factor optimal proof assuming without loss generality let observe one replace point amounts assume let sup min min maximization respect shows follows ral optimality follows considering case next lemma involves norm hereafter denoted lemma let function finite variation exists partition intervals function piecewise constant element partition besides exists partition intervals length larger results hold nondecreasing functions proof clearly results remain valid replace almost everywhere respect lebesgue measure since exists admits countable number discontinuities may therefore assume actually defined starting define recursively sup hence since particular necessarily krd implies process therefore results finite number distinct points also follows definition let set note piecewise constant partition intervals since follows lemma moreover jensen inequality implies shows proves first part lemma second part define follows element length larger divide intervals length larger process results new partition thinner cardinality larger since construction property also true elements partition thinner functions change lemma given two probability densities respect particular element probability density respect scalar proof notice two vectors norm one product cos implies cos orthogonal projection linear space generated hence inf cos sin inequality follows fact cos last result obtained proof proposition apply lemma complete resulting function nonnegative satisfies kfi setting kfi element may apply last part lemma gives conclusion follows letting converge proof proposition let based vij let positive integers intervals equal intervals one apply lemma find approximation monotone piecewise constant pieces satisfies according therefore derive lemma moreover always assume modifying negligeable set necessary belongs given formal minimization respect condition leads taking account fact belong finally set implies corresponding function belongs conclusion follows letting converge vij proof proposition relies following approximation lemma lemma let continuous either convex concave function derivative satisfying affine function defined satisfies sup factor optimal proof changing may assume concave particular since continuous satisfies exists sup function consequently constant improved since reached let function lemma lemma one partition intervals length larger vjj using partition approximate piecewise affine function pieces applying lemma derive sup hence note concave convex opposite case since satisfies assumptions lemma intervals partition defines zero two extremal intervals may use previous approximation method interval get approximation pieces renormalizing lemma conclude exists belongs mimic proof proposition optimize get finally corresponding function belongs conclusion follows letting converge vij references baraud estimator selection respect risks probab theory related fields baraud bounding expectation supremum empirical process weak class appear electron stat baraud sart new method estimation model selection http bellec sharp oracle inequalities least squares estimators shape restricted regression http estimating density order restrictions nonasymptotic minimax risk ann grenander estimator nonasymptotic approach ann model selection via testing alternative penalized maximum likelihood estimators ann inst probab massart minimum contrast estimators sieves exponential bounds rates convergence bernoulli brown low zhao superefficiency nonparametric function estimation ann chatterjee guntuboyina sen risk bounds isotonic shape restricted regression problems ann cule samworth theoretical properties maximum likelihood estimator multidimensional density electron doss wellner global rates convergence mles logconcave densities appear ann statist rufibach maximum likelihood estimation density distribution function basic properties uniform consistency bernoulli grenander abstract inference john wiley sons new york wiley series probability mathematical statistics groeneboom estimating monotone density proceedings berkeley conference honor jerzy neyman jack kiefer vol berkeley wadsworth pages wadsworth belmont groeneboom jongbloed wellner estimation convex function characterizations asymptotic theory ann groeneboom wellner information bounds nonparametric maximum likelihood estimation volume dmv seminar verlag basel guntuboyina sen global risk bounds adaptation univariate convex regression probab theory related fields huber robust statistics john wiley sons new york wiley series probability mathematical statistics cam convergence estimates dimensionality restrictions ann cam asymptotic methods statistical decision theory springer series statistics new york van geer certain nonparametric maximum likelihood estimators ann van geer applications empirical process theory volume cambridge series statistical probabilistic mathematics cambridge university press cambridge univ nice sophia antipolis cnrs ljad umr nice france address baraud sorbonne upmc univ paris cnrs umr lpma paris france address
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delivery heterogeneous mobile andreas shantanu yann daniel jan paolo eth department computer science switzerland baertschi lif cnrs france oct darmstadt disser berlin institut mathematik technische berlin germany hackfeld abstract consider problem delivering messages specified pairs weighted undirected graph mobile agents initially located distinct nodes graph agent consumes energy proportional distance travels graph interested optimizing total energy consumption team agents unlike previous related work consider heterogeneous agents different rates energy consumption weights solve delivery problem agents face three major challenges collaboration work together message planning route take coordination assign agents messages first show delivery problem without collaborating best possible show benefit collaboration general also show benefit collaboration single message planning turns approximate even single agent polynomial time agents unit capacities collaborate show coordination nphard even agents unit capacity efficiently solved exactly uniform weights finally give max message delivery unit capacities acm subject classification analysis algorithms problem complexity nonnumerical algorithms problems keywords phrases message delivery mobile agents energy optimization approximation algorithms digital object identifier page introduction recent technological progress robotics allows mass production inexpensive mobile robots used perform variety tasks autonomously without need human intervention gives rise variety algorithmic problems teams autonomous robots hereafter called mobile agents consider delivery problem work partially supported project snf project dfg priority programme algorithms big data grant chalopin das disser graf hackfeld penna licensed creative commons license leibniz international proceedings informatics schloss dagstuhl informatik dagstuhl publishing germany delivery heterogeneous mobile agents moving objects messages various locations mobile agent corresponds automated vehicle pick message source deliver intended destination agent consumes energy proportional distance travels interested operations team agents total energy consumed minimized general agents may identical may energy efficient others use different technologies different sources power assume agent given weight rate energy consumption per unit distance traveled agent moreover agents may start distinct locations thus may sometimes efficient agent carry message intermediate location hand another agent carries towards destination hand agent may carry several messages time finding optimal solution minimizes total energy cost involves scheduling moves agents points pick handover messages study problem called weighteddelivery graph connects sources destinations objective deliver messages specific pairs using agents located arbitrary nodes note problem distinct connectivity problems graphs network flow problems since initial location agents general different sources messages located means need consider cost moving agents sources addition cost moving messages furthermore correspondence agents messages problem previous approaches delivery messages agents focused bottleneck agents limited energy battery power restricts movements decision problem whether single message delivered without exceeding available energy agent known datadelivery problem budgeteddelivery problem shown weakly paths strongly planar graphs model consider undirected graph edge cost length denoted length simple path sum lengths edges distance nodes denoted equal length shortest path mobile agents denoted weights agents initially located arbitrary nodes denote distance initial location node agent move along edges graph time agent traverses edge incurs energy cost furthermore pairs source target nodes message delivered source node target node message picked agent node visits carried node dropped agents given capacity limits number messages agent may carry simultaneously restrictions much agent may travel denote total distance traveled agent weighteddelivery optimization problem minimizing total energy needed deliver messages schedule describes actions agents sequence ordered list actions actions tuple denotes action agent moving current location node node picks message drops message respectively schedule implicitly encodes times easy compute total energy use cost chalopin das disser graf hackfeld penna optimal feasible schedule figure example optimal feasible schedule two messages two agents denote subsequence actions carried agent subsequence actions involving message call schedule feasible every action directly preceded action messages get delivered see figure contribution solving weighteddelivery naturally involves simultaneously solving three subtasks collaboration individual planning coordination first multiple agents work message need collaborate find intermediate locations message secondly agent works one message plan order wants approach subset messages finally coordinate agent works subset messages without collaboration subsets form partition otherwise subsets necessarily pairwise disjoint even though three subtasks interleaved investigate collaboration planning coordination separately next three sections leads approximation algorithm weighteddelivery given section section consider collaboration aspect weighteddelivery first present polynomial time solution weighteddelivery single message algorithm complexity irrespective number agents general show algorithm uses one agent delivering every message achieve approximation ratio better call benefit collaboration boc least show tight boc boc section look planning aspect weighteddelivery individual planning turns planar graphs approximate within factor less positive side give approximation guarantees restricted versions weighteddelivery turn useful analysis section section study coordination aspect weighteddelivery even collaboration planning taken care schedule fixed except assignment agents messages coordination also turns even planar graphs result holds capacity including setting however becomes tractable restricted uniform weights agents section give approximation algorithm weighteddelivery approximation ratio max due limited space proofs deferred appendix related work problem communicating transporting goods sources destinations graph well studied variety models different optimization criteria problem finding smallest subgraph tree connects multiple sources targets graph called connection problem known problem related generalized steiner tree problem also unlike problems maximum flow delivery heterogeneous mobile agents problem network puts limit number messages transported edge makes problem easier allowing polynomial time solutions problems however agents carrying messages problem case single agent moving graph task optimally visiting nodes called traveling salesman problem visiting edges called chinese postman problem studied former known latter solved time metric graphs traveling salesman problem tours paths one fixed endpoint multiple identical agents graph demaine studied problem moving agents form desired configurations connected independent configurations provided approximation algorithms inapproximability results bilo studied similar problems visibility graphs simple polygons showed many motion planning problems hard approximate another optimization criteria minimize maximum energy consumption agent requires partitioning given task among agents frederickson studied uniform weights called problem gave approximation algorithms single agent multiple agents also minmax context problem visiting nodes tree using agents starting single location known anaya studied model agents limited energy budgets presented hardness results trees approximation algorithms arbitrary graphs problem transferring information one agent others broadcast agents one agent convergecast model message delivery single node pair studied chalopin mentioned recent paper shows three problems remain general graphs even agents allowed exchange energy meet collaboration section examine collaboration agents given message agents aix point carry message one needs find locations handovers schedule entries aix note general two action quadruples aij per agent aij single message overall use structural result tie together weighteddelivery collaboration multiple messages however longer holds case analyze benefit lose forgo collaboration deliver message single agent algorithm weighteddelivery single message lemma optimal solution weighteddelivery single message message delivered agents weights order whenever without loss generality hence optimal schedule agent one pair actions proof agent hands message agent solution construct better solution replacing trajectory carrying message trajectory using agent argument may also assume without loss generality weights agents carrying message optimum schedule strictly decreasing since merge trajectories equal weight chalopin das disser graf hackfeld penna example example shows lemma true one message graph shown figure let one agent weight start vertex second agent weight start vertex optimal schedule message starting location first transported destination thus weights agents transporting message increasing case figure example weights agents order transporting message increasing theorem optimal solution weighteddelivery single message graph agents found time proof use properties lemma create auxiliary graph run dijkstra algorithm computing shortest path given original instance weighteddelivery consisting graph obtain auxiliary directed graph follows node agent node vai furthermore contains two additional vertices arc sai cost arc tai cost two arcs uai vai vai uai cost agents arc uai uaj cost note solution weighteddelivery satisfies properties lemma corresponds cost solution equal length path vice versa implies length shortest path cost optimal solution weighteddelivery assuming graph vertices arcs graph constructed time use floyd warshall pair shortest paths algorithm finally compute shortest path time using dijkstra algorithm fibonacci heaps unfortunately structural properties lemma extend multiple messages next two subsections investigate quality optimal solution changes allow every message transported one agent different messages may still transported different agents one agent may also transport multiple messages time long number messages capacity end define benefit collaboration cost ratio optimal schedule opt schedule without collaboration boc mins cost opt lower bound benefit collaboration theorem instances weighteddelivery agent capacity messages algorithm using one agent delivering every message achieve approximation ratio better min delivery heterogeneous mobile agents proof consider graph given figure length every edge means star graph center paths total length messages message needs transported agent weight starting every vertex first show following agent transports messages vij costs least note implies schedule delivering messages agents every message carried one agent satisfies cost let agent transport messages source destination without loss generality let messages picked order construction agent needs travel distance least reach message distance move back distance picking message going back finally needs move distance overall agent therefore travels distance least overall cost agent deliver messages therefore least consider schedule scol agents collaborate agent transports message identify agent transports messages possible choice total cost schedule given cost scol fstep fstep fstep defined giving current cost transporting message fstep interval first integral corresponds first part schedule messages transported separately therefore cost transporting message fstep exactly second part schedule corresponds part messages transported together one agent time observe function satisfies fstep hence cost scol best approximation ratio algorithm transports every message one agent compared algorithm uses arbitrary number agents every message figure lower bound construction benefit collaboration chalopin das disser graf hackfeld penna therefore bounded boc min cost scol observing obtain following corollary corollary schedule weighteddelivery every message delivered single agent achieve approximation ratio better general better single message upper bounds benefit collaboration give tight upper bounds corollary following theorem shows benefit collaboration general remark finding optimal schedule every message transported source destination one agent already shown theorem theorem let opt optimal schedule given instance weighteddelivery exists schedule every message transported one agent cost cost opt proof assume without loss generality optimal schedule opt every message transported simple path starting point destination easily achieved letting agents drop messages intermediate vertices would otherwise transport cycle define directed multigraph follows set vertices original graph every time optimal schedule agent traverses edge carrying set messages add arc label edges set messages write denote labels call edges original edges edges reverse edges say tour agent satisfies edge labels every original edge traversed carrying exact set messages every reverse edge traversed without carrying message show exists eulerian tour satisfying edge labels every connected component let cheapest agent connected component follow respective eulerian tour agent traverses every edge exactly twice often edge traversed optimal schedule opt agents choose cheapest agent connected component obtain schedule cost cost opt considering subset messages subschedule opt may assume strongly connected construction every connected component strongly connected let amin agent minimum cost among agents move opt let set messages currently placed vertex let amin set messages currently transported agent amin first show procedure computetour computes closed tour amin satisfies edge labels afterwards explain iterate procedure obtain eulerian tour satisfying edge labels claim agent amin starts vertex follows tour computed computetour satisfies edge labels every step returns starting location delivery heterogeneous mobile agents algorithm computetour function computetour drop messages edge incident current vertex current vertex pickup messages traverse delete else let current vertex fetchmessage currentvertex else edge traverse delete algorithm fetchmessage function fetchmessage drop messages current vertex edge leaving current vertex traverse delete else give current vertex let edge incident current vertex current vertex amin pickup messages drop messages traverse delete else let current vertex amin fetchmessage currentvertex procedures computetour fetchmessage make sure agent traverses every edge label carrying exact set messages every edge carrying messages first part claim clear need show amin get stuck vertex returning eulerian ignoring labels edges deleted traversed happen call fetchmessage gives vertex means current vertex contain message edge label containing message note amin currently proceeding according call fetchmessage vertex path message takes start destination optimal schedule opt path simple initial assumption also note since edge labels obeyed message ever moves forward along path opt means amin stuck vertex must initially edge incident taken agent earlier deleted agent traverses edges procedure fetchmessage must call fetchmessage agent traversed edge call completed otherwise original edge corresponding would used amin message would already reached since contradicts message path simple agent currently proceeding according call fetchmessage vertex chalopin das disser graf hackfeld penna call fetchmessage complete must vertex message missing vertex carry paths destination call fetchmessage also incomplete iterating argument obtain current stack functions fetchmessage fetchmessage optimal schedule opt message needs transported transported together similarly message needs transported message transported together message moreover edge messages need transported together particular message needs transported together transported optimal schedule messages must transported certain sequence message needs transported messages transported message vice versa thus computetour must terminate amin returning starting location claim completing tour given computetour following holds every message either transported destination vertex path edges containing labels path reverse direction edges containing labels every edge label traversed agent amin carries set messages thus time path current location message destination edges containing label shows first part claim observe completed call fetchmessage yields closed walk agent starts ends moreover first traverses exactly edges path current position message edges path inductively also holds levels recursive calls fetchmessage hence every original edge also corresponding reverse edge traversed call fetchmessage fact also implies edge traversed procedure computetour fetchmessage corresponding edge traversed call fetchmessage inductively therefore argue edges traversed procedure computetour order corresponding edges traversed path current location amin starting vertex shows termination every original edge also corresponding reverse edge traversed claim combining tours returned multiple calls computetour yields eulerian tour satisfies edge labels every step assume tour resulting call computetour traverse edges let starting vertex tour last vertex tour incident edge visited position message tour finished let graph call computetour without edges message position instead want show run computetour amin starting add resulting tour follows first amin follows last time visits follows finally remaining part graph feasible input computetour claim corresponds instance weighteddelivery message transported schedule opt delivery heterogeneous mobile agents figure choosing largest lower bound representing weight agent currently transporting message done tour amin completed agent positions adapted accordingly claim computetour produce tour satisfies edge labels problem occur combining tours therefore following tour fetchmessage called message yet transported tour completed means vertex visited last time visited tour choice edges incident must visited tour particular must message delivered destination tour contain edge label claim iterative applying argument obtain eulerian tour satisfies edge labels every step single message case single message improve upper bound benefit collaboration theorem tight bound theorem algorithm using single agent weighteddelivery proof using dijkstra algorithm determine agent transport message lowest cost need show cost optimum using agents fix optimum solution let agents labeled order transport message optimum solution ignoring unused agents assume lemma scaling assume without loss generality total distance traveled message point along message path agent cost carrying message point optimum schedule define function function step function monotonically decreasing lemma choose largest see figure note let distance traveled agent without message total cost distances traveled agents without message obtain following lower bound optimum solution cost opt chalopin das disser graf hackfeld penna choice functions coincide least one point interval let point agent carrying message point means show costs cost opt agent transport message alone cost agent reach bounded cost transporting message bounded thus cost algorithm using one agent bounded cost alg using finally obtain cost alg cost opt intermediate dropoffs following show upper bound benefit collaboration still holds additional property message carried single agent without intermediate dropoffs make use result later approximation algorithm weighteddelivery section theorem let opt optimal schedule given instance weighteddelivery exists schedule every message transported single agent exactly one one cost cost opt iii every agent returns starting location proof corollary theorem agent infinite capacity well keep message picked simply remove actions message first last need different analysis given optimum schedule opt look trajectories messages connected agents precisely construct auxiliary multigraph follows vertex set consists messages agent look subsequence optimum schedule since capacity subsequence consists alternating followed action add edge length figure edges correspond portions optimum schedule agent travels without carrying message assume without loss amin amin pmin figure left optimal schedule center auxiliary graph right generality connected denote amin agent minimum weight involved opt otherwise look connected component agent minimum weight separately assume first action amin opt move starting position pmin node picks message note potentially lie anywhere trajectory model adding node pmin connecting edge corresponding length pmin take minimum spanning tree remove redundant edges note total length minimum spanning tree lower bound sum distances traveled agents opt without carrying message thus delivery heterogeneous mobile agents schedule amin move exactly twice along trajectory message twice along path corresponding edge minimum spanning tree cost cost cost opt following tour satisfies property delivers message destination immediately picked first let amin walk amin proceeds towards amin reaches picks delivers along trajectory opt amin reaches let return back original position pmin however along way agent visits endpoint path corresponding edge minimum spanning tree first let amin serve adjacent subtree recursively see figure easy see resulting schedule every message directly transported source destination capacity respected times planning look isolation problem ordering messages within schedule agent call planning formally planning aspect weighteddelivery following task given schedule one agents reorder actions way schedule remains feasible costs minimized generally speaking complex schedule many message handovers reordering options single agent might limited first must respect capacity every prefix number actions exceed number actions even reordering might render infeasible conflicts subschedule planning also includes instances single agent delivers messages one straight target thing decided ordering show setting coordination collaboration aspect weighteddelivery already theorem planning weighteddelivery problem capacities even single agent planar graph proof proceed reduction hamiltonian cycles grid graph problem shown itai similar reduction used sorting problem graf given unweighted grid graph add every vertex new vertex edge message start target denote new graph build instance weighteddelivery taking placing single agent arbitrary vertex let set unit edge length edges figure finding hamiltonian cycle via weighteddelivery single agent chalopin das disser graf hackfeld penna edge length edges except see figure edge gets length instead let length shortest path starting agent deliver messages argue hamiltonian cycle see lower bound let distinguish whether agent ends end reach every least also back using sums least end somewhere else use twice hence get schedule cost reach every vertex exactly end removing schedule directly corresponds hamiltonian cycle illustrated figure using similar ideas use recent results approximation hardness metric tsp immediately show planning weighteddelivery approximated arbitrarily well unless theorem approximate planning weighteddelivery within constant approximation ratio less proof build top result karpinski lampis schmied theorem shows symmetric metric traveling salesperson problem hard approximate ratio better reduction take metric undirected graph duplicate vertices put edge single message like theorem figure find suitable length extra edge arbitrary starting vertex consider following traveling salesman tour want agent come back end hence weighteddelivery want agent end achieve setting large enough avoid traveling twice let cost minimum spanning tree clearly optimum traveling salesman path optimum traveling salesman tour cost least also hence setting length extra edge ensures schedule weighteddelivery end thus uses twice cost least optimum schedule delivery cost remains look schedules end extra edge contributes two thirds cost final schedule least one third approximation gap conserved giving approximation planning restricted settings motivated theorem look restricted setting planning feasible schedule know message completely transported agent without intermediate every message must agent allows give approximations planning capacity theorem let feasible schedule given instance weighteddelivery restriction denote opt reordering optimal cost planning algorithm alg gives reordering alg cost alg cost opt cost alg cost opt delivery heterogeneous mobile agents proof given restriction separate planning independently maintains feasibility denote mjx messages appearing sarj define complete undirected auxiliary graph node set sjx tjx edges weight schedule opt corresponds hamiltonian path minimum length starting subject condition message mji visit source sji directly followed visit destination tji lower bound length total length spanning tree follows starting empty graph first add edges sji tji following idea kruskal add edges sjx tjx sjx increasing order lengths disregarding edges would result creation cycle starting visits edge sji tji directions whenever cross edge sji tji add sji mji tji mji suffix current schedule alg get overall cost cost alg cost opt schedule opt corresponds hamiltonian path minimum length starting subject condition message mji source sji visited destination tji approximate opt schedule alg first collects messages mjx delivering start computing minimum spanning tree subgraph consisting nodes sjx starting collects messages returns back cost cost opt next consider subgraph consisting nodes tjx using christofides heuristic metric tsp path version fixed starting point arbitrary endpoint due hoogeveen deliver messages additionally paying cost opt total gives remark assume additional property agent returns starting position example result theorem get better approximation case instead traversing spanning tree twice model problem known coordination section restrict coordination aspect weighteddelivery assume collaboration planning taken care precisely given sequence containing complete fixed schedule actions without assignment agents actions coordination task assigning agents given actions even though coordination appears flavor matching problem turns optimally match agents given actions holds capacity particular latter however solution agents uniform weight planar graphs give reduction planar given planar formula construct instance weighteddelivery allows schedule good energy cost satisfiable chalopin das disser graf hackfeld penna false true false true true false true false figure left restricted plane embedding satisfied true alse alse true right transformation corresponding delivery graph planar let normal form boolean variables clauses clause given subset three literals form define corresponding graph node set consisting clauses variables add edge clause variable contained furthermore add cycle consisting edges pairs consecutive variables mod call planar plane embedding planar problem deciding whether given planar satisfiable known furthermore problem remains variable node plane embedding required arcs representing positive literals one side cycle arcs representing negative literals side use restricted version reduction assume without loss generality graph connected simple graph variable appears every clause building delivery graph first describe way transform planar graph planar delivery graph see figure transform graph five steps first delete edges cycle keep mind variable node positive literal edges lie one side negative literal edges side secondly let degh denote remaining degree variable node surround variable node variable box variable box contains two paths adjacent internally place degh copies one path called henceforth contains nodes adjacent positive literal edge path contains nodes adjacent negative literal edge next step add single node pair node copies previous step fourth step want paths contain number nodes hence fill nodes end path every path contains exactly degh internal nodes thus variable box contains variable node adjacent internal nodes final node respective finally clause node add new node connect new graph polynomial size steps implemented way planar messages agents weights going place one clause message clause nodes literal message paths variable boxes total messages precisely original clause node place exactly one clause message delivered newly created node furthermore place literal message every internal node vtrue set target vtrue set length edges connecting source target next describe locations agents variable box place one variable delivery heterogeneous mobile agents clause variable false true true false variable variable true false figure agent positions weights black messages edge lengths color agent weight variable node length two adjacent edges set furthermore place literal agents path agent placed respectively gets weight remains set length edges clause nodes internal nodes path construction latter starting position agent uniquely defined weight set length edge illustration see figure agent starting location depicted square message depicted colored arrow reduction key idea reduction variable corresponding variable box contains variable agent either deliver messages thus setting variable true deliver messages thus setting variable false assume set true contained clause adjacent node vtrue yet used literal agent intuitively agent freed variable agent thus sent deliver contained corresponding literal sent deliver clause message since needs transport messages along lemma energy cost given satisfiable assignment solution variables feasible schedule sol agents total energy cost cost sol proof given variables clauses satisfiable assignment variables construct schedule assignment follows assume variable set true consider corresponding variable box variable agent weight delivers messages full length energy needed literal agent placed node vfalse transports two messages message source vfalse message source vfalse summing messages need energy hence messages variable boxes energy consumption furthermore since start satisfiable assignment source clause message connected least one yet used agent weight agent adjacent source clause message move source distance pick deliver distance hence clause get energy consumption exactly variables clauses get total energy consumption fixed sequence schedule without agent assignment fix sequence describes schedule constructed lemma allow infer satisfiable chalopin das disser graf hackfeld penna assignment sequence every action one many messages location immediately followed destination hence restricted schedule permutation satisfies following property two messages lie originating variable node meaning lies left precede schedule energy consumption optimal schedules next three lemmata show total energy consumption optimum schedule opt cost opt cost sol every optimum schedule cost opt cost sol variable agent delivers exactly messages either remark true independent whether opt adheres tho fixed schedule without assignments holds capacity case opt exactly properties described proof lemma since thus agent subsequence uniquely determined since message transported single agent without intermediate subsequences merged match prescribed fixed order actions lower bound end first give upper bound cost sol lower bound total energy consumption feasible schedule first since define cost sol lower bound note every agent weight least double count distance traveled agents via distance covered messages hence careful take account agent might carry two messages time want count energy used time twice hence messages count last edge transported towards target message targets thus distance traveled towards disjoint adjacent edges length least additionally agent deliver clause message needs first travel towards source edge crossings counted yet hence add additional distance clause overall get lower bound total energy consumption lemma independence variable boxes optimum schedule opt agent placed variable box moves clause node back variable box necessarily different variable cost opt cost sol proof assume sake contradiction agent leaves variable box walks clause node later back variable box agent delivers clause message return thus walks corresponding clause message distance twice included travel towards clause node yet cases add least another given lower bound yielding cost opt cost sol restrict look feasible schedules agent either stays inside variable box walks deliver clause message stays target clause message next show also assume agents walk left right inside delivery heterogeneous mobile agents lemma agents move left right optimum schedule opt agent moves point schedule right left along cost opt cost sol proof lemma restrict schedules opt message transported edge connecting without loss generality assume message ever leaves interval transported monotonically left right otherwise could adjust schedule accordingly keeping trajectories agents adapting locations first assume sake contradiction opt agent whose trajectory contains moves right left total length least energy needed least moves yet included lower bound adding get cost opt cost sol hence following assume agent moves strictly less right left going show schedule opt transformed schedule smaller cost contradicting optimality opt consider longest consecutive move agent since moves less left must come handover point lying inside edge handover point lying inside edge first case closer previous action schedule must agent picking message must starting position left hence could replace message agent thus strictly decreasing total distance traveled contradicting optimality second case closer agent moves right picking message let denote agent dropped previous remarks know move left next action reach message inside furthermore weights given hardness reduction know therefore move small right thus strictly decreasing contradicting optimality opt assume agents move right left ready prove key relation optimum schedules sol schedules lemma energy cost optimum schedule optimum schedule opt either total energy consumption cost opt cost sol cost opt cost sol latter case variable agent either delivers exactly messages exactly messages furthermore literal agents respective path deliver exactly two literal messages finally clause messages delivered freed literal agents paths chosen variable agents proof lemmata may assume agent travels another variable box agents move left right furthermore seen proof lemma implies every literal message carried source target single agent continuous motion show variable agent deliver either messages adjacent adjacent would get contradiction optimality assume first sake contradiction stays starting location move first internal node contributes additional total energy consumption let agent carrying first literal chalopin das disser graf hackfeld penna message know must weight replace schedule saving least energy already first literal message contradicting optimality schedule assume either deviated internal node path entered deliver clause message stopped internal node let denote agent carrying message placed specified node edge adjacent clause length weight switch remainder schedule potential increase energy cost clause message delivery amounts gained energy next two literal messages least contradicting optimality schedule hence variable agent delivers either messages messages allows give new lower bound energy consumption cost opt variable agent contributes energy consumption total delivery message respective path needs agent starting location coinciding left message source yielding energy contribution least equality literal agent placed vtrue delivers message placed vtrue consecutive message source vtrue finally source clause message reached agent weight edge length hence delivery clause message needs energy least equality summing clauses variable boxes get cost sol remains note first literal agent weight path chosen variable agent walk path deliver clause message however schedule higher energy consumption cost sol literal agent cross two edges length required energy used estimated lower bound cost sol hence conclude theorem coordination weighteddelivery planar graphs capacities even given prescribed collaboration planning algorithm uniform weights unit capacity note coordination even capacity next show setting approachable restrict uniform weights theorem given collaboration planning form complete schedule missing agent assignment coordination weighteddelivery capacity agents uniform weights solved polynomial time proof denote prescribed schedule without agent assignments since agents uniform weight cost cost feasible schedule determined cost hence action much important agent picks message far comes capacity know agent come either starting position preceding action allows model problem weighted bipartite matching see figure center build auxiliary graph maximum matching bipartite graph tell every action agent performs action comes let add edges agent delivery heterogeneous mobile agents figure illustration coordination following schedule left instance messages agents uniform weight center equivalent weighted bipartite matching problem right resulting trajectories agents starting positions weight furthermore also add edges subsequent weight maximum matching minimum cost captures optimal assignment agents messages found solving classic assignment problem special case minimum cost maximum flow problem problems solved polynomial time instance using hungarian method successive shortest path algorithm respectively cost optimum matching corresponds cost agents moving around without messages cost agents carrying messages easily added consider schedule restricted message subsequence sequence pairs actions every pair message brought shortest path add concatenating piecewise shortest paths gives trajectory agent optimum solution illustrated figure right algorithm remotely inspired simpler problem acm icpc world finals official solution austrin wojtaszczyk later sketched bipartite matching solution approximation algorithm putting results previous sections together obtain following approximation algorithm theorem max weighteddelivery capacity algorithm proof start artificially enlarging weight every agent max increase energy cost contribution agent factor thus problem give max approximation original problem assume without loss generality agents uniform weight max let opt optimal schedule instance weighteddelivery uniform agent weights capacities call feasible schedule restricted every message transported single agent without intermediate dropoffs theorem exists restricted schedule cost cost opt let optr optimal restricted schedule total energy consumption cost optr cost cost opt chalopin das disser graf hackfeld penna define complete undirected auxiliary graph agent starting positions well message sources destinations edge length optr natural correspondence path cover optr exactly simple paths following properties path contains exactly one agent starting position namely endpoint destination node adjacent source node lies endpoint path possibly length endpoint starting position corresponds possibly empty schedule optr cost optr optr let denote tree cover minimum total length among tree covers exactly trees satisfy following properties tree contains exactly one agent starting position namely destination node adjacent source node since optr tree cover satisfying two mentioned properties immediately get optr cost opt theorem use component optimum tree cover construct schedule total energy consumption cost cost optr cost opt remains show tree cover found polynomial time analogously theorem start empty graph add edges add edges increasing order lengths disregarding edges would result either creation cycle join two starting positions tree delivery heterogeneous mobile agents references julian anaya chalopin jurek czyzowicz arnaud labourel andrzej pelc yann convergecast broadcast mobile agents algorithmica david applegate robert bixby vasek chvatal william cook traveling salesman problem computational study princeton series applied mathematics princeton university press princeton usa per austrin jakub onufry wojtaszczyk acm icpc world finals solution sketches http chalopin das disser geissmann graf labourel collaborative delivery mobile robots international colloquium structural information communication complexity sirocco chalopin das disser geissmann graf labourel collaborative delivery mobile robots arxiv preprint davide yann disser luciano guido proietti peter widmayer motion planning problems international symposium algorithms experiments sensor systems wireless networks distributed robotics algosensors pages chalopin shantanu das paolo penna peter widmayer data delivery mobile agents international symposium algorithms experiments sensor systems wireless networks distributed robotics algosensors pages chalopin riko jacob peter widmayer data delivery mobile agents line international colloquium automata languages programming icalp pages nicos christofides analysis new heuristic travelling salesman problem technical report graduate school industrial administration university pittsburgh february jurek czyzowicz krzysztof diks jean moussi wojciech rytter communication problems mobile agents exchanging energy international colloquium structural information communication complexity sirocco erik demaine mohammadtaghi hajiaghayi hamid mahini amin shayan oveisgharan morteza zadimoghaddam minimizing movement acm transactions algorithms talg jack edmonds ellis johnson matching euler tours chinese postman mathematical programming jack edmonds richard karp theoretical improvements algorithmic efficiency network flow problems journal acm jacm robert floyd algorithm shortest path communications acm fraigniaud gasieniec kowalski pelc collective tree exploration latin american theoretical informatics symposium latin pages greg frederickson matthew hecht chul kim approximation algorithms routing problems proceedings annual symposium foundations computer science focs daniel graf sort walking tree european symposium algorithms esa pages chalopin das disser graf hackfeld penna hoogeveen analysis christofides heuristic paths difficult cycles operations research letters july acm icpc world finals problems task catering https may icpcnews acm icpc problem catering https may alon itai christos papadimitriou jayme luiz szwarcfiter hamilton paths grid graphs siam journal computing marek karpinski michael lampis richard schmied new inapproximability bounds tsp journal computer system sciences joseph kruskal shortest spanning subtree graph traveling salesman problem proceedings american mathematical society harold kuhn hungarian method assignment problem naval research logistics quarterly thomas mccormick david delivery connection problems complexity algorithms discrete applied mathematics david lichtenstein planar formulae uses siam journal computing stephen warshall theorem boolean matrices journal acm jacm pawel winter steiner problem networks survey networks
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fundamental groups strongly controllable group system group code including group shift linear block code feb kenneth mackenthun email february abstract finite group defines signature group normal chain complete set coset representatives normal chain arranged triangular form representatives row put correspondence certain sets subtriangles representatives form isomorphic groups subtriangle multiplication induced multiplication show strongly controllable time invariant group system group shift group code associated signature group conversely given signature group construct strongly controllable time invariant group system results show extent study strongly controllable time invariant group systems reduced study signature groups general strongly controllable group system may vary time show signature sequence groups similar properties signature group general strongly controllable group system restricted finite interval case becomes block group system describe additive structure generator vectors comprise block group system new result applies mathematical structure defined finite number coordinates closed group addition coordinate linear block code group field vector space ring module finally give algorithms construct strongly controllable group system time invariant version block version introduction idea group shifts group codes important several areas mathematics engineering symbolic dynamics linear systems theory coding theory research area started work kitchens willems forney trott loeliger mittelholzer kitchens introduced idea group shift showed group shift finite memory shift finite type using work willems linear systems forney trott describe state group state code set sequences group property term group code time invariant group code essentially group shift show group code complete global constraints determined locally see wholely specified sequence connected labeled group trellis sections may vary time form group trellis explain important idea shortest length code sequences generators generator code sequence combination shorter sequences strongly controllable group code nontrivial portions generators bounded length give encoder whose inputs generators whose outputs codewords group code time finite set generators used give symbol codeword loeliger mittelholzer obtain analog derivation forney trott starting group trellis instead group sequences derive encoder use intersection paths split merge identity path trellis analog group trellis quotient group code sequences granule used addition loeliger mittelholzer characterize properties group branch group strongly controllable time invariant group code properties given terms condition set normal subgroups branch group related paths split merge identity path trellis paper find another group signature group characterizes strongly controllable time invariant group code characterize signature group terms properties complete set coset representatives normal chain branch group signature group component time group tensors give similar characterization general strongly controllable group system may vary time encoder analog much global approach finding encoder respectively group code normal chain subcode generators coset representatives coset decomposition chain normal chain encoder steps subcodes step product generator representatives time find encoder based local per unit time approach time encoder output product generator representatives coset decomposition chain time nevertheless enocders use generators encoders shift register structure global encoders forney trott loeliger mittelholzer form time convolution local encoder form time convolution reason call encoder time domain encoder forney trott encoder spectral domain encoder compare outputs time spectral domain encoders generator inputs time runs forward backward show abelian group system four cases give output input nonabelian group system outputs related symmetry tensor set also compare outputs four encoders different bases group system results show time harmonic theory group systems paper regarded giving dual results example work gives results spectral domain work gives results time domain work uses kcontrollable subcodes paper use quotient group systems whose normal subgroups group systems consider time invariant group systems time varying group systems time invariant group systems group shifts therefore results also apply group shifts general time varying group system defined finite interval case block code group therefore results applied block codes gives new information group structure potentially new search algorithms decoding algorithms forney trott suggest term group system alternative group code paper generally use term group system rather group code results analogues classical systems theory harmonic analysis forney trott shown group system reduced group trellis whose vertices states group system states defined using group theoretic construction quotient groups component trellis trellis section collection branches forms branch group time call group trellis first canonic form group system consider strongly controllable group systems fixed integer time sequence exists valid path length sequence alternatively state reached state branches material reviewed section product two paths another path path sequence branches branch group time product two paths product branches paths respectively group trellis sequence branches split identity path merge identity path form two normal chains schreier refinement theorem applied two normal chains obtain another normal chain refinement two chains call schreier series schreier series normal chain branch group time group trellis schreier series written form matrix rows columns determined branches splitting merging trellis paths group trellis strongly controllable matrix reduces triangular form called static matrix static matrix echo matrix ideas used classical linear systems analysis taking column static matrices times form another matrix called shift matrix shift matrix defined time interval also triangular form show shift matrix natural shift property fact shift matrix part group trellis truncation ray paths splitting identity path time discussed section show rows shift matrix used form quotient groups nontrivial generator vectors generator sequences forney trott transversal quotient groups coset representatives generator vectors transversal also triangular form shift matrix call generator matrix time components generator vectors form complete set coset representatives schreier series decomposition branch group static matrices generator matrices arranged form tensor tensor sequence generator matrices set tensors set time component tensor matrix static matrix therefore triangular form also denoted static matrix formed generator matrices times representatives representatives time set generator vectors form coset representative chain branch discussed section selection set generator vectors time necessary sufficient generate sequence component bases forms basis basis gives unique tensor set vice versa component basis constant basis time invariant constant basis always found shown time domain encoder path encoded path gives correspondence material included section two paths product another path let correspondence define operation correspondence set tensors operation forms group regard second canonic form group system second canonic form obtained first canonic form believe revealing structure group system group trellis components tensors time form set think set triangular forms denoted set representatives forms complete set coset representatives normal chain decomposition branch group time domain encoder encodes branch gives correspondence operation induces operation correspondence set components operation forms group called induced group believe plays central role understanding group systems groups discussed section entries triangular form representatives rows columns seen induces group set triangular forms upper coordinates set set subtriangles finite group upper coordinates complete set coset representatives form group induced group multiplication means set subtriangles upper coordinates forms group multiplication induced shifts subtriangles contain representatives set generator vectors representatives means representative put corrrespondence representative belong time shifts generator vector correspondence set subtriangles forms group isomorphic results give necessary sufficient conditions induced group signature group say group signature group group restricted forms group groups isomorphic congruence similar discussed branch group strongly controllable time invariant group system induced group signature group conversely finite group induced group signature group used construct strongly controllable time invariant group system branch group therefore sense study strongly controllable time invariant group systems study signature groups signature group characterizes group system terms generator vectors basis chosen constant signature group contains description generator vectors group system results discussed section time invariant group system special case general group system may vary time section give similar results general case show general group system characterized signature sequence groups slightly general versions signature group used time invariant group system special case general time varying group system group system nontrivial finite time interval call block group system block group system includes linear block code group forney trott show block group system decomposed set generator vectors arranged group trellis paper characterize additive structure group generator vectors comprise block group system linear block code group embedded linear block code field vector space ring module result additive structure applies cases well generally result applies mathematical structure defined finite number coordinates closed group addition coordinate use signature sequence isomorphic signature sequence defined using generator vectors use sequence define group describes additive structure generator vectors example given done section forney trott show group code homomorphism input sequences general however section show homomorphism permutation groups sets generator vectors generators selected representatives subtriangles used define signature group signature sequence section give example homomorphism extended hamming code first order reed muller code section use isomorphic version signature sequence give algorithm construct group system time invariant group system block group system algorithms essentially construction generator vectors group system appears first construction algorithm given general group system possibly general block group system section show group system reduced trivial group system sequence group systems dual derived sequence shifts used symbolic dynamics sequence derivative codes group systems section gives brief review fundamental concepts introduces definitions used follow notation forney trott closely possible one significant difference subscript denotes time use integer place notation superscript used exclusively indicate time thus always appears superscript notation use notation generic sense identity group particular group clear context forney trott study collection sequences time axis defined set integers whose components taken alphabet group alphabet time set sequences group componentwise addition call group system group code sequence given component time group system assumed complete important consequence local behavior sufficient describe global behavior completeness closure symbolic dynamics therefore time invariant complete group system thing group shift symbolic dynamics paper use language associated group systems rather group shifts define set codewords identity define set codewords group system satisfies axiom state whenever two sequences pass state given time concatenation past either future valid sequence canonic state space time defined def canonic state space unique figure definition state system sequence time determined natural map homomorphism therefore well defined state sequence associated well defined state code associated canonic realization group system set pairs sequences state sequence state spaces canonic realization canonic realization minimal realization group system element canonic realization denoted component given canonic state time canonic state time say left state use notation say right state use notation path given clear must equivalently let state code sequences states theorem group isomorphism given correspondence time assignment group isomorphism proof well defined state sequence associated means assigned well defined assignment time map bijection since assigned must time interested canonic realization rather group system remainder paper loss generality considering rather correspondence isomorphism canonic realization described graph minimal realization graph isomorphic canonic realization think component branch trellis section element branch group trellis section bipartite graph left vertices states right vertices states label branch state state group branches subdirect product subgroup direct product group clearly branch label two vertices described group trellis connected sequence trellis sections joined together using common states refer group trellis regard group trellis first canonic form group system states describe state groups isomorphic consider projection map onto left states given assignment homomorphism kernel subgroup elements form identity first homomophism theorem also consider projection map onto right states given assignment homomorphism kernel subgroup elements form identity first homomophism theorem thus branch form results show state group isomorphism time let group trellis let trellis path define time intervals say trellis path segment length one time interval general say trellis path segment length time interval using define projection map time assignment define projection map assignment trellis path segment length say codeword span identity define set trellis paths identity outside time interval integer say group trellis time epoch pair states trellis path segment length connecting two states group trellis strongly controllable integer least integer group trellis strongly controllable denoted paper study case preceding definition controllable terms states equivalent consistent definition controllable terms trellis path segment connecting two half infinite paths static matrix shift matrix section write coset decomposition chain group matrix call matrix chain study matrix chain called static matrix another matrix group elements called shift matrix use matrices construct tensor like manner define set codewords identity component time define set codewords integers define def xjt note consistent definition previously given section xjt integers define def yit note consistent definition previously given section yit clear xjt yit time integer groups xjt yit first introduced group intersect groups used give abstract tions xjt characterization branch group group trellis set define set right states define set left states sets define concatenation valid trellis path segments length two first component second component integers yit note xjt sets trellis path segments length two next result follows directly proposition using notation proposition group trellis equivalently time group two normal series chief series denote normal series xjt yit schreier refinement theorem used prove theorem shows obtain refinement xjt inserting yit call forward schreier series xjt yit since xjt yit chief series forward schreier series xjt yit chief series equation written forward schreier series matrix columns rows note terms bottom row form terms top row form sequence thus indeed refinement sequence normal series xjt call matrix chain forward schreier series xjt yit proposition group trellis xjt xjt proof group trellis proposition notation xjt xjt means rewrite general work considers time invariant group trellis seen proposition proof apply also group trellis diagonal terms matrix chain proposition shows diagonal terms satisfy group trellis means column terms diagonal term diagonal term reduce matrix chain triangular form shown triangle formed two ways depending whether columns shifted shifted columns since useful call static matrix static matrix defined time interval typical entry matrix theorem static matrix description normal chain chief series branch group group trellis proof xjt yit normal chains branch group schreier refinement theorem forward schreier series normal chain replace group trellis denote resulting structure note tensor since coset decomposition chain description coset structure group trellis path traverses sequence cosets note first column description think input remaining columns description isomorphic state thus columns static matrix contain information input state therefore isomorphic copy state code embedded xjt xjt xjt xjt number columns using column static matrix form matrix shown shape longer inclusion one column next however coset decomposition within column preserved call shift matrix shift matrix composite single column static matrices notice shift matrix defined time interval typical entry matrix column static matrix column shift matrix time thus static matrix composite columns shift matrices show shift matrix kind shift property preliminary results discussion show shift matrix physical interpretation paths split identity path time branch define following branch set set branches follow next time epoch valid trellis paths words branch following branch set represents contraction correspondence expansion given correspondence given state group isomorphism clear however note function domain range think relation relation think assignment set branch proposition following branch set branch coset assignment define following branch set set set union set always consists cosets particular xjt integers set integer define composition define set trellis branches time epoch set trellis branches time epoch path trellis note set integer define set trellis path segments time interval start branch proposition subsets proof clear follows proposition subsets matrix contains terms form consider terms form column row example lower left term column row terms shorthand column row thus shift matrix columns rows note row length terms show shift matrix preserves shifts shift property proposition fix fix shift matrix shift property term column row shift term column row proof fix fix fourth equality follows dedekind law subgroups group first column shift matrix important define ykt show row shift matrix trellis path segments time interval start branch theorem fix row shift matrix terms proof prove induction assume true use show true shows row shift matrix terms find matrix quotient groups using shift matrix using theorem represent quotient group adjacent terms column shift matrix two equivalent ways proposition groups proof ykt group trellis path segments group proposition result theorem proof projection onto homomorphism kernel projection onto homomorphism kernel therefore first homomorphism theorem use correspondence theorem third isomorphism theorem complete proof proposition groups proof see note projection group time interval proposition proof see theorem proof projection onto homomorphism kernel path segments identity time projection onto homomorphism kernel path segments identity time therefore first homomorphism theorem show first show let path segment identity time must since therefore show let path segment identity time must since therefore shown use correspondence theorem third isomorphism theorem complete proof note proof breaks try words show define def corollary components transversal transversal proof theorem shows projection gives correspondence cosets cosets therefore projection transversal transversal corollary remark result regarded rectangle criterion shift matrix corners rectangle similar spirit quadrangle criterion latin square configuration theorem net fact rectangle condition generalized starting groups general results needed use corollary create tensor fix define column vector quotient groups vector quotient groups formed groups normal chain center column using def obtain column vectors used form shift matrix def second example shift matrix row shift matrix shift vector shift vector components defined time interval columns shift matrix rows think shift matrix matrix quotient groups shift matrix corollary preserves isomorphism quotient groups shows shift matrix shift quotient group row gives next quotient group natural shift row therefore regard shift matrix structure strongly controllable group system fix define xjt column vector quotient groups vector quotient groups formed groups normal chain center column used form static obtain column vectors xjt matrix def xjt note definition xjt consistent since defined using defined using consistent note xjt think definition xjt time defined parentheses term also think static matrix def clear term one different shift matrices using relate static matrix shift matrix tensor description shown time increases move page vectors shift matrix vectors along diagonal vectors static matrix vectors row superscript parentheses terms like indicate terms belong shift matrix example diagonal terms belong shift matrix starting time center row reduces static matrix let denote tensor say chain tensor given group trellis one chain tensor tensor description coset structure group trellis tensor dual nature shift matrices static matrices natural important way understand look along diagonals terms shift matrices theorem time diagonals description quotient groups next section show recover paths generators representatives coset structure described generators generator matrix review results forney trott define kcontrollable subcode group code transcribe approach group trellis used subcode group trellis defined set combinations code sequences span less show normal series code granule theorem show isomorphic direct product quotient groups defined def called granule coset representative called generator coset representative always taken identity sequence case isomorphic identity group identity sequence coset representative nonidentity generator element span exactly thus every nonidentity generator codeword expressed combination shorter codewords quotient group let denote set coset representatives transversal let transversal follows set set coset representatives cosets know set coset representatives granule set generators means generator theorem every code sequence uniquely expressed product generators thus every code sequence product sequence generators conversely every sequence generators corresponds code sequence basis minimal set shortest length generators sufficient generate group system follows basis set coset representatives paper time let component basis set generators transversals time sequence component bases gives basis component basis constant basis show projection generators also transversal therefore basis found using representatives either lemma set paths formed concatenation groups proof proof proposition means set paths formed concatenation groups well defined branch branch trellis path segment length two paths consist sequences split identity state time merge identity state time therefore path must fix integer let sequence show must similarly must since contains code sequences whose component lemma example means proof know proof lemma show shows branch branch trellis path segment length two continue argument branch branch trellis path segment length two continuing argument shows branch trellis path segment length merges identity state time argument works reverse time well branch branch trellis path segment length two thus see sequence thus shown lemma proof holds follows lemma holds lemma theorem isomorphism correspondence cosets given proof using lemma since shows properly define correspondence cosets given forney trott define input chain def projection fkt using lemma gives fkt input granule theorem forney trott show fkt fkt combining theorem gives following correspondences given input granule theorem theorem shows isomorphism given theorem let set generators transversal transversal theorem shows set generators equally well found tranversal quotient group another way denote transversal besides set coset representatives transversal let denote transversal fix let generator representative lemma know component element theorem corollary know representative pick set generators transversal induces transversal theorem let set generators transversal component generators forms transversal zassenhaus lemma crucial step proof schreier refinement theorem time invariant case use zassenhaus lemma give second proof theorem theorems viewed generalization zassenhaus lemma pick generator arrange nontrivial components generators matrix shown called shift matrix also generator matrix time denoted row matrix shift vector also called generator vector denoted def generator vector nontrivial components generator define column vector def rewrite another related form shown called static matrix element seen component components static matrix occur time generator matrix first column specifies matrix completely static matrix first column determine static matrix uniquely rewrite rtj relate static matrix generator matrix using tensor description shown time increases move page vectors generator matrix vectors along diagonal vectors static matrix vectors row superscript parentheses terms like indicate terms belong generator matrix example belong diagonal terms generator matrix starting time center row entry column reduces rtj static matrix notice term one different shift matrices theorem fix time finite sequence generator matrices times uniquely determines static matrix column generator matrix denoted column static matrix denoted rtj proof center row reduces static matrix entry column generator matrix time let denote tensor say representative tensor regard two different ways sequence static matrices sequence shift matrices first way write static matrix set static matrices denoted therefore equivalent seen theorem determined shift matrices tensor also determined sequence shift matrices denote interpretation using notation shift matrix set possible shift matrices denoted define tensor set set representative tensors determined cartesian product possible shift matrices show correspondence tensors paths section tensor selection one generator vector generator component basis basis sequence component bases forms basis basis gives unique tensor set tensor set corresponds unique basis tensor gives one shift matrix one static matrix time theorem regard selection one coset representative generator vector single coset thus quotient groups shift matrix similarly form time indices selection one coset representative single coset quotient group thus form time indices static matrix explains tensor form tensor given produces sequence shift matrices sequence static matrices sequence shift matrices corresponds uniquely determines sequence static matrices arbitrary sequence static matrices may correspond valid sequence generator vectors therefore following lemma essentially change variable theorem lemma fix fix let set generators transversal component generators forms transversal proof fix examine time fix pick set generators transversal denoted induces transversal choose transversal note set transversals coset representatives cosets quotient groups column column shift matrix words set transversals coset representatives cosets quotient groups column static selecting transversals matrix column xjt column static matrix obtain complete set coset representatives normal chain given static matrix gives following result theorem let set generators transversal component generators forms transversal set forms complete set transversals coset representatives normal chain given static matrix transversal selecting one coset representative obtain coset representative chain normal chain given static matrix branch written using elements coset representative chain convention used equation evaluated note product terms static matrix inner product parentheses product terms column using written equivalent forms shown time find branch using selected set generators times however shown construct path way show next section time domain encoder know shift matrix column vector column vector column completely determined column think shift let set columns possible shift matrices define column shift map assignment assignment given note defined since shifts abbreviate notation column index subscript time index superscript left notational simplicity ambiguity resolved looking arguments define def characterize paths theorem let arbitrary sequence necessarily tensor time tensor time new input element shift proof first assume know formed sequence shift matrices consider fix shift matrix know column rtj column preceding discussion shifts know column shift matrix column conversely assume fix since holds columns form shift matrix shown sequence shift matrices therefore tensor encoder group trellis finite state machine given sequence inputs produce path group trellis forney trott first showed group code encoder shift structure approach thought spectral domain approach give encoder shift structure uses time domain approach theorem shows tensor set natural shift structure remainder section show path encoding gives correspondence encoder sliding block structure uses generator vectors encoder given useful think equivalent forms encoder version useful following discussion assume found basis found generators fix time nontrivial components selected generators encoder form generator matrix theorem generator matrices uniquely determine static matrix column gen erator matrix column static matrix rtj see form sliding block encoder time select new generator matrix whose column vectors shown along diagonals column vectors generator matrix time generator matrix time column vectors shown along diagonals time increases slide along infinite matrix left right time output branch sliding block encoder calculated static matrix rtj whose terms shown center row first term center row new input first column vector new generator matrix selected time remaining terms previous generator matrices lected times respectively calculate branch time sliding block encoder uses time window therefore encoder causal show use implement sliding block encoder need following useful lemma lemma let encoder encodes trellis path segment length group trellis words proof using encoding know form shift since means rewrite terms generators see fix note change limits last double product rewrite changes becomes note term involves generators vector basis terms involve generators vector bases first consider case let encoding since components identity rewrite note involve generators vector bases pair terms product multiplication inner square bracket component component note valid trellis path segment length product valid trellis path segments length hence properties group trellis valid trellis path segment length means encoding consider case let using branch means notice think encoder estimator encoding gives initial estimate time use new input find correct initial estimate theorem tensor encoded path encoding using proof theorem lemma path lemma fix let encoding using correspondence set pairs length set trellis path segments correspondence given encoding using proof let art set pairs length let abt set trellis path segments know lemma encodes length therefore trellis path segment maps art abt map art abt since encodes encodes must know uniquely determined specifically therefore possible possible map art abt onto theorem encoding using proof fix let lemma know exists encoding using sequence pairs encoding sequence pairs must since left right term equalities encode branch encode therefore sequence pairs reduced sequence encoding path theorem tensor set depends choice basis basis fixed fixed theorem given basis fixed correspondence given encoding using proof theorem theorem encoding onto show assume exists encode using impossible since encoding encoding signature sequence signature group generator group groups let let correspondence def define operation tensor set correspondence theorem operation group system theorem group system isomorphic group system correspondence proof definition correspondence gives isomorphism give result multiplication theorem multiplication corresponds multiplication changes generator vector another time length corollary let let correspondence multiplication corresponds multiplication changes generator vector another time length consider component multiplication component multiplication easy product given define component multiplication agrees definition operation component multiplication gives insight structure group system component multiplication time component define multiplication follows let define correspondence branch encoding using emphasize matrix triangular form components marix representative representatives either nontrivial identity define correspondence let let branch encoding using let branch encoding using define operation def correspondence show operation defined consistent operation defined lemma let proof left hand side right hand side elements prove left hand side right hand side equal correspond branch correspondence let let correspondence defined correspondence correspondence theorem time component encoding using time component correspondence since theorem encoding using encoding using definition know correspondence left hand side right hand side correspondence branch therefore must theorem set operation forms group theorem correspondence rtb encoding rtb using say induced group first consider general group systems may vary time therefore component basis may different time branch group may different time therefore component group may different time let encoding using similarly encoding using let assume encoding using definition know give general result products form consider finite group normal chain normal let complete set coset representatives normal chain written using subset complete set coset representative chain lemma let assume let consider product solely product upper coordinates determined upper coordinates upper coordinates proof let since know therfore last result means complete set therefor coset representatives solely determined upper fore upper coordinates upper coordinates coordinates apply result use following notation note static matrix triangular form introduce triangle notation describe certain subsets representatives let representatives specified upper vertices triangle lower vertex representatives satisfy def let set possible triangles define except missing representative likewise define see except missing set representatives group may also written applying lemma gives following lemma fix time fix let product representatives uniquely determined representatives theorem let let fix time fix generator vector uniquely determined representatives also generator vector uniquely determined generator vectors representatives proof generator vector uniquely determined representat apply lemma tive fix time fix know let define operation def lemma operation well defined proof let let show operation well defined need show clear lemma theorem set operation forms group proof first show operation associative need show let using therefore operation associative since operation group associative let identity show identity using let element show element using together results show group last chapter thesis group codes mitchell trott lists several open problems group codes among mentions surely must orthogonal decomposition group codes ideally one would like individual generator vectors form orthogonal basis find true special case nontrivial generator vectors length orthogonal decomposition using groups however general groups show dependence among generator vectors give orthogonal decomposition think group two different ways view think group multiplication upper coordinates upper coordinates find upper coordinates second interpretation group devoid meaning multiplication representatives think permutation elements set permutation operation note subtriangle thus permutation subtriangle set subtriangle set obtain subtriangle set note group group isomorphic generalize theorem proof know define homomorphism gives homomorphism assignment assignment gives correspondence seen corresponassigned unique coset dence gives isomorphism triangles congruent shapes identify representatives congruent positions triangles example representative triangle triangle gruent position representative lemma fix fix representatives congruent positions triangles generator vector lemma fix time fix fix let product representatives uniquely determined representatives proof fix time fix let lemma representatives uniquely determined representatives means generator vectors representatives uniquely determined generator vectors representatives generator vectors representatives used lemma note representatives set generator vectors similarly note representatives set generator vectors representatives set generator vectors therefore representatives uniquely determined representatives use lemma theorem let let fix time fix generator vector uniquely determined representatives also generator vector uniquely determined generator vectors representatives proof generator vector uniquely determined representat apply lemma tive fix time fix know let define operation def well defined lemma operation proof let let show operation well defined need show clear lemma theorem set operation forms group proof proof theorem think group two different ways view think group plication multiply obtain product illustrates coordinates complicated function coordinates coordinates addition lemma shows must two facts place severe restrictions structure example show must type commutative property consider second interpretation group thought permutation group thus permutation subtriangle set subtriangle set obtain subtriangle set second interpretation shown give much revealing explanation group structure henceforth think groups like permutation groups relate groups theorem fix fix define correspondence assignment representatives congruent positions correspondence correspondence isomorphism proof fix note representatives congruent positions generator vector restate theorem slightly terms shifts corollary fix fix define correspondence assignment representatives congruent positions correspondence correspondence shift isomorphism proof fix note representatives congruent positions generator vector corollary gives viewed note shift viewed shift shown see therefore result gives explanation restatement state group isomorphism clearer form see change time variable theorem gives following result theorem fix time fix set operation forms group study homomorphism related groups theorem theorem fix time fix projection map given assignment defines homomorphism proof let consider projections lemma follows therefore homomorphism group group theorem projection map given assignment defines homomorphism proof define projection map assignment show well defined let suppose since agree must agree therefore well defined let consider projections want show corollary projection map given assignment defines homomorphism proof corollary follows directly theorem study homomorphisms theorem projection map given assignment defines homomorphism proof let consider projections lemma follows therefore homomorphism group group following generalization result fix sequence indices times may sequence times may integers may finite infinite subset consider sequence restricted times let set sequences define projection map assignment let define operation def theorem set operation forms group proof show since exists similarly exists know definition time component time component lemma know means last result shows operation associative since operation group associative time let identity show identity let pick using let show inverse pick using therefore inverse together results show group corollary set operation forms group system proof group system since group defined componentwise group addition time group theorem projection map given assignment defines homomorphism proof use corollary projection map given assignment defines homomorphism correspondence isomorphism consider second sequence indices times consider sequence restricted times let set sequences define projection map assignment theorem projection map given assignment defines homomorphism proof proof similar proof theorem signature sequence sequence finite groups group sequence give necessary sufficient conditions group sequence sequence branch groups group trellis group system sections branch group know must normal chain complete set coset representatives normal chain arranged triangular form static matrix given must least one generator span means must static matrices nontrivial representative additional requirements given aside requirements restrictions example rows identities addition row different sets representatives size let fix consider set representatives set representatives suppose two sets define correof size spondence two sets suppose define correspondence sets triangles let set subtriangles fix consider set triangles let triangle triangle triangles congruent identify representatives congruent positions tritriangles example representative angle congruent position representative triangle let triangle using correspondence find unique triangle representatives congruent positions correspondence indicate saying example way correspondence gives correspondence triangle elements set set written show induces group similar manner correspondence given written using representatives coset representative chain let let correspondence def define operation clear operation group call induced group note reused notation defined previously operation necessarily operation defined previously group necessarily like also group sequence say sequence induced groups induced sequence consider group sequence induced sequence say induced sequence signature sequence satisfies following three conditions operation set well defined def forms group note correspondence described iii correspondence triangle elements respectively given isomorphism theorem sequence branch groups group system group shift group code induced sequence signature sequence proof know find basis according previous work paper basis gives tensor set find group component group time show signature sequence condition follows theorem think shift mapping correspondence given condition iii follows corollary proves statement theorem theorem given group sequence signature sequence construct group system sequence branch groups proof form cartesian product sets groups signature sequence sequence form give conditions sequence tensor tensor set theorem know sequence tensor time input shift use conditions iii show conditions form let iii know isomorphism given correspondence triangles representatives congruent positions satisfy correspondence therefore define tensor tensor set satisfy following case form def def input thought shift tensor set set sequences satisfy let define operation paths def defined component component multiplication group show operation forms group show show sufficient show isomorphism condition iii therefore shown using combining shown note need start sequence groups sequence groups form easy find sequence groups induces signature group specialize section time invariant case give necessary sufficient conditions finite group branch group group trellis time invariant group system sections branch group know must normal chain complete set coset representatives normal chain arranged triangular form static matrix given since time invariant may choose constant basis henceforth assume case basis constant aside time indexing since time invariant must generator span means top row contains nontrivial representatives generators general case rows identities contrast general case see sets representatives row must size let fix consider set representatives suppose two sets set representatives size define correspont dence two sets suppose define correspondence sets triangles let set subtriangles consider set triangles let triangle triangle triangles congruent identify representatives congruent positions triangles example representative triangle congruent position representative triangle let triangle using correspondence find unique triangle representatives congruent positions correspondence example indicate saying way correspondence gives correspondence triangle elements set set written show induces group similar manner correspondence given written using representatives coset representative chain let let correspondence def define operation clear operation group call induced group reused notation defined previously operation necessarily operation defined previously group necessarily consider finite group induced group say induced group signature group satisfies following three conditions operation set well defined def forms group note correspondence described iii correspondence triangle elements respectively given isomorphism theorem branch group time invariant group system group shift group code induced group signature group proof theorem group system sequence branch groups signature sequence since time invariant since time invariant may choose constant basis constant may use operations place show satisfies conditions iii definition signature group property signature sequence correspondence assignment know time invariance define correspondence assignment verifies condition definition signature group property iii signature sequence correspondence triangle elements respectively given isomorphism using definition correspondence triangle elements given isomorphism verifies condition iii definition signature group condition satisfied trivially therefore induced group properties iii signature sequence sets row different size representatives signature group since holds sets representatives row must size corollary theorem signature group contains copy generator vectors proof theorem group system signature sequence since time invariant may choose constant basis correspondence know viewed shift mapping fix consider representative representative generator vector since shift correspondence definition component gives assignment correspondence mapping therefore essentially mapping first component generator vector second mappings continuing way shows give representatives components since contains copy result holds generator vectors theorem given finite group signature group construct time invariant group system sequence branch groups proof consider sequence groups sequence signature groups show satisfies conditions iii definition signature sequence property signature group correspondence assignment know time invariance define correspondence assignment verifies condition definition signature sequence property iii signature group correspondence triangle elements respectively given isomorphism using definition correspondence triangle elements given isomorphism verifies condition iii definition signature sequence condition satisfied trivially therefore group sequence signature sequence theorem construct group system sequence branch groups means time invariant block group system section showed general strongly controllable group system time invariant system signature sequence reduces single group signature group section specialize general time varying group system finite time interval let set possible branches time say group system support trivial nontrivial case say block group system block group system isomorphic set finite sequences defined componentwise group addition usually referred engineering literature coding literature linear block code study important case block group system linear block code case general case straightforward extension case work forney trott shows block group system decomposed set generator vectors arranged group trellis section describe form signature sequence next section find signature sequence isomorphic use define group characterizes additive structure group generator vectors block group system assume group system support work know block group system decomposed generator vectors generator vectors length must start time form definition must least one nontrivial generator vector length means must nontrivial generator vectors length representative must start times form general generator vectors length must start times form knowing decomposition generator vectors describe form signature sequence matrices trivial entries identity matrix trivial except first column matrix trivial matrix trivial except first two trivial column columns matrix shifted column shift column note entry new input note entry trivial since one generator vector length general composed two trivial triangles nontrivial parallelogram triangle trivial triangle trivial nontrivial entries form parallelogram shape whose corners representatives words columns nontrivial except column entries triangle columns trivial thus column contains nontrivial entries lastly matrix trivial except last diagonal representatives lemma group system support matrices trivial least one nontrivial generator vector means nontrivial representative nontrivial entries lie parallelogram shape whose corners representatives parallelogram trivially line using lemma restate theorems section block group system follows theorem sequence branch groups group system support induced sequence signature sequence form given lemma theorem given group sequence signature sequence form given lemma construct group system support sequence branch groups isomorphic signature sequence homomorphism generators previously constructed tensor set using generator vectors generator vector first component alternatively last given specifies generator vector uniquely replace component generator vector shift vector first component repeated given times assignment tensor becomes tensor let set tensors obtained way correspondence given assignment operation determines operation forms group replacement component given becomes component shown representa replaced representative tive generator vector place matrix matrix coset representatives coset representatives correspondence assignment matrix form matrix following reuse notation example note representatives specified triangle lower vertex upper vertices let define operation set def gives group isomorphic say induced group induced group group sequence call induced sequence clear induced sequence isomorphic version induced sequence consider group sequence induced sequence say induced sequence signature sequence satisfies following two conditions operation set well defined def forms group note correspondence triangle elements respectively given isomorphism using place eliminated need correspondence definition signature sequence following restatement theorems signature sequence theorem sequence branch groups group system induced sequence signature sequence theorem given group sequence signature sequence construct group system sequence branch groups following analogs theorems corollary theorem fix time fix projection map given assignment defines homomorphism theorem projection map given assignment defines homomorphism corollary projection map given assignment defines homomorphism fix sequence indices times may sequence times may integers may finite infinite subset consider sequence restricted times let set sequences define projection map assignment let define operation def theorem set operation forms group proof proof analogous proof theorem use isomorphism corollary set operation forms group system proof group system since group defined componentwise group addition time group lemma projection map given assignment defines homomorphism proof use result analogous theorem projection map given assignment defines homomorphism correspondence isomorphism correspondence isomorphism consider second sequence indices times consider sequence restricted times let set sequences define projection map assignment theorem projection map given assignment defines homomorphism proof proof similar proof theorem view group tensors slightly different way note representatives common however note representatives representatives slide past one column representatives overlap fix let sliding operation slides past one column operation sequence becomes sequence let set formed way correspondence given mapping formed assignment appears shift vector representative times therefore collapsed shift vector single point lemma set description shift vectors generator vectors correspondence correspondence point shift vector generator vector note also thought sequence columns inputs reverse time order let uts portion shown def uts uts define uts def uts uts let consider sliding operation finite infinite sequence nonoverlapping triangles form becomes finite infinite sequence triangles form uts triangles uts may overlap share representative triangles overlap form polygon shapes define sequence isolated triangles polygon shapes formed overlapping triangles let set formed way let mapping given assignment definitions shown commutative diagram figure since correspondence given assignment correspondence given assignment figure definition theorem group acts therefore acts correspondence proof clear group acts correspondence shows acts let shift vector group action let take shift vector taken shift vector group action takes components taken components one one representative group action takes singular representative therefore view singular representative action group equivalently action set generator vectors indicated singular representatives well equivalently action set generators indicated representatives let group permutations set action group set induces homomorphism given permutation associated theorem group acts let homomorphism given assignment permutation associated given coset group bijection permutation becomes permutation uus define mapping sus assignment uus since homomorphism homomorphism say extension using correspondence gives following theorem homomorphism sus sus extension homomorphism homomorphism given assignment words permutation group set generator vectors indicated homomorphism permutation group generator vector corresponds generator contains equivalently permutation group set generators indicated homomorphism permutation group theorem homomorphism sus proof combine theorems use preceding results study additive structure block group system terms generator vectors remains describe sequence columns block group system block group system support know columns trivial except columns columns following result using lemma lemma columns block group system form representatives identity remaining representatives may trivial nontrivial must colt remaining representatives umn nontrivial representative may trivial nontrivial representatives may nontrivial shown representatives outside triangle shape pyramid shape must trivial example example study block group system case becomes identify nontrivial groups form first find simplest groups considering sequences one term groups triangles case know uts sets form uts block group system nontrivial elements uts uts listed first column table sets respectively shown third column table sets associated groups respectively shown fourth column using see group group elements shown second column table example set uts set cartesian product sets representatriples trivial set containing tives identity groups table shown stacked according representatives define set representatives set uts uts uts set find remaining sets involve index sequences two terms polygon shapes formed multiple triangles form overlap triangles form nontrivial overlapping triangles shown second column table example overlap form polygon shape whose representatives union representatives nontrivial representatives polygon shape shown first column table group corresponds shown third column table example group corresponding def table sets group actions block group system group def def def def def say denoted union tables give nontrivial groups isomorphism groups tables may descriptions isomorphic example isomorphic group listed table nontrivial representatives polygon set set table remaining sets group actions block group system following results subgroup every group alternatively top contained every group row set bottom row set contained one group groups contain groups subgroups subgroups group found triangle shape subgroups example group group subgroup another group projection map later former defines homomorphism application theorem intersection two groups another group subgroup groups union two groups different times group group application theorem example groups shown table union three groups different times group group application theorem example group shown table union three groups different times isomorphic union two groups shown table collection groups group time largest group contains groups time subgroups union largest groups different times group therefore collection groups isomorphic group listed table table representative set table determines generator vector therefore use study additive structure block group system terms generator vectors follows group isomorphic permutation group generator vectors length including identity generator vector length group group isomorphic permutation group generator vectors length isomorphic permutation group group generator vectors length last permutation group isomorphic block group system results application theorem homomorphism union groups application lemma homomorphism block group system union groups theorem union subgroups subgroup another union groups projection map later former defines homomorphism application theorem results example surprising merely extension block group system happens single group single group set highest order coset representatives forms permutation group next set highest order second highest order coset representatives forms permutation group block group system set longest generator vectors forms permutation group two sets second longest generator vectors starting time starting time set starting time forms permutation group set longest generator vectors also set starting time forms permutation group set longest generator vectors union two groups also forms group permutation group longest second longest generator vectors intersection two groups also group permutation group longest generator vectors groups table table give simple direct explanation group structure block group system terms generator vectors example theory group systems developed applied block codes study binary extended hamming code first order code used example code linear trellis diagram code shown figure granule representatives generators code follows code becomes generator vectors length generator vectors length generator vectors length reason two rows identities identify nontrivial groups form way example first find simplest groups considering sequences one term groups def triangles example shown table simplest groups shown stacked according representatives define set representatives set set group def table set group action binary extended hamming code give example show application theorems code let finite sequence time finite sequence example shown table theorem shows homomorphism group permutation group self evident since theorem shows homomorphism example verify theorem showing homomorphism extended hamming code theorem shows homomorphism given map using assignment correspondence isomorphism correspondence isomorphism example therefore assignment becomes determine assignment start work left shown table representatives correspond nontrivial generators code sets choice representatives sets obtain elements shown column four shift vector table representative representative shift vector code generator vectors shown third column table generator vectors column three correspond generators column two codeword code decomposed selecting one generator four transversals shown choice two generators second column table four codewords generators decomposition gives assignment first column table second completes chain assignments remains verify assignment gives homomorphism verify sufficient show assignment first second column table gives homomorphism permutation group four pairs generators shown second column show sufficient show quartet first column forms quotient group first quartet normal subgroup easy show using linearity code verify theorem concatenating examples theorems codeword selected generators decomposition generator vectors table mapping codewords binary extended hamming code homomorphism bottom row note groups groups respectively phic groups respectively sets used construct trellis diagram give code therefore groups scription group structure trellis figure time epoch construction section construct group system may vary time time invariant group system block group system construct group system sufficient construct signature sequence time invariant group system sufficient construct signature group block group system sufficient construct signature sequence support first construct signature sequence specialize construction signature group block group system slightly easier construct isomorphic version signature sequence sequence first give easy way evaluate operation group signature sequence give construction use corollary evaluate operation since corollary holds means may evaluate recursively order using triangles step first step evaluate gives elements row continue way steps order general step evaluate note evaluation depends evaluation already done previous step evaluation greatly simplified since new element need find step finding elements row general use approach construct group signature sequence suppose found groups need find groups homomorphism groups process begins simplest groups terminates group want find definition signature sequence operation set defined def definition operation uses contrast construction signature sequence available priori define operation group therefore construction interpret group different way use define operation group simply define operation set subtriangles forms group set subtriangles formed construction complete vertices known show operation group property definition along lines note definition requires edge representative also available priori tion construction interpret point permutation group devoid meaning representative example construction consider elements set distinct integers defined subtriangle integers construct group satisfies condition definition signature sequence algorithm construct signature sequence define group operation endfor counting order define group projection map given assignment defines homomorphism endfor endfor def define enddo lemma completion algorithm projection map given assignment defines homomorphism proof result trivial consider remaining cases fix fix group three cases consider algorithm group projection map defines homomorphism algorithm group projection map defines homomorphism algorithm two groups projection map defines homomorphism projection map defines homomorphism continuing process construct ladder groups ending given contiguous pair projection map defines homomorphism general many ladders besides fix ladder shown assumption projection map homomorphism pair pair follows projection map homomorphism concatenation continuing process ladder show projection map homomorphism concatenation projection map concatenation ladder besides therefore homomorphism ladders theorem group operation set satisfies proof let lemma projection map projection map means corollary algorithm sufficient satisfy condition group signature sequence proof theorem shows algorithm sufficient satisfy condition algorithm asks group homomorphism makes choices group limited permutation already determined therefore define group need define mutation give algorithm find signature sequence already given algorithm find arbitrary fixed time next give algorithm obtain signature sequence extending right half infinite sequence left half infinite sequence extend right shift obtain group complete definition group satisfies condition need define groups along first column done way construction repeat process successive time epoch extend left find inverse shift obtain group complete definition group satisfies condition need define groups along last diagonal done way construction repeat process previous time epoch algorithm construct signature sequence construct using algorithm enddo extend right time define group shift words correspondence triangle elements respectively given isomorphism counting order define group projection map given assignment defines homomorphism endfor def define endfor repeat times enddo extend left time define group inverse shift words correspondence triangle elements respectively given isomorphism counting order define group projection map given assignment defines homomorphism endfor def define endfor repeat times enddo specialize construction signature sequence construct signature group define isomorphic version similar way use correspondence assignment defined previously let define operation set def gives group isomorphic also say induced group matrix form matrix following reuse notation consider finite group induced group say induced group signature group satisfies following two conditions operation set well defined def forms group note correspondence triangle elements respectively given isomorphism algorithm already constructed group satisfies condition signature sequence construct signature group need satisfy conditions following algorithm modification algorithm includes condition algorithm sufficient satisfy algorithm construct signature group define group operation counting order define group words correspondence triangle elements respectively given isomorphism endfor endfor counting order define group projection map given assignment defines homomorphism counting order define group words correspondence triangle elements respectively given isomorphism endfor endfor def define enddo algorithm simplified follows algorithm construct signature group define group operation define group words correspondence triangle elements respectively given isomorphism endfor counting order define group projection map given assignment defines homomorphism define group words correspondence triangle elements respectively given isomorphism endfor def define enddo specialize construction signature sequence construct block group system construct block group system sufficient construct signature sequence support form given lemma modify algorithm follows time define identity group extend right way extended right algorithm time define group shift include columns form given lemma form group procedure done algorithm time general time extend right shifting group include columns form given lemma form group time define identity sequence group systems section show group system reduced trivial group system sequence group systems find sequence group systems using theorem corollary theorem define sequence indices times consider sequence let set sequences define projection map assignment let define operation def following analogs theorem corollary theorem theorem operation forms group corollary operation forms group system theorem projection map given assignment defines homomorphism corollary homomorphism mapping given correspondence isomorphism proof isomorphism use theorem corollary isomorphism correspondence given correspondence isomorphism proof theorem know homomorphism mapping first homomorphism theorem isomorphism correspondence rkr kernel homomorphism kernel homomorphism set map identity identity sequence set maps identity set condition fixes generator vectors identity identity corresponds subcode correspondence isomorphism therefore since isomorphism correspondence correspondence isomorphism isomorphism correspondence given correspondence isomorphism shown group system strongly controllable fact group system branch group sequence whose terms time components signature sequence whose terms time components results show group system associated natural sequence group systems sequence subcodes denoted used find shift register encoder work also mention related complementary approach finding shift register encoder sequence derivative codes given loeliger sequence derivative codes code state code previous code sequence terminates trivial code sequence derivative codes essentially derived sequence shifts used symbolic dynamics derivative codes derived sequence used construction time invariant group code time invariant homogeneous shift respectively may also consider sequence quotient groups systems defined identity sequence denote sequence corollary know therefore sequence quotient group systems isomorphic sequence group systems general terms sequence terms sequence two sequences isomorphic think sequences dual sequence dual derivative codes derived sequence shifts construction time invariant group system given algorithm regarded dual construction appears dual result literature contruction general group system given algorithm unable obtain loeliger thesis zurich library written request english references kitchens expansive dynamics groups ergodic theory dynamical systems willems models dynamics dynamics reported vol kirchgraber walther new york john wiley forney trott dynamics group codes state spaces trellis diagrams canonical encoders ieee trans inform theory vol loeliger mittelholzer convolutional codes groups ieee trans inform theory part vol trott algebraic structure trellis codes thesis stanford forney coset binary lattices related codes ieee trans inform theory vol forney geometrically uniform codes ieee trans inform theory vol forney minimal realizations linear systems shortest basis approach aug sarvis trott useful groups trellis codes proc ieee int symp inform theory whistler canada sindhushayana marcus trott homogeneous shifts ima math contr vol williams classification subshifts finite type annals mathematics vol erratum annals mathematics vol rotman introduction theory groups edition springer new york hall theory groups chelsea new york lind marcus introduction symbolic dynamics coding cambridge univ press new york keedwell latin squares applications budapest kiado jungnickel latin squares geometries groups survey coding theory design theory part vol ima volumes mathematics applications springer mackenthun groups shift structure schreier matrix algorithm annual conf information sciences systems baltimore march mackenthun group codes schreier matrix form jun mackenthun time harmonic study strongly controllable group systems group shifts group codes may mackenthun fundamental group strongly controllable group system group shift group code sept
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wayla generating images eye movements bingqing centre intelligent machines mcgill university nov james clark centre intelligent machines mcgill university clark abstract one shortcomings aforementioned image enhancement approaches consider problem generating image higher resolution using given image however many instances one even lower resolution image generate image case obviously additional information needed one approach quite popular recently train network generate image based verbal description paper look another situation wherein someone looking image want generate image similar identical one viewed assume information available viewers gaze patterns eye movement trajectories present method reconstructing images viewed observers based eye movements exploring relationships gaze patterns image stimuli looking wayla system learns synthesize images similar original pictures viewed wayla approach based conditional generative adversarial network conditional gan translation technique isola consider two specific applications first reconstructing newspaper images gaze heat maps second detailed reconstruction images containing text newspaper image reconstruction process divided two translation operations first mapping gaze heat maps image segmentations second mapping generated segmentation newspaper image validate performance approach using various evaluation metrics along human visual inspection results confirm ability network perform image generation tasks using eye tracking data immediately obvious whether enough information eye tracking data allow reconstruction viewed link image content gaze patterns stated many studies related visual attention viewers scan interesting locations given image controlling attention trajectory attention explored tracking eye movements viewers examples literature include clark ferrier computational model saliency used robotic system demonstrate relationships exist image saliency eye movements itti visual attention model also showed ability eye fixation data represent viewing behaviors image salience characteristics context attention tracking text reading researchers carried various studies developed numerous theories models order describe eyes move reading example regan demonstrated single word recognition task exists initial fixation location minimizes probability refixation total fixation duration word context introduction image generation always one primary topics field computer vision due various limitations lack source image information often happens available image datasets either insufficient quantity defective quality therefore variety image synthesis methods developed generate images useful valuable subsequent image processing tasks past years various linear nonlinear methods explored improve image quality via interpolation recently researchers focused using superresolution approaches generating detailed images uous reading extended notion preferred viewing location fixation locations depend words read mcconkie discovered optimal fixation position sentence reading slightly left word center duration influenced various factors complexity word moreover found fixation durations longest target word center grammatical factors also impact functional words prepositions often higher skipping rate sum reading behaviors fixation characteristics strongly influenced target locations various lexical variables reading material suggests particular shape structure eye movement trajectories constrain unknown extent text read similar conclusions drawn respect general image viewing eye trajectory information constrain extent contents image viewed proposed method investigate problem generating images eye movements propose use deep learning create neural network takes eye fixation data gaze heat maps generates output images similar original scenes displayed viewers call approach wayla stands looking feasibility approach enhanced presence numerous datasets images corresponding eye tracking annotations published recent years although possible construct image generation model scratch benefit deep learning models perform eye movements image prediction task propose use network based conditional generative adversarial network details presented section instead using solely idea traditional gan network generate images minimizing euclidean distance output images ground truth stimuli use architecture conditional gan way network learn generate synthetic image conditioning corresponding eye fixation heat map result goal becomes minimize difference patch combining eye fixation heat map generated image patch combining eye fixation heat map ground truth image architecture proposed wayla network uses translation model presented isola model study focuses specific application transforming image one version another however focus exploring solution generation problem mainly focused applying model two eye tracking datasets one contains various scanned images newspapers magazines associated gaze heat maps contains eye tracking data obtained text reading network takes eye fixation heat maps provided datasets input produces two sorts output images first type output simplified semantic segmentation second type output detailed photorealistic image particular network able synthesize segmented version newspaper appearances simplified version realistic newspapers consistent data filled semantically labeled regions representing picture text taking step network also effective generating images higher detail level way possible obtain concrete representation people looking reading dataset provided vilkin contains numerous highly detailed newspaper images corresponding segmented versions since database associated eye track data used saliency model mlnet generate eye fixation heat maps ground truth newspaper images heat maps used model people may look viewing newspapers serve eye fixation data used input network training also applied wayla approach another dataset named geco ghent corpus geco dataset contains eye fixation data collected reading text pages novels displayed computer screen employ conditional generative adversarial network implementation eye fixation data used condition operation generator discriminator serves distinguish fake segmentations images real ones therefore case training real examples discriminator input consists image combination eye fixation data ground truth images ground truth images newspaper images various detail levels vilkin dataset images geco dataset however case fake examples combination concatenation eye fixation data output images produced generator way training network examples described learns generate synthetic images similar ground truth images architecture training data preparation training network learn mapping eye movement data newspaper images provided vilkin dataset needed produce eye fixation data figure illustration image generation pipeline approach formulates problem generating detailed newspaper images eye fixation data process network needs fed eye fixation heat maps input trained segmented images ground truth utilize segmented images along eye fixation data train network highly detailed images ground truth figure left ground truth image generated geco dataset right corresponding eye fixation heat map generated fixation data geco dataset feed input model purpose used mlnet model build eye fixation heat map dataset obtain mlnet saliency predictions newspaper dataset fed mlnet network original scanned newspapers used vilkin used saliency model corresponding salience heat map input stimulus allows get fixation heat maps strongly agreement real gaze locations way using generated eye fixation data image datasets provided vilkin dataset train model output newspaper images various detail levels shown figure broke image generation process two phases first phase goal generating semantic segmentations newspaper images second phase used generate detailed newspaper images segmentations also investigated effectiveness model reconstructing text images used obtaining geco eye movement dataset dataset contains records eye fixation positions durations individual reading session enabled directly use eye fixation data input model therefore shown figure able generate eye fixation heat maps correspond different parts novel read participants generating grayscale eye fixation heat maps geco dataset observer time fixation made specific position specific word place bright point grayscale heat map position corresponds recorded fixation location brightness point modulated recorded percentage time spent specific location total trial time completed observer maximum value recorded percentage value percent fixation points fixation duration percentage less value represented less bright point heat maps maximum pixel value synthetic heat map corresponding fixation point duration value percent fixation occupies percent total trial time pixel value becomes possible several fixations made within one word case distinct bright points corresponding distinct fixation positions added heat map however fixation points belonging word chose modulate brightness fixation points using total percentage trial time specific word assumption global duration value useful estimating importance word compared words reading material ground truth images serve targets training network sophisticated data preparation process undertaken ground truth segmented newspaper images ground truth detailed newspaper images simply taken dataset provided vilkin regarding geco dataset chose break reading material parts generating rgb images containing printed text text image size rgb image red channel encoding constant background green channel encoding text content blue channel set zero everywhere found experimentally arrangement provided better training stability reduces possibility divergence allows faster convergence compared using single channel containing text content image contains words arranged rows row contains words one following another generating eye fixation heat maps geco dataset locations saliency points adapted locations generated images ensured correspondence agreement data provided geco dataset network training stated earlier relationships eye movements viewing material examined numerous studies giving confidence develop figure illustration conditional gan architecture used wayla image synthesis method able synthesize images eye track data similar extent images actually viewed recent years generative adversarial networks extensions led advances field image synthesis given enough training samples produce photorealistic images close ground truth pictures therefore chose build system based architecture conditional gan proven efficient way translation various computer vision tasks image segmentation image translation overall architecture based wayla system built illustrated figure input data gaze heat maps obtained aforementioned data preparation steps fed heat maps input layer neural network using architecture conditional gan described work isola modifications made published conditional gan structure presented remaining part section individual training two phases newspaper generation proposed wayla model generate image content based eye fixation data since vilkin dataset provided segmented detailed newspaper images formulated image generation task twophase process first phase consists training network synthesis second phase consists training network generate newspaper images higher level detail image segmentation section present generated newspaper images training wayla model separately independently two phases next section present slightly different approach joins two training phases yield implementation generating newspaper images training network first phase generator fed eye fixation heat maps produced using mlnet training generator figure illustration implementation trains network separately generating segmented images detailed images top part figure shows training process first phase model takes eye fixation data input trained segmented newspaper images ground truth bottom part figures shows training process second phase model takes segmented images vilkin dataset input trained detailed newspaper images ground truth timized produce outputs similar possible ground truth segmented newspaper images discriminator fed image patch concatenates input eye fixation heat maps generated images produced generator receiving kind patches discriminator trained recognize fake images real image case discriminator receives patches concatenate eye fixation heat maps ground truth segmented newspaper images illustration first phase training shown upper part figure second phase trained network synthesize detailed newspaper images based segmented newspaper images lower part figure illustrates input output settings second phase training second phase input layer generator fed segmented images provided vilkin dataset generator optimized produce outputs similar possible ground truth highly detailed newspaper images case discriminator fed image patches concatenate segmented images detailed images task distinguish synthesized data ground truth data design pipeline newspaper generation also explored possibility network learn generate highly detailed newspaper images eye fixation heat maps disposal interesting case achieve goal applied training process system started feeding input layer generator eye fixation heat maps training system generate segmented images using segmented newspapers vilkin dataset ground figure illustration implementation initially model takes eye fixation data input trained segmented newspaper images ground truth model takes input image patches concatenate eye fixation data generated segmented images trained highly detailed newspaper images ground truth truth stage operation exactly first phase training presented section however difference implementation mentioned section implementation indicated input output settings used remaining part training process finishing training system produce segmented images system fed input layer generator new kind input different segmented images used input second phase training presented section time segmented images generated generator trained previously concatenated eye fixation heat maps form new set input rgb images fed system generator takes inputs optimized output images similar possible ground truth highly detailed newspaper images provided vilkin dataset worth mentioning concatenation done way new red channel formed adding pixel values eye fixation heat maps pixels values red channel generated segmented images new blue channel formed taking pixel values blue channel generated segmented images new green channel formed setting values except locations three channels generated segmented equal case green channel pixel remains form white color along two channels design discriminator aforementioned designs receives image patches distinguishes whether belong true image pairs fake image pairs way explored feasibility generating highly detailed newspaper images eye fixation heat maps available illustration design shown figure training geco dataset also applied wayla geco dataset order investigate effectiveness model generate images based eye fixation data generator input eye fixation heat maps created geco dataset generator trained produce images ground truth text embedded images target discriminator trained distinguish generator outputs fake images fake image case discriminator receives input combination eye fixation heat maps generator outputs however real image case discriminator receives eye fixation heat maps concatenated ground truth text embedded images input loss functions used network presented following applied training phases datasets involved study discriminator whose task classify real fake pairs uses following binary cross entropy loss loss function logd logd equation input generator represents ground truth images generator target generator since stated mixing gan loss another standard content loss euclidean loss improve training deep neural networks chose use distance additional loss combine adversarial loss described construct loss function generator distance represents difference outputs generator ground truth images thus overall loss function generator defined set value decision taken based observation experiments analysis made states loss weighted time larger gan loss fewer artifacts produced output generator layers network need trained scratch weights randomly initialized using uniform distribution always preserve percent total samples testing network trained updating generator discriminator alternatively gan loss backpropagated discriminator update weights keeping discriminator weights constant combine loss loss backpropagate error update generator weights minibatches samples per batch used training rmsprop optimizer used optimize generator discriminator learning rate decay rate momentum dropout layers batch normalization used network accelerate convergence segmented newspapers detailed newspapers individual training detailed newspapers ssim table ssim scores obtained comparing synthesized images ground truth evaluation discussion among large body work done evaluating perceptual quality images losses distances dominant performance metrics domain computer vision however order address drawback basic metrics inability indicate structural information carried images chose use structural similarity ssim index compute perceived similarity generated images ground truth images way evaluation metric may provide better understanding overall quality generated images since similarity scores terms pixel values also terms image structure table similarity scores reported segmented detailed newspaper image generation computing scores generated segmented newspaper images ground truth segmented newspapers evaluate ability network prediction computing similarity scores generated detailed newspaper images ground truth detailed newspapers evaluate ability network transformation methods generating images eye movement data investigate significance contribution synthesizing newspaper images looked ssim scores reported various papers concerning image generation comparing ssim scores studies ssim scores indicate synthesized images similar ground truth images mihaela stated autoencoding gan generate images ssim mean value compared ground truth images another paper uses gan structure implement image shown generated image ssim score similar original image human perception addition zhou introduced concept ssim presented image disturbed gaussian noise ssim score noisy version still preserves texture structure characteristics original image figure example results wayla obtained training two phases separately independently qualitative results wayla testing set compared ground truth figure example results wayla obtained implementing training pipeline qualitative results wayla testing set compared ground truth still understandable recognized human inspection ssim score obtained generated detailed newspaper images using implementation situated slightly ssim scores presented table however understandable implementation trains network output highly detailed images without using ground truth segmented images synthesize images limited quality compared case two phases trained individually visually inspecting qualitative results comparing ground truth confirmed effectiveness model synthesizing real images eye data figures allow qualitatively evaluation performance image generation pipeline observed generated segmented images produce patterns visually appear close ground truth also worth noticing wayla applied figure illustration synthesized versus ground truth textembedded images along segmentation process used obtain word segment length distribution figure qualitative results obtained training wayla generate images using geco dataset figure comparison word length distribution obtained ground truth images versus word length distribution obtained generated images horizontal axis word segment length unit pixel vertical axis indicates frequency occurrence different word segment length values specific dataset case newspaper dataset trained generating segmented images network able extract meaningful information eye movement data way structure generated images fits specific dataset well taking first row figure example despite fact numerous eye fixation points eye fixation image generated segmented image part eye fixation locations converted picture areas rest eye fixation locations successfully identified text areas generated result visually inspecting detailed images generated wayla image patterns generated images easily related image patterns ground truth scanned newspapers presented original dataset evaluating performance approach applied geco dataset first provide example results generated images figure qualitative evaluation done important remember although network designed perform image generation without using natural language processing training proven extremely successful sense generated images display areas furthermore numerous valid english words observed generated images also utilized various text analysis metrics combination human inspection order evaluate quality outputs optical character recognition ocr used extract content synthetic images generated network total alphabet characters retrieved synthetic images occurrences ocr engine able convert scanned character valid alphabet letter addition presented figure also used text segmentation attempt divide text content synthesized images segments various lengths segment considered individual word system generates based input eye fixation data worth mentioning illustration purpose display qualitative results images using black text color white background color although used red green encoding method facilitate training network generated results easily converted black white afterwards order similar real image content viewed readers figure compares histogram summarizes length distribution segments obtained synthetic images versus histogram generated using ground truth images observed shape two distributions similar segments lengths ranging pixels means ground truth generated images words constructed using characters thus observe close similarity generated images ground truth images serves demonstration network ability generate highly image content conclusion paper explored possibility inverting relationships image stimuli eye movements successfully developed approach synthesize content viewed images based eye tracking information presented wayla technique utilizes conditional generative adversarial nets generate newspaper images different detail levels eye fixation heat maps moreover system able reconstruct images containing text using solely eye tracking data work proved feasibility generating image content gaze data never demonstrated results showed deep convolutional neural network used effective performing image generation tasks achieving high similarity generated results ground truth although improvements could achieved enhance quality generated images idea inverting path starting viewed content ending eye tracking information widely applied various settings example many previous works use baysian inference standard approach infer visual task eye movements proposed wayla approach used improve performance traditional methods furthermore previously studies related image generation using generative adversarial networks derivatives done without considering gaze information input introducing possibility inferring image content based gaze data wayla approach could open doors diversified image generation models future references boulogne warner neil yager image processing python carpenter visual origins ocular motility vision visual function clark ferrier modal control attentive vision system iccv pages cop dirix drieghe duyck presenting geco eyetracking corpus monolingual bilingual sentence reading behavior research methods cornia baraldi serra cucchiara deep network saliency prediction pattern recognition icpr international conference pages ieee farsiu robinson elad milanfar fast robust multiframe super resolution ieee transactions image processing goodfellow mirza ozair courville bengio generative adversarial nets advances neural information processing systems pages hou andrews cubic splines image interpolation digital filtering ieee transactions acoustics speech signal processing isola zhu zhou efros translation conditional adversarial networks arxiv preprint itti koch niebur model visual attention rapid scene analysis ieee transactions pattern analysis machine intelligence ledig theis caballero cunningham acosta aitken tejani totz wang shi single image using generative adversarial network arxiv preprint liu pan yang learning recursive filters vision via hybrid neural network european conference computer vision pages springer mcconkie kerr reddix zola eye movement control reading location initial eye fixations words vision research mirza osindero conditional generative adversarial nets arxiv preprint regan strategy tactics word recognition reading pathak krahenbuhl donahue darrell efros context encoders feature learning inpainting proceedings ieee conference computer vision pattern recognition pages radford metz chintala unsupervised representation learning deep convolutional generative adversarial networks arxiv preprint rosca lakshminarayanan mohamed variational approaches autoencoding generative adversarial networks arxiv preprint smith overview tesseract ocr engine document analysis recognition icdar ninth international conference volume pages ieee unser splines perfect fit signal image processing ieee signal processing magazine vilkin safonov egorova algorithm segmentation documents based texture features pattern recognition image analysis vitu mcconkie kerr regan fixation location effects fixation durations reading inverted optimal viewing position effect vision research wang bovik mean squared error love leave new look signal fidelity measures ieee signal processing magazine winkler subramanian overview eye tracking datasets quality multimedia experience qomex fifth international workshop pages ieee wotschack eye movements reading strategies reading strategies modulate effects distributed processing oculomotor control volume potsdam zhang image interpolation algorithm via directional filtering data fusion ieee transactions image processing
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continuous circadian phase estimation using adaptive notch filter wei kyle agung bernard john smart lighting engineering research center rensselaer polytechnic institute troy biology department skidmore college saratoga mar abstract actigraphy widely used analysis circadian rhythm current practice applies regression analysis data multiple days estimate circadian phase paper presents filtering method online processing biometric data estimate circadian phase apply proposed method actigraphy data fruit flies drosophila melanogaster ntroduction circadian rhythms unique features life terrestrial species evolved response cyclic environment earth surface reflects natural light cycle example humans molecular oscillations biological clock drive rhythmic physiological behavioral functions heart beat blood pressure body temperature cycle metabolism locomotor activity disruption circadian rhythm caused lack synchrony circadian clock brain external environment may produce serious detriments human health result various issues ranging increased sleepiness decreased attention span lower productivity health problems increased risk cancer diabetes obesity cardiovascular disorders disruption may often observed people irregular sleep patterns artificial deprivation light submariners mine workers frequently shifted cycles night nurses travelers cross multiple time zones therefore control circadian rhythm great importance circadian research among many organisms drosophila widely studied investigation circadian rhythms since starting frontier work pittendrigh pittendrigh studies mostly focused eclosion rhythms drosophila emergence larva pupa case recent decades genetic analysis based biomolecular interaction studies attracted many researchers investigate circadian rhythm drosophila experimentally led unveiling molecular interactions per tim proteins mrna based drosophila circadian models proposed however high cost complexity protocols prohibit continuous measurement circadian rhythm outputs thus another behavioral genotype locomotor activity often chosen output circadian clock although accurate genetic analysis advantages locomotor activity based circadian research continuous measurement integration fully automated control measuring system different types drosophila activity recording system developed past decades widely used method commercial trikinetics enables fully automated recording system traditional method circadian phase estimation mostly based phase extracted onset time acrophase peak fitted cosine curve line despite inaccurate nature method phase estimation daily basis provide continuous phase estimation would obstacle many practices obtaining phase response curve especially feedback control problems relies circadian phase problem paper adopt circadian phase estimation scheme using modified adaptive notch filter anf developed group previously inspired design set experiments obtain phase response curve anf continuously estimate circadian phase based locomotor activity drosophila tool enables continuous circadian phase estimation vital importance circadian related problems especially big step towards closed loop circadian rhythm control drosophila helpful tool complex study human circadian rhythm show paper anf method estimates circadian phase accurately continuously also offers filtering effects successfully estimate circadian phase set data traditional method fails paper organized follows preliminary results circadian phase estimation anf introduced section experiment setup protocols detailed presented section iii experiment design obtaining prc introduced section followed results discussion section concluding remarks given section reliminaries section drosophila locomotor activity affected external light discussed preliminary results circadian phase estimation introduced external light drosophila locomotor activity similar humans drosophila circadian system sensitive blue light almost completely insensitive red light cry cryptochromes drosophila deep brain circadian photorecepter regulates impact light circadian system circadian pacemaker affected cry regulates expression neuropeptide pdf affects locomotor activity therefore drosophila locomotor activity used prediction circadian phase however activity solely controlled circadian pacemaker also affected visual pathway light perception compound eye eye structures drosophila observed experiments locomotor activity peaks right light turns even red light continued adaptive notch filter locomotor activity based circadian rhythm estimation measuring circadian phase experimentally real time challenging reliable method genetic analysis drosophila onset melatonin secretion dim light conditions dim light melatonin onset dlmo human lab test saliva plasma samples however measurements inconvenient time consuming expensive physiological signals activity heart rate body surface temperature possible measured high sampling rate show rhythmic patterns suitable real time circadian phase estimation therefore paper focus drosophila locomotor activity circadian clock output although simple easy measure signal usually noisy corrupted therefore filter required extract circadian phase noisy physiological measurements circadian phase estimation scheme using modified adaptive notch filter anf developed group previously inspired also accommodate mean waveform circadian signal introduced follows assume following form circadian signal sin constant bias white noise proposed modified anf given aanf banf fanf parameters tuning procedures found anf produces estimate argument fundamental harmonic used circadian phase estimate cited work local stability robustness properties modified anf algorithm established effectiveness demonstrated synthesized periodic signals drosophila activity data example shown figure fig example anf algorithm estimating circadian phase drosophila activity data top drosophila activity black anf estimation red middle top periodic blue light middle bottom anf extracts estimated signal argument bottom estimated phase obtained subtracting linearly increasing part argument iii xperiment etup rotocol materials drosophila culture kits carolina biological supply company used breeding preparation experiments kit includes culture vials flies bred stored instant drosophila medium food sustains flies culture vials sorting brushes flynap anesthetic kit also carolina used sedation flies kit includes vial flynap anesthetic several anesthetic transfer containers chemicals application flynap anesthetic mixture triethylamine methanol neutralizer fragrance ethanol flynap poured transfer container enough anesthetize flies least minutes flies canton wild type flies selected inclusion experiment undergone eclosion three days experiment began experimental preparation experimental flies flies incubator placed tubes one tube per fly contained sufficient food experiments duration one side food standard agar food contain pymetrozine tubes loaded nine separate trikinetics drosophila activity monitor dam activity monitors temperature kept degree celsius using temperature control system equipment trikinetics drosophila activity monitor dam monitors shine beam center tube count number times beam broken fly order calculate many times fly move along tube monitors sum count data minute bins send data culturing incubators culture flies placed incubators philips rebel leds cool white light light dark cycles experiment incubators dam activity monitors placed incubators philips rebel leds different wavelengths light sources mirrors mounted inner surface incubators positions leds optimized optical simulator zemax ensure light uniformity monitor temperature control system high pressure air humidifier used ventilation keeping food drying temperature sensors heaters mounted incubator maintain temperature light stimulus hrs light strategy dark table blue light stimulus designated circadian time incubator degree celsius light control system leds different wavelengths mounted top incubators controlled designated light patterns controllable light intensity software use matlab data handling analysis implementation algorithms actogramj used obtain acrophase raw data order compare results based algorithm xperimental esign data nalysis section experiment designed order obtain prc drosophila string using anf method well traditional based method experimental design experiment lab securely locked prevent external stimulus except designated led light noise ceiling light affect results experiment designed flies entrained phase external light stimulus applied entrainment phase order investigate influence light stimulus circadian phase need ensure flies entrained phase given light stimulus therefore three days imposed incubators loaded flies light stimulus three days entrainment incubators put darkness light stimulus programmed designated circadian time lux blue light duration hour table data analysis raw data form text files transferred stores many times fly move along tube within minute data logging time light intensity information data points channel fly total channels raw data analyzed following manner exclude arrhythmic flies use periodogram analysis detect rhythmicity fly examine activity plot exclude inactive flies hands detect average flies lowest incubator show approximately rhythmic used next step obtain phase response curve utilize anf estimate circadian phase incubator screened data analyzed using anf algorithm estimated circadian phase used determine phase shift comparison traditional method using cosinor double actogram determine phase shift acrophase peak cosine wave fitted raw data used comparison determining phase shift actogramj software beginning subjective night defined circadian phase amplitude amplitude period activity activity time min fly passed periodogram analysis derived activity records individual flies complete darkness top panel shows periodogram bottom panel shows raw activity records period time min fly failed periodogram analysis derived activity records individual flies complete darkness top panel shows periodogram bottom panel shows raw activity records fig periodogram analysis esults experiments designed paper standard widely adopted discussed literature intention paper demonstrate new circadian phase estimation tool anf accurate continuous robust noise first demonstrate validity experiment prc obtained using anf compares results literature well traditional method based prc acrophase finally demonstrate filtering effect anf noise added raw data anf successfully rejects noise traditional method shows large error periodogram exclusion arrhythmic flies raw activity data individual fly days entrainment designated light stimulus table obtained analyzed order exclude arrhythmic inactive flies periodogram first applied exclude flies irregular period data sets analyzed periodogram activity records individual flies day day flies complete darkness example shown figure figure flies regular period irregular period shown respectively averaged inclusion ratio incubator comparison anf acrophase based phase estimation excluding arrhythmic flies use selected data set obtain phase response curve raw activity data selected flies incubator first averaged analyzed anf algorithm section determine phase averaged flies incubator results two incubators shown figure figure days entrainment followed free running light pulse respectively top panel shows raw activity black anf estimation red panel shows light pattern bottom panel shows anf estimation circadian phase notice anf provides continuous period adaptation circadian phase estimation useful tool circadian related research question remains accurate anf based method compare anf based method traditional based method using data set traditional method determining circadian phase mostly based daily peak activity identified using eye fit acrophase peak cosine fitted curve use actogramj software obtain acrophase based raw activity data selected flies obtained section control incubator shown figure blue triangles estimated daily periodogram nonparametric estimate power spectral density psd estimate signal uses traditional fourier techniques fast fourier transform fft incubator days free run incubator days light pulse lux intensity duration fig anf analysis averaged data incubator top panel shows raw activity bold black standard deviation light black anf estimation red panel shows light pattern bottom panel shows anf period adaptation double actogram acrophase estimation using actogramj blue triangles estimated daily acrophase blue lines regression fit periodogram last light using actogramj fig obtaining periodogram acrophase based double actogram using actogramj days free run acrophase regression fitted line periodogram day shown figure notice period anf based period estimation trying adapt shown figure bottom panel analysis methods ready move obtaining phase response curve note circadian phase relative concept especially obtaining prc period normally exactly hrs therefore control incubator days entrainment followed used reference incubator figure agrees standard analysis literature phase shift defined phase period another control incubator difference target incubator control incubator analysis applied anf actogramj based method two prcs methods shown figure based phase difference target incubator control incubator day light stimulus given table notice results almost identical demonstrating accuracy anf based method meets standard also compare results obtained literature black stars figure anf obtains similar results compared accepted literature actoj anf errorbar literature fig actogramj based prc red dots interpolated red curve prc obtained anf blue dots interpolated blue curve prc obtained shown black stars filtering effect anf validity anf based circadian phase estimation demonstrated result almost identical obtained based method matches prc obtained researchers well words anf meets accepted standard literature also provides continuous circadian phase estimation yet another important feature anf robustness higher order harmonics noise order demonstrate feature intentionally corrupt raw activity data selected section gaussian white noise variance var two examples shown figure procedure applied obtain anf based prc based prc actojgramj result shown figure notice anf based method successfully rejects noise effect error original prc small average error hrs based method shows large error average error hrs onclusion acknowledgment work supported primarily army research office grant number also gratefully acknowledge support national science foundation nsf smart lighting engineering research center center automation technologies systems cats block grant new york state empire state development division science technology innovation nystar contract authors would also like thank drs mark rea mariana figueiro andrew bierman rpi light research center introducing topic authors standard deviation anf based method individual fly incubator shown error bar calculated square root sum variance target control incubator incubator days free run incubator days light pulse lux intensity duration fig anf analysis corrupted data noise incubator top panel shows raw activity bold black standard deviation light black anf estimation red panel shows light pattern bottom panel shows anf estimation circadian phase actoj noncorrupted data actoj corrupted data anf noncorrupted data anf corrupted data fig prc obtained corrupted data noise anf blue dots interpolated blue curve actogramj red dots interpolated red curve original based data prcs shown dotted curves eferences kripke mullaney atkinson wolf circadian rhythm disorders biological psychiatry knutsson health disorders shift workers occupational medicine sephton spiegel circadian disruption cancer pathway stress disease brain behavior immunity stevens circadian disruption breast cancer melatonin clock genes epidemiology rea bierman figueiro bullough new approach understanding impact circadian disruption human health journal circadian rhythm kelly neri grill ryman hunt dijk shanahan czeisler nonentrained circadian rhythms melatonin submariners scheduled day journal biological rhythms mills circadian rhythms three months solitude underground journal physiology harrington location location location important circadian loops journal clinical investigation july leloup gonze goldbeter limit cycle models circadian rhythms based transcriptional regulation drosophila neurospora biological rhythms zhang wen julius adaptive circadian rhythm estimator application circadian rhythm control american control conference washington june hsu ortega damm globally convergent frequency estimator ieee trans automat control frank zimmerman action spectra phase shifts circadian rhythm drosophila science klarsfeld leloup rouyer circadian rhythms locomotor activity drosophila behavioural processes emery stanewsky forster drosophila cry deep brain circadian photoreceptor neuron schmid yoshii new imagej actogramj chronobiological analysis journal biological rhythms
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magic universal quantum computing permutations apr michel abstract role permutation gates universal quantum computing investigated magic computation clarified permutation gates eigenstates wootters discrete wigner function contextuality following many contributions subject first classification resulting magic states low dimensions performed pacs msc codes introduction quantum physics universal computing considered deeply interrelated relatively new idea science owes much feynman deutsch shor bravyi kitaev mention popular landmarks time line quantum computing also includes important marks many last two decades dedicated quantum computing platforms feynman already understood simulation quantum system classical computer would need exponential resources later deutsch proposed universal quantum computer made quantum gates way simulating quantum system polynomial overhead know shor quantum computer able factor large integers polynomial time exploiting parallelism implementation quantum fourier transform quantum error correction may circumvent undesirable effects due decoherence use quantum codes bravyi kitaev introduced principle magic state distillation universal quantum computation may realized thanks stabilizer formalism clifford group unitaries preparations measurements ability prepare ancilla appropriate single qubit mixed state within frame universal quantum computation based stabilizer formalism actively discussed exist critical let mention intel ntt spin qubits semiconductors google ibm superconducting qubits lockheed infineon trapped ions hardware efforts accomplished university laboratories linear optics atoms cavity qed quantum dots impurity spins solids etc michel resources responsible power quantum computation remarkably odd dimensions contextuality magic states seems magic ingredient addition contextuality states witnessed negative entries distribution discrete wigner function dwt even dimensions situation obscure since contextuality correlated negativity dwt occurs stabilizer states filtering quantum states involved computation based full state tomography proposed finally according contextuality required measurement procedures preparation procedures well order two notions nonclassicality revealed equivalent wigner function recognized contain permutation symmetry structure interestingly enough experimental implementation simple quantum algorithm determining parity permutation performed claim paper permutation symmetry sometimes considered lying beneath concepts magic states contextuality responsible universality efficiency quantum computation permutation gates clifford group whose important elements cnot gate toffoli ccnot gate reveal states eigenvectors focus contribution use permutation groups seen sets permutation gates source quantum states stabilizer magic arising eigenstates groups interest general different ones encountered previous investigation theorem contextuality involved quantum computational universality magic state distillation quantum codes possibly computational already emphasized sec remind standard results generalized pauli group operators wootters discrete wigner function link quantum contextuality sec derive several types magic states arising permutation matrices small dimensions explicit wigner function obtain proofs contextuality based existence pentagons appropriate states stabilizer magic sec describes open vistas study permutations quantum gates unusual recognize relationship permutations quantum gates intend work permutation admits binary matrix representation exactly one entry row column elsewhere magic permutation matrices showing one magic universal quantum computing permutations entry main diagonals well known permutation pauli gate acts one two three qubits respectively similarly permutation gates may act qudits shift gate acting qutrits sec focus magic groups generated two magic permutation gates exist soon sec generalized pauli group discrete wigner function contextuality remind standard tools useful calculations discussion generalized pauli group let prime number qudit pauli group generated shift clock operators follows mod exp dth root unity dimension pauli spin matrices general pauli also called operator form ijm particules one takes kronecker product qudit elements times stabilizer states defined eigenstates pauli group discrete wigner function associated hilbert space prime discrete phase space array points set phase point operators discrete phase space defined see also wootters relations follow phase point operators built satisfy properties analogous continuous phase space context continuous wigner function exp ipx position momentum density matrix particle coordinate pure state wave function required operators satisfy point hermitian michel two points iii taking complete set parallel lines called striation construct average line operators form set mutually orthogonal projectors sum identity operator phase point operators linearly independent form basis space hermitian operators acting hilbert space result density operator developed real coefficients explicitely given wootters discrete wigner function unlike continuous wigner function discrete wigner function quasi probability distribution may take negative values shown hilbert space odd dimension pure states possess discrete wigner function stabilizer states contrary pure state called magic state definition follows corollary establishes pure state one six pauli eigenstates together clifford group operations pauli eigenstate preparation measurement allows universal quantum computation arbitrary prime dimensions magic state distillation investigated multiple qubits see corollary pure state stabilizer state allows universal quantum computation quantum contextuality quantum contextuality forbids theories revealing values observables test specific experimental measuring observables taken account one way characterize quantum contextuality use approach involving sets quantum observables mermin square special subsets eigenstates shared mutually commuting operators smallest proof contextuality kochenspecker set needs rays hilbert space contextuality obtained within stabilizer formalism multiple qubits manifested qudits starting point proof quantum contextuality consists set binary tests represented rank one projectors hvi tests compatible magic universal quantum computing permutations projectors mutually orthogonal tests associated exclusivity graph wherein vertex corresponds projectors edge corresponds compatible projectors witness operator defined follows required value assigned one projector joint measurement compatible observables located selected edge hidden variable theory one expects results tests imax independence number graph cardinality largest set vertices two elements connected edge quantum contextual theory may bypass bound imax upper bound lovasz number exclusivity graph calculated maximum taken unit vectors simplest proof quantum contextuality corresponds cyclic graph also called pentagon originally obtained quantum system qutrit exclusivity graph allows proof quantum contextuality imax every state proof contextuality summarize graphs may considered proof quantum contextuality appropriately chosen projectors state following inequality violated contextuality needed universality quantum computational qudits odd dimension states lying outside stabilizer polytope manifest negativity wigner function simultaneously violate inequality appropriate twoqudit projectors hence display contextuality theorem thus answer yes even dimensions pure states single multiple qubits magic neither condition negativity criterion contextuality sufficient promote states computational universality exist well characterized sets however shown recent paper contextuality needed whenever two conditions satisfied contextuality one retains filtered set quantum states able ensure full state tomography michel state eigenstate table wootters discrete wigner function pure states magic states quantum contextuality groups permutation gates already know permutation symmetry exists discrete wigner function goal section shift attention set clifford gates subset whose elements permutation gates leading either stabilizer states classes magic states related contextuality investigated mind tools described sec restricting existence pentagons appropriate states dimensions larger eigenstates permutation gates consideration living field may different cyclotomic field exp paper restrict classification magic states entries corresponding eigenvalues qubit magic states fault tolerant quantum computing protocols based stabilizer states complemented magic states reach quantum universality two distillation protocols based single qubit magic states first described cos sin cos exp sin cos magic state hadamard matrix belongs single qubit clifford group magic state matrix phase exp gate also belongs clifford group stabilizer states prepared actions clifford group distilled actions dirty mixed state neat pure state thanks appropriate quantum codes matrix elements wootters discrete wigner function single qubit pure states shown table first three rows correspond pauli spin matrices last two magic universal quantum computing permutations rows magic states magic state possesses negative entry unlike magic state latter case negativity wigner function coincide universality magic states qutrit permutation gates smallest dimension occurrence magic states associated permutations three isomorphism two two distinct generators permutation groups three letters permutation group isomorphic contains permutation matrices pauli group shift matrix eigenstates mutually orthogonal stabilizer states third root unity figure triples mutually orthogonal rays arising qutrit permutation gates three pentagons originating rays big black bullets stabilizer states small bullets magic states permutation group isomorphic symmetric group exists three copies one generated contains elements found three extra ones lie pauli group parts clifford group mutually orthogonal triples eigenstates pictured fig apart stabilizer states magic states type identified norrell states strange states michel respectively wigner function expected magic states contains negative entries wigner matrix fig shown three pentagons part orthogonality relations taking rank associated exclusivity graph attached pentagon also pentagon allows proof contextuality one observes magic states strange type involved magic states permutation gates restrict permutation groups whose two generators magic gates happens two groups isomorphic alternating group one copy follows figure maximum cliques orthogonal rays permutation gates thin lines triples thick lines exp missing coordinates straightforward recover maximum cliques orthogonal rays permutation gates thin lines thick lines big black bullets stabilizer states small black bullets magic states magic universal quantum computing permutations looking joined eigenstates shared least two commuting gates set states derived whose orthogonality graph pictured fig wigner functions magic states two types types wigner matrices contain negative entries qutrit case particular case stabilizer states selected permutation gates shown orthogonality graph contains pentagons thus proofs contextuality surprisingly vertices pentagons either stabilizer states magic states second type analogous qutrit strange states magic states permutation gates restrict restrict permutation groups whose two generators magic gates happens permutation groups isomorphic semidirect product symmetric group respectively first case taking triples mutually permutation gates one gets set eigenstates organized maximum cliques shown fig magic missing coordinates cyclotomic field exp symbol means symbol means means group one gets eigenstates shared maximum cliques mutually compatible gates show organization eigenstates mention one arrives types magic states form contextuality may revealed pentagons built states magic states permutation gates smallest permutation group generated two magic permutation gates michel dimension alternating group exist maximum cliques mutually compatible permutation gates whose size giving rise shared eigenstates stabilizer magic defined cyclotomic field exp expected even dimensions negativity entries wigner function sign magicity state stabilizer state one gets representative magic state wigner function notation means means means another permutation group generated two magic permutation gates alternating group giving rise magic states one gets notation means means magic states permutation gates smallest permutation group generated two magic permutation gates dimension isomorphic representative magic state wigner function cos cos cos positive negative sign front matrix entries negative entry wigner function next another permutation group generated two magic gates isomorphic order one finds three types magic magic universal quantum computing permutations states whose entries wigner functions follows higher dimensions table summarizes magic states found permutation gates small dimensions dimensions magic states entries considered column provides sum negative entries wigner matrix shown absolute value sum negative entries discrete wigner matrix computable magic monotone quantum computing resource increase stabilizer operations similarly mana log easily computable additive magic monotone according theorem stabilizer protocol succeeds probabilistically produce copies target state least copies state average conclusion described leading role played permutations shaping type universal quantum computation based magic states living outside stabilizer polytope defined generalized pauli group sec derived main magic states defined eigenstates gates magic permutation groups subset clifford group small dimensions explicitly computed sum negativity discrete wigner matrix estimating value magic state resource universal quantum computation observed state dependent quantum contextuality building block pentagons occurs dimensions appropriate sets stabilizer magic states desirable extend calculations higher dimensions check possible asymptotic trend dimension amount magicity contextuality clarify relation magic permutation groups unitary singled finally relate derived magic states distillation procedures error correcting codes later dim michel magic state sum negative entries positive cos cos cos cos cos cos cos cos cos remark norrell strange table magic states column sum negative entries wigner matrix column column provides permutation group consideration group isomorphic eight element dihedral group paper devoted povms obtained pauli group action magic states references feynman simulating physics computers int theor phys deutsch quantum theory principle universal quantum computer proceedings royal society london nielsen chuang quantum computation quantum information cambridge university press bravyi kitaev universal quantum computation ideal clifford gates noisy ancillas phys rev quantum information processing communication strategic report current status visions goals research europe june also zoller eur phys magic universal quantum computing permutations quantum information science technology roadmap part quantum computation report quantum information science technology experts panel available gottesman theory fault tolerant quantum computation phys rev howard wakkman veitch emerson nature delfosse browne okay raussendorf contextuality resource qubit quantum computation preprint spekkens negativity contextuality equivalent notions nonclassicality prl zhu permutation symmetry determines discrete wigner function phys rev lett gedik silva karpat vidoto soarespinto deazevedo fanchini computational single qudit sci planat geometry contextuality grothendieck coset space quantum information processing wootters formulation quantum mechanics ann phys ferrie representations quantum theory applications quantum information science prog phys mann mello revzen family transforms discrete variables defined hilbert space quant stud math found press gross hudson theorem finite dimensional quantum systems math phys reichardt quantum universality magic states distillation applied css codes quant inf proc meier eastin knill distillation fourqubit code quant inf comp campbell anwar browne distillation prime dimensions using quantum codes phys rev howard vala qudit versions gate phys rev veitch ferrie gross emerson negative resource quantum computation new phys planat small proofs theorem two three four qubits eur phys plus cabello kleinmann portillo quantum contextuality requires rays phys math theor cabello severini winter contextuality physical theories axiom preprint klyachko binicioglu shumovsky simple test hidden variables systems prl cormick galvao gottesman paz pittenger classicality discrete wigner function phys rev veitch mousavian gottesman emerson resource theory stabilizer computation new phys michel planat gedik magic informationally complete povms permutations preprint cnrs avenue des montboucons france address nehru centre advanced scientific research jakkur bengaluru india address mrhaq
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preprint submitted energy buildings november pseudo dynamic transitional modeling building heating energy demand using artificial neural network subodh paudel mohamed elmtiri wil kling olivier corre bruno department energy system environment ecole des mines nantes gepea cnrs umr france environnement recherche innovation veolia france department electrical engineering technische universiteit eindhoven netherlands corresponding author tel abstract paper presents building heating demand prediction model occupancy profile operational heating power level characteristics short time horizon couple days using artificial neural network addition novel pseudo dynamic transitional model introduced consider time dependent attributes operational power level characteristics effect overall model performance outlined pseudo dynamic model applied case study french institution building compared results static pseudo dynamic neural network models results show coefficients correlation static pseudo dynamic neural network model energy consumption error learning phase prediction phase respectively orthogonal array design applied pseudo dynamic model check schedule occupancy profile operational heating power level characteristics results show new schedule provide robust design pseudo dynamic model due prediction short time horizon finds application energy services company escos manage heating load dynamic control heat production system keywords building energy prediction short term building energy forecasting operational heating characteristics occupancy profile artificial neural network orthogonal arrays introduction global concerns climate change regulation energy emissions drawn attention towards researchers industries design implementation energy systems low energy buildings according iea statistics total energy use globally accounts around mtoe mega tonnes oil equivalents residential commercial buildings consume final energy use world european countries consume energy towards thermal comfort buildings small deviations design parameters buildings could bring large adverse effect energy efficiency additionally results huge emissions buildings estimated improvement energy efficiency buildings european union result saving least billion euro annually research active driving towards energy buildings order accomplish ensure thermal comfort essential know energy flows energy demand buildings control heating cooling energy production plant systems energy demand building system thus depends physical geometrical parameters buildings operational characteristics heating cooling energy plant systems weather conditions appliances characteristics internal gains various approaches predict building energy demand based physical methods methods statistical regression methods artificial intelligence methods mentioned zhao physical methods based physical engineering methods uses thermodynamics heat transfer characteristics determine energy demand building numerous physical simulation tools developed energyplus ibpt simbad trnsys carnot compute building energy demand simplified physical model based physical geometrical climatic occupant model presented duanmu bridge complexities collecting physical data required simulation tools possible approaches building energy prediction models like response factor method transfer function method frequency analysis method lumped method though methodologies adapted estimate energy demand buildings different physical models highly parameterized addition physical parameters buildings always known even sometimes data missing also models computationally expensive energy services company escos manage heating cooling loads control applications approaches predict building energy demand limited physical parameters methods strongly dependent measurements historical data statistical regression methods seem feasible predict building energy demand limited physical parameters statistical approaches widely used girardin determine best model parameters fitting actual data different approaches physical behaviour characteristics based statistical data presented yao bridge gap statistical methods work statistical daily load profile grounded energy consumption per capita human behaviour factor method based thermal resistance capacitance network nevertheless statistical models used linear characteristics input output variables evaluate building parameters adapted energy demand behavior regression models also used predict energy demand accurate enough represent short term horizon couple days hourly couple minutes sampling time energy demand prediction order find best fitting actual data kind models requires significant effort time recent years growth research work field artificial intelligence like artificial neural network support vector machines methods known solving complex function energy demand models limited physical parameters neural network method shown better performances physical statistical regression methods authors used static neural network predict energy demand building compared results physical models instance kalogirou used climate variables mean maximum solar radiation wind speed parameters wall roof type coupled artificial neural network ann predict daily heating cooling load buildings work results obtained using ann similar given physical modelling tool trnsys neto presented comparison neural network approach physical simulation tool energyplus work authors used climate variables external dry temperature relative humidity solar radiation input variables predict daily consumption building results showed neural network slightly accurate energyplus comparing real data static neural network model proposed shilin consider climate variables dry bulb temperature information regarding schedule holiday predict cooling power residential buildings dong used support vector machine svm predict monthly building energy consumption using dry bulb temperature relative humidity global solar radiation performance svm neural network model wee compared results show svm better neural network prediction various authors performed hourly building energy prediction using ann mihalakakou performed hourly prediction residential buildings solar radiation multiple delays air temperature predictions input variables ekici used building parameters window transmittivity building orientation insulation thickness dombayci used time series information hour day month energy consumption previous hour predict hourly heating energy consumptions gonzalez used time series information hour day current energy consumption predicted values temperature input variables predict hourly energy consumption building system popescu used climate variables solar radiation wind speed outside temperature previous hours variables mass flow rate hot water previous hours hot water temperature exit plant system predict space hourly heat consumptions buildings used svm predict hourly cooling load office building using climate variables solar radiation humidity outdoor temperature work svm compared static neural network result showed svm better static neural network terms model performance dynamic neural network method includes time dependence presented kato predict heating load district heating cooling system based maximum minimum air temperature kalogirou used jordan elman recurrent dynamic network predict energy consumption passive solar building system based seasonal information masonry thickness thermal insulation many authors occupancy profile significant impact building energy consumption sun mentioned occupancy profile period significant impact initial temperature requirement building morning work reference day targeted day prediction depends previous day beginning following day based occupancy profile period calculated based occupancy profile period addition value correlated weather data prediction errors previous hours used input variables predict hourly cooling load yun used arx autoregressive exogeneous external inputs time temperature indexed model occupancy profile predict hourly heating cooling load building system compared results given neural network results showed occupancy profile significant contribution determination auto regressive terms different intervals time showed variation building heating cooling energy consumption proposed arx model showed similar performance neural network sensitivity analysis heating cooling hot water equipment lighting energy consumption based occupancy profile performed azar different sizes office buildings work found heating energy consumption highest sensitivity compared cooling hot water equipment lighting energy consumption small size buildings also results showed heating energy consumption highly influenced occupancy profile medium small buildings occupancy period moreover literatures focused operational power level characteristics schedule heating cooling energy manage energy production plant system example leung used climate variables operational characteristics electrical power demand power information lighting office equipment implicitly depends occupancy schedule electrical power demand predict hourly daily building cooling load using neural network conclusion reiterated physical models though give precise prediction building energy highly parameterized computationally expensive manage energy control applications escos methods depend measurement historical data effective early stage building operation construction since measurement data available stages building energy data available methods considered measurement data accurate reliable kind models sensitive quality measured data sensitivity accuracy data driven models thus depends measurement data models based statistical regression methods precisely represent short time horizon couple days hourly couple minutes sampling time prediction though perform prediction energy consumptions buildings limited physical parameters also require significant efforts time compute best fitting actual data static neural network models used daily prediction used hourly prediction buildings energy consumptions though dynamic neural network model gives better precision compared static neural network consider occupancy profile operational power level characteristics plant system therefore adapted escos manage energy production control applications important features like transition time dependent attributes operational power level characteristics plant system still missing though authors consider occupancy profile author considers operational characteristics electrical power demand detailed variables application models developed literature reviews summarized table table summary variables application models literature input variable model climate variables author year type model outside tempeature inner temperature ambient dry bulb wet bulb girardin yao catalina wan statistical thermal statistical regression horizon forecast annually daily regression static mihalakakou ekici dombayci gonzalez popescu kato kalogirou static static static static static dynamic dynamic svm regression autoregressive exogeneous monthly monthly yearly monthly daily daily daily static physical occupancy operational profile characteristics parameters shilin leung duanmu relative humidity svm static static yun wind speed dong kalogirou neto sun global solar radiation hourly hourly hourly hourly hourly hourly hourly hourly hourly hourly hourly daily hourly type applications buildings residential heating cooling residential space heating residential office heating cooling buildings total energy consumptions buildings heating cooling office residential cooling power residential heating energy buildings residential heating energy electrical load buildings district heating energy passive solar buildings office building library cooling load high rise buildings small building heating load office space electrical power demand cooling load buildings remarks nominal temperature heating cooling hot water system threshold heating cooling temperature appliances model climate index based principal component multiple lag output predictions ambient air temperature transmittivity orientation insulation thickness heating degree hour method predict value temperature present electricity load hour day outside temperature mass flow rate previous hour hot water temperature highest lowest open air temperature season insulation wall thickness heat transfer coefficient multiple lag dry bulb temperature solar radiation reference day day based occupancy schedule correlated weather data based reference day accuracy calibrated prediction error previous hours occupancy profile represented space electrical power demand clearness sky rainfall cloudiness conditions physical geometrical parameters hourly cooling load factor none studies evaluated transition time dependent effects operational power level characteristics heating plant system predicted building heating energy demand short time horizon couple days short term prediction important escos dynamic control heat plant system paper bridges gap static dynamic neural network methods occupancy profile operational power level characteristics heating plant system introduces novel pseudo dynamic model incorporates time dependent attributes operational power level characteristics effects neural network model performances compared static neural network building heating demand orthogonal arrays applied proposed pseudo dynamic model robust design confirmed new schedule occupancy profile operational heating power level characteristics obtained escos proposed method allows short term horizon prediction around days sampling interval minutes make decision management wood power plant escos next section describes methodology including scope study design transitional pseudo dynamic characteristics neural network model orthogonal arrays finally case study presented results discussion drawn analyze performance different static pseudo dynamic models along robustness proposed pseudo dynamic model heating demand prediction building methodology collection climate data collection building heating energy data operational heating power level characteristics settling steady state time control system transitional pseudo dynamic model occupancy profiles artificial neural network static pseudo dynamic model heating demand prediction development implementation models proposed work based collection real building heating demand operational heating power level characteristics climate variables approximated occupancy profile data see appendix selection relevant input variables outline methodology presented paper shown figure input methodology form climate building heating energy data inputs data occupancy profile operational heating power level characteristics working hours dynamics building heating demand also input methodology includes settling steady state time estimated real building data based operational heating power level dynamics building characteristics transitional pseudo dynamic models designed finally neural networks static pseudo dynamic models designed predict heating demand short time horizon couple days robustness pseudo dynamic model occupancy profile operational power level characteristics analyzed different time intervals confirm occupancy schedule profile operation plant system orthogonal arrays pseudo dynamic model optimum orthogonal arrays design used final prediction building heating demand scope study details transitional pseudo dynamic model neural network model orthogonal arrays described section figure outline proposed methodology heating demand prediction scope study scope paper heating demand prediction short time horizon large building overall objective make energy services decisions management wood power plant escos assumptions carried study highlighted winter period studied existing building considered space heating demand building fed heat network central substation domestic hot water dhw scope heating demand data recorded data acquisition system database thermal comfort inside building performed database thus effects ventilation heating already included database simple occupancy profile building anticipated approximately assist escos schedule heat production system system individual occupant behavior precise occupancy profile considered thus modeling constraints closer operational condition escos estimate heat demand wind speed direction taken consideration due fact present weather variables data taken data acquisition system future weather variables values coming atmospheric modeling system mesh size arpege see aladin see arome see case wind impact heating demand prediction specific building located inside mesh difficult even impossible consider precise effect heating energy demand highly dependent outside temperature climate variables less significant impact heat energy transitional pseudo dynamic model operational heating power level characteristics gives operational features plant system however give abstract information transition attributes operational heating power level illustrated example figure represents set power level production system represents operation schedule state state transition transition transition transition power level state state state hour figure operational heating power level characteristics plant system day figure operational power levels identified different states transition levels level significant effects operational power level characteristics state means consistency power level one operation schedule another transition means change power level one operation schedule another heat production system transition level similar feature transitional power level characteristics overall operational performance however power level required transition point point point point different level power level state operational heating power level characteristics represented point power required transition point represented transitional characteristics shown figure thus power level transition transitional characteristics corresponding operational characteristics written represents initial power level step size transition power level absolute values respectively level represents transitional level depends power level operational characteristics transition level transition level pdl hour figure transitional pseudo dynamic characteristics day transitional characteristics explicate power transition level operational characteristics however dynamic information power level attributes still lacking means power content operational characteristics figure point equal point equal equal equal equal dynamic transition information thus necessary model considers dynamic characteristics building simple first order dynamics building characteristic shown figure represents time constant amplitude heating power tsteady delay time constant figure dynamics building characteristics figure delay represents time takes plant system reach building heating operation power sufficient provide heating demand represents power transferred building heating system plant system dynamics incorporate settling time time elapsed heating power reach remain within specified error band equal almost similar behavior like steady state time steady state time corresponds thus settling time steady state time tsteady gives information dynamic characteristics heating demand dynamic information building thus depends transitional attributes power level information totally dynamic pertaining appearance dynamic behavior pseudo dynamic name chosen thus pseudo dynamic lag transitional attribute information depends time constant range settling steady state dynamic building heating characteristics simplified pseudo dynamic lag pdl calculated equation represents sampling time building data represents new unknown time lies settling steady state time concise value depends dynamics heating demand pseudo dynamic characteristics seen figure pdl pseudo dynamic lag tsteady tsteady pdl neural network model neural network consists neurons interconnect inputs model parameters activation function interconnection neurons represents model parameters output mapping neural network based linear activation function input targeted data model parameters adjusted minimize error difference actual values predicted values produced network data repeated significant change model parameters stops training type learning approach called supervised learning since predicted value model guided actual values numerous ann model like multilayer perceptron mlp radial basis function rbf network recurrent network maps som networks learning algorithm learn generalize network paper mlp taken neural network model since pseudo dynamic model fully dynamic time behavior two ways learning mechanism neural network sequential learning batch learning sequential learning cost function computed model parameters adjusted input applied network batch learning inputs fed network model parameters updated batch learning model parameter adjustment done end epoch one complete representation learning process paper batch learning carried mlp network consists three layers input layer hidden layer output layer exist one hidden layer however according kolmogorov theorem single hidden layer sufficient map function provided suitable hidden neurons paper single hidden layer used shown figure hidden layer assists solve separable problems figure neural network architecture figure neuron varies lag represents input neuron varies hidden output neuron respectively signifies transition signifies transition lag corresponding pdl maximum value max equals pdl max pdl mlp uses logistic function hyperbolic tangent threshold function hidden layer identified empirically network using logistic functions tends converge slower hyperbolic tangent activation function hidden layer learning phase hyperbolic tangent activation functions chosen hidden layer pure linear activation function chosen output layer paper hyperbolic tangent function shown equation represents model parameter transpose matrix division input output data learning validation testing gives generalization model learning data sets used learn behavior input data adjust model parameters validation data used minimize overfitting used adjust model parameter used verify increase accuracy learning dataset actually yields increase accuracy dataset learned network testing data sets used confirm actual prediction neural network model unknown neural network paper data divided learning validation testing sets normalization input data also important faster convergence achieve desire performance goal input data poorly scaled learning process risk inaccuracy slower convergence thus essential standardize input data applying neural network various methods normalization input output variable paper normalization zero mean unit standard deviation done shown equation equation represents mean input variable overall vector input variable number datasets respectively thus applies similarly output variable cost function mlp network computed equation represents predicted values produced network actual values given datasets individual data number datasets cost function neural network model respectively network computed order update model parameters higher degree approximation unknown nonlinear function learning process different methods gradient descent newton method gradient descent slow convergence takes time compute hessian matrix newton method well algorithm used paper takes approximation hessian matrix form newton method model parameter update equation given equation hessian matrix approximated parameter gradient computed jacobian matrix vector cost function initial model suitable chosen scalar identity matrix update model parameter thus depends cost function scalar value stopping criteria different criteria stopping neural network model paper stopping criteria depend number epochs learn network performance goal maximum range maximum failures validation performance goal given maximum failures validation accuracy validation datasets defined stop learning process accuracy learning datasets increase validation accuracy stays decrease model performance performances models characterized mean square error mse coefficient correlation mse calculated mse degree freedom adjustment one issues neural network model learning network increase hidden neurons model performance increased lead neural network learning validation accuracy degree freedom dof adjustments done paper avoid fitting number learning equations model could deliver given equation learning equations network length vector output neurons case equal since heating demand load number model parameters single hidden layer mlp neural network given equation represents number model parameters vector length input neurons vector length hidden neurons respectively dof neural network model difference number learning equations number model parameters network always depends optimum size hidden neurons dof maximum hidden neurons given equation represents scalar constant value depends dof required design wmax maximum hidden neurons dof wmax modified performance goal according degree freedom adjustment given dof model performance also modified based degree freedom adjustment modified mse calculated mse mod ified dof mod ified dof hidden neurons optimal mse mod ified maximum mod ified learning validation calculated different initialized random parameters different number hidden neurons mod ified mse mod ified model performed learning validation based optimal configuration model identified final prediction orthogonal arrays essential know whether schedule occupancy profile operational characteristics obtained escos reliable robust design pseudo dynamic model occupancy profile operational characteristics transition period thus plays important role model performance transition period consider finding best robust model takes long time compute orthogonal arrays identify main effects minimum number trials find best design applied various fields mechanical aerospace engineering electromagnetic propagation signal processing robust design model orthogonal array allows effect several parameters find best design given different levels parameters defined matrix column representing number parameters different settings studied rows representing number experiments orthogonal arrays parameters called factors parameter settings called levels general used represent orthogonal arrays represents number experiments number design parameters number levels strength different methods latin square juxtaposition finite geometries etc create orthogonal arrays different strength levels orthogonal arrays different number design parameter level strength available databases libraries orthogonal arrays used paper taken library case study methodology applied case study ecole des mines nantes french institution building floor area students employees building consists research administration rooms class rooms laboratories seminar halls class rooms different sizes accommodate students big seminar halls occupied students small seminar halls occupied students floor area laboratory data taken data acquisition system consists solar radiation outside air temperature heating demand mid january february sampling interval minutes data outside temperature solar radiation heating demand shown figure used learning phase mathematical equation neural network see section equivalent days minute sampling time data days minute sampling time used validation testing phase outside temperature taken study minimum average maximum value respectively global solar radiation average maximum value respectively occupancy profile operational heating power level characteristics working hours shown figure working day day number occupants hour figure occupancy profiles working working day day power level hour figure operational heating power level characteristics working power demand occupancy profile working day depicted figure figure occupancy profile almost gives information power demand characteristics however hour onwards power demand characteristics accordance occupancy profile thus shows simplified occupancy profile enough characterize heating demand working day day power level hour figure heating power demand occupancy profile working days different neural network models designed based climate variables outside temperature solar radiation day information occupancy profile operational characteristics shown figure case study represent working day represent day information day input neural network model static neural network model consists operational characteristics occupancy profile external temperature solar radiation input variables heating power demand output variable thus vector length input neurons equation equals model comprises additional transitional characteristics model vector length input neurons equation equal case study sampling time real building data minutes settling time estimated approximately minutes steady state time tsteady approximately hour pdl thus calculated equation pdl corresponds settling steady state time nearly equal respectively since pseudo dynamic model depends transition lag operational heating power level building dynamic characteristics pdl varied understand phenomena pseudo dynamic lag pdl varied model comprises model additional parameters one pdl equals model consists model additional parameters two pdl model includes model additional parameters three pdl equals equals model comprises model additional parameters four pdl transitional characteristics equals transitional pseudo dynamic characteristic four lags working day shown figure transition level figure calculated equation case study chosen figure lag means static model contains transition attributes lag means pseudo dynamic model transition lag lag means pseudo dynamic model transition lag effects transitional pseudo dynamic effects heating demand understood figure clear information hidden heating demand climate variables could answer justify transitional pseudo dynamic attributes operational characteristics summary models shown table lag lag lag lag lag transition level hour figure transitional pseudo dynamic characteristics working day heating demand lag lag lag lag lag pseudo dynamic transitional effects heating demand transition effects pdl hour figure pseudo dynamic transitional effects heating demand table summary models model type model input variables remarks model static climates occupancy profile operational characteristics lag model static model transitional characteristics lag model pseudo dynamic model pseudo dynamic transition dead band lag model pseudo dynamic model pseudo dynamic transition lag model pseudo dynamic model pseudo dynamic transition settling time lag model pseudo dynamic model pseudo dynamic transition steady state time lag model cost function equation computed iteratively minimum maximum number hidden neurons maximum number hidden neurons calculated equation chosen gives flexibility degree model parameters thus three minimum hidden neurons chosen case study hidden neurons length thus varied wmax performance model iteration number epochs computed equation model parameters updated based equation initial value chosen value increased factor decreased factor maximum value chosen neural network model study stopped number epochs reached performance goal reached value given equation scope study see subsection accuracy number occupants relevant however essential know inside sampling time staff students come leaves buildings necessary check occupancy operational power level characteristics provided escos right robust design model main controlling factors robust design model transition schedule occupancy operational characteristics figure clear transition occupancy transition occupancy interval hour hour hour hour represented factors respectively similarly transition operational characteristics working day shown figure transition factors represented working day hour hour hour hour day hour hour since sampling interval taken case study minutes three levels used orthogonal arrays model represent minutes ahead occupancy operational characteristics schedule period summary control factors levels shown table osw represents occupancy schedule work day ocsw represents operational characteristics schedule work day ocso represent operational characteristics schedule day table summary control factors levels levels factors osw hour min min osw hour min min osw hour min min osw hour min min ocsw hour min min ocsw hour min min ocsw hour min min ocsw hour min min ocso hour min min ocso hour min min thus factors levels govern robustness model full factorials used generalize model takes experiments orthogonal arrays reduce number experiments strengths applied proposed pseudo dynamic model case study result discussion optimal configuration model based maximum mod ified minimum mse mod ified different random initialized parameters hidden neurons model five random initialized parameters assigned learning phase based neurons minimum mse mod ified maximum mod ified learning validation chosen random initialized parameters optimal configuration model chosen maximum minimum mod ified mse mod ified model performance learning validation datasets different hidden neurons figure shows mod ified mse mod ified performance learning validation testing different hidden neurons sizes model optimal configuration chosen best performance model clear figure maximum minimum mod ified mse mod ified performance achieved hidden neuron size optimal configuration model also noticed although testing performance increases hidden neuron size validation learning increase optimally model example similarly process repeated model find optimal configuration neural network model optimal configurations different neural network model summarized table learning validation testing best performance coefficient correlation hidden neurons figure coefficient correlation performance model learning validation testing mean square error best performance hidden neurons figure mean square error performance model table optimal configuration models model hidden coefficient correlation neurons learning validation testing mean square error learning validation testing model model model model model model table shows static neural network model best mod ified learning validation obtained clear occupancy profile operational characteristics enough determine generalize unknown function building heating demand transitional attributes operational characteristic introduced model mod ified model performance increases significantly learning phase mse mod ified decreases contrast model pseudo dynamic transitional attributes model time constant model leads increase validation phase correspondingly model performance dynamics settling time steady state plays important role characterizing neural network model seen mod ified performance increases learning validation model compare model although transition attributes introduce model addition hidden neuron size also reduces moreover distinguish learning validation performances remained model compared model optimal choice model thus lies settling steady state time view model model show reasonable consistent model performances however minimum hidden neuron size maximum learning criteria essential overall network generalization since hidden neurons size decreases model performance mod ified remained model comparing model model chosen best configuration overall models optimal choice model model delineated error percentage energy consumption kwh actual prediction learning validation phase heating energy consumption error actual prediction learning phase model compare model validation phase heating energy consumption error model compare model energy consumption error clear small heating energy consumption error model compare model learning validation phase one conclude model chosen optimal configuration overall model model thus bridges gap static dynamic neural network model sense better static model increases performance comparable dynamic neural network model robustness pseudo dynamic model orthogonal arrays applied determine highest coefficient correlation learning validation optimum hidden neuron size model table shows coefficient correlation learning validation phase clear table schedule taken escos experiment orthogonal arrays optimal schedule fits best model experiment orthogonal arrays thus ensures transition occupancy hour hour hour hour instead hour hour hour hour period existing case respectively also transition hour hour hour hour instead hour hour hour hour working day hour hour instead hour hour days operational characteristics coefficient correlation orthogonal array design learning validation training phase nevertheless issue overall model difficult increase coefficient correlation beyond due sampling time minutes short sampling time difficult learn datasets changes minutes sample nonetheless good generalization model mod ified value learning phase always acceptable coefficient correlation linear regression obtained neural network model actual prediction heating demand learning validation testing phase model optimum orthogonal array design respectively prediction heating demand model optimum orthogonal array design validation phase shown figure prediction gives power heating demand area curve gives heating energy demand figure clear heating demand tremendously increases approximately third fourth day pseudo dynamic model able predict learn behavior however fluctuation power demand morning consecutive days difficult learn datasets transits rapidly actual power demand prediction heating demand model testing phase optimum orthogonal array design shown figure vivid pseudo dynamic model able predict heating demand however third day pseudo dynamic model able meet heating demand due fact neural network learn threshold maximum heating demand learning phase kind information available database data thus needs improved learning phase feature extraction techniques nonetheless pseudo dynamic model model prediction accordance actual target except rapid transits actual target sum pseudo dynamic transition attributes model orthogonal array design leads best prediction heating demand table coefficient correlation learning validation model element experiment coefficient correlation learning validation testing validation phase actual predict heating demand hour figure prediction heating demand model validation phase optimum orthogonal array design test phase actual predict heating demand hour figure prediction heating demand model testing phase optimum orthogonal array design conclusion paper introduces pseudo dynamic transitional model building heating demand prediction short time horizon using artificial neural network occupancy profile operational heating power level characteristics included model dynamic characteristic building included model determination pseudo dynamic transition lag settling time steady state time heating demand give increment precision model however choice model depends actual time settling steady state results based case study occupancy profile already known results may vary fluctuating occupancy buildings coefficient correlation increases learning validation testing pseudo dynamic comparing static neural network model also size hidden neuron reduced reduces complexities increases generalization model moreover minimum energy consumption error achieved pseudo dynamic model learning validation phase orthogonal array applied optimal pseudo dynamic model confirm schedule occupancy profile operational level characteristics robustness model orthogonal array design leads increases coefficient correlation pseudo dynamic model confirmed new schedule occupancy profile operational level characteristics major contribution paper thus introduction transition novel time dependent attributes operational heating power level characteristics dominant factor building heating demand also orthogonal array design model makes flexibility cross checking schedule occupancy profile operational heating power level characteristics obtained escos design robust model prediction short time horizon days sampling interval minutes thus useful dynamic control building heating demand research focused towards feature extraction data learning phase neural network abnormalities data corrected learning phase also adaptive real time learning criteria seasonal behaviour studied acknowledgement research done collaboration ecole des mines nantes technische universiteit eindhoven veolia environnement recherche innovation funded erasmus mundus joint doctoral programme support gratefully acknowledged references laustsen energy efficiency 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method predict hourly building cooling load urban energy planning energy buildings underwood yik modeling methods energy buildings blackwell science girardin marechal dubuis favrat energis geographical information based system evaluation integrated energy conversion systems urban areas energy yao steemers method formulating energy load profile domestic buildings energy buildings catalina virgone blanco development validation regression models predict monthly heating demand residental buildings energy buildings wan liu lam future trends building heating cooling loads energy consumption different climates building environment yokoyama wakui satake prediction energy demands using neural network model identification global optimization energy conversion management dong cao lee applying support vector machines predict building energy consumption tropical region energy buildings meng development applications hourly building cooling load prediction model international conference advances energy engineering ieee china june http kalogirou florides neocleous schizas estimation daily heating cooling loads using artificial neural networks world congress napoli september http access neto fiorelli comparison detailed model simulation artificial neural network forecasting building energy consumption energy buildings shilin zhifeng neural network prediction urban building energy consumption based matlab application international conference computer modeling simulation ieee china january mihalakakou santamouris tsangrassoulis energy consumptions residential buildings energy buildings ekici aksoy prediction building energy consumption using artificial neural network advances engineering software dombayci prediction heating energy consumption model house using artificial neural networks advances engineering software gonzalez zamarreno prediction hourly energy consumption buildings based feedback artificial neural network energy buildings popescu ungureanu simulation models analysis space heat consumption buildings energy kato sakawa ishimaru ushiro shibano heat load prediction recurrent neural network district heating cooling systems international conference systems man cybernetics smc ieee singapore october kalogirou bojic artificial neural networks prediction energy consumption passive solar building energy sun wang xiao development validation simplified online cooling load prediction strategy super building hongkong energy conversion management yun luck mago cho building hourly thermal load predictions using indexed arx model energy buildings azar menassa comprehensive analysis impact occupancy parameters energy simulation office buildings energy buildings leung tse lai chow use occupancy space electrical power demand building cooling load prediction energy buildings bompay evaluation response etex release atmospheric environment voyant muselli paoli nivet numerical weather prediction nwp hybrid model predict global radiation energy arome modeling system http chen wang ren selection climate variables time scales future weather preparation building heating cooling energy predictions energy buildings haykin neural networks comprehensive foundation second edition pearson education inc zhang advances neural networks isnn international symposium neural networks heidelberg new york hsieh machine learning methods environmental sciences neural networks kernels cambridge university press leung optimization biodiesel production camelina oil using orthogonal experiment applied energy weng yang elsherbeni linear antenna arrays synthesis using tgauchi methods novel optimization technique electromagnetics ieee transactions antennas propagation franek jiang orthogonal design experiments parameter learning image segementation signal processing nguyen new constructions strength mixed orthogonal arrays journal statistical planning inference suen construction mixed orthogonal arrays juxtaposition statistics proabability letters suen dey construction asymmetric orthogonal arrays finite geometries journal statistical planning inference sloane library orthogonal arrays online http access appendix influence input variables model output evaluated based correlation analysis correlation measures strength weakness linear relationship two variables several coefficients measure correlation degree pearson correlation coefficient used determine input variables relevance paper pearson correlation coefficient calculated dividing covariance two variables product standard deviation shown equation represents pearson correlation coefficient equations cov covariance represents strength linear relationship two variables mean values variables standard deviations variables number data cov cov correlation coefficients range perfect positive linear correlation perfect negative linear correlation small positive linear correlation medium positive linear correlation strong positive linear correlation negative linear correlation climatic conditions outside temperature solar radiation operational power level characteristics approximate occupancy profile used evaluate relevance variables affect building heat demand based case study data variables pseudo dynamic transitional attributes signifies dynamics building characteristics consider relevance variable determination since signifies time phase interval heating power transition results show linear coefficient correlation outside air temperature solar radiations occupancy profile operational power level characteristics heat load respectively results thus signifies climatic conditions outside temperature solar radiations relevant input variables predict heat load also clearer occupancy profile operational power level characteristics medium positive correlation heat load shows relevance characterize heat demand behaviour
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ixel econvolutional etworks hongyang gao washington state university hao yuan washington state university zhengyang wang washington state university nov shuiwang washington state university sji bstract deconvolutional layers widely used variety deep models including networks semantic segmentation deep generative models unsupervised learning one key limitations deconvolutional operations result checkerboard problem caused fact direct relationship exists among adjacent pixels output feature map address problem propose pixel deconvolutional layer pixeldcl establish direct relationships among adjacent pixels feature map method based fresh interpretation regular deconvolution operation resulting pixeldcl used replace deconvolutional layer manner without compromising fully trainable capabilities original models proposed pixeldcl may result slight decrease efficiency overcome implementation trick experimental results semantic segmentation demonstrate pixeldcl consider spatial features edges shapes yields accurate segmentation outputs deconvolutional layers used image generation tasks pixeldcl largely overcome checkerboard problem suffered regular deconvolution operations ntroduction deep learning methods shown great promise variety artificial intelligence tasks image classification krizhevsky simonyan zisserman semantic segmentation noh shelhamer ronneberger natural image generation goodfellow kingma welling oord key network layers convolutional layers lecun pooling layers fully connected layers deconvolutional layers frequently used create deep models different tasks deconvolutional layers also known transposed convolutional layers vedaldi lenc initially proposed zeiler primarily used deep models require feature maps generative models radford makhzani frey rezende architectures ronneberger noh although deconvolutional layers capable producing larger feature maps smaller ones suffer problem checkerboard artifacts odena greatly limits deep model capabilities generating images producing smooth outputs semantic segmentation date little efforts devoted improving deconvolution operation work propose simple efficient yet effective method known pixel deconvolutional layer pixeldcl address checkerboard problem suffered deconvolution operations method motivated fresh interpretation deconvolution operations clearly pinpoints root checkerboard artifacts feature map generated deconvolution considered result periodical shuffling multiple intermediate feature maps computed input feature map independent convolutions result adjacent pixels output feature map directly related leading checkerboard artifacts overcome figure comparison semantic segmentation results first second rows images ground true labels respectively third fourth rows results using regular deconvolution proposed pixel deconvolution pixeldcl respectively problem propose pixel deconvolutional operation used pixeldcl new layer intermediate feature maps generated sequentially feature maps generated later stage required depend previously generated ones way direct relationships among adjacent pixels output feature map established sequential generation intermediate feature maps pixeldcl may result slight decrease computational efficiency show largely overcome implementation trick experimental results semantic segmentation samples figure image generation tasks demonstrate proposed pixeldcl effectively overcome checkerboard problem improve predictive generative performance work related pixel recurrent neural networks pixelrnns oord pixelcnns van den oord reed generative models consider relationship among units feature map belong general class autoregressive methods probability density estimation germain gregor larochelle murray using masked convolutions training training time pixelrnns pixelcnns comparable generative models generative adversarial networks gans goodfellow reed variational autoencoders vaes kingma welling johnson however prediction time pixelrnns pixelcnns slow since generate images pixel pixel contrast pixeldcl used replace deconvolutional layer manner slight decrease efficiency largely overcome implementation trick ixel econvolutional ayers etworks introduce deconvolutional layers analyze cause checkerboard artifacts section propose pixel deconvolutional layers implementation trick improve efficiency econvolutional ayers deconvolutional networks deconvolutional layers proposed zeiler widely used deep models applications semantic segmentation noh generative models kingma welling goodfellow oord many architectures use deconvolutional layers decoders one way understanding deconvolutional operations output feature map obtained periodical shuffling multiple intermediate feature maps obtained applying multiple convolutional operations input feature maps shi kernel intermediate feature maps intermediate feature maps output feature map output feature map figure illustration deconvolutional operation deconvolutional layer feature map feature map left figure shows input unit passes kernel output feature map obtained sum values column seen figure purple outputs related entries kernel orange outputs related entries kernel therefore deconvolution decomposed two convolutional operations shown right figure two intermediate feature maps generated convolutional operations dilated combined obtain final output indicates standard deconvolutional operation decomposed multiple convolutional operations output feature map intermediate feature maps input feature map figure illustration deconvolutional operation deconvolutional layer feature map feature map four intermediate feature maps purple orange blue red generated using four different convolutional kernels four intermediate feature maps shuffled combined produce final feature map note four intermediate feature maps rely input feature map direct relationship among interpretation deconvolution illustrated figures respectively clear illustrations standard deconvolutional operation decomposed several convolutional operations depending factor following assume factor two though deconvolution operations applied generic settings formally given input feature map fin deconvolutional layer used generate output fout follows fin fin fin fout fin denotes convolutional operation denotes periodical shuffling combination operation figure intermediate feature map generated corresponding convolutional kernel clear interpretation deconvolution direct relationship among intermediate feature maps since generated independent convolutional kernels although pixels position intermediate feature maps depend receptive field input feature map directly related due periodical shuffling operation adjacent pixels output feature map different intermediate feature maps figure illustration checkerboard problem semantic segmentation using deconvolutional layers first second rows original images semantic segmentation results respectively implies values adjacent pixels significantly different resulting problem checkerboard artifacts odena illustrated figure one way alleviate checkerboard artifacts apply smoothing adds additional complexity network makes entire network fully trainable work propose pixel deconvolutional operation add direct dependencies among intermediate feature maps thereby making values adjacent pixels close effectively solving checkerboard artifact problem addition pixel deconvolutional layers easily used replace deconvolutional layers without compromising fully trainable capability ixel econvolutional ayers solve checkerboard problem deconvolutional layers propose pixel deconvolutional layers pixeldcl add dependencies among intermediate feature maps adjacent pixels different intermediate feature maps pixeldcl build direct relationships among thus solving checkerboard problem method intermediate feature maps generated sequentially instead simultaneously intermediate feature maps generated later stage required depend previously generated ones primary purpose sequential generation add dependencies among intermediate feature maps thus adjacent pixels final output feature maps finally intermediate feature maps shuffled combined produce final output feature maps compared eqn fout obtained follows fin fin fout fin fin denotes juxtaposition feature maps note eqn denotes set kernels involves convolution juxtaposition multiple feature maps since intermediate feature maps eqn depend input feature map previously generated ones term input pixel deconvolutional layer ipixeldcl process pixels output feature maps conditioned input feature maps also adjacent pixels since direct relationships among intermediate feature maps adjacent pixels ipixeldcl expected solve checkerboard problem extent note relationships among intermediate feature maps flexible intermediate feature maps generated later rely part previously generated intermediate feature maps depends design pixel dependencies final output feature maps figure illustrates specific design sequential dependencies among intermediate feature maps ipixeldcl add dependencies among generated intermediate feature maps thereby making adjacent pixels final output feature maps directly related process information input feature map repeatedly used generating intermediate feature maps generating intermediate feature maps information input feature map previous intermediate feature maps used since previous intermediate feature maps already contain information input feature map dependencies input feature map removed intermediate feature maps output feature map input feature map figure illustration ipixeldcl pixeldcl described section ipixeldcl additional dependencies among intermediate feature maps specifically four intermediate feature maps generated sequentially purple feature map generated input feature map blue orange feature map conditioned input feature map purple feature map generated previously way green feature map relies input feature map purple orange intermediate feature maps red feature map generated based input feature map purple orange green intermediate feature maps also propose move one step allow first intermediate feature map depend input feature map gives rise pixeldcl connections indicated dashed lines removed avoid repeated influence input feature map way first feature map generated input feature maps directly rely input pixeldcl orange feature map depends purple feature map green feature map relies purple orange feature maps red feature map conditioned purple orange green feature maps information input feature map delivered intermediate feature maps first intermediate feature map purple removing dependencies intermediate feature maps improve computational efficiency also reduce number trainable parameters deep models simplified pixel deconvolutional layer first intermediate feature map depend input feature map intermediate feature maps generated afterwards depend previously generated intermediate feature maps simplify dependencies among pixels final output feature map work use pixeldcl denote simplified design experimental results show pixeldcl yields better performance ipixeldcl regular deconvolution compared eqn fout pixeldcl obtained follows fin fout pixeldcl illustrated figure removing connections denoted dash lines analyzing relationships pixels output feature maps clear pixel still rely adjacent pixels therefore checkerboard problem solved even better computational efficiency meanwhile experimental results demonstrate performance models simplified dependencies even better complete connections demonstrates repeated dependencies input may necessary ixel econvolutional etworks pixel deconvolutional layers applied replace deconvolutional layers various models involving operations ronneberger vaes kingma welling gans goodfellow replacing deconvolutional layers pixel deconvolutional layers deconvolutional networks become pixel deconvolutional networks pixeldcn semantic segmentation pixel deconvolutional layers used upsample feature maps ones vaes applied decoders image reconstruction generator networks gans typically use deep model radford thus employ pixel deconvolutional layers generate large images figure efficient implementation pixel deconvolutional layer layer feature map feature map purple feature map generated convolutional operation input feature map step another convolutional operation applied purple feature map produce orange feature map step purple orange feature maps dilated added together form larger feature map step since relationship last two intermediate feature maps apply masked convolutional operation instead two separate convolutional operations step finally two large feature maps combined generate final output feature map step experiments evaluate pixel deconvolutional layers vaes results show performance pixel deconvolutional layers outperforms deconvolutional layers networks practice frequently used operation increase height width input feature maps factor two case pixels output feature maps divided four groups eqn dependencies defined figure implementing pixel deconvolutional layers design simplified version reduce sequential dependencies better parallel computation training efficiency illustrated figure design four intermediate feature maps first intermediate feature map depends input feature map second intermediate feature map relies first intermediate feature map third fourth intermediate feature maps based first second feature maps simplified relationships enable parallel computation third fourth intermediate feature maps since dependency addition masked convolutional operation used generate last two intermediate feature maps mentioned already variety different dependencies relations imposed intermediate feature maps simplified design achieves reasonable balance efficiency performance code publicly xperimental tudies section evaluate proposed pixel deconvolutional methods semantic segmentation image generation tasks comparison regular deconvolution method results show use new pixel deconvolutional layers improves performance consistently supervised unsupervised learning settings emantic egmentation experimental setup use pascal segmentation dataset everingham mscoco detection dataset lin evaluate proposed pixel deconvolutional methods semantic segmentation tasks datasets images resized batch training models directly predict label pixel without examine models two ways training scratch model training scratch experiments use architecture ronneberger base model successfully applied various image segmentation tasks https figure sample segmentation results pascal segmentation dataset using training scratch models first second rows original images corresponding ground truth respectively third fourth fifth rows segmentation results models using deconvolutional layers ipixeldcl pixeldcl respectively network consists four blocks encoder path four corresponding blocks decoder path within decoder block deconvolutional layer followed two convolutional layers final output layer adjusted based number classes dataset pascal segmentation dataset classes mscoco detection dataset classes mscoco detection dataset classes pascal segmentation dataset number feature maps layer dataset doubled accommodate output channels baseline model employs deconvolutional layers within decoder path feature maps replace deconvolutional layers proposed pixel deconvolutional layers ipixeldcl simplified version pixeldcl keeping variables unchanged kernel size dcl number parameters ipixeldcl sets kernels parameters pixeldcl sets set kernels enable evaluate new pixel deconvolutional layers regular deconvolutional layers controlling factors experiments models based architecture deeplabresnet chen model also use external data training strategy using external training data finetuning classic greatly boosts performance model accuracy mean iou output eight times smaller input image height width dimensions order recover original dimensions add three blocks feature maps factor block deconvolutional layer followed convolutional layer employing strategy replace deconvolutional layer pixeldcl ipixeldcl using kernels size training scratch experiments analysis results sample segmentation results using deconvolutional layers dcl ipixeldcl pixeldcl pascal segmentation dataset mscoco detection dataset given figures respectively see models using ipixeldcl pixeldcl better capture local information images base model using regular deconvolutional layers using pixel deconvolutional layers spacial features edges shapes considered predicting labels adjacent pixels moreover semantic segmentation results demonstrate proposed models tend produce smoother outputs model using deconvolution also observe training epoch small epochs model employs pixeldcl better segmentation outputs model using ipixeldcl training epoch large enough epochs similar performance though pixeldcl still outperforms ipixeldcl cases indicates pixeldcl efficient effective since much fewer parameters learn table shows evaluation results terms pixel accuracy mean iou two datasets models using ipixeldcl pixeldcl yield better performance base figure sample segmentation results mscoco detection dataset using training scratch models first second rows original images corresponding ground truth respectively third fourth fifth rows segmentation results models using deconvolutional layers ipixeldcl pixeldcl respectively table semantic segmentation results pascal segmentation dataset mscoco detection dataset compare base model using three different methods decoders namely regular deconvolution layer dcl proposed input pixel deconvolutional layer ipixeldcl pixel deconvolutional layer pixeldcl pixel accuracy mean iou used performance measures dataset model pixel accuracy mean iou dcl pascal ipixeldcl pixeldcl dcl mscoco ipixeldcl pixeldcl dcl pascal ipixeldcl pixeldcl model using regular deconvolution model using pixeldcl slightly outperforms model using ipixeldcl models models using ipixeldcl pixeldcl better performance model using dcl ipixeldcl performs best semantic segmentation mean iou accuracy evaluation measure pixel accuracy everingham models using pixel deconvolution better evaluation results mean iou base model using deconvolution mage eneration experimental setup dataset used image generation celebfaces attributes celeba dataset liu avoid influence background images preprocessed facial information retained image generation task reconstruct faces excluding backgrounds training images size images use standard variational vae kingma welling base model image generation decoder part standard vae employs deconvolutional layers apply proposed pixeldcl replace deconvolutional layers decoder keeping components kernel size dcl parameters pixeldcl sets set kernels analysis results figure shows generated faces using vaes regular deconvolution baseline pixeldcl decoders images generated baseline model suffer figure sample face images generated vaes trained celeba dataset first two rows images generated standard vae deconvolutional layers last two rows generated vae model using pixeldcl table training prediction time semantic segmentation using pascal segmentation dataset tesla gpu compare training time epochs prediction time images base model using three different methods decoders namely dcl ipixeldcl pixeldcl model training time prediction time dcl ipixeldcl pixeldcl apparent checkerboard artifacts none found images generated model pixeldcl demonstrates proposed pixel deconvolutional layers able establish direct relationships among adjacent pixels generated feature maps images thereby effectively overcoming checkerboard problem results demonstrate pixeldcl useful generative models since consider local spatial information produce images without checkerboard problem iming omparison table shows comparison training prediction time models using dcl ipixeldcl pixeldcl see models using ipixeldcl pixeldcl take slightly time training prediction model using dcl since intermediate feature maps generated sequentially model using pixeldcl efficient due reduced dependencies efficient implementation discussed section overall increase training prediction time dramatic thus expect major bottleneck proposed methods onclusion work propose pixel deconvolutional layers solve checkerboard problem deconvolutional layers checkerboard problem caused fact direct relationship among intermediate feature maps generated deconvolutional layers pixeldcl proposed try add direct dependencies among generated intermediate feature maps pixeldcl generates intermediate feature maps sequentially intermediate feature maps generated later stage required depend previously generated ones establishment dependencies pixeldcl ensure adjacent pixels output feature maps directly related experimental results semantic segmentation image generation tasks show pixeldcl effective overcoming checkerboard artifacts results semantic segmentation also show pixeldcl able consider local spatial features edges shapes leading better segmentation results future plan employ pixeldcl broader class models generative adversarial networks gans eferences chen george papandreou iasonas kokkinos kevin murphy alan yuille deeplab semantic image segmentation deep convolutional nets atrous convolution fully connected crfs mark everingham luc van gool christopher williams john winn andrew zisserman pascal visual object classes voc challenge international journal computer vision mathieu germain karol gregor iain murray hugo larochelle made masked autoencoder distribution estimation proceedings international conference machine learning ian goodfellow jean mehdi mirza bing david sherjil ozair aaron courville yoshua bengio generative adversarial nets advances neural information processing systems karol gregor ivo danihelka alex graves danilo rezende daan wierstra draw recurrent neural network image generation proceedings international conference machine learning kaiming xiangyu zhang shaoqing ren jian sun deep residual learning image recognition proceedings ieee conference computer vision pattern recognition matthew johnson david duvenaud alex wiltschko ryan adams sandeep datta composing graphical models neural networks structured representations fast inference advances neural information processing systems diederik kingma max welling stochastic gradient variational second international conference learning representations iclr alex krizhevsky ilya sutskever geoffrey hinton imagenet classification deep convolutional neural networks advances neural information processing systems hugo larochelle iain murray neural autoregressive distribution estimator international conference artificial intelligence statistics yann lecun bottou yoshua bengio patrick haffner learning applied document recognition proceedings ieee steven xie simple checkerboard suppression algorithm evolutionary structural optimization structural multidisciplinary optimization lin michael maire serge belongie james hays pietro perona deva ramanan piotr lawrence zitnick microsoft coco common objects context european conference computer vision springer ziwei liu ping luo xiaogang wang xiaoou tang deep learning face attributes wild proceedings international conference computer vision iccv alireza makhzani brendan frey autoencoders advances neural information processing systems hyeonwoo noh seunghoon hong bohyung han learning deconvolution network semantic segmentation ieee international conference computer vision augustus odena vincent dumoulin chris olah deconvolution checkerboard artifacts distill doi url http aaron van den oord nal kalchbrenner koray kavukcuoglu pixel recurrent neural networks proceedings international conference machine learning alec radford luke metz soumith chintala unsupervised representation learning deep convolutional generative adversarial networks arxiv preprint scott reed zeynep akata xinchen yan lajanugen logeswaran bernt schiele honglak lee generative adversarial text image synthesis proceedings international conference machine learning volume scott reed van den oord nal kalchbrenner sergio colmenarejo ziyu wang dan belov nando freitas parallel multiscale autoregressive density estimation arxiv preprint danilo jimenez rezende shakir mohamed daan wierstra stochastic backpropagation approximate inference deep generative models proceedings international conference machine learning olaf ronneberger philipp fischer thomas brox convolutional networks biomedical image segmentation international conference medical image computing intervention springer evan shelhamer jonathon long trevor darrell fully convolutional networks semantic segmentation ieee transactions pattern analysis machine intelligence wenzhe shi jose caballero ferenc johannes totz andrew aitken rob bishop daniel rueckert zehan wang single image video using efficient convolutional neural network proceedings ieee conference computer vision pattern recognition karen simonyan andrew zisserman deep convolutional networks image recognition arxiv preprint aaron van den oord nal kalchbrenner lasse espeholt oriol vinyals alex graves conditional image generation pixelcnn decoders advances neural information processing systems andrea vedaldi karel lenc matconvnet convolutional neural networks matlab proceedings acm international conference multimedia acm matthew zeiler dilip krishnan graham taylor rob fergus deconvolutional networks computer vision pattern recognition cvpr ieee conference ieee matthew zeiler graham taylor rob fergus adaptive deconvolutional networks mid high level feature learning computer vision iccv ieee international conference ieee
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jun bernoulli doi conditions process stay positive near probability ross maller school finance actuarial studies statistics australian national university canberra act australia necessary sufficient condition process stay positive probability near arises studies laws near given terms characteristics keywords process staying positive introduction let real valued process canonical triplet thus characteristic function characteristic exponent borel measure condition lim inf shown wee imply law attention drawn recent paper aurzada savov give extended refined versions chung law using quite different approach wee difference conditions imposed aurzada clear though based examples suggest weaker aim paper give necessary sufficient conditions stay positive near stay negative near hence characterise electronic reprint original article published bernoulli vol reprint differs original pagination typographic detail maller need notation positive negative tails restriction denoted define interested small time behaviour eliminate compound poisson case assuming throughout define truncated winsorised moments functions defined finite virtue property measure behaviour relevant integration parts shows doney lemma gives following version decomposition caters positive negative jumps separately take constants functions behaviour small values relevant keep note compensated sum small positive jumps lim process staying positive near compensated sum small negative jumps lim almost sure limits exist processes positive negative big jumps thus finally standard brownian motion independent jump processes independent motivate approach quote part result due doney gives equivalence remain positive small times probability approaching terms functions negative tail condition reflects positivity small times function remains positive small values dominates negative tail certain way theorem suppose suppose also lim lim suppose alternatively spectrally positive equivalent small happens subordinator furthermore small remarks equivalences theorem doney remark following theorem assumes priori necessary follows inequality lim sup whenever proved buchmann fan maller maller compound poisson behaviour near simply determined sign shift constant eliminate case throughout next section contains main result essentially subsequential version theorem staying positive near subsequential version denote jump process define max max theorem assume suppose following equivalent sequence xtk sequence xtk lim sup suppose alternatively spectrally positive equivalent thus equivalently subordinator iii suppose xtk sequence lim sup remarks ratio well defined sato page shows continuous function xtk replaced xtk without changing result similarly theorem process staying positive near iii assuming contrapositive shows sequence xtk equivalently lim inf lim sup symmetrical argument lim inf lim inf combining gives following corollary assume holds lim inf lim sup one infinite zero conditions also read theorem random walk version theorem kesten maller andrew theorem results related theorem including equivalence inequalities distribution proof theorem lemmas needed first gives bound small jump component proof rather similar lemma bertoin doney maller omit details lemma fix let small jump martingale obtained compensated sum jumps magnitudes lim interpret integrals intervals form define absolute moments assume maller bound absolute constant standard normal next use lemma develop useful bounds define next lemma signs taken together interpret similarly replacing lemma suppose constants satisfying max absolute constant suppose satisfy assume constants satisfying suppose poisson expectation remains true iii suppose suppose constants satisfying process staying positive near proof give proof signs fix take constants satisfying assume apply bound rin lemma measure restricted noting gives substitute get inequality implies since satisfies apply chebychev inequality noting variance get mean also choice giving inequality holds holds general mass sign remains valid sense side equals proves sign argument goes place maller use representation fix take constants satisfying let constants choose satisfy small jump processes bounds note remain true big positive jumps exceeds till time equation remains true probability side last inequality used twice larger independence processes jump upper bound number jumps less equal size occur time distributed poisson expectation note implies poisson distribution stochastically monotone sense poisson rvs means letting poisson expectation using arrive take terms get terms iii assume case define still assume write process staying positive near negative jump components amalgamated compound poisson process comprised jumps term term absent using write gives proof theorem part assume throughout part assume implies neighbourhood assume choose implies also means large without loss generality may assume let since also addition maller set still recall use decomposition write compensated small jump process positive negative big jump processes case suppose since jump least lower bound poisson expectation variance using substituting get xtk xtk since xtk xtk xtk also xtk hence xtk thus holds process staying positive near case alternatively omit term containing follows obtain hence continuing previous argument implies sup using also holds holds obvious assume well holds suppose fails choose obtain contradiction note implies normal assume follows consider cases case assume fact situation introduce quantile versions define function inf set since analogously define inf since set choose satisfy together shows must observe assumption used part proof trivial case included interpret holding maller positive numbers used inequality take set defined holds sequence let max equation implies adding quantity sides gives left quantity smaller right adding sides gives left see cancellation arrive stage helpful assume continuous function follows also thus deduce next write process staying positive near giving substituting obtain choose small enough first expression brackets side positive choose large enough second expression brackets side positive gives large max inequality implies giving contradiction since proves case continuous complete proof part case theorem remove assumption continuity made deriving done using following lemma lemma let measure exists sequence measures absolutely continuous respect lebesgue measure strictly positive satisfying proof refers vague convergence see example chapter kallenberg extend borel measure setting assume observe defines borel probability measure convolved probability measure admits strictly positive density normal expectation variance set easily verified sequence measures desired properties maller complete proof part assume xtk general measure using lemma construct sequence approximating measures converging vaguely positive tails continuous negative side let let processes measures characteristics define replacing subscript functions converge original functions points continuity latter characteristic exponent given replacing assumption xtk xtk arbitrary large enough thus xtk holds subscript quantities probability deterministic holds fact subscript quantities probability whenever proof using continuity shows holds replaced letting shows holds stated get contradiction thus complete proof implies case case assume proof case take satisfy define write set take may since true replacing follow proof case get replaced thus estimating along lines find side smaller choose large enough get hence place hence case assume define together shows must large write inequality implies process staying positive near subtracting quantity sides gives left quantity smaller right adding sides gives see stage assume continuous follows large thus deduce large enough using substituting get large choose small enough expression brackets side positive gives since assumed implies contradiction remove continuity assumption proved maller part deal case assume continuous working case still valid negative jump process absent gives using last inequality since continuous choosing get letting shows since negative jumps deduce bounded variation drift see doney maller theorem remark nonnegative thus subordinator drift follows proved assuming continuity assumption removed together implies theorem hence part iii finally suppose holds choose following exactly proof part get implies xtk since conversely suppose xtk sequence xtk every xtm triplet consequently holds replaced modified version implies completes part iii proof theorem process staying positive near acknowledgements grateful referee close reading paper helpful suggestions boris buchmann supplying lemma research partially supported arc grant references andrew limiting behaviour processes zero probab theory related fields aurzada savov small time lil processes bernoulli bertoin doney maller passage processes across power law boundaries small times ann probab buchmann fan maller distributional representations dominance process maximal jump processes bernoulli appear available doney behaviour processes electron probab doney maller stability attraction normality processes zero infinity theoret probab kallenberg foundations modern probability probability applications new york new york springer kesten maller divergence random walk deterministic random subsequences theoret probab sato processes infinitely divisible distributions cambridge cambridge univ press wee lower functions processes stationary independent increments probab theory related fields received september
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extending prolog incomplete fuzzy information arxiv aug susana claudio departamento lenguajes sistemas del software facultad universidad madrid campus montegancedo madrid spain susana departamento ciencias facultad universidad nacional del comahue universidad madrid buenos aires neuquen argentina vaucheret abstract incomplete information problem many aspects actual environments furthermore many sceneries knowledge represented crisp way common find fuzzy concepts problems level uncertainty many practical systems handle fuzziness uncertainty examples find used minority extend popular system many programmers using ability combining crisp fuzzy knowledge representations seems interesting issue first work fuzzy prolog language models fuzzy logic borel algebra truth value represented using unions intervals real numbers work general truth value representation propagation previous works interpreter language using constraint logic programming real numbers clp implemented available ciao system enhance former approach using default knowledge represent incomplete information logic programming also provide implementation new framework new release fuzzy prolog handles incomplete information complete semantics previous one incomplete prolog moreover able combine crisp fuzzy logic prolog programs therefore new fuzzy prolog expressive represent real world fuzzy prolog inherited prolog incompleteness incorporation default reasoning fuzzy prolog removes problem requires richer semantics discuss keywords incomplete knowledge fuzzy prolog modeling uncertainty fuzzy logic programming constraint programming application implementation fuzzy prolog introduction world information represented crisp way representation imperfect fuzzy management uncertainty important knowledge representation multiple frameworks incorporating uncertainty logic programming fuzzy set theory probability theory logic possibilistic logic general framework proposed generalizes many previous approaches time analogous theoretical framework provided prototype prolog implemented basically rule form assignment certainties taken certainty lattice certainty computed taking set certainties propagated using function aggregation operator flexible approach practical examples prolog framework presented work extend approach arbitrary assignments default certainty values default assumptions usual semantics logic programs obtained unique computation method using different assumptions uniform way assign default truthvalue atoms well known assumptions closed world assumption cwa asserts atom whose inferred facts clauses program supposed false certainty used stable models semantics open world assumption owa asserts atom whose inferred facts clauses program supposed undefined unknown certainty used also approaches assumptions combined atoms interpreted assuming cwa others follows owa anyway seems really interesting combine assumptions generalize use default value aim working incomplete information guarantees rest paper organized follows section introduces fuzzy prolog language complete description new semantics fuzzy prolog provided section section completes details improved implementation using clp extension handle default knowledge illustrating examples provided section finally conclude discuss future work section research partly supported spanish mcyt project fuzzy prolog section going summarize main characteristics fuzzy prolog proposed basis work presented fuzzy prolog general previous approaches introduce fuzziness prolog respects truth value finite union closed represented borel algebra algebra considers intervals interval special case union one element unique truth value particular case interval one element truth value propagated rules means aggregation operator definition aggregation operator general sense subsumes conjunctive operators triangular norms like min prod etc disjunctive operators triangular like max sum etc average operators like arithmetic average average etc hybrid operators combinations operators declarative procedural semantics fuzzy logic programs given equivalence proven implementation proposed language presented fuzzy program finite set fuzzy facts atom truth value element expressed constraints domain fuzzy clauses atoms operator induces definition truth values represented constraints domain obtain information program fuzzy queries fuzzy goals atom variable possibly instantiated represents truth value programs defined usual handling truth values borel algebra real interval deals unions intervals represented constraints refer example expressions represent truth value lot everyday situations represented general representation truth value examples truth value goal depend truth value subgoals body clauses definition fuzzy prolog uses aggregation operators order propagate truth value means fuzzy rules fuzzy sets aggregation done using application numeric operator form aggregation operator must verify addition monotonic continuous deal definition fuzzy sets intervals necessary generalize aggregation operators numbers aggregation operators intervals following theorem proven nguyen walker extend intervals propose following definitions definition given aggregation defined follows xln xun xln xun actually work union intervals propose definition definition given defined intervals defined union intervals follows presentation theory possibility zadeh considers fuzzy sets act elastic constraint values variable fuzzy inference constraint propagation furthermore extension presented paper truth values result aggregations represented constraints constraint signature contains real numbers binary function symbols binary predicate symbols constraint solution domain real numbers interval consistent denoted solvable semantics section contains reformulation semantics fuzzy prolog new semantics complete thanks inclusion default value least model semantics herbrand universe set ground terms made constants function symbols program herbrand base set ground atoms formed using predicate symbols program ground terms herbrand universe arguments definition default value assume function default implement default knowledge assumptions assigns element element herbrand base closed world assumption used default herbrand base open world assumption used instead default herbrand base definition interpretation interpretation hbi consists following subset herbrand base mapping assign truth value element default belong definition interval inclusion given two intervals definition borel inclusion given two unions intervals partitioned intervals jil set border elements intervals except lower limit upper limit jil jik jjk borel algebra complete lattice borel inclusion herbrand base complete lattice set inclusion set interpretations forms complete lattice relation defined follows notice redefined interpretation borel inclusion respect definitions also redefine operational semantics therefore internal implementation fuzzy prolog library sections completely new uniformity reasons kept syntax used fuzzy programs definition interpretation inclusion let hbi hbi interpretations definition valuation valuation atom assignment elements variables ground atom herbrand context valuation substitution definition model given interpretation hbi model fuzzy fact valuations model clause following holds valuations union aggregation obtained model fuzzy program model facts clauses program every program least model usually regarded intended interpretation program since conservative model let appears following theorem meet operator lattice interpretations prove following result theorem model intersection property let models fuzzy program model proof let hbm since models models fact clause valuations facts hence therefore model clauses since hence since monotonic hence therefore model model remark least model semantic let set models program intersection models model least model denote least model program semantics semantics present based consequence operator least lfp declarative meaning program equal include clarity reasons although let fuzzy program herbrand base mapping interpretations defined follows let hbi fuzzy interpretation hbi cond cond cond ground instance fact solvable ground instance clause solvable note since must interpretation default set interpretations forms complete lattice continuous recall definition ordinal powers function complete lattice limit ordinal successor ordinal dually limit ordinal successor ordinal since first limit ordinal follows particular bottom element lattice top element kleene fixed point theorem know least continuous operator reached first infinite ordinal hence lemma let fuzzy program model proof let hbm hbtp vtp first prove direction let element herbrand base btp definition exists ground instance fact ground instance clause since model vtp btp vtp default analogously direction ground instance btp vtp btp vtp therefore model given relationship straightforward prove least model program also least theorem let fuzzy program lfp proof model lemma lfp fixpoint theorem operational semantics improvement fuzzy prolog remarkable new procedural semantics interpreted sequence transitions different states system represent state transition system computation tuple goal substitution representing instantiation variables needed get state initial one constraint represents truth value goal state computation starts initial goal true neither previous instantiations initial constraints get state first argument empty finished computation two arguments represent answer definition transition transition transition system defined fact program mgu truth value solvable rule program mgu constraint represents truth value obtained applying unionaggregation truth values solvable none applicable solvable default definition success set success set collects answers simple goals defined follows truei set elements herbrand base instantiated succeeded truei solution set truth values elements union got backtracking truth values obtained set constraints provided program query computed order prove equivalence operational semantic fixedpoint semantic useful introduce type canonical evaluation strategy strategy literals reduced step derivation obvious reasons derivation called definition transition given following set valid transitions transition defined literals reduced one step theorem given ordinal number hbtpn vtpn successful derivation lengh less equal program iff btpn solvable vtpn proof proof induction base case literals reduced using first type transitions last one literal exits fact mgu truth variable solvable default definition general case consider successful derivation transition literal reduced using fact rule result base case otherwise clause bmi mgu bji induction hypothesis solvable vji vji bji bji definition btpn solvable vtpn theorem program successful derivation truei iff solution proof follows fact theorem theorem fuzzy program three semantics equivalent proof first equivalence follows theorem second theorem implementation syntax clp constraint logic programming began natural merging two declarative paradigms constraint solving logic programming combination helps make clp programs expressive flexible cases efficient kinds logic programs clp linear arithmetic constraints computes real numbers fuzzy prolog implemented syntactic extension clp system clp incorporated library ciao prolog fuzzy library package ciao prolog terminology implements interpreter fuzzy prolog language modified handle default reasoning ciao system including fuzzy prolog implementation downloaded http syntax let recall syntax fuzzy prolog fuzzy prolog clause additional argument head represents truth value terms truth values subgoals body clause fact represented fuzzy prolog fact describes range values union intervals interval even real number particular cases following examples illustrate concrete syntax programs youth youth tall john tall john swift john swift john good player tall good player tall swift swift clauses expanded compilation time constrained clauses managed clp predicates ciao clp operators representing constraint inequalities use code predicates definitions use common operators theoretical definitions example first fuzzy fact expanded prolog clauses constraints youth youth fuzzy clause good player min tall swift expanded tall swift minim predicate included code library function adding constraints truth value variables order implement min minim minim minim min min minim min min min implemented several aggregation operators prod max luka lukasievicz operator etc similar way operator added system without effort system extensible user simply adding code new aggregation operators library combining crisp fuzzy logic example teenager student order use definitions fuzzy predicates include crisp subgoals must define properly semantics respect prolog close world assumption cwa going present motivating example fuzzy clauses usually use crisp predicate calls requirements data satisfy verify definition level superior crisp predicates ussually tests data satisfy body fuzzy clauses example say teenager student student whose age define fuzzy predicate teenager fuzzy prolog student note face risk unsoundness unless semantics crisp fuzzy predicates properly defined cwa means information false predicate definition student john student peter goal student succeeds john peter fails value different student john yes student nick means john student nick semantics prolog one going adopt crisp predicates want system compatible conventional prolog reasoning fuzzy predicates according human reasoning assume owa non explicit information unknown consider following definition age goal age succeeds john susan therefore know age peter know nick age definitely way introduce crisp subgoals body fuzzy clauses translating crisp predicate respective fuzzy predicate example obtain following prolog definition default truth value crisp predicate student nevertheless consider age teenager default value unknown whole interval observe following consults john nick peter means john age nick age data peter age expect behavior fuzzy predicate teenager john susan peter john teenager student student age susan teenager student student know value maturity peter student although student know age example timetable compatibility another real example could problem compatibility couple shifts work place example teachers work different class timetables telephone operators etc imagine company work divided shifts hours per week many workers combine couple shifts week predicate necessary check two shifts day hour day hour day hour day hour fig timetable compatible obtain couples shifts compatible two shifts compatible correct working days monday friday hours repetitions hour shift addition shifts disjoint compatible disjoint many compatible combinations shifts would useful define concept compatibility fuzzy way instead crisp way defined would express two shifts could incompatible one correct disjoint compatible less compatible level compatibility two shifts compatible working hours concentrated employee work days week also two shifts compatible free hours busy hours working days timetable therefore handling crisp concepts correct shif besides fuzzy concepts without definitions represented figure expressed language simple way using operator function definitions reserved word fuzzy predicate simple implementation fuzzy prolog combining types predicates could days hours fig fuzzy predicates without compatible min gives total weekly timetable hours joining two shifts number obtains total number working days weekly timetable number ree returns number free gaps weekly timetable working days corresponding fuzzified crisp predicates aggregation operator min aggregate value checking equal otherwise fails observe timetables figure obtain compatibility couple shifts represented timetable asking subgoal compatible result timetable timetable timetable shifts incompatible regarding compatibility shifts weekly timetable going ask questions shifts timetable figure one hour fixed yet note days week slice time one hour time beginning till one hour week timetable pair day hour one shift list hours week want know complete shift given level compatibility higher obtain slice wednesday monday morning compatible conclusions future work extending expressivity programming systems important knowledge representation chosen practical extended language knowledge representation prolog fuzzy prolog presented implemented prolog instead implementing new resolution system gives good potential efficiency simplicity flexibility example aggregation operators added almost effort extension prolog realized interpreting fuzzy reasoning set constraints translating fuzzy predicates clp clauses rest computation resolved compiler paper propose enrich prolog expressivity adding default reasoning therefore possibility handling incomplete information one worrying characteristics data information need usually available one part information available anyway searches calculations etc done information developed complete sound semantics handling incomplete fuzzy information also provided real implementation based former fuzzy prolog approach managed combine crisp information cwa fuzzy information owa default program great advantage lets model many problems using fuzzy programs extended expressivity language possibility applying solve real problems information defined fuzzy incomplete presently working several related issues obtaining constructive answers negative goals constructing syntax work discrete fuzzy sets applications recently published implementing representation model using unions instead using backtracking introducing domains fuzzy sets using types seems easy task considering using modern prolog types available implementing expansion systems studing advantages implementation xsb system tabling used using approach engine robots robocup league joint project universities references cabeza hermenegildo new module system prolog number lnai pages july cao annotated fuzzy logic programs fuzzy sets systems clark negation failure gallaire minker editors logic data bases pages new york plenum press dubois lang prade towards possibilistic logic programming proc pages mit press fitting bilattices semantics logic programming journal logic programmig fuhr probabilistic datalog implementing logical information retrieval advanced applications journal american society information science gelfond lifschitz stable model semantics logic programming fifth international conference symposium logic programming pages gelfond lifschitz logic programs classical negation seventh international conference logic programming pages jerusalem israel mit press extended abstract complete version new generation computing guadarrama vaucheret fuzzy prolog new approach using soft constraints propagation fuzzy sets systems fss issn hermenegildo bueno cabeza carro banda puebla ciao compiler system experimentation workbench future systems parallelism implementation logic constraint logic programming pages nova science commack usa april jaffar lassez constraint logic programming acm symp principles programming languages pages acm jaffar michaylov stuckey yap clp language system acm transactions programming languages systems kifer semantics expert systems uncertainty proc number lncs pages kifer subrahmanian theory generalized annotated logic programming applications journal logic programming klement mesiar pap triangular norms kluwer academic publishers lakshmanan epistemic foundation logic programming uncertainty lncs lakshmanan shiri probabilistic deductive databases int logic programming symposium pages lakshmanan shiri parametric approach deductive databases uncertainty ieee transactions knowledge data engineering loyer straccia uncertainty partial assumptions parametric deductive databases proc volume lncs pages lukasiewicz fixpoint characterizations disjunctive logic programs probabilistic semantics proc volume pages solving collaborative fuzzy agents problems clp manuel hermenegildo daniel cabeza editors international symposium practical aspects declarative languages padl volume lncs pages long beach usa springerverlag subrahmanian stable model semantics probabilistic deductive databases proc number lncs pages subrahmanian probabilistic logic programming information computation nguyen walker first course fuzzy logic chapman pradera trillas calvo general class triangular aggregation operators operators international journal approximate reasoning subrahmanian semantics quantitative logic programs proc ieee symp logic programming pages computer society press tarski fixpoint theorem applications pacific journal mathematics trillas cubillo castro conjunction disjunction fuzzy sets systems trillas pradera cubillo mathematical model fuzzy connectives application operators behavioural study yager zadeh editors information uncertainty fusion volume pages kluwer academic publishers series kluwer international series engineering computer sciences van emden quantitative duduction fixpoint theory journal logic programming vaucheret guadarrama fuzzy prolog simple implementation using clp constraints uncertainty paphos cyprus workshop http vaucheret guadarrama fuzzy prolog simple general implementation using clp baaz voronkov editors logic programming artificial intelligence reasoning lpar number lnai pages tbilisi georgia october wagner logical reconstruction fuzzy inference databases logic programs proc prague wagner negation fuzzy possibilistic logic programs logic programming soft computing research studies press ehud shapiro logic programs uncertainties tool implementing systems ijcai pages zadeh fuzzy sets basis theory possibility fuzzy sets systems
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oct growth deviations adam boocher alessio grifo jonathan alessio sammartano abstract deviations graded algebra sequence integers determine series residue field arise number generators certain algebras sense deviations measure far ring complete intersection paper study extremal deviations among algebras fixed hilbert series setting prove like betti numbers deviations decrease passing initial ideal maximized ideal also prove deviations grow exponentially golod rings certain quadratic monomial algebras introduction let polynomial ring field homogeneous ideal pair associate two important sets betti numbers namely dimk torsi dimk torr equivalently numbers given ranks free modules appearing minimal free resolutions respectively avoid confusion emphasize latter sequence typically infinite often refer instead series pkr much work devoted studying growth betti numbers various classes rings instance theorem pkr polynomial regular polynomial ring next simplest case sequence polynomial growth complete intersection finally complete intersection exponential growth paper study betti numbers point view related invariant set deviations since pkr integer coefficients constant term equal exist uniquely determined integers called deviations following infinite product expansion holds pkr sense explained remark deviations measure complexity play crucial role avramov proof complete intersection property localizes addition appear naturally number mathematics subject classification primary secondary key words phrases deviations algebra resolution series hilbert series betti numbers koszul algebra golod ring initial ideal ideal edge ideal generators degree acyclic closure number generators degree minimal model ranks components homotopy lie algebra section first results concern deviations algebras fixed hilbert function theorem prove term order algebra presented larger deviations moreover show theorem algebra presented ideal largest deviations generalization result peeva states ideal attains largest values pkr among hilbert function peeva theorem turn relies analogue due bigatti hulett pardue methods use techniques provide proof characteristics characteristic zero also present simpler proof using golod rings golod rings rings whose series maximal among ideals fixed embedding dimension set betti numbers see list golod rings particular golod ring pkr rational function determined edim second half paper turn analysis asymptotic behavior deviations result babenko shows radius convergence generating function coincides pkr case infinitely often theorem furthermore exist subsequences sparse exponential lower bound provided complete intersection theorem theorem prove golod rings certain koszul algebras sequence deviations asymptotically equal preliminaries introduce notions acyclic closures minimal models encode deviations set notation generally follow throughout paper although results therein proved local rings extended graded case minor changes proofs see also let algebra noetherian standard graded algebra field assume internal homological degree denote homological degree internal degree deg differential deg follows leibniz rule cycles form subalgebra boundaries form ideal therefore homology also algebra structure cycle let new variable denote ahyi unique isomorphism algebra extension differential satisfies exterior variable even divided power variable odd proper ideal construct algebra resolution ahy factor ring finite set variables homological degree minimally generates modulo homology classes minimally generate short algebra ahy obtained way called acyclic closure algebra variables called notice homology vanishes theorem proved independently gulliksen schoeller states acyclic closure rhy algebra fact minimal resolution one easily deduce deviation equal number homological degree card follows particular note koszul complex respect basic homological properties characterized terms vanishing deviations illustrated following discussion explains use word deviation remark first deviation equal embedding dimension thus following conditions equivalent field iii every exactness koszul complex gives following equivalent conditions regular iii every finally theorems assmus halperin tate give following equivalent conditions complete intersection iii every following notation construction acyclic closures denote homogeneous maximal ideal given homogeneous ideal starting build another algebra free resolution following steps adjoining polynomial variables instead divided powers kill cycles odd degrees algebra resolution obtained way called minimal model require generate minimally resolution obtained simply called model analogously minimal free resolutions equivalent condition model minimal pxn minimal models always exist unique isomorphism complete intersection equal characteristic zero isomorphic algebras however differ general following theorem due avramov shows minimal models also carry information deviations include statement standard graded case theorem avramov let let homogeneous ideal let model card every furthermore equality occurs every minimal model hence possible compute computing minimal model applications fact finer analysis sequence deviations edge ideals special graphs given finally third context deviations arise naturally homotopy lie algebras homotopy lie algebra graded lie algebra rhy rhy rhy acyclic closure rhy rhy denotes module universal enveloping algebra extr dimension graded component result due avramov states golod free graded lie algebra generated vector space homk denotes homology koszul complex refer chapter details subject extremal deviations goal section study extremal deviations among ideals given hilbert series first main result section theorem show deviations equal term order second main result theorem show among rings hilbert series ring largest deviations denotes ideal modify standard deformation argument apply minimal models integer function set called weight extend arbitrary monomials tnvn given polynomial denote highest weight monomial support sum terms weight equal let new variable define ideal notice upon setting obtain whereas upon setting obtain term order always weight theorem let weight homogeneous ideal every particular term order every proof result holds extend weight setting homogeneous ideal grading induced let minimal model since regular follows free sresolutions respectively hence models inherit algebra structure since positively graded grading induced direct adaptation proof allows conclude hence model minimal model theorem conclude every card serre showed following coefficientwise inequality formal power series holds pkr ring called golod equality achieved refer reader chapter thorough treatment golod rings two homogeneous ideals golod inequality holds proof next proposition extend result sequence deviations proposition let two homogeneous ideals golod rings every proof let golod ring homology koszul complex homotopy lie algebra free graded lie algebra generated vector space homk dimk every section since dimk every desired inequality follows remark remove assumption golod rings result true indeed one take ideals share betti numbers let term order exists zariski open subset gln ideal every denotes image change coordinates defined ideal called generic initial ideal respect denoted gin important property gin fixed action borel subgroup gln char means gin strongly stable thus golod ring theorem let hfi denote hilbert function ideal ideal vector space spanned lexicographically first hfi monomials degree shown ideal hfj hfl every note ideal strongly stable therefore golod ring present second main result section provide two proofs although second one works characteristic zero present order show application proposition theorem let let homogeneous ideal ideal every proof follow construction pardue approach similar one proof main result pardue proof shows ideal hilbert scheme connected ideal sequence deformations degenerations suffices show step deviations decrease result clear let generic changes coordinates since deviations depend isomorphism class generic changes coordinates preserve deviations passing initial ideal follows theorem polarization factoring generic hyperplane sections well known ideal polarization related via regular sequence linear forms finally generic hyperplane sections pardue employs always regular sequence deviations decrease second one proof char may assume let generic change coordinates proposition since gin golod rings hfgin hfi hfl conclusion follows proposition remark deduce written function deviations using sums products binomial coefficients see also thus pointwise inequality deviations implies pointwise inequality betti numbers way theorem recover fact series residue field grow larger passing initial ideal whereas theorem follows considering ideal gives largest series among ideals hilbert series fact originally proved however next example shows pointwise inequality betti numbers necessarily imply pointwise inequality deviations example let denote quotients generic quadrics respectively determine series following two rings property thus series satisfy equation hsr hsa pkr subalgebra yoneda algebra extr generated enveloping algebra graded lie algebra generated degree therefore hilbert series given hsa dimk notice deviations hilbert series two rings order determine proceed show computations case quadrics case analogous series hsa equal diagonal power series comparing find divide hsa find multiply get hsa hence since generated degree conclude obtain way obtain using expressions compute first deviations find whereas verify finally using partial fraction decompositions obtain estimates betti numbers showing thus exponential growth deviations goal section prove complete intersection deviations grow exponentially golod theorem koszul algebra hilbert series ring presented edge ideal graph theorem briefly introduce koszul algebras referring survey topic ring koszul algebra minimal free linear whenever koszul generated quadrics however two conditions equivalent instance classical theorem koszul quadratic monomial ideal denote hilbert series hsr dimk koszul following identity occurs pkr hsr following proposition provides compact formula deviations series certain form proposition let homogeneous ideal assume exist complex numbers pkr every function proof set assumption obtain proceeding apply natural logarithm sides coefficient corresponding maclaurin series obtain compare result follows applying inversion formula arithmetic function using fact every note assumption proposition satisfied several classes rings including complete intersections monomial rings golod rings section koszul algebras follows given two numerical sequences expression stands asymptotic equality abii example consider koszul algebra holds instance edge ideal complete graph proposition present next first main result section show deviations grow exponentially golod rings complete intersections similar techniques applied betti numbers see theorem let let homogeneous ideal golod ring exists real number proof let radius convergence pkr ring regular hence power series pkr infinite since golod hypersurface complete intersection see remark therefore moreover since golod pkr inside since pkr rational function singularities absolute value since nonnegative real coefficients conclude singularity pkr hence root show simple root root boundary result follow proposition otherwise set since therefore pprincipal simple root hence thus root proof completed let graph vertices denote edge edge ideal theorem algebra presented edge ideal koszul graph four vertices edges called claw see figure simple graph said claw appears induced subgraph note complete graphs figure claw following theorem shows asymptotic behavior deviations observed example holds generally theorem let graph exist proof let radius convergence pkr pkr inside dim since complete intersection conclude proof theorem root let multiplicity root result follow proof theorem via proposition show root boundary let independence complex simplicial complex ideal real roots graph hence two polynomials every since root satisfy relation see result follows remark theorem provides another class graphs whose deviations grow exponentially namely whose complementary graph chordal graphs precisely ones linear resolution note case characteristic field relevant thus golod theorem example examine detail deviations paths cycles vertices see also consider independence complex graphs roots determined explicitly respectively jnk cos cos relation root root note cases gives root minimum modulus note let another root map strictly decreasing follows root minimum modulus image root minimum modulus via map root converges goes infinity applying theorem get cos cos goes infinity expressions rhs approach observe integer theorem arbitrary example consists connected components copy hence theorem remark theorem also used produce examples koszul algebras exponential growth deviations two koszul algebras must deviations conclude asking combinatorial question question let graph let unique necessarily simple root minimum modulus note since already know exists real negative root minimum modulus affirmative answer question would imply exponential growth deviations graph complete intersection moreover already known independence complex admits unique necessarily simple root minimum modulus acknowledgements would like thank organizers lecturers participants pragmatic project started thank especially srikanth iyengar aldo conca introducing topic enlightening discussions last four authors also thank advisors aldo conca craig huneke bernd ulrich giulio caviglia respectively acknowledge use software authors also grateful referee helpful suggestions references alikhani peng independence roots independence fractals certain graphs appl math comput assmus homology local rings illinois math avramov flat morphisms complete intersections soviet math dokl translated dokl akad nauk sssr russian avramov local algebra rational homotopy homotopie locale luminy lemaire thomas eds soc math france paris avramov infinite free resolutions six lectures commutative algebra bellaterra progr basel avramov conca iyengar free resolutions commutative koszul algebras math res lett babenko analytical properties series loop space math notes translated mat zametki russian backelin les anneaux locaux relations monomiales ont des rationelles acad paris backelin koszul algebras veronese subrings rings linear resolutions rev roumaine math pures appl bigatti upper bounds betti numbers given hilbert function comm algebra boocher grifo sammartano edge ideals algebra resolutions matematiche chudnovsky seymour roots independence polynomial clawfree graph combin theory ser conca koszul algebras syzygies combinatorial algebraic geometry lecture notes mathematics springer note smallest root independence polynomial combin probab comput felix thomas radius convergence series loop spaces invent math determination class series math scand rings banach cent publ goldwurm santini clique polynomials unique root smallest modulus inform process lett grayson stillman software system research algebraic geometry available gulliksen proof existence minimal resolutions acta math halperin nonvanishing deviations local ring comment math helv herzog hibi monomial ideals springer london herzog reiner welker componentwise linear ideals golod rings michigan math hulett maximum betti numbers homogeneous ideals given hilbert function comm algebra mccullough peeva infinite graded free resolutions appear commutative algebra noncommutative algebraic geometry eisenbud iyengar singh stafford van den bergh eds math sci res inst cambridge university press pardue deformation classes graded modules maximal betti numbers illinois math peeva fixed ideals algebra peeva consecutive cancellations betti numbers proc amer math soc polishchuk positselski quadratic algebras univ lecture ser amer math providence roos research homological algebra mathematics computers simulation schoeller homologie des anneaux locaux acad sci paris sun growth betti numbers modules local rings small embedding codimension small linkage number pure appl algebra tate homology noetherian rings local rings illinois math uliczka remarks hilbert series graded modules polynomial rings manuscripta math adam boocher school mathematics university edinburgh james clerk maxwell building mayfield road edinburgh scotland address alessio dipartimento matematica degli studi genova via dodecaneso genova italy address dali grifo department mathematics university virginia cabell drive kerchof hall charlottesville usa address jonathan department mathematics university kansas snow hall jayhawk blvd lawrence address jmontano alessio sammartano department mathematics purdue university north university street west lafayette usa address asammart
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minimax signal detection weak noise nov marteau univ lyon claude bernard lyon cnrs umr institut camille jordan blvd novembre villeurbanne cedex france email marteau theofanis sapatinas department mathematics statistics university cyprus box nicosia cyprus email fanis november abstract consider minimax signal detection sequence model working certain ellipsoids space sequences real numbers ball positive radius removed obtain upper lower bounds minimax separation radius framework fixed value involved noise level use weak assumptions noise fourth moments assumed uniformly bounded particular use kind gaussianity independence assumption noise shown established minimax separation rates faster ones obtained classical sequence model independent standard gaussian noise surprisingly order minimax estimation rates classical setting additional condition noise classical minimax separation rates also retrieved benchmark inverse problems ams subject classifications keywords phrases ellipsoids inverse problems minimax signal detection inverse problems work supported labex milyon lyon within program investissements avenir operated french national research agency anr introduction consider following sequence model either known positive sequence unknown signal interest sequence random variables noise known parameter noise level observations given sequence joint law denoted denotes space sequence real numbers let known fixed constant concerning noise assume sup arises many situations consider instance stochastic differential equation known bounded linear operator acting unknown response function one wants detect estimate given stochastic process known parameter noise level sake simplicity consider case injective meaning trivial nullspace let standard wiener process identity operator retrieve fourier domain independent standard gaussian random variables direct problem operator retrieve independent standard gaussian random variables inverse problems compact operator retrieve since injective independent standard gaussian random variables inverse problems details regarding models refer let truncated fractional brownian motion let identity operator retrieve spline domain standard gaussian random variables details refer inverse regression problem also provides observations form indeed consider model known injective bounded linear operator acting unknown response function one wants detect estimate sequence independent identically distributed random variables zero mean variance one finite fourth moment given appropriate bases even tight frame see retrieve identity operator see chapter compact operator retrieve approximation fixed sequence depends see minimax signal detection considered literature last two decades refer contributions consider classical gaussian sequence model independent standard gaussian random variables refer survey available results discussion link asymptotic noise level assumed tend zero noise level assumed fixed approaches minimax signal detection aim work obtain upper lower bounds minimax separation radius framework general model weak assumptions noise set introduced particular use kind gaussianity independence assumption noise prove minimax separation rates faster ones obtained classical sequence model see surprisingly order minimax estimation rates classical setting moreover additional conditions noise show classical minimax separation rates retrieved benchmark inverse problems throughout paper use following notations given two sequences real numbers means exist resp means resp also min minimax signal detection given observations consider signal detection problem aim test versus given sequence positive real numbers radius set defined set seen condition decay cases sequence increases fast correspond signal small coefficients case corresponding signal considered smooth sequence fixed main issue minimax signal detection problem characterize values radius hypotheses called null hypothesis called alternative hypothesis separable following test defined measurable function observation values set convention rejected rejected given test investigate type first kind error probability defined sup measures worst probability rejecting true defined often constrained bounded prescribed level type second kind error probability defined sup measures worst possible probability rejecting true defined one would like ensure bounded prescribed level emphasize classical minimax signal detection problem protection possible noise distributions required since noise distribution completely known however general setting consider order produce kind robustness adapted definitions type type error probabilities accommodate possible uncertainty noise let given let test definition separation radius test class defined inf sup sense separation radius corresponds smallest possible value available signal separated test prescribed type type error probabilities respectively definition minimax separation radius class defined inf infimum taken tests minimax separation radius corresponds smallest radius exists test type error probability greater worth mentioning definitions valid fixed required performances given test easy handle sense type error probability bounded test dependence minimax separation radius respect given precisely described control upper lower bounds spectral test control upper bound define spectral test model defined first show test obtain upper bound type error probability given bandwidth consider following spectral test denotes threshold depending easily seen assumption guarantees variance finite every proposition let given consider spectral test defined sup soon proof proposition postponed section remarks using simple bounds easily seen sup sup since hence choice max ensures satisfied spectral test defined test classical setting independent gaussian noise threshold chosen variable case since uniform bound fourth moment sequence available proposition let given consider spectral test defined select threshold sup radius solution equation proof proposition postponed section remark practical purposes solution equation chosen particular exists hence ensuring satisfied small enough control lower bound propose lower bound minimax type error probability defined sequel term inf corresponds infimum taken possible tests proposition let fixed sup inf proof proposition postponed section main difficulty construct appropriate distribution allow one obtain largest possible lower bound minimax separation radius following theorem provides upper lower bounds minimax separation radius defined theorem let given minimax separation radius satisfies inf sup solution equation proof theorem postponed section remark sequences satisfy constants easily seen upper lower bounds minimax separation radius established theorem order follows easily working along lines proof proposition note also condition satisfied various combinations interest among mildly inverse problems ordinary smooth functions severely inverse problems ordinary smooth functions iii mildly inverse problems functions eks among possible situations condition satisfied one mention instance behaviors see also remark remark note upper lower bounds minimax separation radius established theorem quite different compared classical minimax separation radii available literature obtained independent standard gaussian noise see although bias terms coincide corresponding variance terms differ particular defined variance term gausiian noise order qpfor independent standard variance term order greater stress term entails minimax separation rates faster compared ones obtained classical model also worth mentioning surprisingly bias variance terms defined order corresponding terms classical minimax estimation setting particular minimax separation rates general setting coincide minimax estimation rates obtained classical estimation setting illustrative purposes table see also table provides minimax separation rates benchmark problems mildly severely problems ellipsoids ordinary smooth supersmooth sequences minimax separation rate mildly severely exp exp table minimax separation rates defined remark supremum possible noise distributions considered definition type type error probabilities easily seen upper bound type error probability obtained proposition still holds true however corresponding lower bound obtained proposition true gaussianity implies minimax separation rates displayed table still valid standard gaussian noise additional condition noise obtain classical minimax separations rates section demonstrated additional condition noise one able retrieve classical minimax separation rates benchmark inverse problems recall equation displayed proof proposition variance written cov classical setting independent standard gaussian noise hence order retrieve classical minimax separation rates defined needs order achieve separately null alternative hypotheses benchmark problems mildly severely inverse problems stress section deal upper bounds indeed lower bounds established previously literature see theorem independent standard gaussian noise still valid mildly inverse problems assume refers inverse problems refers mildly inverse problems start discussion null hypothesis recall var cov using simple calculations see aim exhibit condition least order assumption let defined let bivariate gaussian random vector moreover exists due isserlis theorem see seen thanks assumption cov results allow propose sharp control variance null hypothesis proposition assume assumption holds proof proposition postponed section propose similar analysis alternative hypothesis proposition assume assumption holds proof proposition postponed section starting using propositions get provided holds soon last inequality provides classical condition already discussed theorem specific case noise assumed independent standard gaussian entails assumption suffices retrieve classical minimax separation rates mildly inverse problems severely inverse problems assume section since minimax estimation minimax separation rates classical setting order see tables stress deteriorate classical minimax separation rates words independent standard gaussian assumption noise needed get classical minimax separation rates severely inverse problems concluding remarks established minimax separation rates general gaussian sequence model noise need neither independent standard gaussian rates faster ones obtained classical setting independent standard gaussian noise surprisingly order minimax estimation rates classical setting involved spectral test depends unknown smoothness parameter signal alternative hypothesis therefore paramount importance practical applications provide minimax testing procedures explicitly depend associated smoothness parameter usually referred adaptation problem however investigation needs careful attention beyond scope present work particular dependency involved constant respect level intricate form one involved classical setting appendix proof proposition let fixed using markov inequality get provided proof proposition let fixed using markov inequality obtain implicitly assumed need upper bound variance term first remark cov calculation using simple algebra get hence using last equality obtain max constant introduced sup sup note using first inequality inequality see get max combining inequalities obtain max calculation first remark noting get hence using inequality expectation expression obtain using first inequality inequality get hence combining obtain using inequality get hence since easily seen using choosing get provided solution equation conclude proof since remark inequality satisfied provided proof proposition gaussian write instead instead denotes associated covariance matrix also define refers test let fixed values made precise later sup inf inf sup inf sup inf sup inf likelihood ratio probability measures last inequality refer particular find inf sup let fixed impose following conditions let remaining submatrix slight abuse notation simple algebra denote exp exp exp hence exp exp easily seen exp exp exp exp exp exp exp hence exp select follows define note kvk easily seen exp first construct specific let denotes sequence independent standard gaussian random variables real sequence obviously since gaussian max max sup need bound expression using get note also since unit vector max since largest eigenvalue smaller using get since hence exp provided conclude proof need ensure constructed belongs remark since increasing sequence inf sup provided hence soon proof theorem proposition proved exists test sup radius satisfying setting arg inf denoting associated test get sup radius satisfying inf hence inf similarly using proposition sup inf radius results occurs hence sup inf radius sup entails sup proof proposition remark assumption particular sum finite whatever value provided hence assumption get proof proposition recall using max moreover using note using proposition using immediately see order conclude using inequalities get cov summarizing computations obtain max used inequality see references baraud minimax rates testing signal detection bernoulli bissantz claeskens holzmann munk testing lack fit inverse applications biophotonic imaging journal royal statistical society series castillo rafeiro introductory course lebesgue spaces cms books smc springer cavalier estimation problem fractional integration inverse problems cavalier inverse problems statistics inverse problems estimation volume lect notes stat pages springer heidelberg ingster sapatinas suslina minimax nonparametric testing problem related radon transform mathematical methods statistics ingster sapatinas suslina minimax signal detection inverse problems annals statistics ingster suslina nonparametric testing gaussian models volume lecture notes statistics new york isserlis formula coefficient order normal frequency distribution number variables biometrika johnstone wavelet shrinkage correlated data inverse problems adaptivity results statistica sinica laurent loubes marteau testing inverse problems direct indirect problem journal statistical planning inference laurent loubes marteau non asymptotic minimax rates testing signal detection heterogeneous variances electronic journal statistics mallat wavelet tour signal processing edition academic press san diego marteau general regularization schemes signal detection inverse problems mathematical methods statistics marteau sapatinas unified treatment asymptotic approaches minimax signal detection statistics surveys tsybakov introduction nonparametric estimation springer series statistics springer new york revised extended french original translated vladimir zaiats
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efficient robust pedestrian detection using deep learning navigation mateus david ribeiro pedro miraldo jacinto nascimento sep instituto sistemas larsys instituto superior lisboa torre norte piso pais lisboa portugal corresponding author abstract paper addresses problem navigation han using multi camera sensors implement person tracking system main contributions paper follows novel efficient deep learning person detection standardization constraints first stage approach propose cascade aggregate channel features acf detector deep convolutional neural network cnn achieve fast accurate pedestrian detection regarding human awareness defined constraints associated robot motion use mixture asymmetric gaussian functions define cost functions associated constraint methods proposed herein evaluated individually measure impact components final solution including proposed pedestrian detection constraints tested typical domestic indoor scenario four distinct experiments results show robot able cope constraints defined common proxemics social rules keywords pedestrian detection convolutional neural network navigation introduction robots interact naturally humans social environments must ability plan motion accounting typical social norms paper address robot navigation presence humans resorting multi cameras static outside onboard cameras person tracking system one research focus robotics interaction role social environments people think robot interacting person comes mind robot speak hand object however motion great importance social context robot requested fetch item simply normal navigation behavior needs adjusted according proxemics rules disturb people study robot navigation presence people called humanaware navigation han approaches han literature use sensors onboard robot even though sensors bring advantage useful external sensors add information environment terms coverage space also terms precision estimation person position hence paper addition onboard camera external cameras mounted ceiling used pedestrian detection task setup ensures broader perception environment capable humans robots time preprint submitted elsevier robotics autonomous systems furthermore paper address importance incorporating deep learning han based architecture integration provides efficiency robustness pedestrian detection task detailed next traditionally detection task usually accomplished sliding window paradigm based exhaustive search image find object positions search features window location extracted computed possibly multiple scales later evaluated classifier however procedure easily become intractable due substantial number windows complexity features processing thus use richer expensive features necessary constrain computations restricted number locations considering certain regions interest accordingly propose cascade acf detector cnn acf allows obtain selective search identifies promising image regions presence pedestrians proposals alleviates cnn task since number windows proposals classify substantially reduced advantages cascade twofold firstly use expensive computations cnn classification promising acf proposals consequently operating regions interest paradigm allows speed detection procedure perform pedestrian detection real time requirements methodology previously described contributes efficiency goal secondly first stage cascade acf provides october large number false positives number fps drastically reduced application cnn maintaining true positives means errors corresponding fps provided acf solved within cnn processing approach allows achieve robustness goal cnn model results first large object dataset dataset transfer learning procedure adopted improves final model order guanratee independence particular application han context performed although cnn could specific han dataset finally also propose novel solution han resorting multiple cameras onboard offboard people state estimation coupled aforementioned strategy costmap computed combining several constraints associated han time robot receives new goal computes path costmap compared approaches main contributions presented paper novel efficient technique people detection standardization constraints moreover since computations performed using cpu methodology economically viable solution reaches intended runtime figures accuracy requirements therefore mentioned approach integrated robots onboard cpus gpus solution tested simulated realistic scenarios results show proposed solution fulfils aforementioned goals paper proposes novel extensions authors previous works namely specifically han constraints introduced algorithm proposed comparing present paper additional outside cameras used han constraints introduced tested additional details included method evaluation related work classifier locally decorrelated channel features ldcf accurate variant acf also slower decorrelates previously mentioned image channels resorting linear filters idea adding filtering step features studied work nevertheless application deep compositional architectures namely convolutional neural networks tasks image classification localization detection significantly boosted application deep learning arises natural forthcoming step indeed deep learning based architectures learn hierarchical features make possible reach better classification performance using handcrafted ones avoid overfitting training cnn models dataset must substantial size contain corresponding annotations however apparent limitation since problem addressed transferring parameters already trained cnn model datasets belonging tasks example generic object classification model interest cnn concerning model interest retrained dataset corresponding specific problem computations associated cnn expensive compared ones required methods using handcrafted features therefore improve detector speed hybrid solution adopted cascading faster shallower method based handcrafted features deep cnn handcrafted approach generates proposals promising regions pedestrians locations whose classification refined cnn accuracy enhanced removing false positives currently various deep learning approaches developed literature mainly extending main pipeline successful object detectors fast faster pipeline comprises combination proposal extraction cnn evaluation identical hybrid scheme generic example acf used conjunction fast architecture two cnn branches small large scales work able improve proposal extraction module faster scheme resorting diverse network outputs order detect objects different scales work presented uses region proposal network cnn based faster refines detections resorting cnn features boosted decision trees approach adopt similar architecture follows pipeline however apply bounding box regression contrasting besides since concerned speed introduce acf score rejection threshold value proposals eliminated processed cnn allowing achieve faster running time figures main focus devoted integration computer vision module method han module order achieve accurate fast enough system give better comprehensive understanding contributions presented herein next review related work han respectively pedestrian detection one main goals framework achieve accurate fast pedestrian detection algorithm one major topics addressed computer vision community surveys available classically problem addressed using conventional handcrafted features image gradient hog wavelets etc plateaued recent years popular handcrafted methods include aggregate channel features acf individual pixel lookups extracted concatenation luv histogram oriented gradients gradient magnitude image channels pixels serve features applied boosted decision trees navigation regarding han planner approach found work focuses human comfort addressed three criteria preventing personal space invasions navigating humans field view fov preventing sudden appearances fov humans criteria modeled cost functions costmap path planning performed algorithm even though han planner accounts replanning people move adapt personal space motion mind two extensions han proposed prediction cost function increasing cost front moving human decreases probability robot entering area concept compatible paths means two paths compatible agents follow paths reaching goal position without deadlocks figure illustration proposed methodology cascading acf nondeep detector deep cnn first acf detector performs selective search identifying promising image regions might contain pedestrians generates pedestrian proposals see green rectangles third image figure candidate proposals rgb feature map forwarded cnn accurately classified see text alternative approach proposed differs han planner considered constraints formulation instead focusing simply human comfort constraints concerning social rules navigate right side narrow passages human navigation behavior face direction movement also taken account another important issue related human comfort social context interference humans interacting humans objects issue tackled besides considering proxemics back space person constraint included model space interacting entities important work presented authors presented framework planning smooth path set milestones added deleted modified based static dynamic components environment recently defined three goals humanaware navigation human comfort space people keep different contexts known theory proxemics velocity robots navigate close humans respect social rules mimic human behavior present results complete framework han section concludes paper people detection deep learning paper follows strategy mentioned previous section providing following contributions first adopt deep learning based approach using pretrained models second able drastically decrease computational effort associated exhaustive search performed sliding window process accomplish cascade acf detector cascade strategy twofold first provides selective search approach significantly improves computational efficiency since output proposals acf taken account cascading cnn able boost performance acf detector improve classification accuracy acf proposals reducing number false positives fig illustrates proposed approach task outline paper paper organized follows section introduces main stages proposed framework section describes methodology section details cnn model operations used section related deep learning methodology integrated navigation setup accomplish cnn training must performed section well adaptation cnn section experimental evaluation conducted section evaluate performance proposed methodology inria dataset section two real scenarios comprising corridor mbot sequences section section presents han constraints used proposed framework section evaluate han constraints section methodology section formalize adopted methodology first let consider available following training set denotes input image denoting image size class label defined denotes absence presence pedestrian ith image training dataset input acf detector input image detector recall exist several detectors regionlets ldcf choose acf detector since fast paper rgb feature map considered image bias number classes consideration output cnn mentioned seen approximation input data represented equation convolution mentioned formally defined acf ldcf used provide candidate windows proposals along scores confidences formalized following output set set represents set bounding boxes coordinates denoting point width height enclosing pedestrian denote content proposals image delimited bounding boxes acf detector confidence scores assigned proposals mentioned generalization ability cnn boosted resorting models instead using random initialization therefore use proposed vgg cnn model imagenet formally dataset cnn number classes model imagenet case number classes section detail transfer learning accomplished classification problem stands convolution input region addresses convolutional filters represented weight matrix bias vector notice input obtained following structure convolution activation operations preceding layer represents input image convolutional layers followed sequence fully connected layers perform particular instance convolution entire vectorised input denotes final stage fully connected layers followed classification layer defined function follows see fout softmax wout cnn model formalize main ingredients cnn architecture basically type deep networks comprises several processing stages stage characterized two types layers namely convolutional layer containing nonlinear activation function subsampling layer former convolutional filter applied input latter size reduction input achieved two stages typically followed several fully connected layers multinomial logistic regression layer see details formally convolutional neural network analytically represented following mapping represents image space represents classification space function defined exp exp fout represents output inference process takes input representing number classes case input proposals acf detector represents two output classes pedestrian thus written similarly fout ffc denotes composition operator represents convolutional layer represents model parameters comprising input weight matrices rkl bias vector rnl layer representing size filters layer input channels represents activation layer rectified linear unit relu details see layer function allows obtain denotes subsampling function pools using mean max functions values region input data ffc layer containing weights representing connections fully connected layers biases also belong model parameters fout multinomial logistic regression layer containing weights wout fout ffc inputs proposals image content rgb feature map delimited bounding boxes denoted see fig main idea take proposals processed cnn produce classification probability given proposal contains pedestrian proposals classified non pedestrians discarded allowing eliminate false positives ones regarded pedestrians kept including original acf detector score case cnn prediction output formally represented basic operations cnn subsampling layer necessarily need present every case details also available http trained using binary loss training set indexed follows training images negative images containing pedestrians positive images containing pedestrians test images experimental setup build training set also validation set cnn model obtain positive set use ground truth positive training bounding boxes proposals corresponding image content delimited bpos data augmentation performed positive samples using following two steps log log let cnn represented model process cnn defined following three steps training stages see block cnn illustrated fig convolutional subsampling layers represented parameters horizontal flipping set bpos resulting new set bpos including also bpos random deformations including translation scale range pixels beginning end applied previous set bpos allows tain new set bpos training fully connected layers see green block cnn illustrated fig represented parameters training one multinomial logistic regression layer parameters see red block cnn illustrated fig minimizing loss function dataset build negative set bneg extract negative windows proposals negative images using strategy mentioned employing trained version ldcf result set negative windows obtained defining negative proposals per image strategy allows acquire total proposals used training validation worth mention transferring large number layers cnn key achieve best classification results transfer learning problems following strategy first take layers initialize new model see since changed cnn input size reduce computational expense layers randomly initialized gaussian distribution dimensions compatible inference possible finally introduce new binomial logistic regression layer parameters randomly initialized gaussian distribution adapted two classes pedestrian afterwards cnn model minimizing crossentropy loss function using pedestrian training set adaptation cnn model mentioned sec use cnn model necessary following describe required steps first detailing model used describing modifications needed model used network select vgg deep architecture specifically configuration cnn architecture receives input images network comprises convolutional layers three fully connected layers multinomial logistic regression layer see sec five operations nonlinear subsampling operate region reducing input factor two operations appear layers rectified linear unit relu selected applied convolutional fully connected layers furthermore size receptive field convolutional layers however number filters different described materials methods section addresses implementation details adopted deep learning methodology integrated navigation setup first sec describe experimental setup used train deep cnn sec addresses adaptation model task outcome sections final cnn architecture finetuned subsequently used testing purposes described sec cnn training inria dataset train cnn model use inria dataset common benchmark used research work detection pedestrians dataset comprises filters layers filters layers see additional details http details found http filters layers table logarithmic average miss rate frames per second fps number true positives false positives false negatives inria dataset acf detector filters layers filters layers dataset filters layer filters associated different ilsvrc class layer function inria baseline mentioned model performed imagenet classes million training thousand validation thousand test images inria threshold adaptation deep network order decrease computational effort required model expected input size cnn reduced original nevertheless due change input size inference possible first fully connected layer moreover task requires use two classes order represent presence absence pedestrian therefore layer must adjusted accordingly order solve aforementioned issues choose randomly initialize parameters three fully connected layers specifically obtained gaussian distribution zero mean variance changed network using previously mentioned sec positive negative sets obtained inria pedestrian dataset process network acts regularization procedure similar data augmentation regarding hyperparameters use epochs minibatch samples learning rate momentum special effort dedicated mentioned hyperparameters test run acf detector inria test images order obtain proposals regions potentially containing pedestrians run proposals cnn order classify pedestrians experiments detailed next see sec tabs obtained matlab running cpu mode ghz intel core ram bit architecture piotr computer vision matlab toolbox version employed execute acf method perform performance evaluation regarding implementation cnn framework matconvnet toolbox utilized metrics fps fps acf proposals sec evaluate performance method real scenarios comprising two sequences termed herein corridor mbot special attention given details achieve requirements detection performance evaluation inria dataset test accuracy module integrated navigation setup first assess performance inria dataset adopted evaluation metric log average miss rate proposed metric obtained nine values false positives per image fppi interval since miss rate lower values correspond superior performances improved accuracies demonstrate effectiveness proposed cascade detector acf cnn architecture also present tab field inria baseline results acf alone results cascade order notice improvement achieved experiments conducted achieve log average miss rate acf detector alone cascading acf cnn able reach tab shows number true positives false positives false negatives use cnn notice number significantly reduced maintaining indicates cnn successfully discarding allowing reach performance improvements justifying observed gain important concern building algorithm verify ensure runtime figures satisfy realtime requirements tab shows acf running times frames per second fps obtained inria dataset results conclude system able perform fps improved introduction cnn pipeline evaluation people detection section provides testing results evaluation proposed method section divided two parts sec describe performance evaluation concerning accuracy runtime figures method inria dataset used approach proposed herein shown fig general applied detector regionlets ldcf spatial pooling detectors acf method resulted significant runtime decrease fps fps improve mentioned runtime figures achieving much accuracy possible two strategies used reduce original images size discard acf proposals confidence score certain threshold work latter option adopted notice reducing size images may jeopardize quality detections furthermore confidence scores outputted acf detector constitute relevant indicator filter proposals achieved applying threshold proposals ones certain score kept processed cnn experiments threshold value set confidence score threshold accordance value suggested proposed thresholding technique based upper lower bounds confidence scores acf another important remark gain speed results fact cnn classify smaller portion acf proposals instead therefore easier false positives discarded threshold operation harder false positives discarded cnn threshold value controls potential accuracy loss speed gain selecting later case threshold operation able improve runtime overall detector compared results obtained baseline results shown tab field inria threshold seen frame rate increases fps baseline fps threshold order assess speed metrics suited realtime applications perform experiments real han scenarios sec table runtime figures top baseline bottom threshold threshold operation applied acf proposals using overall method dataset baseline threshold data seq corridor data seq mbot total time sec total time sec avg det acf avg det acf avg det avg det acf time sec acf time sec cnn time sec cnn time sec frame rate fps frame rate fps total time sec total time sec avg det acf avg det acf avg det avg det acf time sec acf time sec cnn time sec cnn time sec frame rate fps frame rate fps number detections avg acf application cnn also cnn time data sequence top two columns field named baseline applying threshold operation described sec possible obtain approximately fps datasets shown tab field threshold suited applications navigation method described previous section get bounding boxes images representing pedestrians multiple instances proposed applied images several cameras time assuming set images single image also used goal section firstly project position pedestrians image world environment fuse information given different imaging sensors define han constraints included conventional path planer computing position pedestrian world coordinate systems including estimation pedestrian velocity methods used paper module shown sec main goal section define robot path environment humans may interact possible three goals considered namely human comfort respect social rules naturalness fulfill requirements following constraints taken account performance evaluation real scenarios perform experiments real scenarios acquired two indoor datasets evaluate task two datasets considered experiments corridor dataset containing images mbot dataset containing images size frame image sequences results obtained two datasets shown fig bounding box score showing confidence containing pedestrian total time perform dataset corridor frames mbot frames roughly seconds seconds respectively running time figures per frame shown tab table field baseline refers metrics final detector detailed sec algorithm reaches fps fps corridor mbot sequences respectively runtime longer mbot dataset number acf detections processed cnn larger comparison corridor dataset observation depicted tab comparing average take least effort path naturalness sec keep distance static obstacles naturalness sec respect personal spaces human comfort sec pedestrian detection corridor sequence pedestrian detection mbot sequence figure real scenarios cameras mounted ceiling robot test used two sequences images acquired possible real scenario camera locations figs shown three images corridor mbot sequences respectively avoid navigating behind sitting humans human comfort sec create remove signaled tracks people tracking performed world coordinate system instead image plane purpose two assumptions considered people standing walking given person feet always ground plane thus point represents person feet line bottom left right corners respective bounding box since person times center bounding box good estimate position feet middle point considered line segment projection performed transforming selected point homography computed priori transformation image plane floor plane positions associated targets used measurements kalman filters one person perform tracking constant velocity motion model considered prediction step filter association measurements targets performed nnjpda method hard assignment method one measurement assigned target using maximum posteriori map approach association complete condition checks assignment distance higher threshold case track measurement unassigned assignment three cases considered interfere interactions human comfort sec overtake people left social rule sec first two constraints related navigation problems method used paper described sec whereas remaining four constraints han details constraints shown sec people tracking world explained previous sections module returns bounding boxes representing people scene sent module uses nearest neighbour joint probabilistic data association nnjpda array kalman filters track people one person performs following steps project middle point bottom left right corners bounding box ground plane world coordinates predict new state track associate measurements existing tracks determine tracks need created removed update prediction respective measurement experiments also considered seated person even though method designed type cases results satisfactory variables define space around person assignment exists track assigned represent person position measurement assigned graphical representation parameters shown fig personal space walking person modelled successful assignment proceeds correction step filter assigned measurement track assigned two possibilities track increases inactivity flag inactivity threshold reached track deleted measurement assigned iterations new kalman filter created measurement velocity estimates required han constraints given kalman filter state estimates considering person state position velocity asymgauss max person orientation speed graphical representation person walking along direction velocity presented fig regarding walking person makes sense personal space front larger back ensure robot pass front person decreasing risk collision hand person standing consider personal space defined using previous formulation robot may pass behind close person causing discomfort thus case propose personal space modelled circular gaussian exp path planner obstacle avoidance first han constraint addressed path planner see list beginning sec work algorithm used ensuring minimum cost path long heuristic admissible total cost node given sum cost reaching node heuristic cost latter considered euclidean distance initial goal position since environment dynamic people may appear walking scene planner computes path periodically goal second constraint prevent robot passing close obstacles solved attributing high cost area surrounding obstacles next subsection define remaining constraints included path planner navigation standard deviation direction respectively formulation also considered seated person object hand required robot able enter personal space arm length however robot allowed enter random direction instead allowed approach person front thus solution open region front person degrees distance personal space hand scenario depicted figure navigation cost functions cost functions used work based previous approaches however reformulated namely cost functions associated constraints order better integrated approach standardize formulation visibility constraint constraint concerns preventing discomfort passing behind seated person reformulated problem asymmetric gaussian personal space cost functions third constraint list accounts personal space models presented next consider three different situations person standing walking seated case walking person used formulation proposed authors call asymmetric gaussian asymgauss asymgauss interaction constraint fifth constraint prevents robot interfering person interacting object represented interaction set modelled circle pex pey otherwise orientation function variance direction middle position interacting entities denoted pex pey radius half distance entities interactions considered importance factor varies variance direction variance direction personal space object hand seated person walking person figure representation cost functions associated different people postures fig represents cost function personal space person walking direction fig shows cost function person standing oriented direction object hand fig represents cost function case person seated fig shows total cost function walking person including social rule overtaking left personal space robot placed behind person goal position defined person starts walking robot replan path person left robot requested hand object person across room however needs pass seated person starts moving turned prevent interference robot replans around area however yet another person walking behind couch must taken account figure evaluation proposed navigation system using simulated environments environment created using gazebo results shown rviz ros package figs show sequences images representing experiments respectively details regarding experiments given text cases seen costmap current goal position path trajectory image robot platform used experimental results realistic scenarios image camera mounted ceiling image camera mounted ceiling image camera mounted ceiling robot camera image depiction environment rviz showing robot position pedestrian positions cameras figure representation setup used experiments realistic scenario fig shows robot platform figs show images cameras used detect pedestrians seen images already show bounding boxes identifying person environment conclude fig shows environment ros rviz package position cameras position robot pedestrian respective han constraint case pedestrian standing proposed system implemented extension ros navigation stack cost functions described previous section implemented costmap layered structure simulation environment gazebo running machine intel core ram experiments performed overtake constraint constraint represents social rule overtaking people left considered walking persons constraint also represented using asymmetric gaussian asymgauss cost function fusion three possible postures person standing seated walking two one cost function applied must combined since main goal framework maximize comfort humans cost functions combined taking maximum cost value attributed point space first case multiple cost functions affecting space seated person whose personal space given max experiment robot navigating encounters slow walking person must overtake goal verify respects constraints sec experiment robot requested hand object person across room starts moving turned robot replans path however another person going across room behind seated person must overtaken reach goal experiment designed evaluate navigation module regarding constraints sec results experiments shown fig next present experimental results using simulated realistic environments cost function depicted fig second case concerns walking person personal space must combined respective social rule person overtaken left max results complete framework section evaluate proposed framework using proposed han purpose use mbot mobile platform see fig typical indoor work scenario shown fig using five distinct cameras one onboard four cameras fixed ceiling figures showing images captured onboard camera three ceiling cameras shown figs graphical representation cost function shown fig evaluation navigation constraint evaluate proposed constraints han two experiments defined tested simulated environments validation consider following experiments starts moving person turns thus robot needs replan path interfere interaction started seen second image fig pedestrian partially occluded couch onboard sensor thus running sensor able detect person environment however since using another sensor external robot detecting person robot knew person turned robot replaned path according respective constraint last experiment show case robot asked receive object person case robot identify person position given proposed algorithm robot navigate towards position enter personal space able receive object seen fig proposed han work expected even case pedestrian seen onboard sensor video experiments sent supplementary material ros packages han available community experiment first experiments robot must avoid three standing people distributed along main corridor environment performing slalom path avoid entering personal space experiment robot navigating encounters slow walking person must overtake goal verify respects constraints sec experiment person seated couch watching robot wants across room goal test robot respects constraints sec experiment robot navigates towards person hand object goal verify modification personal space constraint sec regarding han throughout experiments robot displayed similar behaviour simulation terms trajectory execution however parameters cost functions needed adjusted values presented previously derived empirically taking account values literature space restrictions real scenario intuition comfort distances next subsection discuss results obtained experiments conclusions work addresses problem navigation robot social context purpose paper derive robust efficient solution using deep learning addition regarding han reformulate respective constraints order standardization constraints validate contributions first use inria dataset evaluate accuracy runtime figures method evaluate performance using real data use two sequences images one acquired using robot onboard camera second sequence acquired using external camera results show method robust fast enough frames per second robot navigation applications evaluate standardization han constraints use simulated environment realistic scenario test modules four distinct scenarios results show robot correct behavior means han working properly discussion experimental results let start experiment results shown fig firstly one see proposed correctly detected pedestrians including onboard sensor able detect people front robot using information robot planned executed correct path towards given goal performing slalom path avoid entering personal space regarding second experiment different setup considered person environment robot starts move person start moving blocking robot path seen fig firstly robot plans path taking account person standing environment person starts walking robot replan path overtake pedestrian left proposed able detect people different sensors including onboard sensor since using multiple sensors detections even onboard camera see pedestrian robot continues right path towards goal results one conclude robot able correctly detect pedestrian able plan path according defined humanaware navigation constraints considering third experiment goal avoid passing front person watching person interacting object results shown fig firstly robot plans motion according less cost path towards given goal position would include passing person robot references references dalal triggs histograms oriented gradients human detection ieee proc computer vision pattern recognition cvpr felzenszwalb girshick mcallester ramanan object detection discriminatively trained part based models ieee proc pattern analysis machine intelligence viola jones robust face detection int computer vision zhu chen yuille freeman latent hierarchical structural learning object detection ieee proc computer vision pattern recognition cvpr dollar appel belongie perona fast feature pyramids object detection ieee trans pattern analysis machine intelligence figure experiment people standing robot movement top bottom show rviz representation environment people robot positions path planned instant detection onboard camera sensor three detections three different cameras mounted ceiling figs one see robot navigation going towards goal position avoiding people respecting personal space constraint figure contrarily previous experiment case person environment instead standing starts moving blocking robot path figure show environment images taken cameras similar fig figs see firstly person identified standing robot starts moving also starts moving thus robot replans path order overtake pedestrian left notice since using multiple sensors even person outside onboard camera fov robot performs well pedestrian identified external sensors figure figure shows results third experiment top bottom show representation environment including people robot positions path planned detections onboard camera detections external camera results shown figs possible observe two paths planned followed robot firstly robot plans path passing couch pedestrian turned robot replans path around couch figure similar scheme shown fig top bottom show environment representation location pedestrian robot well robot path detections onboard camera detections external camera figs possible observe robot follows path planned hand position costmap personal space changed robot reach position girshick donahue darrell malik rich feature hierarchies accurate object detection semantic segmentation ieee proc conf computer vision pattern recognition cvpr hosang omran benenson schiele taking deeper look pedestrians ieee proc conf computer vision pattern recognition cvpr russakovsky deng krause satheesh huang karpathy khosla bernstein berg imagenet large scale visual recognition challenge int computer vision yosinski clune bengio lipson transferable features deep neural networks proc neural information processing systems nips mateus miraldo lima sequeira navigation using external omnidirectional cameras iberian robotics conference ribeiro mateus miraldo nascimento deep learning pedestrian detector robot navigation autonomous robot systems competitions icarsc ieee international conference ieee dollar wojec schiele perona pedestrian detection evaluation state art ieee trans pattern analysis machine intelligence benenson omran hosang schiele computer vision eccv workshops zurich switzerland september proceedings part springer international publishing ten years pedestrian detection learned nam dollar han local decorrelation improved pedestrian detection proc neural information processing systems nips zhang benenson schiele filtered channel features pedestrian detection goodfellow bengio courville deep learning book preparation mit press girshick fast ren girshick sun faster towards object detection region proposal networks liang shen yan fast pedestrian detection corr url http cai fan feris vasconcelos unified deep convolutional neural network fast object detection zhang lin liang faster well pedestrian detection hosang omran benenson schiele taking deeper look pedestrians proceedings ieee conference computer vision pattern recognition cvpr ribeiro nascimento bernardino carneiro improving performance pedestrian detectors using convolutional learning pattern recognition sisbot luis alami simeon human aware mobile robot motion planner ieee trans robotics kruse kirsch sisbot alami exploiting human cooperation robot navigation ieee int symposium robot human interactive communication kruse basili glasauer kirsch legible robot navigation proximity moving humans ieee workshop advanced robotics social impacts arso kirby social robot navigation thesis robotics institute carnegie mellon university scandolo fraichard anthropomorphic navigation scheme dynamic scenarios ieee proc int conf robotics automation icra pandey alami framework towards socially aware mobile robot motion dynamic environment proc int conf intelligent robots systems iros kruse pandey alami kirsch robot navigation survey robotics autonomous systems hall hidden dimension man use space public private bodley head shi collins goldiez donate liu dunlap robot motion planning velocity constraints int symposium collaborative technologies systems cts wang yang zhu lin regionlets generic object detection ieee proc int conf computer vision iccv simonyan zisserman deep convolutional networks largescale image recognition int conf learning representations iclr krizhevsky sutskever hinton imagenet classification deep convolutional neural networks proc neural information processing systems nips piotr computer vision matlab toolbox pmt http vedaldi lenc matconvnet convolutional neural networks matlab acm proc int conf multimedia paisitkriangkrai shen van den hengel strengthening effectiveness pedestrian detection spatially pooled features proc european conf computer vision eccv verma hebbalaguppe vig kumar hassan pedestrian detection via mixture cnn experts thresholded aggregated channel features iccv workshop daum huang probabilistic data association filter estimation presence measurement origin uncertainty ieee control systems magazine hart nilsson raphael formal basis heuristic determination minimum cost paths ieee trans systems science cybernetics marder eppstein berger foote gerkey konolige office marathon robust navigation indoor office environment ieee proc int conf robotics automation icra koay sisbot syrdal walters dautenhahn alami exploratory study robot approaching person context handing aaai spring symposium multidisciplinary collaboration socially assistive robotics hershberger smart layered costmaps contextsensitive navigation proc int conf intelligent robots systems iros messias ventura lima sequeira alvito marques robotic platform edutainment activities pediatric hospital ieee int conf autonomous robot systems competitions icarsc
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consideration publication theory practice logic programming oct improved compilation logic programs iliano cervesato department computer science carnegie mellon university iliano submitted january revised january accepted january abstract prior work showed logic programming compilation given justification generic abstract logic programming languages demonstrated technique case hereditary harrop formulas linear variant compiled clauses logic formulas except presence abstraction atomic goals matching head paper revisit previous results detailed fully logical justification away spurious abstraction refine resulting technique support programs efficiently appear theory practice logic programming keywords compilation abstract logic programming hereditary harrop formulas logic programs introduction cervesato presented general methodology developing compiler associated intermediate language abstract logic programming language alpl miller satisfies basic properties applied abstractly language hereditary harrop formulas linear variant also based concrete implementations twelf pfenning llf cervesato pfenning systems directly methodology identified right sequent rules behave like left rules appear uniform proof used corresponding connectives compilation targets constructs program clauses intermediate language therefore another alpl abstract machine relied like source alpl transformation based duality left right rules proving correctness compilation process amounted simple induction finally horn clauses connectives target alpl corresponded key instructions warren abstract machine wam warren wam essential component commercial prolog systems since many compiled programs run order magnitude faster interpreted notoriously procedural instruction set wam regarded wondrous piece engineering without logical status sharp contrast deep cervesato logical roots prolog words rosenzweig wam resembles intricate puzzle whose many pieces fit tightly together miraculous way result understanding complex spite availability excellent tutorials proving correctness formidable task rosenzweig russinoff adapting logic programming languages major endeavor done clp jaffar nadathur mitchell contrast methodology cervesato simple mostly easily verifiable general applicability technique cervesato however one blemish made use equality atomic formulas together binder atomic goals lacked logical status paper remedy drawback carefully massaging head clauses allows replace constructs equality regular universal quantifications arguments clause head result improved prooftheoretic account compilation logic programs sits squarely within logic also opens doors specializing compilation process programs brings potential away unification favor matching efficient operation many languages present results language hereditary harrop formulas highest level abstraction like cervesato however general terms source alpl level abstraction considered indeed process using implement compiler clf watkins cervesato concurrent linear logic programming language combines backward forward chaining paper organized follows section recalls compilation process cervesato section present improved compilation process section refine support moded programs lay future developments sections background recap section recall compilation process presented cervesato succinctness focus smaller source language corresponds language underlying twelf system pfenning technique first used comment larger languages including examined cervesato section source language take language freely generated atomic propositions intuitionistic implication universal quantification source language expand openended atomic propositions cervesato predicate symbol followed zero terms program sequence closed formulas language call given following grammar formulas atoms programs cervesato leave language terms open require predicative substituting term variable alter outer structure formula improved compilation logic programs uniform provability new imp atm immediate entailment imp atm fig uniform deduction system often write atom predicate symbol sequence terms applied implicitly assume predicate symbol consistently applied number terms throughout program arity write resp substitution term free occurrences variable term resp formula simultaneous substitution denoted abstract logic programming language miller appropriate choices term language indeed expressive power miller nadathur twelf pfenning differs first language discussed cervesato omission conjunction truth see section operational semantics given two judgments uniformly provable immediately entailed defining rules given figure produce uniform proofs miller uniform provability judgment includes right sequent rules goal atomic rule atm calls immediate entailment judgment focuses program formula decomposes prescribed left sequent rules strategy complete respect traditional sequent rules logic miller logic programming perspective connectives appearing goal handled right rules search directives left rules carry preparatory phase target language cervesato target language compilation process distinguished compiled goals compiled clauses compiled goal either atomic proposition hypothetical goal goal solved presence additional clause universal goal goal solved presence new constant compiled clause form variable stood atomic goal resolved present clause could either match head clause invoke goal request variable instantiated term compiled program sequence compiled clauses grammar resulting language call follows goals clauses programs operational semantics compiled program given grammar cervesato goals atm new imp clause instances exists fig search semantics defined basis following two judgments uniformly provable uniformly provable clause instances whose variable instantiated atomic formula operational semantics shown figure observe partial exception atm consists solely right rules means every connective seen search directive dynamic clause preparations embodied left rules turned right search rules static compilation phase compilation compilation process transforms logic program compiled program expressed means following three judgments program compiled clause compiled goal compiled judgments defined rules figure see cervesato details ongoing example consider following two clauses taken type checking specification simply typed clarity write program clauses using reverse implication instead positive formulas app arr lam arr app arr lam arr compiled language sound complete see cervesato formal statements proof directions proceeds straightforward induction contrasts greatly complex proofs soundness correctness previously devised wam rosenzweig russinoff improved compilation logic programs programs empty clause clauses atm imp goals atm imp fig compilation fully logical compilation clauses compiled expressions form language fully logical section consider different compilation target language lies entirely within logic previous section generic horn clause form compiled execution rule atm reduced current atomic goal clause instance note may depend compile horn clause sequence fresh variables distinct equal number arity stands conjunction equalities variable term corresponding position arity zero notice binder gone run time formula resolve atomic goal clause immediately reduces like earlier may depend variables correspond directly argument registers wam closely related permanent variables formula understood uncurried form outer implications transformed conjunctions universals existentials literally would yield formula incorrect occurrences variables within escaped scope instead formula installs fresh variables arguments head predicate adds equality constraints body target language generalize intuition formula horn clauses second target language given following grammar goals clauses residuals programs cervesato goals atm new imp clauses imp residuals true exists fig search semantics compiled goals like section atoms hypothetical goals universal goals compiled clauses form possibly empty outer layer universal quantifiers enclosing implication whose head always consists predicate name applied possibly empty sequence distinct variables body residual residual either equality constraint trivial constraint logical truth like section goal invocation instantiation request notice full result compiling clause operational semantics specified following three judgments uniformly provable immediately entailed uniformly provable differ instantiation variables clause head side equalities respectively operational semantics given figure goals handled exactly way uniform provability top part figure operational reading compiled clauses instance immediate entailment rule imp special case imp isomorphic note rule imp reduces immediately residual head clause matches atomic goal proved rules residuals correspond closely rules clause instances original target language bottom figure rule requires two sides equality indeed equal rule true always satisfied rules figure build uniform proofs miller characteristic abstract logic programming languages operational semantics decomposes goal atomic formula top segment figure selects clause focuses finds matching head middle segment decomposes body bottom segment may eventually expose goals cycle repeats particular atomic goal exposed successful derivation necessarily contain instance rule atm picks clause head many instances rule arity instance rule imp necessary sequence steps captured improved compilation logic programs following derived backchaining rule replacing rules atm imp rule yields system equivalent figure taking primitive amounts replacing construction compiled clauses synthetic connective call therefore accounting structure atomic propositions proper quantification patterns provides fully logical justification clause compilation lacked compilation compilation transforms logic programs compiled logic programs order define auxiliary notion pseudo clause come handy pseudo clauses pseudo clause retains outer structure clause hole place residual general pseudo clause form fully compiled clause variables coincide pseudo clauses generated processing head clause hole needs replaced compiled body residual write operation pseudo clause instantiation formally defined follows often case contextual operations pseudo clause instantiation generally lead variable capture may free occurrences variables within result occurrences bound outer quantifiers compilation expressed means following four judgments program compiled head compiled clause compiled goal compiled defined rules figure wrote conjunctions equalities judgment compiles clause pseudo clause residual assembled clause rules clause imp programs goals otherwise compiled figure clause heads handled differently rule atm invokes auxiliary head compilation judgment compile goal pseudo clause equalities form seed clause residual consider first example clause section head app compiled pseudo clause equality constraints app new variables core equalities cervesato programs empty clause heads new clauses imp atm goals atm imp fig compilation extended compiled body clause arr existential quantifications original variables clause finally wrapped around result embedding hole pseudo clause resulting clause displayed top part figure target language sound complete respect order show need following auxiliary results first statement proved induction structure second induction given derivation lemma length statements soundness completeness follows proof proceeds mutual induction first derivation antecedent theorem soundness compilation theorem completeness compilation conclude section showing figure output compilation procedure two examples seen section stretch source clauses left align corresponding atoms gleaned clauses ample opportunities optimizations compilation process particular constraint mentioning variables sides often eliminated replacing existential variable universal variable rest clause removing existential quantifier exception multiple constraints form leading logical constant makes succinct presentation compilation process plays actual role also eliminated improved compilation logic programs app arr lam arr app arr lam arr fig compilation example interesting rewrite clauses using synthetic connective discussed earlier omitted occurrences readability app arr lam arr support moded programs section specialize compilation process outlined case source program program argument positions predicate symbol designated either input output input arguments guaranteed ground terms time goal called dually output arguments guaranteed made ground time call returns operational benefits working programs interpreter generic program must implement unification programs executed relying uniquely pattern matching variable instantiation desirable matching often behaves better general unification example efficient term languages away decidable term languages general unification stirling development section motivated sound independently cervesato whether program statically enforcing brings operational advantages discussed results section depend source language section assume predicate symbol comes mode declares arguments input written output written simplicity exposition decorate actual arguments atomic propositions symbols term input position atomic proposition written read similarly output position written pronounced amounts revising grammar atomic propositions follows atoms like assume arity predicate symbol remains constant program require atomic propositions marks positions pattern mode actual language would rely explicit mode declarations typographic convenience without loss generality examples assume input positions precede output positions atomic formula written possibly empty sequences terms input resp output positions avoid notational proliferation use markers mode designators symbol decorations like primes subscripts working generic terms therefore indicate possibly different terms similarly term sequences level abstraction rules figure capture operational semantics variant mode annotations simply ignored however moded execution requires two operational choices left open rules resolved using algorithmic strategy order rule imp searches derivations two premises substitution term rule picks assume strategy prolog implement rule imp left right implement rule lazily replacing variable logical variable instantiated incrementally unification allows view atomic goal procedure call program debray warren terms input position seen actual arguments procedure terms output position yield return values section formalize notion see debray warren prolog sarnat twelf refine operational semantics make goal evaluation order unification explicit see pientka instead refine compilation process account mode information produce compiled programs executed without appealing unification target language horn clause compiled execution order forces guess final values output variables goals improved compilation logic programs body fully executed move equality last goal since appear nowhere else residual equality assignment computed instance accordingly write furthermore program clause invoked ground terms input position bound ground terms input equality match variables appropriate subterms reason write expanding goal clause compiled almost follows arrows represent data flow execution note parallels control flow executing atomic goal desirable separate call verification output terms returned caller match expected output terms goal rewriting atomic goal compiled clause formula fresh variables transformation preserves control data flow special provision needs made input arguments variables instantiated ground terms moment call made next generalize intuition formula horn clauses third target language defined following grammar goal matches atomic goals goals programs clauses residuals residuals refine equality predicate matching predicate assignment predicate level abstraction behave like equality execution match predicate form ground term may contain variables bind variables ground subterms thereby realizing matching however presented programs terms assumed ground performs unification assignment predicate called variable term ground term programs simply binds compiled clauses programs like following motivations atomic goal compiled formula form grammar isolated match predicates cervesato goals matches true mtch atomic goals atm exists goals new imp clauses imp residuals assg mtch true exists fig search semantics specify operational semantics means following five judgments provable uniformly provable uniformly provable immediately entailed uniformly provable parallel grammar presented resulting operational semantics shown figure rules clauses unchanged respect language residual rule equality duplicated isomorphic rules matching assignment rules compiled goals instead proliferated due handling terms output position atomic goals observe rule atm essentially combination rule atm rule conjunction rules exists true standard rules existential quantification truth rule mtch combines rules conjunction matching like case rules figure construct proofs uniform miller makes abstract logic programming language successful derivation operational semantics decomposes goal formulas form rules goals segment rules exists mtch true necessarily reduce steps atomic formula similarly left premise rule atm selects clause focuses finds potentially matching head clauses segment proceeds decomposing body residuals segment cycle repeats whatever goals finds improved compilation logic programs noticed atomic goal form necessarily reduced many applications rule exists variables instance atm via right branch similar number uses rules mtch true respectively entails side following display derivable atm factored rule work performed atm degenerates rule side display akin atm system obtained replacing rules well rules indeed equivalent rule set figure rule entices interpret compiled formula atomic goal synthetic operator call invokes clause ground input arguments matches returned values terms output position recovered atomic goals rules carry sequence reasoning steps similar led backchaining rule exposing trailing assignments generic compiled clause form successful derivation rule pick clause applications rule instantiate variables terms next rule imp invoke instantiated residual occur appear formula reduces pushing substitution rule exists instantiate variables terms mention variables pushing substitution yields formula since variables occur neither finally rule assg must equal successful derivation necessary sequence steps captured following derived backchaining carried assignment conclusion rule seen refinement makes use trailing assignment compiled clauses derived inference rules imp become unnecessary system consisting rules goal rules implication universal quantification residual rules equivalent figure taking rule primitive amounts replacing compiled clauses following synthetic connective refines return cervesato programs empty clause heads new new clauses imp atm atomic goals new goals atm imp fig compilation variables interpreted local variables execution clause akin permanent variables wam valid proof system occurrence always immediately followed instance conclusion latter must match premise former fact realizes requirement upon returning call output terms must checked terms output position caller compilation compilation transforms logic programs compiled programs input level detail considered would operationally advantageous refinement semantics figure handles quantifiers lazily make use two auxiliary notions section pseudo clauses encountered already section analogous notion pseudo atomic goal defined follows pseudo clauses pseudo atomic goals like pseudo clauses retain outer structure clause replacing embedded residual hole pseudo atomic goals hole place trailing matches general form pseudo clauses pseudo atomic formulas accounting input output positions section wrote replacement hole residual noted variable capture could generally occur similarly write replacement hole matches compilation process modeled following five judgments reminiscent compilation judgments complex clause compilation improved compilation logic programs needs handle matching assignment opposed generic equality furthermore new judgment needed compile atomic goals program compiled head compiled clause compiled atomic goal compiled goal compiled write conjunction matches compilation terms input position assignments compilation output terms respectively body compiled clause compiled atomic goals write conjunction matches rules compilation define judgments shown figure compiling clause modeled judgment returns pseudo clause residual inclusive input matches output assignments fill hole rules clauses segment build residual starting compilation head displayed heads segment rules therein differ similar inference fact dispatch terms input output positions zones judgment matches assignments respectively residuals assignments plugged hole pseudo clause clause fully compiled seen programs segment rule imp compilation goals differs treatment atomic formulas upon encountering atom compilation appeals new judgment generates pseudo atomic formula matches integrated rule atm zone left turnstile serves accumulator much like compiling heads target language sound complete respect following lemma collects auxiliary results needed prove property first two statements proved induction structure third induction given derivation lemma term sequence length program following soundness completeness theorems cases proof proceeds mutual induction first derivation antecedent theorem soundness compilation theorem completeness compilation conclude section revisit ongoing examples assume cervesato app arr lam arr app arr lam arr fig compilation example mode predicate first argument input second output result compiling two familiar clauses shown figure section moded compilation process offers ample opportunities optimization matches assignments variables side corresponding existential quantification often elided occurrences optimized away instructive rewrite clauses two synthetic connectives introduced earlier omitting readability app call arr call return lam return call return arr larger source languages cervesato illustrated original abstract logical compilation method language hereditary harrop formulas language differs presence conjunction formulas form truth original treatment could handle easily clause position compiled disjunctions falsehood respectively approach taken sections support directly problem soon allow connectives clauses multiple heads even none consider example improved compilation logic programs clause two heads compiled ensure immediacy embodied compilation strategy produces pseudo clause applied residual thereby exposing flattened head compiled clause close top level possible achieve may one head one approach dealing problem observe distributes antecedent example obtain formula observe conjunction clauses compiled section results combined means disjunction approach generalizes full language hereditary harrop formulas pushes conjunctions outside leaving inner formulas resembling clauses conjunction truth goal position left alone problematic clauses head reduced preprocessing steps implemented transformation integrated compilation process abstract logic programming language examined cervesato language linear hereditary harrop formulas found core lolli hodas miller llf cervesato pfenning improved compilation process discussed paper extends directly presence linearity linear hereditary harrop formulas feature form conjunction truth technical device outlined needed obtain workable compiled clauses future work discussion section sets stage nearly functional operational semantics programs indeed given atomic goal ground terms input positions proof search instantiate output positions ground terms succeeds logic programming setting one answer could returned indeed programs clauses predicate implement partial function observation informed choice notation synthetic operators exposed call return believe case programs detailed operational semantics exposes variable manipulations using logical variables explicit substitutions restricts execution order bring functional interpretation surface would provide logical justification natural impulse give programs semantics typical functional programming languages atomic predicates carry input terms terms output position emerge process reduction future work intend carry program giving detailed operational semantics well rules goal perform logical transformations akin paper expose functional semantics programs would also allow prove formally operator section indeed implemented matching rather general unification cervesato acknowledgments work supported qatar national research fund grant nprp grateful frank pfenning carsten robert simmons jorge sacchini many fruitful discussions well anonymous reviewers references aci warren abstract machine tutorial reconstruction mit press rosenzweig wam definition compiler correctness logic programming formal methods practical applications beierle pluemer eds computer science artificial intelligence vol ervesato foundation compilation logic programming languages joint international conference symposium logic programming jicslp jaffar mit press manchester ervesato fenning linear logical framework information computation ervesato fenning walker watkins concurrent logical framework examples applications technical report department computer science carnegie mellon university pittsburgh march revised may ebray warren automatic mode inference logic programs journal logic programming odas iller logic programming fragment intuitionistic linear logic information computation jaffar ichaylov tuckey yap abstract machine clp proceedings sigplan conference programming language design implementation pldi san francisco iller nadathur logic programming proceedings third international logic programming conference shapiro london iller nadathur fenning cedrov uniform proofs foundation logic programming annals pure applied logic nadathur itchell system description teyjus compiler abstract machine based implementation lambda prolog sixteenth conference automated deduction cade ganzinger fenning system description twelf framework deductive systems proceedings international conference automated deduction lnai trento italy ientka tabled logic programming thesis department computer science carnegie mellon university russinoff verified prolog compiler warren abstract machine journal logic programming arnat syntactic finitism metatheory programming languages thesis department computer science yale university tirling decidability matching logical methods computer science warren abstract prolog instruction set technical note sri international menlo park watkins ervesato fenning walker concurrent logical framework judgments properties technical report department computer science carnegie mellon university pittsburgh march revised may
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coordinated operation shaving smoothing hierarchical approach huaiguang jiang member ieee yingchen zhang senior member ieee yuche chen changhong zhao member ieee jin tan member ieee dec national renewable energy laboratory golden rapid adoption distributed photovoltaics pvs certain regions issues lower net load valley day steep ramping demand sunset start challenge normal operations utility companies urban transportation systems also high peak congestion periods steep ramping traffic patterns propose using emerging electric vehicles evs stations cdss coordinate operation power distribution system pds urban transportation system uts therefore operation challenges system mitigated utilizing flexibility system proposed operation approach designed hierarchically consists higher lower level higher level assume integrated operation pds uts target operation minimize social cost meanwhile target evs cdss customers minimize expenditures exists equilibrium two targets determine optimal price lower level temporal spatial models pds uts developed provide detailed analysis system specifically pds built threephase unbalanced power flow model optimal power flow opf problem relaxed semidefinite relaxation programming sdp solved alternating direction method multiplier admm dynamic user equilibrium due problem formulated uts based static user equilibrium sue additional constraints ensure temporally continuous path flow evs cdss function reserves pds uts state charge soc considered optimize schedule reduce impacts pds conducted simulation numerical analysis using ieee pds sioux falls system cars uts two systems simulated jointly demonstrate feasibility effectiveness proposed approach index power distribution system duck curve urban transportation system electrical vehicle station state charge dynamic user equilibrium traffic congestion traffic pattern optimal power flow distributed computation omenclature functions fut fcso fcds fpt fut qtp qtdp qtdut fpt fka xtka parameters pru cka abbreviations pds uts cds sue due opf sdp admm soc power distribution system urban transportation system station static user equilibrium dynamic user equilibrium optimal power flow semidefinite relaxation programming alternating direction method multiplier electrical vehicle state charge objective function utility customer time interval objective function customer time interval objective function smart cds cost function pds time interval cost function uts time interval cost function waiting time netload pds time interval total power cds total evs objective function opf pds travel time function link time interval traffic flow xtka graph uts node set link set set path used connect pair defined vrut vsu vru vsu original node set destination node set respectively traffic flow capacity link graph pds node set link set upper bound lower bound soc weight factor pds uts iteration step coefficient gradient descent electrical power price congestion fee parking ratios cdss capacity cds variables traffic flow link time interval optimal price time interval qrdtu cev number vehicles departing time interval via path time interval set number vehicles assigned path pru departing time interval variable indicate time interval whether trip assigned path pru via link departing time interval complex voltage bus phase complex current bus phase power injection bus iqi complex impedance matrix complex power bus bus bus ancestor bus indicates hermitian transpose kth iterative variables admm distributed opf computation discharging positive charging negative speed time transportation information based locational marginal pricing lmp optimal deployment charging stations proposed evs optimal flow proposed wireless charging technology congestion toll manipulation paper coordinated operation approach proposed address biramp problems pds uts also provides multifunction platform researches emission reduction multisystem integration future city planning main contributions paper considering complexity system coordinate approach designed higher level lower level operate system spatial temporal domain hierarchically higher level pds uts regulated treated together utility minimize social cost evs cdss treated customers minimize expenditure equilibrium exists utility customers determine operation variables optimal price total electrical demand total available evs lower level detailed models pds uts evs cds considered specifically determine variables spatial temporal domains lower level pds built branch flow model equivalent classic businjection model suitable pds radial topology several developed approaches solving optimization problem associated pds usually called opf opf genetic algorithm opf relaxation cone program socp based relax opf sdp threephase unbalanced pds meanwhile several distributed algorithm designed solve opf dual decomposition predictor corrector proximal multiplier pcpm based admm applied decompose opf problem reduce computation time lower level uts due widely used estimate traffic volume congestion time travel cost link based assumption travelers choose best route minimize cost transportation wardrop user equilibrium travel times paths equal pair less unused paths static user equilibrium assignment used integrate pds considering behaviors impacted prices due applied keep temporal domain continuous uts paper evs cdss designed reserves pds uts respectively considering soc optimal schedule designed meet requirements pds uts reduce impacts ntroduction rooftop photovaltaic gained foothold many power systems continuously declining cost brings many benefit costumers reduced energy cost however also presents unprecedented challenges power systems operation control upper plot fig shows duck curve recorded regions california peak solar generation middle day sinks netload lower valley peak load occurs right sunset huge volume energy demand ramps short time frame creates artificial peak ramping costly balance using current power system assets surprising complex systems also suffer similar issues shown lower plot fig transportation system also high peaks steep ramps due fairly constrained traffic patterns rush hours urban areas thanks spawn evs widely located cdss two originally independent systems pds uts coupled together specifically increasing number evs seen significant amount distributed highly manipulatable small reserves used provide demand response enhance system reliability provide services power systems studies geographical information transportation information ignored evs treated aggregated optimal placement charging stations studied geographical information fig main idea problems power distribution system transportation system urban area tion viii pds test bench consists real systems pds ieee distribution system uts sioux falls transportation system evs demonstrate proposed approach based fig acknowledge cybersecurity critical aspect consider proposed system operation also large volume data generated proposed system requires flexible reliable communication network proposed system contains two complex systems many industrial communication network infrastructures pds uts operations need attached wifi integration brings lot challenge monitoring anomaly detection data transmission storage security attack analysis mitigation paper assume messages electrical power price traffic congestion information various control signals transmitted correctly without cybersecurity issues paper organized follows section flowchart proposed approach introduced section iii equilibrium utility customers designed determine optimal system variables section based sue due applied model uts compute dynamic traffic flow section based branch flow model opf problem relaxed sdp solved admm section considering soc evs behaviors optimal schedule designed reduce impacts pds section vii numerical results presented validate proposed approach conclusion presented lowchart roposed pproach proposed operation approach shown fig higher level system consists two major parts utility part taking pds uts whole shown green customer part taking cdss related evs shown yellow time interval duck curve netload recorded renewable energy generation data end user load data pds meanwhile travel data collected traffic monitoring data congestion model uts shown two green blocks pds uts treated regulated utility minimize social cost yellow block evs cdss treated customers minimize expenditures exists equilibrium determine system operation variables optimal prices total demand electrical power total required evs variables fitted lower level inputs lower level system consists three major parts pds part evs cdss part uts part determine system operation variables spatial temporal domains detailed models first uts accurately simulate dynamic traffic system spatial temporal domain due built based sue temporal continuous constraints proposed due problem transfered convex problem solved optimal results second pds objective function designed minimize cost pds built unbalanced branch flow model based sdp relaxation minimized synchronized time unified gis information load data net load duck curve cds cds cds utility pds traffic flow monitoring congestion traffic flow curve upper level renewable energy utility uts customers equilibrium utility customers scada ami demand electrical power required power price pds information sdp relaxation unbalanced power distribution system branch flow model power distribution system road battery soc dynamic traffic assginment due model model evs discharging power optimal power flow admm gis discharging price schedule discharging speed gis total available evs discharging schedule congestion model congestion available evs temporal continuosly constraints user equilibrium evs stations lower level optimal discharging price urban transportation system fig flowchart proposed coordinate operation approach number available evs cds constraints illustrated follows opf problem solved admm third specified electrical power demands available evs cds objective function fcds designed minimize impacts pds considering soc addition paper assumed drivers know traffic information well get optimal price wireless communication immediately uts network reached short time fpt defined convex function determine pds cost time interval qtp netload pds qtdp total power cdss defined iii quilibrium esign etween tilities ustomers igher evel qtdp ncds set cds fut defined convex function determines uts cost time interval qtut traffic load qtdut total evs time cost time considering number evs large qtdut estimated equilibrium design paper focus small city tens thousands cars means single user behavior weak impact equilibrium design higher level utility consists pds uts customer consists cds related evs utility objective minimize social cost shown fut objective function utility designed minimize social cost variables time interval electrical power charging negative discharging positive cds weight coefficient uts qtdut average speed evs waiting cost parking also defined convex fut min fpt qtp qtdp fut dut dut function summary proposed utility objective function fut convex indicate optimal point existed customer objective minimize expenditure based objective function customer cds related evs designed follows fcso min pit dynamic ser quilibrium ssignment ower evel compared sue due accurately describe traffic dynamics set successive time intervals paper due based sue temporal generalization basic model uts uts represented graph node set link set origin destination pair defined vrut vsu set paths pru used connect pair traffic flow pair represented qrdu departing time set time intervals congestion model presented follows fka cka objective function customer consists two parts time consumption cdss electrical power price benefit discharging positive cost charging negative cds discuss objective function customer fcso also convex solutions exists optimal price equilibrium computed fka travel time travel impedance traffic flow link time interval initial travel time cka traffic flow capacity fut qut optimal price depends pds part uts part fut based gradient descent distributed algorithm proposed optimize utility customers jointly equilibrium specifically objective function customer computed independently optimal price iteration qtp dynamic user equilibrium based sue due built follows jdue mint fut qut pit fkta fkta pru pru uts number vehicles path pru departing time interval travel time via path pru variable defined optimization stepsize set algorithm converge small enough means results projected onto set defined summary section optimal price gives computed higher level optimal results upper bound power injection following sections models uts pds cds built provide detailed information temporal spatial domains operate system lower level time interval link path pru departing time otherwise time interval traffic flow link given equals sum traffic flows via link departing time interval path pru pru number vehicles given algorithm due solution step initialization set iteration number threshold update new demand qrdtu using assignment generate equals sum traffic flows assigned path pru pru total time consumption path pru given keeps temporal continuous traffic flow time intervals temporal unique constraint fkta step update travel time fkt beginning loop step step ensures vehicle assigned two links certain time interval step determine descend direction find shortest using assignment generate path fkt shortest route pattern solutions step determine iteration step designed considering temporal continuous traffic flow time interval take initial state paper test bench based sioux falls network departing time time interval time interval due problem transfered sue problem sue based lagrangian generated hessian matrix positive definite indicate convexity optimal travel flow computed kkt conditions therefore solution uts time interval designed algorithm step compare result threshold fulfilled result traffic flow time otherwise back step interval objective function sdp relaxation opf pds objective function contains two major parts generation cost system line loss designed follows fpt iqi indicates power injection bus complex power bus bus set buses common ancestor bus indicates hermitian transpose addition according matrix defined pds represented radial graph similar uts pds also contains temporal characteristics superscript considering unbalanced system phase configuration defined radial distribution system unique ancestor bus bus branch bus named branch flow model time interval defined step update search employed determine short time branch flow model fkta ptimal ower low ower istribution ystem ower evel diag pds two weight coefficients generation cost system line loss respectively constraints voltage current illustrated follows positive semidefinite rank fkta paper injection power plays important role active control reactive control opf min sit sti vit lti min fpt fpt min sit sti vit lti optimal power injection considering relationship uts optimal injection power decides number evs cds cds pds uts explained fig yellow circles according sdp relaxation introduced constraint rank removed relaxed objective problem shown follows pru pru fpt pds arg min arg min opf solution admm system opf computation unbalanced pds bottleneck whole system operation firstly real pds usually contains lot feeders brings high computation loads system secondly pds highly dynamic system requires short computation time therefore admm distributed algorithm proposed solve problem short time based lagrange multiplier additional penalty term augmentation term standard form admm formulated follows arg min variables sit sti vit lit maintained bus seemed local agent admm indicate augmented lagrangian derived detail formulation fpt fpt found iteration variables illustrated follows convex problem solved admm min considering time intervals uts admmbased distributed opf computation architecture built dramatically reduce time consumption proposed coordinated operation approach therefore addition designed user equilibrium higher level detail models uts pds build provide spatial temporal information lower level next section smart strategy proposed evs bidirectional cdss reduce impact pds mart harging discharging ower evel paper evs cdss act reserves pds uts section take cds example cdss contain performances time interval evs parked cds parking time defined real world depends many factors example different habits drivers beyond scope paper determined follows convex convex sets additional penalty term efficient augmented lagrangian function shown follows coefficient convergence quadratic penalty term increase converge speed therefore relaxed objective function formulated min min fpt xti yit xti yit fpt xti yit optimal power injection determined opf solution average speed evs electrical power price congestion fee ratio parking cdss capacity cds parking time stochastic evs impact stability pds considering soc smart evs approach proposed reduce impact pds follows fcds min cev cev cev soc cev discharging positive charging ative speed time pcds cev netload cds equals original load pds minus total power total demand contains soc constraint ratio soc vii umerical imulation simulation flowchart shown fig corresponding description shown algorithm step distributed algorithm used reduce time consumption equilibrium computation step time interval due problem transferred sue problem search employed determine iteration step short time uts evs cdss step test bench based ieee pds admm implemented computed opf distributed manner constraints step step cds independent smart evs computed parallel also helps reduce computation time computation time analysis shown fig two curves indicate average computation times proposed approach less scenarios compared time interval minutes proposed approach quick enough support continuously operations systems flowchart simulation pcds cev fpt fpt congestion fee evs parking ratio parking capacity cds simulations executed using server ghz intel xeon cpu ram software python matlab matlab global optimization toolbox parallel computing toolbox communication network good enough transmit information without cybersecurity issues pcds cev pcds cev cev cev cev xtj yjt pds pds shaving smoothing esults shown fig blue curve netload peak pds orange curve result proposed approach peak load decreases evs discharging yellow curve traffic delay peak uts purple curve reduced traffic delay proposed approach total peak traffic delay time uts decreases hours hours fig also clear netload pds increases total traffic delay uts decreases hours hours fig clear proposed approach benefit pds uts reduce smooth fig detailed traffic congestion information congestion scenario shown traffic maps pair scenario shown table red pink arrows indicate traffic congestions roads traffic congestion information without proposed approach shown fig test bench based ieee pds sioux falls uts cover similar geographical areas assumed small city evs problem duck curve data based traffic behavior data based cdss located node shown yellow circles illustrated pds uts respectively time interval minutes systems pds cost function fpt defined quadratic function uts cost function fut defined similarly fut convex quadratic form indicates utility cost increase fast increasing pds load uts load discussed utility objective function fut convex weight factor equals indicates importance pds uts similarly customer objective function fcso also set convex function electrical power price fig test bench consists pds uts algorithm simulation process step initialization data collection collect data parameters pds uts cdss evs topology information electrical power price congestion fee set simulation time interval min etc objective function utility pds uts objective function customers evs optimal information discharging price step higher level equilibrium computation defined utility part pds uts customer part cdss evs build utility customer objective function compute equilibrium uts due parking evs cdss step lower level due uts receive information build due objective tsu function constraints solve algorithm generate traffic information traffic data information collection pair traffic information cdss smart cds cds optimal power flow step lower level opf pds receive information build opf objective function constraints relax solve generate cds upper bound power injection step lower level smart evs ceive information compute parking evs cds build smart generate feedback pnt pds information uts cev cds optimal discharging price optimal power injection pds opf demand load data collection load fig test bench consists pds uts shown fig red arrows indicate traffic delays roads larger minutes shown fig proposed approach pink arrows indicate traffic delays roads less minutes number pink arrows pink arrows step system update new uts pds information back step thousand computation time traffic computation time power computation time power fig computation time comparison different traffic load factor original netload proposed approach original traffic hours proposed approach power time total traffic delay fig traffic congestion scenario traffic congestion scenario proposed approach total traffic delay original netload proposed approach original traffic hours proposed approach power nodes time fig shaving smoothing pds uts uts compensation pds table pair uts social cost analysis fig cost proposed approach lower clear proposed approach benefit social cost increasing number evs fig red curve cyan curve indicate social cost different power factors number uts fig also clear proposed approach benefit social cost increasing power factor summary prosed approach benefit pds uts different traffic scenarios load factors addition social cost increases increasing pds load number indicates quadratic objective function pds uts design benefits system reduce total social cost fig blue curve circle green curve square one group indicates social costs different number evs road load pds means total load pds smart evs test robustness proposed approach select two continuous intervals load data contains biggest much less number red arrows red arrows also indicates alleviation traffic congestion proposed approach total congestion delay reduces hours hours summary fig fig biramp problems reduced smoothed proposed approach benefits pds uts thousand discharging speeds controlled meet constraint original cost traffic proposed approach traffic original cost power proposed approach power viii onclusion paper hierarchical approach proposed shave smooth problem powertraffic system higher level lower level higher level gives overview whole system lower level gives detailed description part evs cdss function reserves pds uts utilize flexibility optimize operations system higher level pds uts treated together minimize social cost evs cdss treated customers minimize expenditure equilibrium designed determine optimal prices total demand electrical power total required evs lower level considering spatial time domain detail models pds uts built specifically determine power injection evs behaviors cds smart evs approach proposed reduce impacts pds numerical results test bench consists ieee pds sioux falls uts evs cdss used demonstrate feasibility effectiveness proposed approach implementation weather human behavior social issues bring stochastic impacts powertraffic system increase uncertainties bring challenges system operation addition systems natural gas delivery system water system also impact proposed system result nonnegligible consequences next step factors stochastic renewable energies departure time evs cybersecurity system human behaviors taken consideration social cost power fig different social costs comparison different traffic load factors original netload original discharging power discharging power time minutes original netload original discharging power discharging power eferences cheng tsao lin seeds energyefficient distributed server farm ieee transactions systems man cybernetics systems vol cui zhang florita hodge sun solar power ramp events detection using optimized swinging door algorithm proc asme int design eng tech conf comput inf eng conf jiang zhang zhang gao muljadi knowledge discovery smart grid operation control situation big data visualization platform north american power symposium naps ieee katsigiannis georgilakis tsinarakis novel colored fluid stochastic petri net simulation model reliability evaluation small isolated power systems ieee transactions systems man systems humans vol tian zhao review solar collectors thermal energy storage solar thermal applications applied energy vol connolly lund mathiesen leahy review computer tools analysing integration renewable energy various energy systems applied energy vol gonzales daganzo evening commute cars transit duality results user equilibrium combined morning evening peaks behavioral sciences vol time minutes fig normal discharging cds without considering impacts pds smart discharging cds considering impacts pds deviations within minutes fig discharging taken example evs employed test proposed approach shown fig without proposed approach stochastic discharging behaviors increase deviations netload fig proposed smart approach deviations netload smoothed decreases impacts pds discharging time cheng chang urban transportation energy carbon dioxide emission reduction strategies applied energy vol huang tan sun parameterized batch reinforcement learning longitudinal control autonomous land vehicles ieee transactions systems man cybernetics systems foley tyther calnan impacts electric vehicle charging electricity market operations applied energy vol liu wang botterud zhou vyas assessment impacts phev charging patterns scheduling stochastic unit commitment ieee transactions smart grid vol aliprantis ying load scheduling dispatch aggregators electric vehicles ieee transactions smart grid vol sortomme optimal scheduling energy ancillary services ieee transactions smart grid vol jiang zhang zhang gao muljadi auxiliary controller enhance voltage stability distribution system high renewable energy penetration ieee transactions smart grid vol heymans walker young fowler economic analysis second use electric vehicle batteries residential energy storage energy policy vol geng xie learning coupling data support vector machine based approach ieee transactions power systems vol optimal fast charging station placing sizing applied energy vol yin guan optimal deployment public charging stations hybrid electric vehicles transportation research part methodological vol wei mei shahidehpour fang optimal flow urban electrified transportation networks ieee transactions smart grid vol subhonmesh low chandy equivalence branch flow bus injection models communication control 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feb matrix models eigenvector deviations fluctuations joshua cape minh tang carey priebe department applied mathematics statistics johns hopkins university february abstract estimating eigenvectors subspaces central importance numerous problems statistics computer science applied mathematics paper characterizes behavior perturbed eigenvectors range matrix models encountered statistical random matrix theoretic settings prove approximation results sharp deviations well distributional limit theory fluctuations concise methodology considered paper synthesizes tools rooted two core concepts namely deterministic decompositions matrix perturbations probabilistic matrix concentration phenomena illustrate theoretical results via simple simulation examples involving stochastic block model random graphs keywords matrix models eigenvector perturbation asymptotic normality random matrices eigenvector deviations fluctuations cape tang priebe introduction paper considers setting large symmetric matrices representing additive perturbation matrices whose columns orthonormal eigenvectors corresponding leading eigenvalues respectively ask question entrywise close matrices eigenvectors quite general structural assumptions main results address question level sharp deviations level fluctuations theorems quantify entrywise closeness modulo necessary orthogonal transformation theorem subsequently presents multivariate limit theorem rows matrix properly transformed provided exhibits additional structure viewed operator numerous problems statistics consider eigenstructure large symmetric matrices prominent examples problems include spike population covariance matrix estimation johnstone silverstein well principal component analysis jolliffe nadler problems received additional attention windfall result advances random matrix theory bai silverstein benaychgeorges nadakuditi addition examples problem community detection study networks led widespread interest understanding perturbations large graph laplacian adjacency matrices given success spectral clustering methodologies lei rinaldo rohe sarkar bickel tang priebe towards end recent ongoing concurrent effort statistics computer science mathematics communities devoted obtaining precise entrywise bounds eigenvector perturbations abbe cape eldridge fan mao tang paper distinguishes literature presenting deviation fluctuation results within concise yet flexible matrix model framework amenable statistical applications inference employ norm machinery perturbation considerations introduced cape also employed abbe tang order obtain strong eigenvector deviations fluctuations cape tang priebe bounds paper extends results cape tang demonstrate careful analysis within unified framework leads results multivariate distributional limit theory preliminaries real matrices orthonormal columns denoted columns form orthonormal bases subspaces distance subspaces commonly defined via notion canonical angles osine ine matrix decomposition crucially involve singular values matrix specifically writing singular values diagonal matrix canonical angles defined arccos bhatia one frequently encounters matrix sin since commonly considered spectral frobenius matrix norms small values sin indicate small angular separation distance subspaces corresponding importantly canonical angle notion distance subspaces holds row space alignment via orthogonal transformation orthogonal matrix letting denote exist absolute constants choice norm cai zhang min sin paper also focus matrices form instead consider matrix norm defined matrix denote vector norms respectively quantity convenient interpretation maximum euclidean norm rows shown serve useful surrogate maximum absolute entry matrix norm kmkmax maxi among advantages working norm remains eigenvector deviations fluctuations cape tang priebe invariant respect orthogonal matrices see cape additional discussion subsequent analysis shown particularly meaningful exhibits coherence recht assumed delocalized sense decays sufficiently quickly tall thin matrices matrix standard norm relations reveal ktkmax differ small factor depending relationship holds spectral frobenius norms ktk ktkf respectively since necessarily rank contrast may certain cases much smaller ktk factor depending summarized ktkmax ktk ktkf refer reader cape general discussion norm statistical applications main results setting let symmetric matrix block spectral decomposition given diagonal matrix contains nonzero eigenvalues matrix whose orthonormal columns corresponding eigenvectors diagonal matrix contains remaining ordered eigenvalues associated matrix orthonormal eigenvectors let symmetric matrix write perturbation assumption let denote possibly scaling parameter log constants eigenvector deviations fluctuations cape tang priebe assumption exist constants assumption suppose kek concentrated sense exist constants kek probability least assumption introduces sparsity scaling factor additional flexibility paper considers regime dependence sequences matrices suppressed notational convenience assumption specifies magnitude leading signal eigenvalues corresponding leading eigenvectors interest simplicity specificity leading eigenvalues taken prescribed order remaining eigenvalues assumed vanish assumption specifies random matrix concentrated spectral norm classical probabilistic sense concentration holds widely random matrix models case expectation since advantage assumption coupled assumption together application weyl inequality bhatia implicit ratio terms behave explicitly assumption specifies additional probabilistic concentration requirement arises conjunction model flexibility introduced via sparsity scaling factor assumption notation used denote ceiling function assumption suppose exist constants integers dlog log fixed standard basis vector fixed unit vector cek log probability least exp log provided assumption amounts concentration estimate motivated lemma along variants thereof holds broad class random symmetric matrices including wigner matrices whose entries exhibit subexponential decay see definition remark definition therein technical discussion accompanying material addition taking simple union bound collectively eigenvector deviations fluctuations cape tang priebe standard basis vectors columns yields event holds probability least constant sufficiently large function fundamentally though connection sparsity factor satisfies sufficiently large case behavior reflected reduces bernsteintype probabilistic concentration contrast example instead log dlog log log remark regimes functionally correspond regime appropriate rescaling approximation deviations assumptions section spectral norm subspace analysis via sin theorem bhatia yields sufficiently large exists uwk equation serves benchmark bound quantity shown times much smaller order theorem suppose assumptions hold context section suppose addition log log sufficiently large exists min log bound obtained norm methods demonstrably superior bound implied log sufficiently quickly proof theorem first proceeds way refined deterministic matrix decompositions subsequently leverages additional aforementioned probabilistic concentration assumptions proof framework extends permit analysis culminating theorem section process proving theorem also prove theorem extension refinement theorem theorems paper collectively proven section eigenvector deviations fluctuations cape tang priebe theorem suppose assumptions hold assumption holds suppose addition log log sufficiently large exists matrix satisfying max log euk moreover log theorem summarized saying respect norm moreover entrywise sense equation provides collective eigenvector subspace characterization relationship leading eigenvectors via perturbation remark always holds euk kek replaced upon invoking concentration generally rourke suitable choices moreover theorem numerous regimes involving limit theory fluctuations section specify additional structure purpose establishing limit theory assumed strictly positive leading eigenvalues reminiscent spike covariance kernel population matrix setting possible though involved obtain similar results allowed strictly positive strictly negative leading eigenvalues order specifically modifications would give rise considerations involving structured orthogonal matrices well indefinite orthogonal group eigenvector deviations fluctuations cape tang priebe assumption suppose written almost surely symmetric invertible matrix also suppose fixed index theiscaled ith row written eij converges distribution centered multivariate normal random variable second moment matrix theorem suppose assumptions hold assumption holds suppose addition log log max log euk probability let column vectors denoting ith rows respectively exist sequences ofh orthogonali matrices random vector converges distribution centered multivariate normal random variable covariance matrix equation theorem amounts mild additional regularity condition holds example log suitable constant context paper side often shown behave log log remark example matrix structure let probability distribution defined let independent random vectors second moment matrix let exists orthogonal matrix depending strong law large numbers guarantees almost surely eigenvalues order asymptotically almost surely moreover asymptotically almost surely constant suitably controlled imposing additional assumptions taking bounded imposing moment assumptions conditioning yields deterministic choice purposes assumption eigenvector deviations fluctuations cape tang priebe remark example matrix multivariate normality continue discussion remark let entries centered independent identically distributed symmetry common variance given classical multivariate central limit theorem asymptotic normality condition assumption holds slight abuse notation theorem yields similar behavior holds generally entries permitted heterogeneous variances different distributions simulations stochastic block model sbm holland simple yet ubiquitous random graph model vertices assigned one possible communities blocks probability two vertices sharing edge occurs conditionally independently vertices latent community memberships undirected sbm graphs vertices binary symmetric adjacency matrix viewed additive perturbation low rank population edge probability matrix sbms matrix corresponds appropriate dilation block edge probability matrix language paper verified aforementioned assumptions hypotheses hold following sbm examples consider graphs arising stochastic block model balanced equal block sizes block bernoulli edge probabilities given respectively rank eigenvalue multiplicity two specifically figure plots empirical mean confidence interval computed independent realizations simulated adjacency matrices figure also plots function log large captures dominant behavior bound theorem cape tang priebe norm value eigenvector deviations fluctuations figure approximation simulations sbm corresponds number vertices axis corresponds values table marginal sample covariance matrices block sbm final column shows theoretical marginal covariance matrices obtained theorem alternatively consider graphs arising stochastic block model vertices belonging block one block edge probability matrix entries table shows aggregate sample ance matrix estimates centered random vectors together theoretical covariance matrices specified theorem eigenvector deviations fluctuations cape tang priebe appendix proof theorems begin several important observations namely sin exists depending sin particular taken product left right orthogonal factors singular value decomposition additional details concerning eqs may found example cape importantly relation yields matrix sylvester equation spectra disjoint one another high probability assumptions follows written matrix series bhatia second equality holds since rank choice matrix decomposed follows satisfying krw entries satisfy rij hui hui define matrix entrywise according denoting hadamard matrix product eigenvector deviations fluctuations cape tang priebe rightmost matrix factor expanded euu therefore bounded spectral norm using manner euk combining observations together properties matrix norms yields following norm bound kmax euk assumptions application weyl inequality bhatia guarantee exist constants kek high probability sufficiently large therefore considering appropriate matrix series kekk log used fact log sufficiently large assumption high probability kek max cek log since euk kek log log krw log completes proof theorem eigenvector deviations fluctuations cape tang priebe next decompose matrix extending proof techniques order obtain fluctuations particular using matrix series form yields final term satisfies bound log follows assumption holding namely kekk log hand modifying previous analysis used bound yields kku log bound extending pre vious argument used bound entries satisfy rij hui hui define matrix entrywise according denoting hadamard matrix product eigenvector deviations fluctuations cape tang priebe rightmost matrix factor written bounded spectral norm order hence kmax log therefore shown since log residual matrix satisfies max log euk leading term agrees order bound theorem namely log proves theorem intermediate step proving theorem proceed finish assumption exists orthogonal matrix depending hence matrix therefore written multiplication right inverses orthogonal transformations therefore yields matrix relation uwx fixed let column vectors denoting ith rows respectively equation implies kri probability addition almost surely assumption together continuous mapping theorem scaled ith row satisfies assumption combining eigenvector deviations fluctuations cape tang priebe observations together slutsky theorem yields exist sequences orthogonal matrices wri particular completes proof theorem references abbe fan wang zhong entrywise eigenvector analysis random matrices low expected rank arxiv preprint bai silverstein spectral analysis large dimensional random matrices volume springer nadakuditi eigenvalues eigenvectors finite low rank perturbations large random matrices advances mathematics bhatia matrix analysis volume graduate texts mathematics new york cai zhang perturbation bounds singular subspaces applications statistics arxiv preprint appear annals statistics recht exact matrix completion via convex optimization foundations computational mathematics cape tang priebe norm singular subspace geometry applications statistics arxiv preprint eldridge belkin wang unperturbed spectral analysis beyond arxiv preprint eigenvector deviations fluctuations cape tang priebe knowles yau yin spectral statistics graphs local semicircle law annals probability fan wang zhong eigenvector perturbation bound application robust covariance estimation arxiv preprint holland laskey leinhardt stochastic blockmodels first steps social networks johnstone distribution largest eigenvalue principal components analysis annals statistics jolliffe principal component analysis springer lei rinaldo consistency spectral clustering stochastic block models annals statistics mao sarkar chakrabarti estimating mixed memberships sharp eigenvector deviations arxiv preprint nadler finite sample approximation results principal component analysis matrix perturbation approach annals statistics rourke wang random perturbation low rank matrices improving classical bounds arxiv preprint rohe chatterjee spectral clustering stochastic blockmodel annals statistics sarkar bickel role normalization spectral clustering stochastic blockmodels annals statistics silverstein limit theorems eigenvectors large dimensional sample covariance matrices journal multivariate analysis silverstein eigenvectors large dimensional sample covariance matrices journal multivariate analysis eigenvector deviations fluctuations cape tang priebe tang cape priebe asymptotically efficient estimators stochastic blockmodels naive mle mle spectral arxiv preprint tang priebe limit theorems eigenvectors normalized laplacian random graphs arxiv preprint appear annals statistics
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wseas transactions systems balasubramanian system reliability fault tolerance design metrics tradeoffs distributed minority majority voting based redundancy scheme balasubramanian school computer engineering nanyang technological university nanyang avenue singapore balasubramanian abstract distributed minority majority voting based redundancy dmmr scheme recently proposed efficient alternative conventional redundancy nmr scheme physical design circuits systems dmmr scheme enables significant improvements fault tolerance design metrics compared nmr scheme albeit expense slight decrease system reliability context paper studies system reliability fault tolerance design metrics tradeoffs dmmr scheme compared nmr scheme majority logic group dmmr scheme increased size relative minority logic group example dmmr nmr systems realized using cmos process compared results show dmmr systems similar better fault tolerance whilst requiring similar fewer function modules counterpart nmr systems simultaneously achieve optimizations design metrics nevertheless dmmr systems upper hand respect fault tolerance design metrics optimizations comparable nmr dmmr systems regard system reliability nmr systems closely followed dmmr systems closely followed dmmr systems verdict dmmr systems preferable implement higher levels redundancy combined system reliability fault tolerance design metrics perspective realize circuits systems nmr dmmr reliability fault tolerance figure merit digital design asic standard cells realizing specific portions future generation mission circuits systems demand greater fault tolerance context recently shown distributed minority majority voting based redundancy dmmr scheme forms efficient viable alternative conventional nmr scheme implementing mission circuits systems nmr scheme identical function modules used equivalent outputs identical function modules supplied introduction nmr dmmr redundancy nmr widely used design mission circuits systems used space aerospace nuclear defence banking financial industrial applications reference suggests due increasing reliability variability issues nanoelectronics regime selective utilization higher levels redundancy involves use several identical function modules called progressive modular redundancy may needed volume wseas transactions systems balasubramanian referring fig dmmr system least two function modules comprising majority logic group maintain correct operation least one function module amongst minority logic group maintain correct operation concurrently hand referring fig dmmr system least three function modules comprising majority logic group maintain correct operation least one function module amongst minority logic group maintain correct operation concurrently article succinctly study system reliability fault tolerance design metrics aspects example dmmr systems comparison counterpart nmr dmmr systems understand tradeoffs involved majority voter processes produces nmr system output performing majority voting function modules outputs block schematic nmr system shown fig nmr scheme odd least function modules required satisfy majority logic state function modules tolerated nmr scheme majority logic group dmmr voter fig block diagram nmr system dmmr scheme function modules split two groups majority logic group minority logic group majority function modules majority logic group least one function module minority logic group maintain correct operation ensure correct operation dmmr system note generic dmmr system identical function modules constitute majority logic group remaining identical function modules constitute minority logic group positive integers equivalent outputs produced identical function modules majority minority logic groups combined dmmr voter shown fig produce dmmr system output function module function module function module identical inputs supplied function modules outside world function module function module function module maj majority voter dmmr system output min minority logic group majority logic group identical inputs supplied function modules outside world dmmr systems fig shows dmmr system topology majority logic group comprises function modules minority logic group comprises function modules fig shows dmmr system topology majority logic group comprises function modules minority logic group comprises function modules corresponding dmmr voters also shown fig majority voting element present dmmr voter dmmr system majority voter majority voting element present dmmr voter dmmr system majority voter identical inputs supplied function modules outside world identical inputs supplied function modules outside world function module function module function module function module function module function module function module function module dmmr voter majority voter min maj dmmr system output minority logic group fig block diagrams dmmr system dmmr system volume wseas transactions systems balasubramanian system reliabilities counterpart nmr dmmr systems equations give respective system reliability expressions dmmr systems assuming perfect dmmr voters system reliability equations counterpart nmr dmmr dmmr dmmr systems given denotes module reliability denotes system reliability since identical function modules used modules reliabilities assumed equivalent denotes reliability probability correct working function module denotes nonreliability probability state function module simulation results conclusions example nmr dmmr dmmr systems especially targeting higher levels redundancy implemented based cmos process considering array multiplier representative function module structural integrity different redundant systems preserved technology mapping facilitate legitimate comparison corresponding design metrics synthesis simulation mechanism environment typical case pvt test benches supplied time intervals maintained ensure uniformity enable direct correspondence previous results simulation results viz average power critical path delay silicon area different redundant systems estimated given table fom calculated inverse product also given table since power delay area desirable minimized high value fom indicates optimized design signifies reduction cost dmmr dmmr dmmr dmmr systems would able accommodate faulty failure state maximum function modules dmmr dmmr cope faulty failure state maximum function modules respect former dmmr system requires one function module less dmmr systems respect latter dmmr system requires one function module less system one function module dmmr system analysis system reliabilities dmmr dmmr dmmr dmmr systems versus module reliabilities reveals nmr systems feature highest reliability closely followed reliabilities dmmr systems closely followed reliabilities dmmr systems example system reliabilities systems system reliabilities dmmr systems system reliabilities dmmr systems respectively general dmmr systems show improvements system reliability compared system reliabilities dmmr systems whilst featuring degree fault tolerance although requiring similar greater number function modules hence system reliabilities dmmr systems happen lie midway corresponding table power delay area metrics various nmr dmmr dmmr systems fom power delay area system type dmmr dmmr dmmr dmmr compared dmmr dmmr systems implementations report respective improvements fom comparison dmmr dmmr systems implementations report corresponding enhancements fom simulation results show fom dmmr systems lies approximately midway foms counterpart nmr dmmr systems hence inferred giving importance system reliability alone nmr system topology preferable however realize higher levels redundancy nmr scheme volume wseas transactions systems balasubramanian method proc ieee international conference emerging technologies balasubramanian lakshmi narayana chinndadurai power optimized logic circuit design novel synthesis technique proc ieee international conference emerging technologies balasubramanian mastorakis power delay area comparisons majority voters relevant tmr architectures proc international conference circuits systems signal telecommunications balasubramanian prasad mastorakis fault tolerance improved majority voter tmr system architectures wseas transactions circuits systems vol balasubramanian design nmr system health monitor applications springerplus vol pages balasubramanian mastorakis asicbased implementation synchronous sectioncarry based carry lookahead adders proc international conference circuits systems signal telecommunications balasubramanian mastorakis design synchronous based carry lookahead adders improved figure merit wseas transactions circuits systems vol would poor choice taking account system implementation cost comparison dmmr system topology would better choice could considerably reduce implementation cost whilst featuring slightly less system reliability nevertheless combined system reliability fault tolerance design metrics cost weight perspective dmmr system preferable rest whilst associated moderately less system reliability references johnson design analysis fault tolerant digital systems longman publishing company boston usa koren mani krishna systems edition morgan kaufmann california usa dubrova design springer new york usa ban naviner progressive modular redundancy designs nanoelectronics microelectronics reliability vol septembernovember balasubramanian maskell distributed minority majority voting based redundancy scheme microelectronics reliability vol balasubramanian mastorakis standard cell based voter use tmr implementation proc european conference circuits technology devices balasubramanian arabnia standard cell based efficient majority voter design proc international conference embedded systems applications synopsys databook version vai vlsi design crc press florida usa yano sasaki rikino seki topdown logic design ieee journal circuits vol june balasubramanian lakshmi narayana chinnadurai design combinational logic digital circuits using mixed logic synthesis volume
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mar reinforcement learning challenging robotics environments request research matthias plappert marcin andrychowicz alex ray bob mcgrew bowen baker glenn powell jonas schneider josh tobin maciek chociej peter welinder vikash kumar wojciech zaremba openai correspondence matthias marcin abstract purpose technical report first introduces suite challenging continuous control tasks integrated openai gym based currently existing robotics hardware tasks include pushing sliding pick place fetch robotic arm well object manipulation shadow dexterous hand tasks sparse binary rewards follow reinforcement learning framework agent told using additional input second part paper presents set concrete research ideas improving algorithms related hindsight experience replay environments environments released part openai brockman use mujoco todorov physics engine fast accurate simulation video presenting new environments found https fetch environments fetch environments based fetch robotics parallel gripper similar tasks used andrychowicz added additional reaching task pick place task bit fetch tasks goal describes desired position object reaching rewards sparse binary agent obtains reward object target location within tolerance otherwise actions dimensions specify desired gripper movement cartesian coordinates last dimension controls opening closing gripper apply action subsequent simulator steps returning control agent agent action frequency observations include cartesian position gripper linear velocity well position linear velocity robot gripper object present also include object cartesian position rotation using euler angles linear angular velocities well position linear velocities relative gripper https http andrychowicz training task relied starting training episodes state box already grasped necessary successful training target position box sometimes air sometimes table use technique anymore figure four proposed fetch environments fetchreach fetchpush fetchslide fetchpickandplace reaching fetchreach task move gripper target position task easy learn therefore suitable benchmark ensure new idea works pushing fetchpush box placed table front robot task move target location table robot fingers locked prevent grasping learned behavior usually mixture pushing rolling sliding fetchslide puck placed long slippery table target position outside robot reach hit puck force slides stops target location due friction pick place fetchpickandplace task grasp box move target location may located table surface air hand environments environments based shadow dexterous anthropomorphic robotic hand degrees freedom joints controlled independently whereas remaining ones coupled joints hand tasks rewards sparse binary agent obtains reward goal achieved within tolerance otherwise actions use absolute position control joints hand apply action subsequent simulator steps returning control agent agent action frequency observations include positions velocities robot joints case object manipulated also include cartesian position rotation represented quaternion hence well linear angular velocities reaching task include cartesian position fingertips reaching handreach simple task goal contains target cartesian position fingertip hand similarly fetchreach task task relatively easy learn goal considered achieved mean distance fingertips desired position less block manipulation handmanipulateblock block manipulation task block placed palm hand task manipulate block target pose achieved goal includes target position cartesian coordinates target rotation quaternions include multiple variants increasing levels difficulty handmanipulateblockrotatez random target rotation around axis block target position handmanipulateblockrotateparallel random target rotation around axis block target rotations axes target position handmanipulateblockrotatexyz random target rotation axes block target position said found easy even partially broken implementations sometimes learn successful policies conclusions drawn task alone https figure four proposed shadow dexterous hand environments handmanipulateblock handmanipulateegg handmanipulatepen handreach handmanipulateblockfull random target rotation axes block random target position goal considered achieved distance block position desired position less applicable full variant difference rotation less rad egg manipulation handmanipulateegg objective similar block task instead block object used find object geometry makes significant differences hard problem egg probably easiest object goal includes target position cartesian coordinates target rotation quaternions include multiple variants increasing levels difficulty handmanipulateeggrotate random target rotation axes egg target position handmanipulateeggfull random target rotation axes egg random target position goal considered achieved distance egg position desired position less applicable full variant difference rotation less rad pen manipulation handmanipulatepen another manipulation time using pen instead block egg grasping pen quite hard since easily falls hand easily collide get stuck fingers goal includes target position cartesian coordinates target rotation quaternions include multiple variants increasing levels difficulty handmanipulatepenrotate random target rotation axes pen target rotation around axis target position handmanipulatepenfull random target rotation axes pen target rotation around axis random target position goal considered achieved distance pen position desired position less applicable full variant difference rotation ignoring less rad environment interface environments use goals describe desired outcome task example fetchreach task desired target position described goal environments fully compatible openai gym api slightly extend upon support new type environment environments extend newly introduced observation space first enforces constraint observation space concretely requires observation space type space least following three keys observation actual observation environment example robot state position objects axis pen parallel body goes tip opposite end median test success rate median test success rate sparse rewards ddpg sparse rewards dense rewards ddpg dense rewards sparse rewards ddpg sparse rewards dense rewards ddpg dense rewards epoch median test success rate median test success rate sparse rewards ddpg sparse rewards dense rewards ddpg dense rewards epoch sparse rewards ddpg sparse rewards dense rewards ddpg dense rewards epoch epoch figure median test success rate line interquartile range shaded area four fetch environments goal agent achieve case fetchreach would target position goal agent currently achieved instead case fetchreach position robots end effector ideally would quickly possible exposed reward function second expose reward function way allows recomputing reward different goals necessary requirement algorithms substitute goals detailed example available appendix compatibility standard algorithms since openai gym commonly supported algorithm frameworks tools like openai baselines dhariwal include simple wrapper converts new goal observation space common array representation detailed example available appendix benchmark results evaluate performance ddpg without hindsight experience replay andrychowicz environments variants compare following four configurations sparse rewards dense rewards ddpg sparse rewards ddpg dense rewards detailed hyperparameters found appendix environments train single machine cpu cores core generates experience using two parallel rollouts uses mpi synchronization fetchreach fetchpush fetchslide fetchpickandplace handreach train epochs one epoch consists full episodes amounts total timesteps remaining median test success rate median test success rate sparse rewards ddpg sparse rewards dense rewards ddpg dense rewards sparse rewards ddpg sparse rewards dense rewards ddpg dense rewards epoch sparse rewards ddpg sparse rewards dense rewards ddpg dense rewards median test success rate median test success rate sparse rewards ddpg sparse rewards dense rewards ddpg dense rewards epoch epoch epoch figure median test success rate line interquartile range shaded area four fetch environments environments train epochs amounts total timesteps evaluate performance epoch performing deterministic test rollouts per mpi worker compute test success rate averaging across rollouts mpi workers implementation available part openai dhariwal cases repeat experiment different random seeds report results computing median test success rate well interquartile range figure depicts median test success rate four fetch environments fetchreach clearly simple environment easily solved four configurations remaining environments clearly outperforms configurations interestingly performs best reward structure sparse also able successfully learn dense rewards vanilla ddpg typically easier learn dense rewards sparse rewards challenging figure depicts median test success rate four hand environments similar fetch environments significantly outperforms ddpg baseline fact baseline often able learn problem similar sparse reward structure works significantly better dense reward using able learn partly successful policies environments especially handmanipulatepen especially challenging able fully solve note depict results variants four environments figure complete set plots environments variants found appendix believe reason typically performs better sparse rewards mainly due following two reasons learning critic much simpler sparse rewards dense case critic approximate highly function includes euclidean distance positions difference two quaternions rotations hand learning sparse return much simpler since critic differentiate successful failed states https dense reward biases policy towards specific strategy instance may beneficial first grasp object properly start rotating towards desired goal dense reward however encourages policy chose strategy achieves desired goal directly request research deciding problem worth working probably hardest part research present set research problems believe lead improvements problem propose least one potential solution solving many require inventing new ideas make tracking progress work ideas easier would like ask authors cite report publishing related research automatic hindsight goals generation andrychowicz goals used generated using heuristic replaying goal achieved random future timestep episode instead could learn goals valuable replay could chosen goals achieved seen training generated separate neural network given transition input biggest question judge goals valuable replay one option would train generator maximize bellman error bears lot similarity prioritized experience replay schaul expect techniques paper may useful unbiased changes joint distribution replayed state action goal tuples unprincipled way could theory make training impossible extremely stochastic environment albeit noticed practice consider environment special action takes agent random state episode ends action would seem perfect hindsight replay goal achieved agent future avoid problem one potential approach would use importance sampling cancel sampling bias would probably lead prohibitively high variance gradient hierarchical levy showed promising results applying hierarchical setup one possible extension work would replace hindsight goals also actions higher level asked lower level reach state state reached could replay episode replacing action could allow higher level learn even lower level policy bad principled could make training unstable richer value functions uvfa schaul extended value functions multiple goals tdm pong extended different time horizons innovations make training easier despite fact learned function complicated else could fed value function improve discount factor success threshold binary rewards faster information propagation algorithms use target networks stabilize training dqn mnih ddpg lillicrap however comes price limiting maximum learning speed algorithm target network update sends information returns one step backward time bootstrapping used noticed learning speed early stages training often proportional frequency target network excessive target network updates leads unstable training worse final performance adapt frequency target network updates moving average coefficient used update network maximize training speed better ways update target network simple replacement moving average time ways stabilize training limit learning speed clipped objective similar one used ppo schulman target networks computed using moving average main network parameters returns generates data extremely therefore multistep returns used unless employ correction factors like importance sampling many solutions dealing data munos clear would perform well setup training data far another approach would use optimality tightening inequalities using returns beneficial decreased frequency bootstraping lead less biased gradients moreover accelerates transfer information returns backwards time accordingly experiment often limiting factor training compare previous paragraph combine algorithms like ppo schulman preliminary results vanilla policy gradients presented rauber approach needs tested challenging environments like ones proposed report one possible option would also use techniques similar ones employed ipg combine recent improvements would interesting see recent improvements perform combined list potential improvements long prioritized experience replay schaul distributional bellemare schulman reverse curriculum generation florensa frequent actions algorithms sensitive frequency taking actions frame skip technique usually used atari mnih continuous control domains performance goes zero frequency taking actions goes infinity caused two factors inconsistent exploration necessity bootstrap times propagate information returns backward time design algorithm retain performance even frequency taking actions goes infinity problem exploration addressed using parameters noise exploration plappert faster information propagation could achieved employing returns approach could adaptive learnable frame skip references andrychowicz wolski ray schneider fong welinder mcgrew tobin abbeel zaremba hindsight experience replay advances neural information processing systems pages bellemare dabney munos distributional perspective reinforcement learning arxiv preprint brockman cheung pettersson schneider schulman tang zaremba openai gym arxiv preprint dhariwal hesse klimov nichol plappert radford schulman sidor openai baselines https florensa held wulfmeier abbeel reverse curriculum generation reinforcement learning arxiv preprint lillicrap ghahramani turner levine interpolated policy gradient merging gradient estimation deep reinforcement learning arxiv preprint liu schwing peng learning play day faster deep reinforcement learning optimality tightening arxiv preprint kingma adam method stochastic optimization arxiv preprint notice normally data used algorithms come earlier version policy therefores relatively close data case completely replace goals fed network levy platt saenko hierarchical arxiv preprint lillicrap hunt pritzel heess erez tassa silver wierstra continuous control deep reinforcement learning arxiv preprint mnih kavukcuoglu silver rusu veness bellemare graves riedmiller fidjeland ostrovski control deep reinforcement learning nature munos stepleton harutyunyan bellemare safe efficient reinforcement learning advances neural information processing systems pages plappert houthooft dhariwal sidor chen chen asfour abbeel andrychowicz parameter space noise exploration arxiv preprint pong dalal levine temporal difference models deep control international conference learning representations rauber mutz schmidhuber hindsight policy gradients arxiv preprint schaul horgan gregor silver universal value function approximators proceedings international conference machine learning pages schaul quan antonoglou silver prioritized experience replay arxiv preprint schulman abbeel chen equivalence policy gradients soft arxiv preprint schulman wolski dhariwal radford klimov proximal policy optimization algorithms arxiv preprint todorov erez tassa mujoco physics engine control intelligent robots systems iros international conference pages ieee api examples exposed reward function following example demonstrates exposed reward function used reward substituted goals info dictionary used store additional information may necessary reward independent goal state derived simulation import gym env gym make env obs done env env env obs obs copy env obs compatibility standard algorithms following example demonstrates wrap new environments make observation spaces compatible existing implementations simply wrap environment specify desired keys dictionary would like use import gym env gym make env env gym env env numpy hyperparameters ensure fair comparison perform hyperparameter search following parameters actor learning rate critic learning rate coefficient batch size probability random action scale additive gaussian noise action norm coefficient since searching possible combinations exhaustivley intractable randomly sample combinations train policy environment four configurations sparse dense ddpg sparse ddpg dense picked environment since configurations capable learning environment configuration combination train random seeds average performance across select best hyperparameter combination numerically compute area test success rate curve select combination achieves best performance across tasks experiments paper use following hyperparameters found aforementioned search actor critic networks layers units relu adam optimizer kingma training actor critic buffer size transitions coefficient action norm coefficient observation clipping batch size rollouts per mpi worker number mpi workers cycles per epoch batches per cycle test rollouts per epoch probability random actions scale additive gaussian noise probability experience replay normalized clipping hyperparameters described greater detail andrychowicz full benchmark results median test success rate sparse rewards ddpg sparse rewards dense rewards ddpg dense rewards epoch median test success rate sparse rewards ddpg sparse rewards dense rewards ddpg dense rewards epoch median test success rate sparse rewards ddpg sparse rewards dense rewards ddpg dense rewards epoch sparse rewards ddpg sparse rewards dense rewards ddpg dense rewards median test success rate epoch sparse rewards ddpg sparse rewards dense rewards ddpg dense rewards median test success rate epoch median test success rate sparse rewards ddpg sparse rewards dense rewards ddpg dense rewards epoch median test success rate sparse rewards ddpg sparse rewards dense rewards ddpg dense rewards epoch median test success rate sparse rewards ddpg sparse rewards dense rewards ddpg dense rewards epoch sparse rewards ddpg sparse rewards dense rewards ddpg dense rewards median test success rate epoch median test success rate sparse rewards ddpg sparse rewards dense rewards ddpg dense rewards epoch sparse rewards ddpg sparse rewards dense rewards ddpg dense rewards median test success rate epoch sparse rewards ddpg sparse rewards dense rewards ddpg dense rewards median test success rate epoch
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storage communication load balancing distributed cache networks jun mahdi jafari siavoshani member ieee ali pourmiri seyed pooya shariatpanahi consider load balancing network caching servers delivering contents end users randomized load balancing via power two choices wellknown approach parallel distributed systems framework investigate tension storage resources communication cost load balancing performance end propose randomized load balancing scheme simultaneously considers cache size limitation proximity server redirection process contrast classical power two choices setup since memory limitation proximity constraint cause correlation server selection process may benefit power two choices however prove certain regimes problem parameters scheme results maximum load order log log network size exponential improvement compared scheme assigns request nearest available replica interestingly extra communication cost incurred proposed scheme compared nearest replica strategy small furthermore extensive simulations show trend depend network topology library popularity profile details keywords randomized algorithms distributed caching servers request routing load balancing communication cost content delivery networks ntroduction problem motivation advancement technology leads spread smart communication devices masses causes rapid growth demands data communication although telcos spending hugely telecommunication infrastructures keep data demand explosion caching predictable data network hours near end users proposed promising solution challenge approach used extensively content delivery networks cdns akamai azure amazon cloudfront etc mobile video delivery approach cache network usually referred set caching servers connected network giving content delivery service end users schematic view typical cache network see figure figure shows caching servers connected authors names appear alphabetical order work presented part ipdps jafari siavoshani department computer engineering sharif university technology tehran iran email mjafari pourmiri department computer engineering university isfahan isfahan iran email shariatpanahi school computer science institute research fundamental sciences ipm tehran iran email pooya assigned requests incoming requests backhaul network request redirection fig general distributed cache network backhaul network servers responsible delivering contents requested customers server assign demand responding server could via assignment strategy every cache network three critical parameters namely storage resource communication cost network imbalance storage memory characterizes percentage total content library cached server communication cost amount data transferred inside backhaul network satisfy content requests finally network imbalance characterizes uniformly requested contents loads distributed among different responding servers usually measured comparing load busiest server average load servers request assignments every request assignment strategy fact leads three quantities practical viewpoint interested distributed server selection strategies scalable large networks randomized load balancing via power two choices paradigm parallel distributed settings approach upon arrival request corresponding user query current loads two independently random chosen servers allocates request least loaded server considering load balancing perspective berenbrink showed scheme allocating balls requests tasks etc bins servers machines etc maximum number balls assigned bin called maximum load log log high probability deviates log log average load deviation depends number servers although power two choices strategy addresses load balancing issue distributed manner consider role two important quantities namely memory communication cost goal paper extend power two choices framework order characterize three quantities results show maximum load communication cost servers memory three entangled parameters thus previous studies sufficient designing practical load balancing strategy cache networks problem setting contributions paper consider general cache network model entails basic characteristics many practical scenarios consider network servers library size files server cache files network lowtraffic hours let assume popularity distribution library assume cache content placement server proportional popularity distribution hours sequential file requests library distributed among servers uniformly random every server either serves requests redirects via assignment scheme nodes cached files define maximum load assignment scheme maximum number allocations single server assigning requests communication cost average number hops required deliver requested file request origin baseline assignment scheme consider request arrived every server dispatched nearest file replica scheme results minimum communication cost ignoring maximum load servers show grid topology every constant uniform distribution scheme result maximum load interval log log log log high moreover every constant maximum load log also investigate communication cost incurred scheme uniform zipf popularity distributions particular derive communication cost order uniform distribution grid topology contrast propose new scheme considers memory maximum load communication cost simultaneously request scheme chooses two random candidate servers cached request putting constraint distance requesting node proximity constraint due cache size limitation proximity constraint current results balanced allocation literature carried setting basically show two chosen servers become correlated might diminish power two choices since correlation arises memory limitation proximity constraint main challenge address paper high probability refers event happens probability constant power two choices achievable fig suppose shaded area shows region power two choices asymptotically achievable details refer theorem characterizing regimes benefit power two choices time low communication cost particular suppose two constants let uniform distribution grid topology provided log log log maximum load log log communication cost therefore set log log achieve power two choices communication cost order log communication cost log factor communicationpcost achieved nearest replica strategy figure shows region parameters power two choices asymptotically achievable noted theoretical results derived grid networks main reason assuming grid presentation clarity results extended topologies theoretical results derived large networks asymptotic analysis simulation results show validity even finite sized networks also simulations investigate problem diverse settings considering network topologies related work load balancing focus many papers cache networks among distributed approaches attracted lot attention see randomized load balancing via power two choices popular approach direction chen consider two choices selection process second choice next neighbor first choice xia use length common prefix lcp replication arrive recursive balls bins problem authors benefit metaphor power two choices design algorithms randomized load balancing contrast works follow theoretical approach derive provable results cache networks theoretical works investigating power two choices cache networks consider role two parameters among memory resource load balancing tion cost works summarized three categories follow memory resource communication cost investigated many works non papers considered load balancing issue cache networks important practice load balancing memory resource investigated authors consider supermarket model performance evaluation cdns although work considers memory limitation account consider proximity principle central issue paper liu study setting clients compare servers terms web applications video applications choose favourite ones setup objectives different consider moreover considered effect randomized load balancing scheme communication cost communication cost load balancing investigated without considering effect cache size limitation although works mentioned non provides rigorous analysis contrast standard balls bins model works introduced effect proximity constraint ball bins framework standard balls bins model ball request picks two bins servers independently uniformly random allocated one lesser load however many settings selecting two random servers might infeasible costly words proximity constraint translates correlation two choices balls bins model related choices related work paper kenthapadi panigrahi proposed model bins connected graph corresponding ball node chosen uniformly random first candidate one neighbours chosen uniformly random second candidate ball allocated one minimum load assumption proved graph sufficiently dense average degree log log log allocating balls maximum load log log although model used considers proximity principle assigning request origin neighbors directly applied cache network setup first consider communication practice communication done fashion second model accommodate cache size limitation servers cache size limitation introduces notion cache content placement based popularity profile addition limitation introduce new source correlation choices considered organization paper follows section present notation problem setup section iii nearest replica strategy investigated baseline scheme section propose analyze proximityaware two choices strategy time considers memory limitation proximity constraint benefits power two choices section performance two schemes investigated via extensive simulations finally discussions concluding remarks presented section otation roblem etting notation throughout paper high probability refers event happens probability constant let graph vertex set edge set let denote degree every pair nodes denotes length shortest path neighborhood distance defined finally use denote poisson distribution parameter problem setting consider cache network consisting caching servers also called nodes connecting neighboring servers forming grid direct communication possible adjacent nodes communications carried fashion remark throughout paper sake presentation clarity may consider torus nodes helps avoid boundary effects grid asymptotic results hold grid well suppose cache network responsible handling library files whereas popularity profile follows known distribution network operates two phases namely cache content placement content delivery cache content placement phase node caches files randomly library according popularity distribution replacement independent nodes also note throughout paper assume unless otherwise stated consider time block files requested servers sequentially chosen uniformly random let denote number requests demands arrived server large library popularity profile consider two probability distributions namely uniform zipf parameter uniform distribution considers equal popularity files zipf distribution request probability popular file inversely proportional rank follows confirmed case many practical applications given cache content placement assignment strategy determines request mapped server let denote number requests assigned server end mapping process strategy define following metrics respect positive number define four areas follows shown fig easy see four areas size communication cost strategy average number hops requesting node serving node denoted maximum load strategy maximum number requests assigned single node denoted iii earest eplica trategy simplest strategy assigning requests servers allocate request nearest node cached file strategy formally defined leads minimum communication cost try reduce maximum load definition strategy nearest replica strategy strategy request assigned nearest node sense graph shortest path cached requested file multiple choices ties broken randomly consider set nodes cached file say according strategy demand node file served arg induces voronoi tessellation torus corresponding file denote alternatively define strategy assigning request file corresponding voronoi cell center order analyze maximum load imposed node investigate size voronoi regions following lemma direction lemma uniform popularity distribution maximum cell size number nodes inside cell log particular every voronoi cell centered node contained log furthermore grid size constant exists voronoi cell size log proof lemma upper bound fix node assume denoted pair torus definition communication cost maximum load let fix arbitrary every node define indicator random variable taking value cached file node cached file otherwise first term determines probability caches second one determines probabilitypthat nodes cache setting log applying inequality log follows moreover using approximation therefore applying union bound nodes files implies every exists least one node shares common file supported choose log similarly prove argument suppose cached file want find upper bound size voronoi cell centered corresponding order let define nearest node file similarly define assume origin nodes system define show voronoi cell contained thus size upper bound size also know var cov last equality holds indicator random variables easy see every cov cache content placement different nodes independent processes consider pairs pair nodes three following cases considered case fig demonstration regions used upper bound proof lemma voronoi cell consider fig let consider node complement assume puw shortest path passes node definition know length shortest path shows closer definition belong voronoi cell centered similarly show arbitrary node closer either rather since arbitrarily choose contains every voronoi cell centered given size voronoi cell centered arbitrary node bounded log lower bound let define indicator random variable every fixed taking value cached file cached file otherwise note setting log using similar approximations used let following claim claim every probability claim shows exists least voronoi cell size log concludes proof order prove claim note definition indicator random variables case let split summation based follows var cov cov cov applying results yields var log log use fact log applying chebychev inequality leads var log log therefore concentrates around mean proves claim probability node caches hence every ready present main results section characterize maximum load strategy two different parameter regimes theorems theorem suppose constant uniform distribution strategy achieves maximum load log proof consider node cached set distinct files say applying lemma shows voronoi cells centered corresponding cached files contained size log also round every arbitrary node requests file probability request randomly chooses origin type hence union bound node may request file probability log log since requests expected number requests imposed node log using chernoff bound see appendix shows handle log requests hand establish lower bound maximum load proceed follows lemma shows exits voronoi cell center node handle requests least log nodes also node cell may request file probability average log requests imposed cell center similarly chernoff bound one see node experiences load log concludes proof constant remark noted result log maximum load also proved zipf distribution content placement distribution chosen proportional file popularity distribution consequently result insensitive however proof involves lengthy technical discussions omit paper numerical investigation remark refer section theorem suppose uniform distribution maximum load interval log log log log proof theorem establish upper bound log maximum load follow first part proof theorem obtain lower bound consider arbitrary server cached file set distinct files note lemma every node high probability let define indicator random variable taking nearest replica outside zero otherwise easy see correlated example consider set files wjt every conditioned event node cached files subset library size hence constant let using chernoff bound moderately correlated indicator random variables see lemma implies therefore contain replica least fraction files cached namely using union bound nodes deduce similar statement every node probability least therefore every severs request follows since requests easy see load server bounded poisson distribution constant hand known see maximum number taken poisson distribution log log log hence lower bound proved next investigate communication cost strategy following theorem theorem uniform popularity distribution strategy achieves communication cost every zipf popularity distribution achieves log log proof theorem assume request arbitrary node file probability file cached another node cache content placement different nodes independent thus number nodes probed geometric random variable success probability results average trials leads expected distance averaged different files cached requested file request assigned node lesser load ties broken randomly sake illustration first consider examples following uniform distribution zipf distribution example example node store library constraint proximity mentioned section number files handled node random variable case according strategy two random nodes chosen network nodes request assigned node lesser load therefore terms maximum load problem reduced standard power two choices model balanced allocations literature model bins sequential balls randomly allocated bins every round ball picks two random bins uniformly allocated bin lesser load shown maximum load network maxi log log exponential improvement compared strategy define every hand known every see log inserting equations completes proof theorem shows file popularity reduces communication cost skew file popularity determined parameter affect communication cost example communication cost similar uniform distribution becomes independent since strategy assigned request nearest replica theorem characterizes minimum communication cost one achieve however theorems show logarithmic growth maximum load function network size imbalance network load strategy request assignment consider current load servers natural question whether request allocation one use limited information servers current load order reduce maximum load also one ask affect communication cost address questions following section hoices trategy strategy introduced previous section result minimum communication cost maximum load strategy order log log log section investigate strategy result exponential decrease maximum load reduces maximum load log log formally defined follows definition two choices strategy request born arbitrary node consider two uniformly random chosen nodes however contrast example cache networks usually node store subset files makes problem different standard balls bins model considered due memory constraint node choices much limited case words case related choices related choices scenario event choosing second choice correlated first choice correlation may annihilate effect power two choices demonstrated example example regime subset library say whose files cached inside network hand file type requested times hence file requested log log log times see since file replicated times requests file distributed among nodes thus maximum load corresponding nodes least log log log hence due memory limitation benefit power two choices although example shows memory limitation annihilate power two choices always case example shows even scenarios achieve log log example every constant popularity distribution strategy achieves maximum load log log also notice uniform zipf distributions satisfy requirement whenever zipf parameter roughly speaking may partition servers based cached file hence equivalent noted including thus paper use alternatively joint subsets servers similarly request types request addressed corresponding subset servers thus disjoint balls bins maximum load determines maximum load original setup reason contrast example benefit power two choices assumption proof example easy see number caching servers specific file say denoted distributed bin thus applying chernoff bound see appendix implies exp moreover let denote number requests file sum bin random variables applying chernoff bound see appendix poisson random variables yields exp notice npj suppose denotes event exp also define event model bins caching servers balls requests achieves maximum load log log shown event happens probability every constant see exp exp since disjoint subsystems union bound subsystems shows two choice model achieve desired maximum load probability npj concludes proof due example assumption popularity profile show uniform zipf distributions satisfy example assumption uniform distribution files setting npj setting using fact applied examples bring attention following question question view memory limitation server cache networks regimes terms problem parameters one benefit power two choices balance load addressing question general case challenging example needs completely different approach simplicity case interaction balls bins hand consider request say allocated server load two candidate bins cached compared however load file types also accounted comparison flow load information different subproblems makes entangled examples considered proximity constraint results fairly high communication cost however general since parameter controls communication cost chosen much less network diameter proximity awareness introduces another source correlation memory limitation two choices thus considering proximity constraint may annihilate power two choices even large memory cases demonstrated following example example example request arrives server server chooses two random choices among neighbours request allocated one lesser load since exists server maxi log log log requests arrive maximum load network maxi least log log log thus similar question regarding memory limitation effect one pose following question regarding proximity principle also zipf distribution depending consider two cases every log used question view proximity constraint scheme regimes terms problem parameters one benefit power two choices balance load order completely analyze load balancing performance scheme one consider sources correlation simultaneously case examples end following investigate two memory regimes namely main result presented following theorem theorem suppose two constants let log log log uniform popularity distribution strategy achieves maximum load log log communication cost remark accessible proof theorem assumed note proof techniques also extended case order prove theorem let first present interesting result shown follows theorem given almost edges nodes representing bins balls thrown bins choosing random edge probability placing smaller two bins connected edge maximum load log log log log remark note original theorem presented assumed edge chosen uniformly among edges graph however slightly generalize result edge chosen probability proof follows original proof idea modifications computation parts due lack space omit order apply theorem first need define new graph follows definition configuration graph given parameter configuration graph defined graph whose nodes represent servers two nodes say connected cached common file torus every two servers let set distinct files cached nodes also denote define number distinct cached files let define goodness placement strategy follows definition goodness property every positive constant say file placement strategy every lemma proportional cache placement strategy introduced section proof clearly every set cached files every node replacement mapped integral solution equation expresses number times file cached node combinatorial argument shows equation integer solutions graph said almost vertex degree first fixed set file indexes size say count number integral solutions equation order bound every also recall assumed hence every thus choosing every last equality follows union bound nodes yields similar argument every thus constant write applying union bound pairs servers every hence putting together concludes proof following lemma presents useful properties strategy lemma conditioning goodness file placement assuming log log log almost mkr request strategy samples edge two servers probability proof consider arbitrary node distinct files definition every node hand connected addition therefore every given node distinct cached files degree binomial distribution bin hence applying chernoff bound implies probability conditioning goodness file placement also symmetry torus every high probability every concludes proof part remains show strategy picks edge probability first notice recall file cached every node probability independently given node file let number nodes distance cached file binomial distribution bin every since log log log log every applying chernoff bound implies probability concentrates around mean hence every consider edge define set nodes may pick pair randomly strategy hard see picked strategy conditioned goodness every simplified picked strategy last equality follows proof theorem applying lemma shows configuration graph almost graph moreover step every edge chosen randomly probability hence apply theorem conclude maximum load log log log log log log follows log log log hence let present next result regarding regime theorem suppose uniform distribution file library strategy achieves load log log communication cost log log log proof let choose log log log assumption configuration graph corresponding graph two nodes connected since network symmetric every hence regular graph also hard see strategy equivalent choosing edge uniformly applying theorem results maximum load log log addition choosing two random nodes results communication cost point theorem endure log log log benefit luxury power two choices encouraging result imulations section demonstrate simulation results two strategies introduced previous sections namely nearest replica proximity aware two choices strategies simulations implemented python code available online figure shows maximum load strategy function number servers different curves correspond different cache sizes network graph torus files uniform popularity placed uniformly random node point average simulation runs figure agreement logarithmic growth maximum load asymptotically proved section iii even intermediate values makes results section iii general comparing different curves reveals fact larger cache size setting balanced network happens enlarging cache sizes results uniform voronoi tessellation cells smaller variation size furthermore figure shows communication cost strategy function cache size different curves maximum load maximum load cache size cache size cache size cache size strat cache size strat cache size strat cache size strat cache size servers servers fig maximum load versus number servers strategy network topology torus file popularity uniform average cost hops library size library size library size cache size files fig communication cost versus cache size strategy network topology torus size file popularity uniform correspond different library sizes network graph torus size point average simulation runs figure agreement result theorem order simulate strategy first set study effect cache size maximum load cache size cache size cache size cache size maximum load servers fig maximum load versus number servers strategy network topology torus file popularity uniform library size moreover assume fig maximum load versus number servers strategies network topology torus file popularity uniform library size moreover assume communication cost consider effect limited performance system figure shows maximum load network versus number servers curve demonstrates different cache size network graph torus files uniform popularity placed uniformly random node point average simulation runs curve since cache size number files fixed increasing number servers translates increasing file replication figure demonstrates system performance large system sizes however get better understanding network behavior let compare load balancing performance strategies figure file library size figure file replication low due high correlation two choices strategy power two choices expected reflected figure example curve corresponding fast growth maximum load mimics load balancing performance strategy see trend curve corresponding however assuming since enough file replication network load balancing performance greatly improved due power two choices accordance lessons learned section also transition region mixed behavior observed observations made figure important practical implication since employing strategy beneficial networks high file replication situations limited cache size less sophisticated strategy proper choice figure draws communication cost versus number servers various cache sizes similar setting used figure since figure constraint proximity communication cost growth order simulations presented far considered case order investigate effect parameter performance system figure simulated network operation different values results average cost hops maximum load cache size cache size cache size cache size servers strat zipf strat zipf strat zipf strat zipf strat zipf strat zipf servers fig communication cost versus number servers strategy settings similar figure cache size cache size cache size cache size cache size cache size cache size average cost hops fig maximum load communication cost strategy network topology torus size file popularity uniform library size maximum load communication cost shown figure consider torus servers files uniform popularity placed uniformly random node point average simulation runs figure like figure observe two performance regimes based file replication network high memory regime curves corresponding achieve power two choices sacrificing negligible communication cost hand low memory regime decrease maximum load even expense high communication cost values intermediate values clearly observe maximum load communication cost simulations investigated performance networks torus topology uniform file popularity distribution agreement theoretical results indications however one may ask sensitive findings network topology file popularity choices thus following examine network performance zipf file popularity profile practical network topologies namely random geometric graph rgg power law random graph model maximum load maximum load fig maximum load versus number servers strategies network topology torus file popularity zipf library size cache size moreover assume point average independent simulation runs strat zipf strat zipf strat zipf strat zipf strat zipf strat zipf servers fig maximum load versus number servers strategies network topology rgg remaining settings similar figure figures show maximum load versus number servers different network topologies namely torus rgg power law model zipf distribution also figures demonstrate performance maximum load communication cost different network topologies namely torus random rgg power law model zipf distribution parameter simulations show trends found theoretical results also valid practical network settings convenience summary simulation parameters stated table iscussion oncluding emarks section first discuss three important practical issues related proposed scheme conclude paper note zipf distribution corresponds uniform distribution maximum load maximum load requests strat zipf strat zipf strat zipf strat zipf strat zipf strat zipf gamma gamma gamma gamma fig maximum load versus number servers strategies network topology power law random graph remaining settings similar figure maximum load requests gamma gamma gamma gamma average cost hops fig maximum load communication cost strategy network topology torus size file popularity zipf library size cache size point average independent simulation runs maximum load requests gamma gamma gamma gamma average cost hops fig maximum load communication cost strategy network topology rgg size remaining settings similar figure fig maximum load communication cost strategy network topology power law random graph size remaining settings similar figure fig number average cost hops servers net topology torus torus torus torus torus rgg power law torus rgg power law strategy strategy strategy strategy strategy varying strategy strategy strategy strategy varying strategy varying strategy varying popularity uniform uniform uniform uniform zipf zipf zipf zipf zipf zipf table summary simulation parameters figure theoretical results sections iii stated topology main reason assuming rather unrealistic topology developing main idea paper clearly however noticed approach extended general graph models expense lengthy proofs calculations example mentioned section size ball radius around node main feature affects results suppose instead assuming consider graph rdim parameter dim appear results instead dim special case example term statement theorem would generalized dim generally even basic technicalities extended however paper investigate network topologies random geometric graphs power law networks via extensive simulations section discussed main trends also valid topologies well thus findings cover wider class graph models similar cdn network topologies proposed two choices scheme implemented distributed manner see notice upon arrival request server strategy needs two kinds information redirect request information provided requesting server without need centralized authority following way first one cache content users neighborhood radius since cache content dynamic servers much slower requests arrival done periodic polling nearby servers without introducing much overhead also see distributed hash table dht schemes second type information queue length information two randomly chosen nodes inside neighborhood radius also efficiently done distributed manner polling piggybacking practice request arrivals servers operation happen continuous time needs queuing theory based performance analysis however shown behaviour load balancing schemes continuous time known supermarket model static balls bins problems closely related thus conjecture proposed scheme also performance queuing theory based model postpone rigorous analysis scenario future work summary work considered problem randomized load balancing tension communication cost memory resources cache networks proposing two request assignment schemes investigated analytically moreover extensive simulation results support theoretical findings provide practical design guidelines acknowledgment authors would like thank seyed abolfazl motahari omid etesami thomas sauerwald farzad parvaresh helpful discussions feedback eferences jafari siavoshani pourmiri shariatpanahi proximityaware balanced allocations cache networks ieee international parallel distributed processing symposium ipdps cisco cisco visual networking index global mobile data traffic forecast update white paper zhang lin caching information centric networking survey computer network vol nygren sitaraman sun akamai network platform internet applications operating systems review vol golrezaei shanmugam dimakis molisch caire femtocaching wireless video content delivery distributed caching helpers infocom proceedings ieee march azar broder karlin upfal balanced allocations siam vol mitzenmacher power two choices randomized load balancing ieee trans parallel distrib vol adler chakrabarti mitzenmacher rasmussen parallel randomized load balancing random struct algorithms vol lenzen wattenhofer tight bounds parallel randomized load balancing distributed computing vol berenbrink czumaj steger balanced allocations heavily loaded case siam vol peng cdn content distribution network corr vol online 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llorca caire rate caching coded multicasting random demands corr vol online available http online available https bins albert statistical mechanics complex networks reviews modern physics vol bauer hurley waldvogel replica placement location using distributed hash tables annual ieee conference local computer networks lcn october clontarf castle dublin ireland proceedings karger lehman leighton panigrahy levine lewin consistent hashing random trees distributed caching protocols relieving hot spots world wide web proceedings annual acm symposium theory computing paso texas usa may mitzenmacher richa sitaraman power two random choices survey technique results handbook randomized computation volume dubhashi panconesi concentration measure analysis randomized algorithms cambridge university press auger doerr theory randomized search heuristics foundations recent developments river edge usa world scientific publishing ppendix ome tail ounds theorem chernoff bounds suppose independent random variables pnn let every following inequalities hold exp exp particular exp proof see deal moderate independency state following lemma lemma deviation bounds moderate independency see lemma let arbitrary binary random variables let binary random variables mutually independent independent assume latter term bounded deviation bound independent random variables
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online auctions online learning nikhil zhiyi rad may microsoft research redmond department computer science university hong kong department computer science cornell university may abstract consider revenue maximization online auctions pricing seller sells identical item period new buyer new set buyers online posted pricing problem show regret bounds scale best fixed price rather range values also show regret bounds almost scale free match offline sample complexity comparing benchmark requires lower bound market share results obtained generalizing classical learning experts bandit problems versions version reward action different range regret given action scales range rather maximum range introduction consider following revenue maximization problem repeated setting called online posted pricing problem period seller single item sell new prospective buyer seller offers sell item buyer given price buyer buys item price private valuation item private valuation buyer never revealed seller monopolistic seller iteratively set prices wishes maximize revenue also cares market share estimating price sensitivities demand models order optimize revenue market share bedrock econometrics emergence online marketplaces enabled sellers costlessly change prices well collect huge amounts data renewed interest understanding best practices data driven pricing extreme case price updated buyer online pricing problem described one always use less frequent price updates moreover problem intimately related classical experimentation estimation procedures problem studied online learning perspective variant multiarmed bandit problem revenue pricing algorithm compared revenue best fixed posted price hindsight difference two called regret analyzed assumption made distribution values regret bounds required hold worst case sequence values blum assume buyer valuations show zhiyi huang supported rgc grant rad niazadeh supported nsf grant google fellowship part work completed zhiyi visiting rad internship microsoft research redmond following multiplicative additive bound regret regret times revenue optimal price log log log blum hartline show additive factor made log log trading log factor extra factor undesirable aspect bounds scale linearly particularly problematic estimate might set generous upper bound range prices wish consider typical use case algorithm used many different products widely varying price ranges may able manually tune range product separately dependency seems unavoidable reflected lower bounds problem lower bounds discussed later introduction yet somewhat surprisingly first contribution paper show replace best fixed used definition benchmark particular show additive bound made log best fixed price hindsight allows use generous estimate lose log factor algorithm balances exploration probabilities different prices carefully automatically zooms relevant price range violate known lower bounds since instances close blum blum hartline also consider full information version problem call online auction problem valuations buyers revealed algorithm buyer made decision information may available context buyers bid items awarded item bid hidden price case additive term improved log tight show replaced particular show additive term made log purely multiplicative bounds sample complexity regret bounds mentioned turned purely multiplicative factor following way algorithm guaranteed get fraction best fixed price revenue provided number periods additive term regret bounds follows observation revenue lower bound best fixed price revenue call number periods required get multiplicative approximation function convergence rate algorithm multiplicative factor also target recent line work sample complexity auctions started dhangwatnotai cole roughgarden give comprehensive discussion line work section samples valuations given fixed unknown distribution goal find price revenue hidden distribution fraction optimum revenue distribution sample complexity minimum number samples needed guarantee function sample complexity convergence rate full information setting closely related sample complexity always smaller convergence rate problem easier following valuations case sample complexity whereas arbitrary worst case case convergence rate standard bounds allow regret depend loss best action instead worst case loss however even bounds still depend linearly range losses thus would allow replace best fixed price sample complexity corresponds offline problem get samples convergence rate corresponds online problem need decide given valuation without knowing valuations arrive future formalized terms online offline reduction folklore shows convergence rate upper bound automatically translated sample complexity upper bound lets convert sample complexity lower bounds lower bounds convergence rate turn lower bounds additive error additive multiplicative regret bound additive error online auction problem hence also posted pricing huang moreover insightful compare convergence rates show best known sample complexity upper bound proving better convergence rates would mean improving bounds well natural target convergence rate problem therefore corresponding sample complexity achieving always trivial interesting version sample complexity bound auctions analogous convergence rate bound version takes account revenue market share surprisingly gets sample complexity bounds scale free dependence means works unbounded valuations best fixed price benchmark relaxed ignore prices whose market share equivalently probability sale fraction increases benchmark lower meaningful benchmark since many cases revenue goal even monopolist reasonable goal maximize revenue subject constraint market share certain threshold gives sample complexity log huang fact set without loss generality values bound matches sample complexity best fixed price revenue addition bound gives precise interpolation target market share increase number samples needed decreases almost linearly second contribution paper show convergence rate almost matches sample complexity full information setting mild dependence rate proportional log log also show near optimal convergence rate posted pricing multiple buyers results full information online auction setting extend multiple buyer model model time period new set buyers compete single item seller runs truthful auction determines winning buyer payment benchmark set mechanisms mechanisms optimal period buyers potentially different types value buyer drawn independently type dependent distribution fact convergence rates also imply new sample complexity bounds problems except computationally efficient various bounds comparison previous work summarized tables conjecture lower bound posted pricing problem worse factor since one needs explore differnet prices values guarantee revenue posting price beat price particular price would sell least times unfortunately yet guarantee online algorithm gets market share although strongly believe showing bounds market share algorithm important avenue future research lower bound online single buyer auction max online posted pricing online multi buyer auction best known sample complexity huang blum kleinberg leighton upper bound best known convergence rate paper thm devanur gonczarowski nisan table number needed get approximation best offline sample complexity offline case samples unknown distribution convergence rate online case worst case sequence sample complexity always larger convergence rate lower bounds hold sample complexity except online posted pricing problem sample complexity version additive multiplicative regret bounds converted convergence rates dividing additive error last row number buyers last column denotes optimal price lower bound sample complexity online single buyer auction online posted pricing online multi buyer auction max huang upper bound paper best known sample complexity thm kleinberg leighton table sample complexity convergence rate opt market share online learning main technical ingredients results variants classical problems learning expert advice bandit introduce versions problems action reward bounded different range third contribution give algorithm problem whose regret certain action scales range rewards particular action contrast regret bounds standard versions scale maximum range expect bounds independent interest versions problems exhibit subtle variations appear standard versions first applications auctions pricing rewards actually makes difference expert bandit versions minimax regret bounds rewards provably better rewards could negative bandit version prove better bound require bound hold best action rather actions rewards various regret bounds comparison standard bounds summarized tables use algorithms based online stochastic mirror descent osmd bubeck weighted negative entropy legendre function framework gives regret bounds terms local norm well initial divergence bound differently version problem technical sections highlight subtle variations arise result different bound paper standard regret bound cmax log cmax log cmax log cmax log upper bound log log log best action log ccmax min log ccmax ccmax min min freund schapire auer lower bound log log ccmax min table regret bounds rewards reward action time symmetric range rewards reward action time suppose time horizon actions set number actions techniques used bound two terms foster also consider online learning problem motivated model selection problem consider additive bounds symmetric case full information bandit feedback regret bounds comparable general bounds better applications consider bounds better application related work online pricing problem also called dynamic pricing much studied topic across disciplines operations research management science talluri van ryzin economics segal marketing course computer science bandit approach pricing particularly popular see den boer recent survey various approaches problem kleinberg leighton consider online pricing problem assumption values considered purely additive factors showed minimax additive regret number periods similar spirit regret bounds scale since one normalize values finer distinction magnitude best fixed price absent work recently syrgkanis also consider online auction problem emphasis notion oracle based computational efficiency assume values consider scaling issue makes contribution orthogonal starting dhangwatnotai spate recent results analyzing sample complexity pricing auction problems cole roughgarden devanur consider multiple buyer auctions regular distributions unbounded valuations give sample complexity bounds polynomial number buyers morgenstern roughgarden consider arbitrary distributions values bounded gave bounds polynomial roughgarden schrijvers huang give improvements versions respectively tables give comparison results bounds problems consider dynamic pricing problem also studied given number copies item sell limited supply agrawal devanur babaioff badanidiyuru besbes zeevi also variants seller interacts buyer repeatedly buyer strategize influence utility future periods amin devanur model main results consider variety online algorithmic problems parts multiscale online learning framework start defining framework expressing results terms regret bounds general problem next investigate different auction design problems covered framework show get multiplicative cum additive approximations problems help learning framework consider competing benchmarks show algorithms get pure multiplicative approximations respect benchmarks translate convergence rate online algorithms sample complexity auctions generalize many sample complexity online adversarial auction settings compare bounds known sample complexity design new algorithms achieving sample complexity bounds offline bayesian auction problem online learning framework online learning framework basically classical learning expert advice problem bandit problem main difference range different could different suppose set actions problem proceeds rounds round algorithm picks action adversary picks reward function simultaneously action reward algorithm gets reward git full information setting algorithm sees entire reward function bandit setting algorithm sees reward git total reward algorithm denoted galg git standard best fixed action benchmark gmax consider bandit setting experts action set countable action set finite size identify reward entire reward function revealed algorithm round bandit learning bandit setting prove regret bounds call also regret guarantees towards end define following quantities regreti galg use notation regret bound action upper bound regreti depends range well prior distribution action set way handle countably many actions let cmin inf cmax applicable minimum maximum range first state version regret bound parameterized bounds stronger type bounds standard theorem exists algorithm experts problem takes input distribution ranges parameter satisfies log regreti compare get using standard analysis experts problem arora second term regret bound log cmax choosing uniform distribution theorem gives log also one compare version bound classic regret bound cmax log experts problem setting log corollary corollary exists algorithm experts problem takes input ranges satisfies regreti log bandit version get similar regret guarantee best action require regret bound hold actions get weaker bound second term instead difference bounds bandit full information setting essentially factor unavoidable theorem exists algorithm online bandits problem takes input ranges parameter satisfies arg log regreti log also one compute versions bounds theorems setting log log resepctively corollary compare regret bound cmax log adversarial bandit problem auer corollary exist algorithms online bandits problem satisfies arg log regreti log online auction design auction design problems consider follows online single buyer auction action set reward function adversary picks value price reward full information setting value revealed algorithm round online posted pricing bandit setting algorithm learns indicator function price picks round online multi buyer auction action set set mechanisms buyers see definition adversary picks valuation vector reward mechanism revenue valuation buyers given denoted revm algorithm sees full vector valuations show get multiplicative cum additive approximations problems gmax benchmark blum blum hartline main improvement results additive term scales best price rather let best fixed price hindsight price achieves gmax theorem algorithms online single buyer auction online posted price auction online multi buyer auction problems take input parameter satsify galg gmax respectively three problems mentioned log log log log log log log log even known upfront still get similar approximation guarantee online single buyer auction online multi buyer auction log log log log bounds sample complexity auctions imply first bound theorem tight log factors lower bound instance best upper bound known log conjecture bound online posted pricing problem tight log factors leave resolving open problem third bound comparable best sample complexity multi buyer auction problem roughgarden schrijvers better large worse smaller also compare corresponding upper bounds first two problems blum blum hartline respectively log log log log log log min competing benchmarks single buyer problem define benchmark benchmark restricted prices sell item least fraction rounds gmax max observed footnote one replace get corresponding guarantees gmax rather gmax however main point results show graceful improvement bounds chosen larger multiple buyers multi buyer auction problem define benchmark follows sequence value vectors let denote largest value least distinct define benchmark gmax maxm revm min min understood applied wise max mechanisms focus purely multiplicative approximation factors competing gmax particular given interested approximation state results terms convergence rate say convergence rate algorithm time horizon guaranteed galg gmax main results follows theorem algorithms online single buyer auction online posted pricing online multi buyer auction problems convergence rates respectively log log log log log log log log even known upfront still get following similar convergence rates online single buyer auction online multi buyer auction respectively log log log log log compare sample compexity bounds first within log log factor best sample complexity upper bound huang lower bound online single buyer auction also best lower bound known pricing online posted pricing problem conjecture right dependence sample complexity bounds problem known fact introduce definition benchmark problem online learning symmetric range standard analysis experts bandit problems holds even range rather assumed contrast subtle differences best acheivable regret bounds symmetric range first show following upper bound full information setting range symmetric bounds follows style regret bounds theorem detailed discussion choice initial distribution affects bound deferred appendix section theorem exists algorithm experts problem symmetric range takes input distribution ranges parameter satisfies regreti log cmin compute version bound theorem similar section log ccmax min setting corollary cole roughgarden show least linear dependence necessary values drawn regular distribution lower bound needs unbounded valuations lower bound probably holds large enough clear holds corollary exists algorithm online experts problem symmetric range takes input ranges satisfies regreti log ccmax min compare regret bound standard cmax log regret boundqfor experts problem see replace dependency cmax standard bound log ccmax min natural ask whether could get rid dependence log show regret bound log like rewards however next theorem shows dependence log bound necessary weak sense constant universal depend ranges lower bound holds small values horizon nonetheless grows theorem exists action set size ranges time horizon algorithms online experts problem symmetric range sequence gain vectors regreti log ccmax min show following upper bound bandit setting range symmetric bound also follows style regret bounds theorem theorem exists algorithm bandits problem symmetric range takes input ranges parameter satisfies cmax regreti ccmax log cmin min also similar qto section compute version bound theorem cmax log cmax setting cmin corollary bound comparable standard regret bound cmax log auer adversarial bandits problem corollary exists algorithm online bandits problem symmetric range satisfies cmax regreti ccmax log cmin min bandit problem following theorem shows bound improved beyond log factors get guarantee like theorem instance theorem exists action set size ranges algorithms online bandit problem symmetric range sufficiently large time horizon sequence gain vectors cmax regreti cmin reason chose include bound table organization start section showing regret upper bounds experts problem rewards theorem corresponding upper bounds bandit version section theorem section show regret bounds theorems imply corresponding bounds problems theorems finally regret upper lower bounds symmetric range discussed section theorems online learning full information section look full information learning problem different experts different ranges exploit structure achieve regret bounds map section section propose algorithm exploits aforementioned structure later section show algorithm online mirror descent weighted negative entropy legendre function instances prove regret bound without dependency log section msmw algorithm propose msmw algorithm update style learning problem algorithm presented algorithm main idea behind algorithm taking account different ranges different experts therefore normalizing reward expert accordingly dividing reward expert projecting updated weights accordingly performing smooth projection simplex described later algorithm msmw input initial distribution learning rate initialize randomly pick action drawn observe exp gic find binary search exp exp end equivalence online mirror descent omd weighted negative entropy possible analyze regret msmw algorithm algorithm using first principles look proof lemma appendix section take different approach show algorithm indeed instance online mirror descent omd algorithm particular choice legendre function preliminaries online mirror descent fix open convex set closure case respectively closedconvex action set case set probability distributions experts heart omd algorithm legendre function strictly convex function admits continuous first order partial derivatives denotes gradient map one think omd member projected gradient descent algorithms gradient update happens dual space rather primal projection defined using bregman divergence associated rather distance definition bregman divergence bubeck given legendre function bregman divergence associated denoted defined definition online mirror descent bubeck suppose legendre function every time online mirror descent algorithm legendre function selects expert drawn distribution updates given rewards gradient update argmin bregman projection called learning rate omd use following standard regret bound omd refer bubeck thorough discussion omd completeness proof also provided appendix section lemma learning rate parameter benchmark distribution omd algorithm legendre function admits following msmw algorithm omd application focus particular choice legendre function captures different learning rates proportional different experts saw earlier algorithm start defining weighted negative entropy function definition given weighted negative entropy defined corollary straightforward see legendre function moreover following lemma shows algorithm indeed omd algorithm equivalent omd algorithm associated lemma msmw algorithm algorithm weighted negative entropy legendre function proof look gradient update step omd equation legendre transform using corollary therefore exp moreover bregman projection step argmin argmin overpa convex set find closed form solution look lagrangian dual function kkt conditions exp unique number exp exp algorithm equivalent omd weighted negative entropy legendre transform lemma initial distribution learning rate parameter benchmark distribution msmw algorithm satisfies galg regret bound rewards proof theorem proof theorem suppose imin action minimum let let lemma imin get note imin galg galg qimin cimin qimin qimin term rhs upper bounded similarly qimin term rhs upper bounded qimin cimin qimin qimin cimin finally note instances lhs lower bounded galg regreti galg putting together get regreti galg galg theorem follows choosing rearranging terms multi scale online learning bandit feedback section look bandit feedback version multi scale online learning inspired online stochastic mirror descent algorithm introduce algorithm algorithm follows standard bandit route using unbiased estimators rewards full information strategy case msmw also mix msmw distribution extra uniform exploration use tailored initial distribution learning setting map section section propose bandit algorithm prove general regret guarantee rewards section show get style regret guarantee best arm weaker guarantee arms bandit multiplicative weight algorithm present bandit algorithm algorithm set actions finite let learning rate exploration probability show following regret bound algorithm input exploration parameter learning rate initialize imin arm minimum range cimin let randomly pick expert drawn observe git let otherwise exp find binary search exp exp end lemma exploration probability learning rate parameter algorithm achieves following regret bound gains regreti log proof define ealg git expectation randomness algorithm ealg galg ealg hence upper bound regreti galg suffices upper bound definition probability algorithm picks arm ealg hence initial distribution ealg ealg ealg next upper bound term rhs note probability choosing experts msmw experts rewards lemma benchmark distribution algorithm satisfies definition equals probability equals otherwise thus fix random coin flips first rounds thus fix take expectation randomness round note upper bounded putting together combining ealg galg let recall recall imin arm minimum range cimin similar expert problem section term rhs upper bounded log hence ealg log galg lhs lower bounded galg galg lemma follows putting back rearranging terms regret bounds rewards proof theorem proof theorem let lemma get expected regret action bounded best arm regret bounded desired regret arbitrary action note galg thus regret bound action lemma upper bounded ealg log theorem follows letting auctions pricing auctions pricing online learning problems online single buyer auction posted pricing recall round algorithm chooses action price adversary picks value algorithm collects reward gpt order obtain approximation optimal revenue suffices consider prices form result reduce online single buyer auction problem online posted pricing problem online learning problem full information bandit feedback respectively actions whose ranges form geometric sequence online multi buyer auction multi buyer auctions consider set discretized myersontype auctions action space start defining auctions definition auctions auction defined virtual value mappings given value profile item given bidder largest virtual value bidder pays minimum value would keep winner myerson shows bidders values drawn independent necessarily identical distributions auction auction devanur lemma observe obtain approximation suffices consider set discretized auctions treat bidder value equal closest power result suffices consider set discretized auctions defined virtual values log real numbers devanur gonczarowski nisan note discretized auction fact completely characterized total ordering actual values matter indeed allocation rule payment rule determined ordering virtual values result action space finite set log actions range action discretized auction largest price ever charged auction largest value form exists proof theorem proof online single buyer auction recall formulation problem online learning problem full information case known follows theorem letting uniform distribution log actions discretized prices known upfront consider countably infinite action space comprised prices form let prior distribution price approximation guarantee follows theorem online posted pricing recall formulation problem online learning problem bandit feedback part follows theorem log actions online multi buyer auction recall formulation problem online learning problem full information case known follows theorem let uniform distribution log actions auctions known upfront consider countably infinite action space follows let log auctions values assume auctions treat values greater choose prior distribution probability mass auction range equal approximation guarantee follows theorem proof theorem proof online single buyer auction known theorem letting uniform distribution log actions discretized prices price recall galg log log optimal price subject selling least rounds therefore log log additive term approximation guarantee theorem holds treatment case known upfront essentially theorem consider countably infinite action space comprised prices form let prior distribution price online posted pricing recall formulation problem online learning problem bandit feedback theorem log actions price galg log log log optimal price subject least rounds therefore log log log additive term approximation guarantee theorem holds online multi buyer auction suppose best auction recall largest value least distinct may assume without loss generality distinguish values greater hence note running auction anonymous reserve auction mapping values less virtual value values greater equal virtual value gets revenue least finally implies obtain approximation suffices consider prices least hence suffices consider auctions given distinguish among values greater distinguish among values smaller log different values given log distinct values considered thus log distinct auctions kind hence total number distinct actions need consider log log known letting uniform distribution actions theorem recall eqn log log log log log galg log log log log log additive term approximation guarantee due eqn theorem holds treatment case known upfront similar theorem known upfront consider countably infinite action space follows let log auctions distinguish among values greater distinguish among values smaller choose prior distribution probability mass auction approximation guarantee follows theorem given equal essentially argument known case remark devanur show values drawn independent regular distributions optimal approximation unguarded optimal convergence rate online multi buyer auction problem theorem implies sample complexity modulo mild log log dependency range almost matching best known sample complexity upper bound regular distributions online learning symmetric range section consider online learning rewards symmetric range look full information bandit setting prove regret upper bounds defer regret lower bound proofs appendix sections expert problem symmetric range recall proof lemma proof requires choosing vector ith coordinate elsewhere action noting get following regret bound corollary lemma corollary initial distribution learning rate parameter msmw algorithm achieves following regret bound log regreti prove regret upper bound theorem using corollary proof theorem proof follows choosing appropriate initial distribution corollary corollary log regreti let imin action minimumprange cimin cmin consider initial distribution cmin putting remaining probability mass imin action imin term rhs upper bounded cmin cmin cimin imin definition log cmin regreti log cmin theorem imin note imin thus follows choosing ccimin theorem holds following min calculation imin case bandit problem symmetric range start showing following regret bound whose proof alteration lemma symmetric range deferred appendix section next prove theorem min learning rate lemma exploration rate min ccmax algorithm algorithm achieves following regret bound regreti log cmax min proof theorem let ccmax lemma theorem follows noting references shipra agrawal nikhil devanur bandits concave rewards convex knapsacks proceedings fifteenth acm conference economics computation acm kareem amin afshin rostamizadeh umar syed learning prices repeated auctions strategic buyers advances neural information processing systems sanjeev arora elad hazan satyen kale multiplicative weights update method applications theory computing peter auer nicolo yoav freund robert schapire gambling rigged casino adversarial bandit problem foundations computer science annual symposium ieee moshe babaioff shaddin dughmi robert kleinberg aleksandrs slivkins dynamic pricing limited supply acm transactions economics computation ashwinkumar badanidiyuru robert kleinberg aleksandrs slivkins bandits knapsacks foundations computer science focs ieee annual symposium ieee ziv kirsten hildrum felix online auctions digital goods proceedings thirteenth annual symposium discrete algorithms society industrial applied mathematics omar besbes assaf zeevi dynamic pricing without knowing demand function risk bounds algorithms operations research avrim blum jason hartline online auctions proceedings sixteenth annual symposium discrete algorithms society industrial applied mathematics avrim blum vijay kumar atri rudra felix online learning online auctions theoretical computer science bubeck introduction online optimization lecture notes richard cole tim roughgarden sample complexity revenue maximization symposium theory computing stoc new york usa may june arnoud den boer dynamic pricing learning historical origins current research new directions surveys operations research management science nikhil devanur zhiyi huang psomas sample complexity auctions side information proceedings annual acm sigact symposium theory computing acm nikhil devanur yuval peres balasubramanian sivan perfect bayesian equilibria repeated sales proceedings annual symposium discrete algorithms siam peerapong dhangwatnotai tim roughgarden qiqi yan revenue maximization single sample games economic behavior dylan foster satyen kale mehryar mohri karthik sridharan personal communication yoav freund robert schapire generalization learning application boosting european conference computational learning theory springer yannai gonczarowski noam nisan efficient empirical revenue maximization singleparameter auction environments proceedings acm stoc zhiyi huang yishay mansour tim roughgarden making samples proceedings sixteenth acm conference economics computation acm robert kleinberg tom leighton value knowing demand curve bounds regret online auctions foundations computer science proceedings annual ieee symposium ieee jamie morgenstern tim roughgarden nearly optimal auctions advances neural information processing systems roger myerson optimal auction design mathematics operations research tim roughgarden okke schrijvers ironing dark proceedings acm conference economics computation acm ilya segal optimal pricing mechanisms unknown demand american economic review vasilis syrgkanis sample complexity measure applications learning optimal auctions arxiv preprint kalyan talluri garrett van ryzin theory practice revenue management vol springer science business media deferred proofs discussions discussion choice bandit symmetric range describe choice initial distribution affects bound given theorem action set finite choose uniform distribution get term log kci recovers standard bound setting cmax choose get log particular form arithmetic progression constant difference infinitely many experts convergent cmin cmin choose gives log min log factor dependence symmetric range proof theorem proof theorem first show online learning algorithm sufficiently large instance two experts log rounds either construct instance rounds adaptively always gain action gain either action proof theorem follows cmin cmax instance let denote probability algorithm picks action round rewards two actions respectively first rounds first show algorithm small regret respect action must upper bounded since adversary may let action cost round large show since upper bounded algorithm must large regret respect action proceed upper bounding concretely show following lemma lemma suppose proof lemma prove induction consider base case suppose contradiction consider instance action always gain case expected gain algorithm even always correctly picks action remaining contradiction assumption instance next suppose lemma holds rounds prior round expected gain algorithm first rounds arm gain suppose contradiction consider instance action gain first rounds afterwards case expected gain algorithm even always correctly picks action round contradiction assumption consider instance action always gain suppose immediate implication lemma algorithm expected gain algorithm upper bounded note instance thus regret action least greater sufficiently large regret symmetric range theorem proof theorem first show online bandits algorithm problem instance thatqhas two arms sufficiently large sufficiently large either prove existence instance looking stochastic setting gain vectors consider two instances admit fixed gain action first instance gain action probability otherwise hence expected gain playing action per round instance second instance gain action probability otherwise hence expected gain playing action two per round instance note proves theorem cmin cmax suppose contradiction algorithm satisfies let denote expected number times algorithm plays action instance expected regret respect action instance assumption next standard calculation get divergence observed rewards single round two instances action played action played divergence observed reward sequences two instances use standard inequality divergences measurable function exp let distribution observed rewards round two instances let action played algorithm inequality bound divergence observed rewards two instances get round probability algorithm plays action instance plus probability algorithm plays action instance least exp round thus expected number times algorithm plays action instance round denoted least second inequality holds sufficiently large therefore expected regret action contradiction assumption instance least proof lemma proof lemma galg applying regret bound omd lemma rhs galg bound first term regret local norm exp exp upper bounded also rewrite second term regret fact set note summing term local norm putting pieces together get desired bound also provide elementary proof lemma using first principles proof lemma first principles based update rule algorithm log wip therefore log log log log log log note due normalization step algorithm log log first summation equal log log log log combining eqn log log log part telescopic sum sum upper bound part follows log get log log log log part telescopic sum sum work part relation get log exp exp upper bounded putting together get log note summing log finally log get log hence log lemma follows choice initial distribution proof omd regret bound order prove omd regret bound need properties bregman divergence lemma properties bregman divergence bubeck suppose legendre function associated bregman divergence defined definition strictly convex convex function choice pythagorean theorem convex set argmin given lemma ready prove lemma proof lemma obtain omd regret bound use due pythagorean theorem lemma summing hand sides symmetric range bandit regret bound proof lemma proof lemma define ealg git expectation randomness algorithm ealg galg ealg hence upper bound regreti galg suffices upper bound definition probability algorithm picks arm reward round least ealg cmax hence benchmark distribution alg cmax alg alg cmax cmax inequality due inequality follows cmax largest possible reward per round next upper bound term rhs note probability choosing experts msmw experts rewards lemma benchmark distribution algorithm satisfies definition equals probability equals otherwise thus fix random coin flips first rounds thus fix take expectation randomness round note upper bounded putting together kcmax combining recall galg kcmax cmax cmax let recall recall imin arm minimum range cimin similar expert problem section term rhs upper bounded log hence log cmax galg lhs lower bounded ealg cmax ealg lemma follows putting back rearranging terms
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wiretap channels random aug parameters alexander bunin ziv goldfeld haim permuter shlomo shamai shitz paul cuff pablo piantanida abstract study secret message secret key rates simultaneously achievable wiretap channel wtc channel state information csi encoder model subsumes instances csi availability special cases calls efficient utilization state sequence reliability security purposes inner bound capacity region derived based superposition coding scheme inspired past work authors region shown attain capacity certain class established virtue two versions strong lemma derived region yields improvement upon previously best known result reported prabhakaran best knowledge upon existing lower bounds either setup even semantic security requirement relaxed weak secrecy demonstrated region strictly larger reported preceding works ntroduction background physical layer security pls rooted principles approach provably secure communication dates back wyner celebrated paper wiretap channel wtc harnessing randomness noisy communication channel combining proper physical layer coding pls guarantees protection eavesdroppers requirement legitimate parties share secret key advance two fundamental questions field pls regard finding best work alexander bunin shlomo shamai supported european union horizon research innovation programme grant agreement work goldfeld permuter supported israel science foundation grant erc starting grant cyber security research grant university negev work paul cuff supported national science foundation grant air force office scientific research grant work presented part second international workshop communication security wcs affiliated eurocrypt bunin shamai department electrical engineering technion israel institute technology haifa israel albun sshlomo goldfeld permuter department electrical computer engineering university negev israel gziv haimp cuff department electrical engineering princeton university princeton usa cuff piantanida laboratory signals systems france achievable transmission rate secret message noisy channel highest attainable rate distributed parties agree upon based correlated observations base model transmission wyner wtc two legitimate parties communicate noisy channel presence eavesdropper secrecy capacity degraded wtc derived result extended general case security analyses relied evaluating particular conditional entropy terms named equivocation technique widely adopted community ever since recently distribution approximation arguments emerged tool choice proving security approach relies lemma scl originated another paper wyner scl states distribution induced randomly selecting codeword appropriately chosen codebook passing memoryless channel asymptotically indistinguishable distribution random noise scl developed years stricter proximity measures distributions achieved based advanced versions one make channel output observed eavesdropper wtc seem like noise particular approximately independent confidential data turn implies security notably focused tight exponents respect relative entropy total variation respectively study agreement pioneered maurer independently ahlswede studied achievable rates based correlated observations terminals communicate via noiseless rate unlimited public link capacity public communication allowed characterized result generalized case public link finite capacity optimal random coding scheme cases combination superposition coding coding encoder controls source rather observing source becomes channel input setup evolves wtc special case model also studied model contributions general framework consider wtc encoder channel state information csi model combines wtc gelfand pinsker channel therefore sometimes referred dependence channel transition probability state sequence accounts possible availability correlated sources terminals similarity transmission agreement tasks makes integration single model natural adhering general framework study rate pairs simultaneously achievable encoder csi scenario studied achievable rate formula established result improved based novel superposition coding agreement focus recently see also references therein combined model considered respective causal scenario recently studied psfrag replacements wsn encoder decoder eavesdropper fig wiretap channel encoder channel state information exploited simultaneous secret message transmission secret key generation prabhakaran derived benchmark inner bound capacity region result optimal several classes propose novel superposition coding scheme combined model subsumes aforementioned achievability results special cases specifically well existing inner bounds transmission agreement known authors captured furthermore inner bound shown achieve strictly higher rates previous results coding scheme used herein inspired namely superposition codebook encodes entire confidential message outer layer utilized using redundancies inner outer layers transmission correlated state sequence means likelihood encoder although redundancy indices chosen part encoding process show true distribution close uniform consequently long certain redundancy index kept secret along confidential message may declared security analysis based constructing inner codebook better observable eavesdropper making inner layer index decodable enhances secrecy resources legitimate parties extract outer layer use secure part redundancy index outer layer latter declared results derived strict metric criterion cryptographic gold standard adapted wtc framework computationally unbounded adversaries noisy observation shown equivalent negligible mutual information confidential information case pair eavesdropper observations maximized possible message distributions proof relies strong scl superposition lemma heterogeneous scl lemma since past secrecy results derived weak secrecy metric vanishing normalized respect uniformly distributed pair achievability outperforms schemes terms achievable rate pairs also upgraded sense security organization paper organized follows section establishes notation definitions sets problem section iii states main result inner bound optimal region section inner bound shown tight certain class channels section discuss past results captured within considered framework illustrate improvement result yields proof main result content section finally section vii summarizes main achievements outlines main insights emerging work reliminaries roblem preliminaries use following notations customary set natural numbers reals define given two real numbers denote set integers calligraphic letters denote sets stands cardinality denotes cartesian product element denoted whenever dimension clear context vectors sequences denoted boldface letters let probability space sample space probability measure random variables denoted uppercase letters conventions random vectors similar deterministic sequences probability event denoted denotes conditional probability given use denote indicator function set probability mass functions pmfs finite set denoted pmfs denoted letters subscript identifies random variable possible conditioning example two discrete correlated random variables probability space use denote respectively marginal pmf joint pmf conditional pmf given particular represents stochastic matrix whose elements given expressions understood accordingly three random variables satisfy form markov chain denoted pmf gives rise probability measure denote accordingly every use denote expectation taken respect similarly use indicate entropy mutual information term calculated respect pmf random vector entries drawn independent identically distributed manner according every write pnx stands power set similarly every write pny conditional product pmf pny given specific sequence denoted pny use empirical pmf sequence denote set sequences length respect pmf number definition total variation let measurable space two probability measures space total variation sup sample space countable probability measures induced respectively reduces problem setup study encoder csi establish novel achievable region semantically secured rate pairs let finite sets discrete memoryless encoder csi shown fig state sequence sampled manner according revealed fashion sender independently observation sender chooses message set maps pair onto channel input sequence key index mapping may random sequence transmitted transition probability output sequences observed receiver eavesdropper respectively based receiver produces pair estimates eavesdropper tries glean whatever pair remark general model considered model general instance noncausal csi known terminals seemingly broadest model one may consider driven triple correlated state random variables wst known transmitter receiver eavesdropper respectively however setting encoder csi defining channel transition probability wsr one recovers aforementioned model csi encoder model also supports existence public private respectively transmitter receiver eavesdropper receiver addition instead noisy channel definition code encoder csi message set key set pair functions stochastic encoder decoding function message distribution induced joint pmf wsn probability measure induced performance evaluated terms rate pair maximal decoding error probability key uniformity independence metric definition error probability error probability max wsn subscript denotes underlying pmf remark operational interpretation error probability error probability defined maximizing set messages maximization respect message rather respect pair choice independent code distribution estimate induced code see similar logic applies subsequent definition key uniformity independence metric definition key uniformity independence metric key uniformity independence message metric max pkn uniform pmf definition information leakage metric information leakage eavesdropper message pmf denotes taken respect metric respect max definition achievability pair called achievable rate pair encoder csi every sufficiently large exists max definition capacity region csem encoder csi convex closure set achievable rate pairs iii esult main result work novel inner bound capacity region encoder csi achievable region least good best known achievability results considered problem strictly larger cases state main result let finite sets define terms calculated respect joint pmf forms markov chain theorem capacity inner bound following inclusion holds csem proof theorem given section based secured superposition coding scheme superposition codebook constructed independently state sequence entire secret message encoded outer layer thus data carried inner layer likelihood encoder uses redundancies inner outer codebooks correlate transmitted codewords observed state sequence upon part correlation index outer layer declared encoder key inner layer designed utilize part channel better observable eavesdropper saturates eavesdropper redundant information leaves insufficient resources extract information pair outer layer legitimate decoder hand decodes layers codebook declares appropriate indices decoded pair remark interpretation theorem get intuitive understanding result theorem examine two different perspectives joint pmf opposite inequality holds third rate bound becomes redundant dominating bounds side rhs total rate reliable secured unsecured communication superposition codebook supports clearly bounds rate may transmitted difference rhs total rate secrecy resources produced outer layer codebook since security pair comes entirely outer layer difference upper bound sum rates opposite case second inequality inactive left interpretation remains understand consider following since approximately rate inner codebook means looking solely inner layer decoder lacks resolution decode however success communication protocol relies decoder reliably decoding layers therefore case rate outer layer allocated convey inner layer index recalling security analysis based revealing inner layer eavesdropper rate allocation effectively results loss secrecy resources outer layer giving rise rate bound remark alternative representations defining see suffices restrict maximization joint pmfs satisfy markov chain furthermore rewriting bounds max evident maximizing joint pmfs satisfying max attains optimality indeed opposite inequality holds one could always choose achieve higher rates adapting theorem framework confidential transmission requires channel resources reliability security lesser two resources therefore limits feasible transmission rates main focus paper utilization residual secrecy resources offers however secrecy lesser resource superior capability channel support reliable communication may utilized considering framework theorem naturally extends inner bound region considered equivocation represents portion message secured eavesdropper intuitively answers question much information eavesdropper lack decoding entire message framework enables communicating rates higher secrecy capacity long full secrecy forfeited since equivocation added value full secrecy channel offers resources reliable communication security simplicity assume formally equivocation rate given uniformly distributed message achievability pair requires existence sequence vanishing error probability equivocation rate satisfies adaptation arguments proof theorem see section shows pair satisfying pmf induces joint distribution achievable prove inner bound follow derivation section replacing message therein pair uniformly distributed messages rates respectively total rate communication ensure distribution approximation arguments lemma error probability analysis hold suffices inequalities hold role satisfy equivocation requirement security analysis carried respect therefore inequality replaced conclude noting securing implies desired equivocation entropy terms taken respect distribution induced extracted psfrag replacements wln encoder decoder eavesdropper wsn fig wtc key reliable secure sequence codes applying elimination remove produces ight ecrecy apacity esults operationally appealing special case considered following assume eavesdropper channel less noisy main channel legitimate parties share wln independent state sequence wsn using secure confidential data setup illustrated fig formally let alphabets key state channel input two channel outputs respectively considered instance whose channel transition matrix factors less noisy less noisy means random variable forms markov chain refer special case wtc key theorem applies since case certain instance encoder csi subsequently shown obtained inner bound tight thus characterizing secrecy capacity region wtc key following corollary states result corollary capacity region capacity region wtc key set rate pairs satisfying max terms respect joint pmf proof corollary relegated appendix note bounds total communication rate function communication channel bounds total secrecy rate depending solely secret source direct consequence corollary established legitimate parties best attainable rate csm min max simple coding scheme achieves secrecy capacity namely using capacity achieving error correction code channel effectively converted reliable legitimate parties compresses results uniform random variable latter used encrypt via pad encrypted message transmitted reliable therefore achievable rate equal minimum capacity channel maxqu rate key scheme may seem natural best knowledge none past achievability results csi prior attain performance section special case setup used demonstrate improvement result previous benchmark achievable region revious esults pecial ases compare result theorem related past works previously best known inner bound region attainable considered theorem next subsection restates inner bound shows theorem strictly outperform afterwards provide comparison best past achievability results transmission agreement achievability result captures previous lower bounds secrecy capacity achievability results subsume previous lower bounds generation rate relating one another three benchmarks use evaluate performance theorem note recovers imply one another remark another result generation csi found theorem therein seemingly attains higher rates schemes inner bound incorrect region suggested theorem account inner layer codeword decoded legitimate decoder see second case remark explanation reason chose benchmark generation problem following steps proof theorem appears another constraint assumed without explicitly stated following notations missing constraint seems would assure decodability inner code layer legitimate receiver without relying outer layer taking additional constraint consideration inner bound theorem recovers amended theorem follows use denote inner layer outer layer channel input encoder csi observations legitimate receiver eavesdropper respectively theorem originally denoted respectively adjust model identify theorem random variable representing input outputs public communication link order comply rate restriction public link restrict distribution finally set independent maximal entropy respect substituting maximizing distributions satisfy produces amended version theorem conclude discussion theorem original form appendix provides specific example shows rates achievability formula exceeding capacity region result theorem recovers previously best known achievable region encoder csi theorem following region established rper rper rper terms taken respect independent forms markov chain first note theorem recovers rper restricting independent since independent pair always holds consequently third rate bound becomes redundant rper recovered result derived weak secrecy metric vanishing normalized pair eavesdropper observation sequence message assumed uniform achievability hand ensures theorem therefore improves upon theorem rates achieves sense security provides psfrag replacements wln encoder memory stuck faults bec wsn decoder eavesdropper fig section example setup achieving strictly higher rates since theorem allows inner layer coding random variables independent state coding generally requires correlating supported inner layer instead shannon strategies coding operates independent allowed latter optimal encoder observes state causally generally encoder csi available demonstrate improvement theorem exploit aforementioned limitation scheme therein along observation beneficial exploit part considered better observable eavesdropper transmit inner layer code let consider wtc key defined section shown fig whose transition probability key state defined three parameters follows independent random variables ber ber joint distribution denoted memory msaf deterministic channel driven ternary state binary input output symbols respectively related function given output msaf channel fed binary erasure channel erasure probability abbreviated bec input ternary output bec related means erasure random variable function eavesdropper noiselessly observes transmitted symbol state random variable respect definitions transition matrix channel possible interpretation communication scenario legitimate parties communicate public database memory faults known transmitter receiver database faults assumed known full eavesdropper secure communication legitimate parties share denote corresponding channel furthermore let rper denote maximal achievable secrecy rates attained theorem theorem respectively virtue corollary specifically theorem tight considered channel stated following proposition rper strictly capacity proposition exist rper proposition proven appendix proof relies observation rper full utilization key implies upper bounded capacity considered channel causal csi turn capacity upper bounded capacity msaf causal csi choosing parameters channel strictly causal msaf capacity superiority scheme compared theorem established remark example actually demonstrates theorem special case theorem achieves strictly higher rates theorem transmission theorem lower bound established transmission considered csem transmission lower bounded csem rgcp max rgcp rgcp min terms taken respect rgcp projection theorem axis main difference coding scheme superposition code additional index outer layer codebook also encodes along redundancy indices used correlate transmission observed state sequence via likelihood encoder based distribution approximation arguments show approximately independent message approximately uniform pair known transmitter reliably decoded receiver finally securing along analysis established intuition behind construction unlike message key independent state sequence chosen user therefore padding ensures correlation state sequence valid key long secured observing portion allocated favor implies also achievable region replaced outperforms rgcp settings external random source wln observed legitimate parties eavesdropper capacity communication channel zero say setup legitimate parties may use random source generate rate theorem supports strategy rgcp nullifies case see let state channel output observed legitimate receiver respectively inserting first term inside minimum produces agreement two achievable schemes proposed agreement wtc terminals access correlated sources results imply one another difference theorem based source channel separation theorem relies joint coding setup consists three correlated sources observed encoder decoder eavesdropper respectively triple plays role state general framework defined state distribution setting recovers model see remark first scheme theorem operates assumption decomposes wsy product two wtcs one independent state given input one depends thus legitimate receiver respectively eavesdropper observes output respectively wtc also respectively noisy version state sequence drawn according corresponding conditional marginal wsy scheme shows capacity csk lower bounded separate csk rbps max maximization give rise joint pmf wsx satisfying respect distribution form markov chains independent independence essence separation uses channel two purposes carrying communication agreement based sources securing part communication using wiretap coding setting theorem limiting union joint pmfs satisfy keeping distribution recovers joint coding scheme theorem rely aforementioned decomposition lower bounds csk joint csk rbps max maximization give rise joint pmf wsx satisfying setting joint theorem valid auxiliary pair rbps recovers shown cases scheme achieves strictly higher rates joint separate coding scheme rbps joint rbps theorem captures results unifies two schemes joint particular outperforms rbps furthermore since results derived weak secrecy metric theorem also upgrades roof heorem subsequently presented proof follows lines similar proof theorem several claims herein recovered corresponding assertions identifying index pair scheme proofs claims omitted reader referred fix conditional pmf let message distribution first show exists sequence codes key distribution approximately uniform conditioned message vanishing average error probability use expurgation technique theorem ensure vanishing maximal error probability done without harming statistical properties key since hold message original message set codebook use superposition codebook outer layer carries codebook constructed independently sufficient redundancy enable correlating transmission define index sets let codebook set random vectors length according denoted describe outer layer codebook fix random inner layer every outcome let collection random vectors length distribution qvn outcome given denoted also set denote realizations finally random superposition codebook given denotes fixed codebook let set possible outcomes codebook construction induces pmf codebook ensemble every qvn encoder decoder described next superposition codebook encoder encoding function based turn allows approximate induced joint distribution simple distribution use analysis given encoder randomly chooses according ple conditional marginal defined every encoder declares chosen index key channel input sequence generated feeding chosen along state sequence channel sampled random vector accordingly stochastic encoding function given decoder plen upon observing decoder searches unique tuple unique quadruple found set quadruple otherwise defined respect codebook message distribution codebook induced joint distribution wsn plen pmn message distribution uniform write instead approximating distribution show high probability close total variation another distribution lends simpler reliability security analyses stands pmn following lemma states sufficient conditions good approximation total variation certainty lemma sufficient conditions approximation exist large enough max particular also holds log subscript indicates probability measure expectation taken respect random codebook lemma essentially restates lemma index therein replaced pair proof lemma relies strong scl superposition codes basic properties total variation due similarity lemma omit proof reader referred lemma key analyzing performance proposed code reliability analysis presented next exploits convergence expected value show average error probability made arbitrarily small expurgation method theorem used later stage proof upgrade vanishing maximal error probability average error probability analysis average error associated codebook next step establish expected value codebook ensemble approximately expected average error probability analyzed shown converge zero due simple structure analysis requires nothing standard typicality arguments use two following lemmas lemma average error probability following relation holds shorthand lemma average error probability rate tuple satisfies proof lemma found average error probability analysis part section lemma also proven reference standard typicality decoding arguments stress conditions ensure reliable decoding four indices particular pair combining claims lemmas lemma long satisfied key analysis structure implies pkn adopting abuse notation used reliability analysis use lemma upper bound slightly abuse notation actually functions code rather codebook however since uniquely defines prefer presentation sake simplicity probability decay exponentially fast zero grows therefore assuming holds exists max max max lemma proceed security analysis security analysis part mainly deals analyzing metric distribution following lemma explains reason states attained codebook also attained lemma induced approximating distribution let sufficiently large independent exist large enough values independent subscripts indicate mutual information term calculated respect corresponding pmf proof lemma extends lemma provided appendix hypothesis essentially follows lemma thus holds account max max implies codebook rhs small fix consider see relative entropy chain rule follows relative entropy conditional marginal distribution maximizing sides message distributions max max max max max max inserting holds max two following lemmas state conditions probability rhs vanishes exponentially quickly close lemma total variation dominates relative entropy let finite sets let qyn absolutely continuous respect qyn qyn qyn log log qyn log min lemma lemma proof omitted readily verified combining lemma see codebook rates satisfying exists max sufficiently large exists grows lemma sufficient conditions rate tuple satisfies exist sufficiently large max lemma follows security analysis role therein combining lemma deduce exist sufficiently large code extraction derivation shows simultaneously satisfied ebn sufficiently large also selection lemma lemma implies existence sequence superposition codebooks outcome random codebook sequence since indicator functions take values large enough account satisfies target key statistics reliable respect average error probability last step upgrade small maximal error probability standard step uses expurgation technique see theorem namely pushing average error probability least half messages result probability error throwing away rest messages ensures maximal error probability inflicting negligible rate loss discarding messages harm key uniformity independence metric thus producing new sequence codes satisfies applying elimination shows rate pair achievable concludes proof vii ummary oncluding emarks studied rates simultaneously achievable encoder csi model subsumes instances csi availability special cases inner bound capacity region derived based superposition coding scheme likelihood encoder arguments inspired presented class inner bound achieves capacity demonstrated class previously best known region prabhakaran strictly furthermore showed inner bound derived recovers best lower bounds either rate achievable considered derivations ensure thus upgrading security standard past results derived weak secrecy metric capacity region setup remains open problem finding good outer bounds particular interest extensions multiple terminals action dependent states source reconstruction models examined well ppendix roof orollary recall wtc key whose transition matrix satisfies condition induces joint distribution given proceed direct converse proofs direct fix structure implies evaluating bounds theorem respect setting using independent combining two bounds sum one similarly implied independence finally due joint distribution produces achievable region satisfies hence term zero maximizing concludes proof converse get notice secret communication rate setup exceed total reliable communication rate therefore upper bound secrecy capacity given channel capacity formula max underlying joint pmf thus max max max max max max follows independent see follows recasting bound enhance channel allowing encoder control state secret source yet constraining original statistics obtained channel equivalent wtc input outputs legitimate receiver eavesdropper respectively therefore channel secrecy capacity upper bound sum thus max note property channel implies concludes converse proof ppendix emonstration ncorrectness heorem first restate theorem notations work theorem stipulates following lower bound capacity csk encoder csk rzib max maximization conditional pmfs satisfying terms taken respect appropriate marginals forms markov chain consider following setup let three ber random variables also set three random vectors whose coordinates copies respectively let deterministic function represents output sequence deterministic memoryless channel fed wtc without state additional secrecy may extracted form key channel model theorem also incorporates public communication link setup restate theorem assuming public communication rate zero let stochastic encoder sequence produces transmits private binary legitimate receiver encoder observes determines binary transmission decoder observes eavesdropper observes stands addition modulo time instance eavesdropper observes mod thus channel use encoder observes two fair coin tosses decoder observes one namely chosen random using third fair coin decoder knows coin observes encoder private encoder decoder enables transmission single noiseless bit time coins flipped legitimate parties wish agree upon key kept secret eavesdropper observes modulo addition two coins time flipped denoting generated legitimate parties induced joint pmf system qan comply notations identify also denoting pair observed decoder valid choice random variables ber independent achieves rzib hence showing capacity proposed setup strictly less contradict achievability rzib theorem rate setup showing vanishing average error probability weak secrecy used definition achievability coexist setup rate attained consider sequence codes achieving rzib setup exists sequence follows definition rate alphabet size since uniform distribution maximizes discrete entropy fano inequality following requirement vanishing decoding error weak secrecy requirement lemma considered setup capacity upper bounded bits per channel use csk lemma follows considered setup without eavesdropper falls within framework common randomness problem model proof theorem shows capacity upper bounded ccr rate communication link transmitter receiver evaluating rhs respect considered setup shows equals bits per channel use upper bound remains valid security requirement introduced since reduce admissible rates lemma guarantees existence sequence following condition may added set another technical lemma need stated next proof relegated appendix lemma hold combining using finally lower bound conditional information leakage term first consider fact apparently strong requirement used including vanishing term uses follows chain rule deterministically defined since ber sequences independent conclude uses evidently contradicts ppendix roof roposition fix set binary entropy function inverse restriction respectively readily verified virtue inner bound theorem attains capacity given see csm min cgp cgp maxqu capacity channel state distribution corollary theorem find cgp obtain csm therefore achievability may also verified directly theorem substituting ber independent show rper fix joint distribution evaluate region replaced distribution factors note independence restriction feasible joint distributions assume contradiction evaluating respect produces rate least high specifically assume consider following upper bound uses markov relation follows independent distribution account single inequality must hold equality happen following argument must hold conditioning removed first positive term implies independent second negative term zero deterministically defined since observing conditioned constant deduce relies independence last equality implies exists deterministic function expanding third negative term respect similar manner presented point obtain establishes markov chain since independent markov relation point implies independent pair observe effectively means inability scheme theorem support coding inner layer implies coding supported proceed analyze deductions consider follows data processing inequality see section since forms markov chain define observe independent since pair forms markov chain since upper bound rhs maximizing conditional distributions satisfy thus max expression rhs capacity msaf causal encoder knowledge state sequence however causal csi useless msaf encoder demonstrated section omitting availability csi msaf encoder channel equivalent binary symmetric channel flip probability see whose capacity equals conclude max contradiction ppendix roof emma fix simplicity notation abbreviate respectively consider log log log log log log log log due triangle inequality uses theorem follows assumption note function log monotone increasing exists finally since upper bounded log log log log plugging gives log bound rhs uniform decays exponentially fast zero grows result lemma follows maximizing sides message distributions ppendix roof emma using basic information identities produces next show combined gives implies required thus complete proof lemma suffices show holds consider following steps follow since independent every evident ber follows since deterministic function uses independence see eferences bloch barros security information theory security engineering cambridge univ press cambridge liu chen wang physical layer security next generation wireless networks theories technologies challenges ieee commun surv first quarter zeng physical layer key generation wireless networks challenges opportunities ieee commun june wyner channel bell sys broadcast channels confidential messages ieee trans inf theory may wyner common information two dependent random variables ieee trans inf theory mar han approximation theory output statistics ieee trans inf theory may hou kramer informational divergence approximations product distributions proc canadian workshop inf theory cwit toronto ontario canada jun goldfeld cuff permuter capacity wiretap channels type ieee trans inf theory jul goldfeld cuff permuter arbitrarily varying wiretap channels type constrained states ieee trans inf theory bastani parizi telatar merhav exact random coding secrecy exponents wiretap channel ieee transactions information theory jan yagli cuff preparation maurer secret key agreement public discussion common information ieee trans inf theory may ahlswede common randomness information theory cryptography part secret sharing ieee trans inf theory jul narayan common randomness secret key generation helper ieee trans inf theory wyner ziv function source coding side information decoder ieee trans inf theory gelfand pinsker coding channel random parameters problemy pered inform problems inf trans chen han vinck wiretap channel side information ieee trans inf theory goldfeld cuff permuter wiretap channel random states available encoder submitted ieee trans inf theory available arxiv https fujita secrecy capacity wiretap channels side information transmitter ieee transactions information forensics security nov han sasaki wiretap channels causal state information strong secrecy arxiv preprint aug available https khisti diggavi wornell agreement channel state information transmitter ieee trans inf forensics security mar bassi piantanida shamai shitz secret key generation noisy channels common randomness arxiv preprint available https prabhakaran eswaran ramchandran secrecy via sources channels ieee trans inf theory song cuff poor likelihood encoder lossy compression ieee trans inf theory apr bellare tessaro vardy cryptographic treatment wiretap channe proc adv crypto crypto santa barbara usa liang poor shamai information theoretic security foundations trends commun inf theory liu chen wiretap channel state information proc asilomar conf signals syst comp page pacific grove chia gamal wiretap channel causal state information ieee trans inf theory may khisti diggavi wornell generation using correlated sources channels ieee trans inf theory zibaeenejad key generation wiretap models side information information forensics security ieee transactions july shannon channels side information transmitter ibm res kuznetsov tsybakov coding memory defective cells problemy pered inform problems inf trans cover thomas elements information theory wiley edition dai han vinck luo tang wiretap channel channel state information entropy ahlswede common randomness information theory cryptography capacity information theory ieee transactions heegaard gamal capasity computer memories defects ieee trans inf theory jafar channel capacity causal noncaudal side information unified view ieee trans inform theory
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datalog bag semantics via set semantics leopoldo georg reinhard mar carleton university ottawa bertossi university oxford wien abstract duplicates data management common problematic work present translation datalog bag semantics extension datalog warded set semantics theoretical point view allows reason bag semantics making use theoretical foundations set semantics practical point view allows handle bag semantics datalog powerful existing query engines required extension datalog moreover translation potential extensions capture bag semantics semantic web query language sparql introduction duplicates common feature data management appear instance relational databases queried means sql rdf data queried means sparql however semantics data operations queries presence duplicates always clear mostly related fact duplicates handled bags multisets whereas common semantics used data management settheoretical making difficult tell apart duplicates use sets alone address problem bag semantics datalog programs proposed call bag semantics dtb semantics intuitively two duplicates tuple intentional predicate accepted syntactically different derivation trees also equivalent formulation given terms evaluation dtb semantics used provide bag semantics sparql dtb semantics two major drawbacks first operational thus losing declarative logicbased semantics datalog second defined via new constructs dtbs thus leaving world query languages losing applicability large body query optimization techniques goal paper instead identify extension datalog allows express bag semantics terms classical set semantics stay within realm query languages bertossi member millenium research institute foundations data chile end show dtb semantics datalog program represented means transformation program way intended model former including duplicates characterized result chase instance latter achieved creating right tuple identifiers tids means existential rules duplicates different tids tuple admissible usual duplicates falling back bag semantics original datalog program establish correspondence dtb semantics programs required task belong class warded programs properly extends datalog tractable conjunctive query answering cqa problem successfully applied represent sparql owl core entailment regime set semantics though see also warded looks promising general language specifying different data management tasks also show datalog stratified negation captured similar transformation wellbehaved class way achieve fully declarative way expressing bag semantics important query language immediately recover full relational algebra including set difference bag semantics terms query language set semantics moreover translation ensures cqa preliminaries assume reader familiar relational data model predicate positions pos denote set positions predicate similarly pos denotes set positions predicates datalog program basic multiset operations follow consider multisets elements domain integer multiplicities mult definition iff mult multiset associate set set consider multiset relations unless stated otherwise assume contain tuples arity say multiset contained denoted iff every mult mult mgu generate root label children label edges root children moreover define following multiset operations multiset union defined mult mult mult multiset selection condition defined multiset containing tuples satisfy multiplicities multiset projection defined follows let elements accordingly consider tik multiset containing tik mult defined sum multiplicities tuples producing multiset natural join assume tuples arity arity simplify presentation assume natural join via last attribute first define following multiset iff mult mult define atoms col root latter case tree set dts atoms atoms multiset col set syntactically different dts bag dtb semantics multiset dtbs atoms col example consider program col following dts five trees etc five trees six trees etc total col contains different trees dtbs multiset difference two definitions considered rrm mult max mult mult alternatively ran mult mult mult otherwise stands multiset intersection contentious operation extending semantics multiset intersection would treat special case join may however intersection treated used bag semantics provide bag semantics datalog program multiset edb via transformation program set edb obtained assume set nulls program partitioned two infinite ordered sets unique global tuple identifiers tids usual nulls programs given multiset edb program instead using colors syntactically different derivation trees use elements identify elements edb tuples resulting applications chase procedure predicate program schema introduce new version extra first argument position accommodate tid null atom appears duplicates create tuples pairwise different nulls tids used identify element obtain set edb multiset ebd given rule introduce appearing positions existential quantifiers rule head formally generate fresh tids rule applies precisely rule form becomes rule fresh different variables bag semantics datalog follow tuples colored tell apart duplicates element extensional database edb via infinite ordered list colors multiset mult copies colored respectively col becomes set multiset col col set colored set col produces multiset stripping tuples colors example col col inverse operation decoloration gives col col multiset consider datalog programs multiset predicates extensions multisets multiset edbs derivation tree wrt tree labeled nodes edges follows edb predicate col col contains single node label rule form tuple dts atoms unify resulting program evaluated according usual set semantics set edb use classical chase instantiated body rule becomes true say new tuple created first new null used yet new null stands tid newly created atom chase variant assumed oblivious chase tgd activated every instantiation rule body makes rule body true rule never activated instantiated body chase instance obtained collecting atoms obtained applying chase final atoms derivation sequence precisely derivation sequence form atoms tgds set atoms contained makes body true created example cont becomes multiset relational algebra mra consists following basic multiset operations section multiset union multiset projection multiset selecm tion condition multiset natural join multiset difference ran positive mra multiset difference excluded theorem immediately obtain proposition pbb semantics applied positive relational algebra coincides mra support full mra cover also multiset difference ran difficulty step shown revisiting example example cont consider modified program rules rule replaced body atom negated rewriting rule positive case yields variable occurring negated body atom positive one making rule longer safe thus introduce auxiliary predicate aux order eliminate variable edb becomes analogously trees example chase produces eleven new atoms total get following chase result chase inspired operation col section introduce multiset merging operations sometimes using double braces emphasize construction intended produce multiset definition set tuples tids multiset set definition given datalog program multiset edb bag semantics pbb semantics assigns multiset aux idea generalized transform safe datalog programs stratified negation denoted rewriting rule form aux keep denoting pbbs bag semantics datalog stratified safe negation referred obtained via rewriting tids bag semantics introduced via dts extending dtb semantics positive datalog extension applies datalog programs stratified negation safe variable rule head negative literal must appear positive literal body rule leads interpretation negation ran operator keep using dtbs denote new dtb semantics assume datalog programs stratified safe following extensions theorem proposition obtained pbbs chase chase pbb semantics coincides dtb semantics following result states immediately obtained lemma theorem datalog program multiset edb dtbs pbbs lemma datalog program multiset edb correspondence dts col root atoms chasesequences end atoms corollary given datalog program multiset edb instance chase minimal model datalog program theorem stratified datalog program multiset edb dtbs pbbs proposition pbb semantics applied relational algebra including multiset difference ran coincides mra work detail application pbb semantics evaluation sparql bag semantics give first experimental results implementation vadalog system forthcoming full version paper recall warded shown expressive enough capture sparql owl core entailment regime set semantics though tackle extension sparql owl core entailment regime bag semantics transformation warded currently also working inexpressibility results related transformation presented recall translation cover multiset intersection moreover multiset difference handled form ran sometimes natural form left conjecture two operations expressible set semantics verification conjecture matter ongoing work properties pbb semantics warded datalog introduced particularly fragment cqa tractable details actually journal version currently review warded datalog extended stratified negation preserving favorable properties warded datalog show program obtained datalog program construction section indeed warded theorem let program datalog let program represents pbb semantics warded conclusion future work work proposed achieved declarative computable specification multiset semantics datalog actually extended specification thus capturing full multiset relational algebra mra including difference resulting programs still good computational properties provably allow tractable conjunctive query answering immediate application results evaluation sparql duplicates via datalog rewriting proposed intermediate step illustrated following example example consider following sparql query https taskforce propertypaths prefix foaf http select name foaf mbox mailto alice example foaf name name references angles multiset semantics sparql patterns proc iswc volume lncs pages arenas gottlob pieris expressive languages querying semantic web proc pods pages acm bellomarini gottlob pieris sallinger swift logic big data knowledge graphs proc ijcai pages gottlob kifer taming infinite chase query answering expressive relational constraints artif intell glimm ogbuji hawke herman parsia polleres seaborne sparql entailment regimes recommendation march https gottlob pieris beyond sparql owl entailment regime rules rescue proc ijcai pages aaai press johnson klug testing containment conjunctive queries functional inclusion dependencies comput syst maher ramakrishnan fixpoints logic programs proc naclp pages mit press mumick pirahesh ramakrishnan magic duplicates aggregates proc vldb pages morgan kaufmann mumick shmueli finiteness properties database queries proc adc pages world scientific query retrieves names people alice knows translation datalog omit prefix use constant alice short mailto alice example keep notation simple moreover assume rdf data query evaluated given relational database single ternary predicate database consists triples following datalog program answer predicate ans equivalent sparql query ans mbox alice knows name apply transformation section suppose query evaluated multiset triples recall transformation would first extend triples quadruples adding null tid actually easily automatized first rule program thus get following program ans mbox alice knows name
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horofunctions graphs linear growth oct matthew tointon ariel yadin abstract prove linear growth graph finitely many horofunctions provides short simple proof finitely generated infinite group linear growth virtually cyclic nous montrons graphe croissance admet nombre fini horofonctions donne une preuve courte simple que chaque groupe infini type fini croissance est virtuellement cyclique introduction let infinite connected locally finite graph denote graph metric let said linear volume growth balls metric grow linearly radius graph said polynomial volume growth balls grow polynomially given element define busemann function via given geodesic ray define horofunction bzn simple fact limit exists note constant fact shows unbounded theorem let infinite connected locally finite graph linear volume growth set horofunctions finite finitely generated group said polynomial respectively linear volume growth hence every cayley graph polynomial respectively linear volume growth remarkable theorem gromov states finitely generated group polynomial volume growth contains nilpotent subgroup finite index application theorem give short argument prove case gromov theorem follows theorem let finitely generated infinite group linear volume growth contains cyclic subgroup finite index fact gromov theorem implies group subquadratic growth virtually cyclic also proved elementary methods justin van den dries wilkie imrich seifter last two giving bounds index cyclic subgroup terms volume growth nonetheless present proof completely different rather short record prove theorem section proof theorem section let mention related probably much difficult question conjecture let cayley graph polynomial volume growth set horofunctions countable matthew tointon ariel yadin proving conjecture would provide alternative proof gromov theorem using variant lemma suggested karlsson one method prove conjecture could using structure finitely generated nilpotent groups relying gromov theorem would somehow miss point example walsh shows nilpotent groups always finite orbit space horofunctions seems extended virtually nilpotent groups well would interesting prove conjecture even quadratic growth case without using gromov theorem since would imply new proof characterization recurrent groups finite extensions horofunctions graph linear growth graph say sequence disjoint sets vertex set neighbours every lie call sets partite sets call path monotone one vertex proof theorem essentially rests following result proposition let graph whose partite sets cardinality exist monotone paths every infinite monotone path infinite intersection recall bipartite graph two finite sets equal cardinality matching subgraph element connected precisely one element vice versa hall marriage theorem states matching exists subset neighbourhood strictly smaller cardinality lemma matching satisfies proposition proof easy see existence matchings implies vertices may partitioned monotone paths sufficient satisfy proposition givensan graph sequence may define new graph placing edge exists monotone path note graph partite sets following immediate lemma exists sequence satisfies proposition conclusion proposition holds well proof proposition proceed induction base case easy assume may also delete every element lie infinite monotone path potential problem may longer cardinality using lemma may fix passing subsequence let sequence consider graph every exists matching graph done combining lemmas thus assume sequence exist specifically hall marriage theorem exists exist every monotone path ends without loss generality assume sets minimal respect properties hence horofunctions graphs linear growth every element lies monotone path passing subsequence infinite sequence size umj fixed subset let consider graph claim satisfies proposition suffice lemma move proving claim every monotone path starting ending must end vmj thus minimality vmj monotone path starting vmj ending must end particular graph infinite monotone path must satisfy following dichotomy either vmj exists vmj let induced subgraph vertex set vmj induced subgraph vertex set vmj note partite sets vmj size also partite sets vmj size thus infinite monotone path induces infinite monotone path either induction exist infinite monotone paths infinite monotone path must intersect one infinitely many times similarly paths thus infinite monotone path must intersect one infinitely many times completes proof proof theorem note linear growth writing ball radius infinite increasing sequence every define bmn define graph joining path length define map set geodesic rays starting set monotone paths sense proposition natural way specifically geodesic ray starting unique monotone path passing elements note surjective onto set monotone paths also infinite intersection let given proposition pick using surjectivity geodesic rays starting geodesic ray tail coincides tail geodesic ray starting see lemma however infinite intersection proposition infinite intersection implies particular complete set horofunctions completeness include short argument following standard lemma lemma geodesic ray starting exists coincides tail geodesic ray starting proof sequence since every triangle inequality also implies bounded sequence therefore eventually constant say combined implies infinite path initial segment geodesic path followed therefore geodesic ray starting matthew tointon ariel yadin case gromov theorem group acts space note busemann function bxz hence horofunction following observation learned anders karlsson lemma set horofunctions group contains finite orbit finiteindex subgroup admitting surjective homomorphism onto proof letting act finite orbit contains subgroup fixes element orbit thus implies homomorphism every particular constant finitely many cosets contradicting fact horofunctions unbounded conclude image subgroup thus admits surjective homomorphism onto remark essentially argument shows generally graph linear growth aut acts transitively vertices subgroup admitting surjective homomorphism onto proof theorem theorem implies finite set horofunctions set horofunctions invariant case contains finite orbit lemma therefore implies exists finite index admits surjective homomorphism onto let kernel homomorphism since finite index finitely generated linear growth since must finite hence standard methods implies also thus contains infinite cyclic subgroup acknowledgements grateful emmanuel breuillard anders karlsson martineau tom meyerovitch ville salo anonymous referee helpful comments discussions supported erc grant getemo supported israel science foundation grant references van den dries wilkie effective bound groups linear growth arch math gromov groups polynomial growth expanding maps publ math ihes hall representatives subsets london math soc imrich seifter bound groups linear growth arch math justin groupes croissance acad sci paris karlsson ergodic theorems noncommuting random products lecture notes http walsh action nilpotent group horofunction boundary finite orbits groups geom dyn laboratoire orsay univ cnrs orsay france address department mathematics university negev israel address yadina
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radical analysis network online handwritten chinese character recognition jianshu zhang yixing zhu jun lirong dai jan national engineering laboratory speech language information processing university science technology china hefei anhui china email xysszjs zyxsa jundu lrdai great progress made online handwritten chinese character recognition due emergence deep learning techniques however previous research mostly treated chinese character one class without explicitly considering inherent structure namely radical components complicated geometry study propose novel radical analysis network tran firstly identify radicals analyze structures among radicals simultaneously recognize chinese characters generating captions based analysis internal radicals proposed tran employs recurrent neural networks rnns encoder decoder rnn encoder makes full use online information directly transforming handwriting trajectory features rnn decoder aims generating caption detecting radicals spatial structures attention model manner treating chinese character composition radicals reduce size vocabulary enable tran possess capability recognizing unseen chinese character classes corresponding radicals seen evaluated database proposed approach significantly outperforms modeling approach relative character error rate cer reduction meanwhile case recognition unseen chinese characters tran achieve character accuracy traditional method capability handle ntroduction machine recognition handwritten chinese characters studied decades challenging problem due large number character classes enormous ambiguities coming handwriting input although conventional approaches obtained great achievements treated character sample whole without considering similarity internal structures among different characters capability dealing unseen character classes however chinese characters decomposed fundamental structure components called radicals intuitive way first extract information radicals embedded chinese characters use knowledge recognition past decades lots efforts made chinese character recognition example proposed matching method chinese character recognition first detected radicals separately employed hierarchical radical matching method compose radicals character analysis structures radicals handwriting input chinese character caption fig illustration tran recognize chinese characters analyzing radicals corresponding structures tried characters candidate radicals proposed way could handle structure brings many difficulties recently proposed learning chinese character recognition turned character class combination several radicals spatial structures generally approaches difficulties dealing radical segmentation analysis structures among radicals flexible besides focus recognizing unseen chinese character classes paper propose novel approach online handwritten chinese character recognition namely radical analysis network tran different mentioned approaches tran radical segmentation structure detection automatically addressed attention model jointly optimized entire network main idea tran decompose chinese character radicals detect spatial structures among radicals describe analysis radicals chinese character caption handwritten chinese character successfully recognized caption matches accessible illustrate tran learning way fig online handwritten chinese character input visualized fig composed four different radicals handwriting input finally recognized chinese character caption structures among radicals detected based analysis radicals proposed tran possesses capability recognizing unseen chinese character classes radicals seen proposed tran improved version based model rnn model extensively applied many applications including machine translation image captioning speech recognition mathematical expression recognition raw data online handwritten chinese character input sequence tran first employs stack bidirectional rnn encode input sequence representations unidirectional rnn decoder converts representations output character captions one symbol time predicted radical coverage based attention model built decoder scans entire input sequence chooses relevant part describe segmented radical structure radicals proposed tran related previous work two main differences focused application ran printed chinese character recognition paper focuses handwritten chinese character recognition interesting investigate performance ran handwritten chinese character recognition handwritten characters much ambiguous due diversity writing styles instead transforming online handwritten characters static images employing convolutional neural network encode choose directly encode raw sequential data employing rnn encoder order fully exploit dynamic trajectory information recovered static images main contributions study follows propose tran online handwritten chinese character recognition size radical vocabulary largely less chinese character vocabulary leading decrease redundancy among output classes improvement recognition performance tran possess ability recognizing unseen newly created chinese characters radicals seen experimentally demonstrate ran performs online handwritten chinese character recognition compared show effectiveness recognizing unseen character classes escription hinese character caption section introduce generate captions chinese characters character caption composed three key components radicals spatial structures pair braces radical represents basic part chinese character often shared different chinese characters compared enormous chinese character categories amount radicals quite limited declared standard published national language committee china nearly radicals consist chinese characters complicated spatial structures among radicals fig single stl str sbl fig graphical representation eleven common spatial structures among radicals different radicals divided internal line illustrates eleven common structures descriptions demonstrated follows sometimes single radical represents chinese character therefore find internal structures characters structure structure stl structure str structure sbl structure structure structure structure surround structure within structure decomposing chinese characters radicals internal spatial structures following use pair braces constrain single structure take stl example captioned stl generation chinese character caption finished radicals included caption iii proposed approach section elaborate proposed tran framework namely generating underlying chinese character caption sequence online handwritten trajectory points illustrated fig first extract trajectory information input feature original trajectory points stack bidirectional rnns employed encoder transform input feature representations since original trajectory points sequence extracted representations also sequence associate representations character caption generate context vector via weighted summing representations unidirectional rnn decoder uses context vector generate character caption one symbol time introduce encoder given feature sequence employ rnn encoder encode representations rnn shown strength processing sequential signals however simple rnn revealed serious problems training namely vanishing gradient exploding gradient therefore improved version rnn named gated recurrent units gru alleviate two problems employed study utilizes update gate reset gate control flow forward information backward gradient gru hidden state encoder computed decoder encoder gru gru function expanded follows fig overall framework tran online handwritten chinese character recognition composed bidirectional rnn encoder unidirectional rnn decoder attention model produce weighting coefficients context vector contain useful trajectory information decoding step feature extraction data acquisition online handwritten chinese character movements pen states stored sequential data length sequence xycoordinates pen movements indicates stroke ith point belongs address issue sampling different writing speed size variations coordinates different potable devices interpolation normalization original trajectory points first conducted according extract feature vector point condition true zero otherwise last two terms flags indicate status pen respectively convenience following sections use denote input sequence encoder wxz uhz wxr uhr tanh wxh urh denotes sigmoid activation function denotes multiplication operator update gate reset gate candidate activation respectively wxz wxr wxh uhz uhr urh related weight matrices nevertheless even unidirectional gru access history input signals ability modeling future context therefore exploit bidirectional gru passing input vectors two gru layers running opposite directions concatenating hidden state vectors encoder use history future information obtain representation encoder stacks multiple gru layers top illustrated fig study encoder consists bidirectional gru layers layer forward backward gru units also add pooling time axes gru layers representations overly precise contain much redundant information decoder needs attend less number encoder output reduces leading improvement performance pooling operation accelerates encoding process pooling applied top gru layer dropping even output time assuming bidirectional gru encoder produces highlevel representation sequence length one pooling operation bidirectional gru encoder representations vector decoder attention model obtaining representations decoder aims make use generate chinese character caption output sequence represented sequence encoded vectors vocabulary size length character caption note length representation sequence length character caption variable address mapping representation sequence character caption attempt compute intermediate vector incorporates useful information representation sequence decoder utilizes vector predict character caption one symbol time contains overall information input sequence call context vector decoding step probability predicted word computed context vector current decoder state previous predicted symbol using perceptron denotes softmax activation function symbols vocabulary denotes maxout activation function let denote dimensions embedding decoder state denotes embedding matrix since context vector needs intuitive way produce summing representation vectors time step however average summing robust leads loss useful information therefore adopt weighted summing weighting coefficients called attention probabilities attention probability performs description tells part representation sequence useful decoding step compute decoder state context vector follows gru eti tanh watt uatt exp eti exp etk gru past attention probabilities coverage vector contains information alignment history shown adopt coverage vector order let attention model know part representation sequence attended let denote attention dimension watt uatt raining esting etails training objective proposed model maximize predicted symbol probability shown use objective function log represents ground truth word time step length output string implementation details gru encoder introduced section decoder uses two layers using forward gru units embedding dimension decoder state dimension attention dimension set convolution kernel size computing coverage vector set convolution operation number convolution filters set utilize adadelta algorithm gradient clipping optimization adadelta hyperparameters set decoding stage aim generate likely character caption given input trajectory arg max log however different training procedure ground truth previous predicted word prevent previous prediction errors inherited next decoding step simple beam search algorithm employed implement decoding procedure maintained set partial hypotheses beginning sos time step partial hypothesis beam expanded every possible word likely beams kept procedure repeated output word becomes eos xperiments see decoder adopts two unidirectional gru layers calculate decoder state gru function one denotes current decoder state prediction eti denotes energy time step conditioned attention probability ith element computed taking eti input softmax function context vector calculated via weighted summing representation vectors attention probabilities employed weighting coefficients computation attention probability also append coverage vector ith vector attention model coverage vector computed based summation section present experiments recognizing seen unseen online handwritten chinese character classes answering following questions tran effective recognizing seen chinese character classes tran effective recognizing unseen chinese character classes tran analyze radicals spatial structures performance recognition seen chinese character classes section show effectiveness tran recognizing seen chinese character classes set character class commonly used chinese characters dataset used training casia dataset including totally samples training samples testing training testing data produced different writers enormous handwriting styles across individuals tatable esults casia dataset online handwritten hinese character recognition methods human performance traditional benchmark tran reference accuracy ble human performance casia test set previous benchmark listed proposed method represents method casia dataset belongs based methods achieved accuracy tran achieved accuracy revealing relative character error rate reduction fairly comparable tran use sequential dropout trick proposed performance good best performance presented explained contributions study section main difference radical based method based method chinese character recognition size radical vocabulary largely less chinese character vocabulary yielding decrease redundancy among output classes improvement recognition performance performance recognition unseen chinese character classes number chinese character classes fixed novel characters created also overall chinese character classes enormous difficult train recognition system covers therefore necessary recognition system possess capability recognizing unseen chinese characters called learning obviously traditional based methods incapable recognizing unseen characters since objective character class never seen training procedure however tran able recognize unseen chinese characters radicals composing unseen characters seen validate performance tran recognizing unseen chinese character classes divide common chinese characters classes classes choose handwritten characters belonging classes original training set new training set choose handwritten characters belonging classes original testing set new testing set testing character classes handwriting variations never seen training explore different size training set train tran ranging eos fig examples attention visualization decoding procedure red color trajectory describes attention probabilities namely lighter color denotes higher attention probabilities darker color denotes lower attention probabilities chinese character classes make sure radicals testing characters covered training set table esults newly divided testing set based casia dataset online handwritten unseen hinese character recognition train classes train samples test accuracy see table recognition accuracy unseen chinese character classes available training set contains chinese character classes believe difficult train tran properly accommodate large handwriting variations number character classes quite small training set contains character classes tran achieves character accuracy relatively pleasant performance compared traditional recognition systems recognize unseen chinese character classes means accuracies definitely performance recognizing unseen chinese character classes good performance presented handwritten chinese characters much ambiguous compared printed chinese characters due large handwriting variations attention visualization section show attention visualization tran able recognize internal radicals analyze spatial structure among radicals fig illustrates example attention visualization dotted line one chinese character class corresponding character caption dotted line images denoting visualization attention probabilities decoding procedure draw trajectory input handwritten chinese character greyscale image visualize attention images corresponding symbols generated decoder decoding step see fig encountering basic radicals attention model generates alignment well corresponding human intuition also mainly focus ending last radical beginning next radical detect spatial structure take example attending ending last radical beginning next radical attention model detects direction therefore topbottom structure analyzed immediately generating spatial structure decoder produces pair braces employed constrain structure chinese character caption onclusion future work study introduce tran online handwritten chinese character recognition proposed tran recognizes chinese character identifying internal radicals analyzing spatial structures among radicals show experimental results tran outperforms method recognition online handwritten chinese characters possesses capability recognizing unseen chinese character categories visualizing learned attention probabilities observe alignments radicals analysis structures correspond well human intuition eferences suen berthod mori automatic recognition handprinted state art proceedings ieee vol plamondon srihari online handwriting recognition comprehensive survey ieee transactions pattern analysis machine intelligence vol liu jaeger nakagawa online recognition chinese characters ieee transactions pattern analysis machine intelligence vol zhang yin zhang liu bengio drawing recognizing chinese characters recurrent neural network ieee transactions pattern analysis machine intelligence yang jin tao xie feng dropsample new training method enhance deep convolutional neural networks largescale unconstrained handwritten chinese character recognition pattern recognition vol zhong jin xie high performance offline handwritten chinese character recognition using googlenet directional feature maps document analysis recognition icdar international conference ieee chang interactive system chinese character generation retrieval ieee transactions systems man cybernetics vol wang fan optical recognition handwritten chinese characters hierarchical radical matching method pattern recognition vol liu new approach online handwritten chinese character recognition pattern recognition icpr international conference ieee wang yin liu chinese character recognition via learning deep residual networks document analysis recognition icdar international conference ieee bahdanau cho bengio neural machine translation jointly learning align translate arxiv preprint graves supervised sequence labelling recurrent neural networks springer vol cho van gulcehre bahdanau bougares schwenk bengio learning phrase representations using rnn statistical machine translation arxiv preprint luong pham manning effective approaches neural machine translation arxiv preprint kiros cho courville salakhudinov zemel bengio show attend tell neural image caption generation visual attention international conference machine learning vinyals toshev bengio erhan show tell neural image caption generator proceedings ieee conference computer vision pattern recognition bahdanau chorowski serdyuk brakel bengio large vocabulary speech recognition acoustics speech signal processing icassp ieee international conference ieee zhang zhang liu wei dai watch attend parse neural network based approach handwritten mathematical expression recognition pattern recognition zhang dai attention dense encoder handwritten mathematical expression recognition arxiv preprint graves mohamed hinton speech recognition deep recurrent neural networks acoustics speech signal processing icassp ieee international conference ieee zhang dai approach attention online handwritten mathematical expression recognition document analysis recognition icdar international conference ieee zhang zhu dai ran radical analysis networks learning chinese characters arxiv preprint krizhevsky sutskever hinton imagenet classification deep convolutional neural networks advances neural information processing systems online available https bengio simard frasconi learning dependencies gradient descent difficult ieee transactions neural networks vol zhang tang dai based interspeech chung gulcehre cho bengio empirical evaluation gated recurrent neural networks sequence modeling arxiv preprint liu liu modeling coverage neural machine translation arxiv preprint zeiler adadelta adaptive learning rate method arxiv preprint cho natural language understanding distributed representation arxiv preprint liu yin wang wang casia online offline chinese 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bivariate distributions gwo dong lin xiaoling dou satoshi kuriki academia sinica taiwan waseda university japan dec institute statistical mathematics japan abstract treat bivariate blm distributions unified approach develop new general properties blm distributions including joint moment generating function product moments dependence structure necessary sufficient conditions survival functions blm distributions totally positive order two given previous results specific blm distributions improved particular show survival copula survival function totally positive orders regardless parameters besides point slepian inequality also holds true blm distributions ams mathematics subject classifications primary key words phrases property bivariate distributions marshall olkin bve block basu bve freund bve likelihood ratio order usual stochastic order hazard rate order survival function copula survival copula positive quadrant dependent total positivity slepian postal addresses gwo dong lin institute statistical science academia sinica taipei taiwan gdlin xiaoling dou waseda university ohkubo shinjuku tokyo japan xiaoling satoshi kuriki institute statistical mathematics midoricho tachikawa tokyo japan kuriki introduction classical univariate property remarkable characterization exponential distribution plays prominent role reliability theory queuing theory applied fields feller fortet galambos kotz recent bivariate property however shared famous marshall olkin block basu well freund bivariate exponential distributions among many others see chapter balakrishnan lai chapter kotz kulkarni bivariate distributions well investigated individually literature main purpose paper however develop unified approach new general properties bivariate blm distributions share bivariate property section first review univariate bivariate properties summarize important known properties blm distributions derive section new general properties blm distributions including joint moment generating function product moments stochastic inequalities dependence structures blm distributions investigated section find necessary sufficient conditions survival functions densities exist blm distributions totally positive order two previous results specific blm distributions improved particular show survival copula survival function totally positive orders regardless parameters section study stochastic comparisons family blm distributions point slepian bivariate normal distributions also holds true blm distributions property first review univariate property let nonnegative random variable distribution function satisfies multiplicative cauchy functional equation degenerates constant denoted exp exponential distribution positive parameter lifetime system positive survival function equivalent means conditional probability system surviving time given surviving time equal unconditional probability system surviving time namely failure performance system depend past given present condition case say distribution lacks memory point called property memoryless property simplicity consider positive random variable property holds true iff exp next consider bivariate property let positive random variables joint distribution marginals namely moreover denote survival function intuitive extension property bivariate case strict blm property lacks memory pair equivalent survival function positive system means conditional probability two components surviving times given surviving times equal unconditional probability two components surviving times one solution marshall olkin namely independent bivariate exponential distribution survival function exp constants words independent random variables exp exp positive parameters pioneering paper marshall olkin considered instead weaker blm property lacks memory equal pair solved functional equation turns given marginals satisfies blm property iff survival function form positive constant see also barlow proschan convenience denote blm blm distribution marginals parameter survival function denote blm family blm distributions namely blm blm marginal distributions theorem summarizes important known properties blm distributions details see marshall olkin block basu block ghurye marshall convenience denote max min theorem let blm blm following statements true marginals densities respectively moreover derivatives exist bounded variation iii increasing nondecreasing exp exp independent vii differentiable remark necessary conditions vii also play sufficient conditions obey blm distribution example addition conditions vii assume marginal densities absolutely continuous bona fide survival function slight modification theorem marshall olkin required instead note conditions different unless consequence iii see corollary ghurye marshall hand condition together continuous marginals also implies blm distribution block interesting recall independent nondegenerate random variables independence characterization distributions suitable conditions see ferguson crawford rao shanbhag namely general blm distributions share independence property independent random variables remark observations least one positive survival function purely singular almost surely iff iff implies absolutely continuous almost surely iff marginal densities together satisfy see ghurye marshall view results survival function blm rewritten convex combination two extreme ones absolutely continuous purely singular survival function exp max clearly parameter regulates hand ghurye marshall section gave interesting random decomposition blm represented integral another bivariate survival function see also ghurye marshall olkin generalizations blm distributions remark kulkarni proposed interesting useful approach construct blm distributions starting marginal failure rate functions first choose two realvalued functions constant satisfying following modified conditions functions absolutely continuous set exp exp way defined bona fide blm distribution conditions together imply conditions vii theorem hold true see remark conversely smoothness conditions setting blm distribution marginal failure rate functions satisfy conditions properties theorem follow immediately kulkarni proposition recall three important blm distributions literature details see chapter balakrishnan lai example marshall olkin bivariate exponential distribution bve marginals exponential blm blm defined reduces bve survival function form exp max positive constants written explicitly absolutely continuous singular bivariate distributions respectively practice bve arises shock model system formally lifetimes two components exp exp exp independent joint survival function defined singular part identified conditional probability exp max absolutely continuous part calculated via see next example example block basu bivariate exponential distribution bve actually absolute continuous part bve joint density form exp exp survival function equal exp max exp max note case marginals exponential rather negative mixtures two exponentials specifically exp exp exp exp example freund bivariate exponential distribution freund bve joint density form exp exp survival function equal exp exp exp exp worths noting choosing freund bve reduces block basu bve new general properties blm distributions let blm blm marginals parameter survival function denote transform resp resp theorem transform blm blm prove theorem need following lemma due lin lemma let defined transform equal dxdy proof theorem calculate double integral dxdy changing variables integration parts dzdy dxdy similarly lemma together completes proof denote moment generating function mgf resp resp following general result theorem let blm blm let real numbers mgf provided expectations mgfs exist prove theorem need instead following lemma due lin lemma let defined let two increasing functions expectation product equal provided expectations exist proof theorem case let esx ety lemma dxdy calculate double integral dxdy lemma together completes proof case case apply lemma set esx ety increasing functions therefore dxdy carry double integral complete proof case case iii set esx ety lemma remaining proof similar case omitted case case treated theorem proof completed next consider product moments blm distributions theorem positive integers product moment blm blm form provided expectations exist first product moment neat representation terms marginal means parameter calculate pearson correlation blm distributions corollary provided expectations exist prove theorem apply following lemma due lin lemma let defined let expectations finite positive real numbers product moment proof theorem calculate double integral dxdy changing variables integration parts dxdy dzdy dzdy similarly finally lemma together completes proof moment generating functions specific blm distributions see chapter kotz product moments distributions see nadarajah next later results need notations reliability theory random variables say smaller usual stochastic order denoted smaller hazard rate order denoted increasing smaller reversed hazard rate order denoted increasing suppose densities respectively say smaller likelihood ratio order denoted increasing definitions related stochastic orders see stoyan shaked shanthikumar lai xie well kayid latter studied stochastic comparisons age replacement models hand distribution define notions increasing failure rate ifr decreasing failure rate dfr increasing failure rate average ifra decreasing failure rate average dfra follows say ifr dfr resp log convex concave resp ifra dfra resp log increasing decreasing resp equivalently resp see barlow proschan chapters bivariate ifra bivariate dfra distributions defined similarly bivariate ifra dfra resp resp see block savits worths mentioning definitions bivariate ifra distributions extend univariate case see esary marshall shaked shanthikumar using reliability language following useful results especially theorem iii means blm family positive bivariate aging plays favor positive univariate aging sense ifra vice versa general true even condition positive dependence lifetimes see bassan spizzichino remark analyzed relations among univariate bivariate agings dependence theorem let blm blm exp iii bivariate ifra distribution iff marginals ifra bivariate dfra distribution iff marginals dfra proof part follows immediately theorem iii see ghurye marshall part follows fact likelihood ratio order stronger usual stochastic order hazard rate order reversed hazard rate order stoyan part iii holds true verifying proof part similar applying stochastic inequalities simplify proof previous known results example corollary let blm blm following statements true hazard rates marginals bounded hence functions log log convex hence differentiable iii let supports marginals densities respectively nonnegative constants positive respectively absolutely continuous hence proof part follows facts exp part due plot characterization see theorem stoyan part iii follows facts theorem latter implies least one left extremities marginal distributions zero finally prove part note theorem see ghurye marshall absolutely continuous hence proof complete dependence structures blm distributions recall bivariate distribution marginals positively quadrant dependent pqd equivalently implies nonnegative covariance hoeffding representation covariance see lin stronger positive dependence property pqd total positivity defined nonnegative function rectangle product two subsets say totally positive order tpr fixed determinant matrix nonnegative function said tpr order karlin total positivity plays important role various concepts bivariate dependence see shaked lee moreover applying total positivity bivariate distribution survival function derive useful probability inequalities among many applications applied fields including statistics reliability economics see gross richards karlin proschan especially latter studied totally positive kernels arise convolutions type distributions characterize property survival functions blm distributions theorem let blm blm survival function iff marginal distributions ifr together satisfy exp proof define ratio definition iff necessity suppose exp implies exp next prove marginal distribution ifr note following statements equivalent increasing decreasing barlow proschan increasing iii increasing fixed ratio latter true case assumption similarly prove ifr ratio sufficiency suppose marginal distributions ifr together satisfy exp want prove ratio without loss generality consider three possible cases remaining cases proved exchanging roles case equivalence relations shown necessity part continuity case ratio exp exp exp assumption recall ifr distribution new better used barlow proschan therefore hence last similarly case exp exp assumptions completes proof recall also bivariate distribution marginals exist copula bivariate distribution uniform marginals survival copula namely links links corollary let blm blm survival copula iff marginal distributions ifr together satisfy exp proof since marginal absolutely continuous support positive density see corollary iii strictly increasing continuous similarly marginal strictly increasing continuous theorem suffices prove iff survival copula recall facts inverse functions respectively iii functions decreasing required result follows immediately see lemma counterpart property reverse regular order two nonnegative function say determinant matrix see asadi examples joint densities survival functions mimicking proof theorem conclude blm blm survival function iff survival copula iff marginal distributions dfr satisfy exp construct blm distribution survival function first consider pareto type distribution lomax distribution density function survival function choose parameters checked defined bona fide survival function seen conditions theorem satisfied bve therefore survival function survival copula bve regardless parameters general result given theorem next characterize different approach property joint densities absolutely continuous blm distributions theorem let blm blm absolutely continuous joint density function suppose marginal density functions three times differentiable finite assume functions joint density function iff marginal densities satisfy exp prove theorem need concept local dependence function following lemma part essentially due holland wang alternative complete proof part provided proof holland wang assumed implicitly integrability local dependence function kemperman gave without proof result continuity smoothness condition see also newman wang proved positive continuous bivariate density cartesian product uniquely determined marginal densities local dependence function latter exists integrable hand jones investigated bivariate distributions constant local dependence lemma let positive function iff log iff local dependence function log provided partial derivatives exist proof part trivial definition functions see nelsen part follows part fact smoothness assumption log iff local dependence function prove part directly note following statements equivalent log log log increasing increasing ratio satisfies function proof complete proof theorem assumptions joint density function form local dependence function log therefore iff property holds true necessity also rectangle rectangular area region hence property holds true lemma observation next property follows fact ratio satisfies exp completes proof necessity part sufficiency suppose want prove crossproduct ratio assumptions rectangle four vertices lies entirely region assumption lemma assumption implies remaining cases apply technique factorization ratio necessary example denote intersection diagonal line boundary rectangle split original rectangle four adding new point calculate ratio factor greater equal one previous results proof complete known bve pqd copula survival copula see barlow proschan moreover due theorem corollary see also nelsen direct proof even lin able extend results following theorem survival function survival copula regardless parameters prove theorem need two useful lemmas lemma see marshall lemma essentially due gantmacher krein see also karlin alternative version lemma let integer tpr nonnegative functions product function tpr tpr increasing decreasing composition function tpr lemma let two positive functions define symmetric function nondecreasing function proof theorem prove first survival function rewrite survival function exp exp exp symmetric function exp exp let exp lemma see function lemma next recall survival copula inverse decreasing functions exp exp respectively therefore lemma well known bivariate distribution density joint survival function see balakrishnan lai general result given follows theorem bivariate distribution tpr density tpr consequently density proof let consider first indicator functions apply theorem gross richards restated example prove tpr property claim determinant nonnegative prove let recall tpr two variables tpr two variables gross richards theorem applies hence tpr similarly survival function tpr proof complete gross richards theorem let integer let bivariate tpr density assume functions tpr two variables matrix totally positive minors orders nonnegative real numbers mentioned balakrishnan lai bve pqd extend result following theorem bve density freund bve density proof take exp exp constants part follows lemma part proved similarly remark approach applies bivariate distributions like pellerey generalized bivariate distribution described marshall olkin shock model assume instead independent general positive random variables limited exponential ones let log hazard function generalized bivariate distribution survival function max exp max pqd pellerey related shock models see marshall olkin ghurye marshall well aven jensen section extend result theorem follows theorem let generalized distribution defined survival function survival copula provided functions strictly decreasing proof write survival function exp exp exp symmetric function exp exp taking exp lemma know function hence survival function lemma proves part prove part note marginal survival functions exp exp two functions strictly increasing conditions turn implies marginal distribution functions strictly increasing hence survival copula quantile function inf see shorack wellner therefore part lemma proof complete stochastic comparisons blm distributions provide information blm distributions study stochastic comparisons blm family usual define notions upper orthant order concordance order laplace transform order follows let marginals denote iii stoyan shaked shanthikumar example following results whose proofs straightforward omitted theorem let blm blm iff equivalently bivariate distribution iff equivalently increasing functions provided expectations exist iii equivalently completely monotone functions provided expectations exist pair marginals theorem reduces corollary following interesting result related famous slepian inequality bivariate normal distributions see discussion remark corollary let blm blm correlation iff equivalently remark corollary consider standard bivariate normal distributions instead blm ones conclusion also holds true necessary part socalled slepian see slepian stoyan general results wikipedia said result true gaussian processes general true random variables however see corollary infinitely many blm distributions sharing slepian inequality bivariate normal ones remark finally compare effects dependence structure blm distributions different coherent systems consider system let two components lifetimes blm lifetime series system composed two components exp lifetime parallel system composed components obeying distribution therefore mean times failure series parallel systems respectively decreasing increasing latter implies compare theorem corollary provided expectations exist see also aven jensen section special cases exponential marginals well lai lin general results acknowledgments authors would like thank two referees helpful comments constructive suggestions improve presentation paper paper presented ibusuki international seminar ibusuki phoenix hotel held march waseda university japan authors thank organizer professor masanobu taniguchi kind invitation audiences comments suggestions references aven jensen stochastic models reliability springer new york balakrishnan lai continuous bivariate distributions springer new york barlow proschan statistical theory reliability life testing probability models begin silver spring bassan spizzichino relations among univariate aging bivariate aging dependence exchangeable lifetimes multivariate block characterization bivariate exponential distribution ann block basu continuous bivariate exponential extension amer statist block savits ifra closure problem ann block savits multivariate increasing failure rate average distributions ann crawford characterization geometric exponential distributions ann math esary marshall multivariate distributions increasing hazard rate average ann asadi hazards model statistics feller introduction probability theory vol wiley new york ferguson characterization exponential distribution ann math ferguson characterization geometric distribution amer math monthly fortet elements probability theory gordon breach new york freund bivariate extension exponential distribution amer statist galambos kotz characterizations probability distributions springer new york gantmacher krein oscillation matrices kernels small vibrations mechanical systems revised edn translation based russian original providence ghurye multivariate lifetime distributions adv appl ghurye marshall shock processes aftereffects multivariate lack memory appl gross richards algebraic methods toward probability inequalities stochastic processes related topics rajput eds boston gross richards algebraic methods toward probability inequalities ann slepian inequality modularity integral orderings high dimensional probability progress probability springer basel holland wang dependence function continuous bivariate densities comm statist theory methods jones local dependence function biometrika jones constant local dependence multivariate karlin total positivity vol stanford university press karlin proschan type distributions convolutions ann math kayid izadkhah alshami laplace transform ordering time failure age replacement models korean statist theory kemperman measures partially ordered space indagationes mathematicae kotz balakrishnan johnson continuous multivariate distributions vol models applications wiley new york kulkarni characterizations modelling multivariate lack memory property metrika lai lin mean time failure systems dependent components appl math lai xie stochastic ageing dependence reliability springer new york lee dependence total positivity ann pellerey generalized distributions related bivariate aging properties multivariate lin dou kuriki huang recent developments construction bivariate distributions fixed marginals journal statistical distributions applications lin lai govindaraju correlation structure olkin bivariate exponential distribution statist methodology marshall olkin multivariate exponential distribution amer statist marshall olkin bivariate life distribution multivariate marshall olkin arnold inequalities theory majorization applications springer new york stoyan comparison methods stochastic models risks wiley new york nadarajah exact distributions bivariate exponential distributions statistics nelsen introduction copulas springer new york newman asymptotic independence limit theorems positively negatively dependent random variables inequalities statistics probability tong ims lecture series vol rao shanbhag type functional equations applications stochastic models wiley new york shaked family concepts positive dependence bivariate distributions amer statist shaked shanthikumar multivariate conditional hazard rates mifra mifr properties appl shaked shanthikumar stochastic orders springer new jersey shorack wellner empirical processes applications statistics wiley new york slepian barrier problem gaussian noise bell system technical journal wang construction continuous bivariate density functions statist sinica
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generating representative executions extended abstract hendrik maarand tarmo uustalu dept software science tallinn university technology analyzing behaviour concurrent program made difficult number possible executions problem alleviated applying theory mazurkiewicz traces focus canonical representatives equivalence classes possible executions program paper presents generic framework allows specify possible behaviours execution environment generate executions program environment discarding abnormal executions generation phase key ingredient mazurkiewicz trace theory dependency relation used framework two roles first part specification executions allowed part normality checking algorithm used discard abnormal executions framework instantiated relaxed memory models sparc hierarchy introduction let consider fragment dekker mutual exclusion algorithm example init observed concurrent program two processors flag variable used communicate wants enter critical section processor may enter critical section notified processor setting flag variable reading flag variable processor checking interested whether possible starting initial state processors see others flag variables meaning processors enter critical section interested mutual exclusion property one processor enter critical section interleaving semantics sequential consistency program following executions abcd cdab acbd cabd acdb cadb six four last executions actually equivalent sense initial state reach final state purposes enough check final state one observe mutual exclusion property satisfied situation different consider possible executions processor like follows total store order tso model tso possible writes reordered later reads processor resulting execution observable bdac satisfy mutual exclusion property paper seek alleviate difficulty analyzing large numbers executions concurrent programs especially relaxed memories generate applying theory mazurkiewicz traces focus type canonical representatives equivalence classes possible executions program present generic framework interpreting concurrent programs different vasconcelos haller eds workshop programming language approaches software places eptcs maarand uustalu work licensed creative commons attribution license generating representative executions semantics executions foata normal form corresponding maximal parallelism generated instantiate framework relaxed memory models sparc hierarchy work vein partial order reduction techniques analysis systems widely used especially model checking also applied relaxed memories zhang novelties different memory models modelled uniformly based flexible notion backlog shadow events using standard normal form trace theory using generalized traces dynamic independency relation able define execution equivalence finely resulting bigger fewer equivalence classes framework prototyped haskell one easily separate phases generating tree symbolic executions program discarding abnormal executions running tree symbolic executions initial state separation made without performance penalty thanks lazy evaluation mazurkiewicz traces execution run sequential program represented sequence symbols record events caused program order occurred sequence string finite alphabet execution concurrent program represented interleaving executions processors involved thereby reducing concurrency choice mazurkiewicz traces traces generalization strings letters string allowed commute allows representation behaviour words traces equivalence classes strings respect congruence relation allows commute certain pairs letters dependency relation reflexive symmetric binary relation events causally related meaning two events happen concurrently complement dependency relation called independency relation strings sabt sbat represent behaviour two strings said mazurkiewicz equivalent transformed finite number exchanges adjacent independent events example trace acbd represents strings acbd cabd acdb cadb purposes standard mazurkiewicz traces enough therefore turn generalized mazurkiewicz traces sassone generalized mazurkiewicz traces dependency relation dynamic depends current context partial execution performed far dependency relation prefix denoted subscript omitted relation static besides reflexive symmetric must satisfy sanity conditions importantly must case setting strings sabt sbat considered equivalent normal forms traces equivalence classes reasonable ask canonical representative normal form trace two normal forms traces lexicographic foata normal forms going look foata normal forms purposes step subset pairwise independent letters foata normal form trace sequence steps individual steps chosen left right maximal cardinality since step consists independent letters step executed parallel meaning foata normal form encodes maximal parallel execution example foata normal form acbd interested checking whether given string normal form according given maarand uustalu dency relation convenience also assume ordering total events independent string foata normal form split sequence steps concatenation steps gives original string following conditions satisfied every every every step letters increasing order wrt definitions consider dependency relation context similarly first condition ensures events step executed parallel second condition ensures every event appears earliest possible step maximal parallelism third condition picks permutation step representative step notice string normal form neither string string prefix normal form means checking string normality scanning left right discard soon discover abnormal prefix framework proceed describing framework generating representative executions program instantiations different memory models going look programs executing machine consists processors shared memory processor also access local memory registers executions investigate symbolic sense look actual values propagating memory abstract actions performed still goal find possible final states program given initial state idea symbolic executions computed canonical executions picked final state needs computed done lazily meaning evaluation particular execution given initial state cancelled immediately discovered execution normal language system consists arithmetic boolean expressions commands arithmetic expression either numeral value register arithmetic operation boolean expression either boolean constant boolean operation comparison arithmetic expressions commands consist assignments registers loads stores shared memory constructs framework defined top events generated system think events occurrences phases actions executing program trigger event thought record pid eid kind act pid identifier processor generated event eid identifier event kind defines whether main shadow event act action performed event action operation registers load store variable assertion registers assertion used record decision made unfolding control structure program example particular execution one true branch conditional taken assertion fails execution evaluated given initial state execution valid initial state since interested modelling different memory models framework parameterized architecture characterizes behavioural aspects system architecture consists four components predicate shadows describes whether action executed single stage two stages generating one main event two events main shadow event relation samedep describes events processor must happen events processor plays role determining possible next events processor also generating representative executions defines events dependent relation diffdep describes two events different processors dependent finally relation orders independent events relations samedep closure diffdep together determine dependency relation sense mazurkiewicz traces relation used totally order events within step previous paragraph mentioned shadow events key ingredients framework modelling intricate behaviours example actions fact needs reflected executions two events main event shadow event tso example described model writes memory first enter processor later flushed memory consider write buffer main event write action flush event shadow event write action two events shadow event globally observable generating normal forms process generating executions program divided two stages lazily generating executions program discarding normal form executions generated follows processors completed complete execution done otherwise pick processor yet completed allow make small step repeat process local configuration processor consists residual program backlog value counter provide identifiers generated events small step either correspond beginning action next instruction according case new main event generated added completing already started case shadow event removed processor backlog added execution step start new action shadows predicate used check whether new shadow event added backlog action completed main event adding new main event shadow events backlog dependent event removed backlog independent according samedep older events backlog conditionals like expanded choice two programs choices correspond branches conditional together assertion condition generation executions described small step rules appendix second stage procedure single normal forms among generated executions done checking normality executions according three conditions given section foata normal forms rules checking normality execution scanning left right given appendix instead generating flat set executions first stage actually generate tree executions prefixes executions shared since process selecting canonical executions precisely discarding ones according conditions foata normal forms fused generation stage discard whole set executions discover current path tree violates normality conditions precisely walking tree keep track current prefix must normal form node check whether event associated node would violate normality conditions added prefix normality condition violated subtree starting node need computed actually require samedep hold least main events eid eid main event shadow event case eid also require samedep hold eid eid eid eid main event maarand uustalu corresponding shadow event assumptions prove total set executions captured generated tree closed equivalence normality checking stage keeps normal forms discards forms follows pruned set executions contains exactly one representative every execution program introduction noted example program six executions interleaving semantics four equivalent executions depicted figure four equivalent executions acbd acdb cabd cadb ones middle program framework would generate acbd four corresponds foata normal form three would discarded precisely normal form extended neither normal form first one fails due condition second one fails due condition node path shaded picture highlight place normality condition violated cabd start checking normality valid neither normal form discard executions start includes cabd cadb subtree node shaded highlight fact figure executions example program instantiation relaxed memory models sequential consistency sequential consistency model execution concurrent program interleaving program order executions component threads specified architecture following way shadows false samedep eid eid diffdep crxw pid pid crxw represents property returns true events access location least one write diffdep also takes two arguments ignored represent backlogs two processors events originate information recovered prefix execution much information generating representative executions need prefix execution memory models consider could also take prefix execution compute necessary information setting shadows always false means instructions execute atomically setting samedep require eid eid means events processor must generated program order reordered reflects definition total store order total store order tso model possible write action reordered later reads meaning writes happen asynchronously time order write actions preserved tso specified following way shadows iswrite samedep ismain ismain eid eid ismain isshadow eid eid isshadow isshadow eid eid diffdep pid pid pid pid eid eid crxw like crxw except considers shadow write events instead main write events global write events read events global access memory need generalized mazurkiewicz traces since pending write location read read action would read value memory thus could dependent events processors consider main event write instruction write buffer shadow event flushing write buffer memory tso thought model every processor shadow processor events every main processor program order events associated shadow processor program order event shadow processor must happen corresponding event main processor example introduction following traces foata normal form tso stands shadow event last one rejected partial store order partial store order pso model allows reorderings tso also possible write reordered later write different location thought separate write buffer every variable pso specified tso exception samedep relation samedep ismain ismain eid eid ismain isshadow eid eid isshadow isshadow eid eid var var intuitively corresponds pso since like tso except dependency relation events processor shadow events dependent location allows one reorder writes different locations relaxed memory order relaxed memory order rmo model enforces program order instruction pairs variable instruction pairs dependency first instruction read dependency instruction pairs means maarand uustalu instructions specify rmo following way shadows true samedep ismain ismain eid eid ismain isshadow eid eid isshadow isshadow eid eid var var iswrite isread iswrite datadep controldep diffdep crxw pid pid pid pid eid eid crxw like crxw except considers shadow reads shadow writes global read write events tso pso shadow read considered global actually reads value memory model happens older shadow write location backlog consider events reads register written consider two events older one conditional newer one write fences models like tso pso rmo allow reordering events becomes necessary able forbid reorderings certain situations rule relaxed behaviour example introduction behave correctly tso possible processors read value avoid situation necessary make sure processors first perform write effects write operation become globally visible may perform read restriction program behaves correctly tso way achieve insert fence write read instructions framework fences described two parameters take values store load indicate events ordering enforced fence instructions ignored since reorderings possible able restore sequentially consistent behaviour tso requires fences pso requires also fences rmo requires four kinds fences tso pso rmo idea fences shadow events samedep relation modified disallow unwanted reorderings example program requires fence read operations appearing fence performed write operations appearing fence completed means samedep must modified consider shadow fence dependent older shadow write events newer read events dependence shadow event prevents fence event removed backlog older dependent events removed also prevents removing newer dependent events fence removed backlog likewise new main read event added execution fence event backlog idea similar types fences related work relaxed memory consistency models specification verification tasks extensive research topic owens showed adheres tso model gave operational generating representative executions axiomatic models alglave defined framework axiomatic style working relaxed memory models also generic sense different memory models represented specifying relations considered global generating possible executions framework turns quite similar executable specification rmo given park dill precisely notion backlog seems correspond reordering box used boudol defined generic operational semantics captures tso pso rmo uses temporary stores similar backlogs however consider partial order reduction set executions program mentioned due interest exploring full set executions constructing explicitly use trace theory foundation partial order reduction work also close methods based model checking like zhang abdulla executable specification also given yang approach based axiomatic specifications execution found searching instantiation satisfies constraints either prolog sat solver conclusion presented generic framework finding canonical representatives equivalence classes possible executions program framework proceeds lazily generating executions given program discards foata normal form framework allows uniformly represent semantics certain class relaxed memory models illustrated encoding models sparc hierarchy terms framework instantiation framework particular model specifies executions occur given program equivalent correspond one generalized mazurkiewicz trace representable normal form plan continue work elaborating formal aspects framework formalized soundness completeness foata normalization standard traces dependently typed functional language string equivalent normal form string equivalent normal form string normal form development scaled generalized traces adapted prove tree filtering algorithm keeps exactly one representative equivalence class executions move formalization specifications memory models acknowledgments research supported estonian ministry education research institutional research grant erdf funded coe project excite references abdulla aronis atig jonsson leonardsson sagonas stateless model checking tso pso baier tinelli editors proc int conf tools algorithms construction analysis systems tacas lect notes comput sci springer alglave shared memory poetics thesis paris available http boudol petri serpette relaxed operational semantics concurrent programming languages luttik reniers editors proc combined wksh expressiveness concurrency wksh structural operational semantics electron proc theor comput sci maarand uustalu cartier foata problemes combinatoires commutation lect notes math springer godefroid methods verification concurrent systems approach problem springer lamport make multiprocessor computer correctly executes multiprocess programs ieee trans comput mazurkiewicz introduction trace theory book traces owens sarkar sewell better memory model berghofer nipkow urban wenzel editors proc int conf theorem proving higher order logics tphols lect notes comput sci springer park dill executable specification analyzer verifier rmo relaxed memory order proc ann acm symp parallel algorithms architectures spaa acm sassone nielsen winskel deterministic behavioural models concurrency borzyszkowski sokolowski editors proc int symp mathematical foundations computer science mfcs lect notes comput sci springer sparc international david weaver sparc architecture manual yang gopalakrishnan lindstrom slind nemos framework axiomatic executable specifications memory consistency models proc int parallel distributed processing symposium ipdps ieee zhang kusano wang dynamic partial order reduction relaxed memory models proc acm sigplan conf principles language design implementation pldi acm semantic rules small steps processor bklg bklg shadows act bklg eid act eid act act prg bklg eid prg eid act bklg eid act bklg eid act eid act act prg bklg eid prg bklg eid older prg newer older eid prg newer older eid prgi bklg eid bklg eid generating representative executions small steps system pid pid pid executions pid normal executions pid pid issdiff pid iss pid pid iss pid iss iss iss iss iss iss iss iss
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dual control approximate bayesian reinforcement learning dual control approximate bayesian reinforcement learning edgar klenske aug intelligent systems germany philipp hennig intelligent systems germany editor manfred opper abstract control dynamical systems uncertain dynamics poses tough elementary case bayesian reinforcement learning reasoning effect actions future observations offers principled solution intractable review extend old approximate approach control problem known dual control context modern regression methods specifically generalized linear regression experiments simulated systems show framework offers useful approximation intractable aspects bayesian producing structured exploration strategies differ standard approaches provide simple examples use framework approximate gaussian process regression feedforward neural networks control exploration keywords inference reinforcement learning control gaussian processes filtering bayesian introduction central problem learning interactive settings learner actions influence future observations episodic settings control problem repeatedly unchanged dynamics comparably simple notions exploration succeed assigning exploration bonus uncertain options macready wolpert audibert acting optimally one sample current probabilistic model environment thompson sampling see thompson chapelle perform well dearden kolter srinivas approaches however model effect actions future beliefs limits potential balancing exploration exploitation issue drastic case control single ongoing trial controller hope returned known states exploration must carefully controlled avoid disaster klenske hennig principled solution problem offered bayesian reinforcement learning duff poupart hennig probabilistic belief dynamics cost environment used simulate plan trajectories also reason changes belief future observations influence future decisions elegant formulation combine physical state parameters probabilistic model augmented dynamical description aim control system due inference augmented system invariably strongly nonlinear dynamics causing prohibitive computational finite state spaces discrete time poupart continuous space time hennig idea augmenting physical state model parameters noted early termed dual control feldbaum seems conceptual standards complexity hindered application exception strand several works meier tse tse tse tse authors developed techniques limiting computational cost dual control modern perspective seen form approximate inference bayesian reinforcement learning bayesian reinforcement learning community certainly aware work duff hennig found widespread attention first purpose paper cast dual control algorithm approximate inference technique bayesian parametric gaussian general regression extend framework ideas contemporary machine learning specifically explain principle formulated gaussian process context investigate simple practical approximations result also give simple example use algorithm dual control environment model constructed feedforward neural network rather gaussian process model notation throughout consider dynamic systems pomdps form cxk state dynamics observation model time state gaussian disturbance control input continuous action denoted simplicity assume scalar throughout measurements observations corrupted gaussian noise generative model thus reads cxk linear map trajectories vectors analogously occasionally use subset notation assume dynamics known described gaussian uncertainty general linear model nonlinear features uncertain matrices simplify notation reshape elements parameter vector vec vec define reshaping transformations dual control approximate bayesian reinforcement learning initialization belief states parameters assumed gaussian control response linear common assumption physical systems nonlinear mappings included generic form complicate following derivations raise issues identifiability simplicity also assume dynamics change time could relaxed autoregressive model would give additive terms derivations throughout assume finite horizon terminal time quadratic cost function state control target trajectory define state control cost goal line standard optimal control reinforcement learning find control sequence minimizes expected cost horizon exk past measurements controls prior information incorporated belief relative expectation calculated effectively serves bounded rationality approximation true information state since equation recursive final element cost defined differently ext without control input future cost optimal control sequence minimizing cost denoted associated cost min exk recursive formulation written amounts alternating minimization expectation steps influences enters latter expectation nonlinearly classic optimal control linear base case known found dynamic programming bellman bertsekas bayesian dual control feldbaum coined term dual control describe idea also known bayesian reinforcement learning machine learning community adaptive control considers past observations dual control also takes future observations account necessary ways deal uncertain parameters substantial klenske hennig drawbacks robust controllers example sacrifice performance due conservative design adaptive controllers based certainty equivalence uncertainty parameters taken account mean estimates show exploration learning purely passive systems obvious excitation leads better estimation also worse control performance attempts finding compromise exploration exploitation generally subsumed term dual control control literature achieved taking future effect current actions account shown optimal dual control practically unsolvable cases aoki examples solutions found simple systems sternby instead large number approximate formulations dual control problem formulated decades since includes introduction perturbation signals jacobs patchell constrained optimization limit minimal control signal maximum variance serial expansion loss function tse modifications loss function filatov unbehauen comprehensive overview dual control methods given wittenmark historical numerous treatments meaning term dual control evolved time applied fundamental concept optimal exploration methods approximate notion varying degree treatment studies one class practical methods aim approximate true dual control solution central observation bayesian dual control states parameters subject uncertainty part uncertainty caused randomness part lack knowledge captured way probability distributions states parameters thus subsumed augmented state feldbaum duff poupart notation optimal probabilistic priors defined written compactly optimal control augmented system new observation model using cost analogous unfortunately dynamics new system nonlinear even original physical system linear inference always nonlinear future states influence future parameter beliefs vice versa first problem unique dual control thus inference analytically tractable even gaussian assumptions aoki standard remedy use approximations popularly linearization extended kalman filter gives sequence approximate gaussian likelihood terms even incorporating gaussian likelihood terms future dynamics still intractable involves expectations rational polynomial functions whose degree increases length prediction horizon following section provides intuition complexity also descriptive power augmented state space aside note several authors kappen hennig previously pointed another possible construction augmented state incorporating actual value parameters state parameters gaussian belief advantage state observed without noise belief parameters follow stochastic differential dual control approximate bayesian reinforcement learning precisely follows ordinary deterministic differential equation follows stochastic differential attempted solve control problem differential equations directly numerical advantage formulation augmented state also drawbacks decided adopt first simplicity directly formalizable sde vanishes pomdp setting state observed without noise state observations corrupted exact belief state gaussian process parameters natural meaning approximate methods used retain gaussian belief dynamics intertwined chosen approximation changing approximation changes dynamics causes additional complication generally speaking entirely natural give differing treatment state parameters state parameters thus treated within framework also allows extending framework case also parameters follow sde toy problem provide intuition sheer complexity optimal dual control consider perhaps simplest possible example linear scalar system axk buk target observations known optimal drive current state zero one step trivially verified oracle abxk let parameter uncertain current belief time option simply replacing parameter current mean estimate known certainty equivalence control dual control literature tse resulting control law used many adaptive control settings practice substantial deficiencies uncertainty large mean good estimate controller might apply completely useless control signals often results large overshoots beginning parameter changes slightly elaborate solution compute expected cost optimize gives optimal feedback cautious control dreyfus dreyfus used term open loop optimal feedback approach term misleading modern readers fact algorithm klenske hennig control law reduces control actions cases high parameter uncertainty mitigates main drawback controller leads another problem since controller decreases control rising uncertainty entirely prevent learning consider posterior observing gaussian chosen controller uncertainty buk shows fully observed axk buk dual effect updated depends large values according new uncertainty system thus never learn act even large known phenomenon aoki however derivation control amounts minimizing myopic controller horizon single step long therefore control indeed optimal case optimality principle bertsekas means optimal solution last step every controller since show form exploration probing tse myopic controller enough show dual properties order expose dual features horizon least length since optimal controller follows bellman equation solution proceeds backwards solution second control action identical solution myopic controller applying first control action belief unknown parameter needs update according resulting inserting gives min min min since already rational function fourth order shows quadratically relevant expectations computed closed form aoki simple case though possible compute optimal dual control performing expectation sampling prior fig shows samples gray one single sample highlighted orange empirical expectation dashed green sample rational function even leading order contrast cost dual cost much narrower leading cautious behavior dual controller average dual cost minima zero either side reflecting optimal amount exploration particular belief state question monte carlo solution remain feasible larger horizons aware successful solutions continuous state spaces cost cost dual control approximate bayesian reinforcement learning figure left computing dual cost simple system costs optimal control sampled parameter thin gray one sample highlighted orange expected dual cost dashed green optimal lies minimum dashed green line right comparison sampling dashed green thin gray samples three approximations red bayesian exploration bonus blue solid green line approximate dual control constructed section see also sec details however see poupart sampling solution bayesian reinforcement learning discrete spaces including notes considerable computational complexity approach next section describes tractable analytic approximation involve samples approximate dual control linear systems tse constructed theory algorithm tse approximate dual control based series expansion related differential dynamic programming control nonlinear dynamic systems mayne separates three conceptual steps described sec together yield contemporary perspective amounts structured gaussian approximation bayesian find optimal trajectory deterministic part system mean model nominal trajectory certainty equivalent control linear systems easy see nonlinear ones poses nontrivial feasible nonlinear model predictive control problem diehl yields nominal trajectory relative following step constructs tractable quadratic expansion around nominal trajectory construct local quadratic expansion approximates effects future observations expansion quadratic optimal control law relative deterministic perturbation control constructed dynamic programming plugging perturbation control residual dynamics approximate quadratic system gives approximation step adds cost uncertainty deterministic control cost klenske hennig compute control initialize predict state covariance given compute next value search compute trajectory covariances make new measurement evaluate simulate run system yes search apply control figure approximate dual control algorithm show overall structure adapted tse left cycle inner loop performing nonlinear optimization current time step perform prediction arbitrary control input opposed analytically computed control input later steps optimize numerically repeated computation steps varying minimize approximate cost three steps explained detail subsequent sections interplay different parts algorithm shown figure abstract introductory work tse relatively general explicit formulation tse applies linear systems since works difficult parse contemporary readers following sections thus first provide short review extend modern concepts section follow transparent case linear system tse augmented state still gives nonlinear system interact multiplicatively parameters assumed deterministic known controller uncertainty captured distribution representing lack knowledge certainty equivalent control gives nominal reference trajectory certainty equivalent model built assumption uncertain coincide likely value mean system propagates deterministically without noise means nominal parameters current mean values dual control approximate bayesian reinforcement learning decouples entirely optimal control finite horizon problem computed dynamic programming aoki yielding optimal linear control law momentarily simplified notation constant defined computed recursively reference trajectory followed controller gives nominal trajectory inputs states current time horizon true future trajectory subject stochasticity uncertainty deterministic nominal trajectory optimal control associated nominal cost provides base relative approximation constructed quadratic expansion around nominal defines cost uncertainty central idea control project nonlinear objective quadratic locally linearizing around nominal trajectory maintaining joint gaussian belief introduce small perturbations around nominal cost states control perturbations arise stochasticity state parameter uncertainty note change state results change control signal optimal control signal step depends state even though origin uncertainties different arises stochasticity lack knowledge modeled joint probability distribution approximate gaussian filtering ensures beliefs remain gaussian note shifting mean nominal trajectory change uncertainty note expected perturbation parameters nil parameters assumed deterministic affected state input calculating gaussian filtering updates principle possible future measurements since violates causality principle glad ljung nonetheless possible use expected measurements simulate effects future measurements uncertainty since effects deterministic sometimes referred preposterior analysis raiffa schlaifer second order around nominal trajectory cost approximated klenske hennig optimal cost nominal system approximate additional cost perturbation although uncertain parameters show explicitly equation step captures dual effects uncertainty trajectory depends via dynamics higher uncertainty time causes higher predictive uncertainty thus increases expectation quadratic term control decreases uncertainty lower approximate cost modeling benefit exploration reason fact still quadratic function closed form solution make tractable tse make ansatz terms expectation written amounts applying dynamic programming perturbed system expectations cost gaussian beliefs computed analytically zero mean linear terms quantities vanish expectation allows analytic minimization approximate optimal cost time step min feasible given explicit description gaussian filtering update important note assuming extended kalman filtering update mean expected future observations nil expect see measurements consistent current mean estimate nonetheless variance changes depending control input dual effect following dynamic programming equations perturbed problem including additional cost uncertainty resulting cost amounts tse neglected effects dynamics recalling dropping constant part dual cost approximated jkd const recursive equation dual control approximate bayesian reinforcement learning defined augmented system blkdiag approximation overall cost used subsequent optimization procedure optimization current control input gives approximate dual control last step amounts outer loop overall algorithm optimization algorithm used find minimum dual cost function every step algorithm proposes control input dual cost evaluated depending approximate filtering carried horizon perturbation control plugged give analytic recursive definition approximation dual cost jkd function current control input nonlinear repetitions steps proposed locations yields approximation optimal dual control conceptually simplest part algorithm outer loop dominates computational cost every location whole machinery evaluated extension contemporary machine learning models preceding section reviewed treatment dual control linear dynamical systems tse section extend approach inference dual control dynamics nonlinear dynamical systems extension guided desire use number popular standard regression frameworks machine learning parametric general regression nonparametric gaussian process regression feedforward neural networks including base case logistic regression parametric nonlinear systems begin generalized linear model mentioned nonlinear features principle function popular choices include sines cosines radial basis functions sigmoids polynomials others caveat structure crucially influences properties model modeling perspective approach quite standard machine learning however dynamical learning setting requires adaptations first allow modeling dynamical systems original states must included gives features form consisting linear representation augmented general features next challenge optimal control nonlinear dynamical systems optimized closed form using dynamic programming even deterministic nominal system instead find nominal reference trajectory using nonlinear model predictive control diehl case begin dynamic programming locally linearized system optimize nonlinearly numerical method across trajectory adds computational cost requires care achieve stable optimization performance specific system setups filtering observations also involved case nonlinear dynamics experiments reported stayed within extended kalman filtering framework klenske hennig retain gaussian beliefs states parameters extensions approach elaborate filtering methods interesting direction future work includes relatively standard options like unscented kalman filtering uhlmann also recent developments machine learning probabilistic control analytic moment propagation features allow deisenroth rasmussen final problem generalization derivations preceding sections nonlinear dynamics take relatively simplistic approach nevertheless turns work well linearization gives locally linear dynamics whose structure closely matches essentially amounts extended kalman filtering augmented state using linearization approximation described sec applied analogously nonparametric gaussian process dynamics models treatment parametric linear models makes comparably easy extend description finitely many feature functions feature space defining gaussian process dynamics model assume true dynamics function draw gaussian process prior prior mean function prior covariance function kernel using widely used notion regression rasmussen williams formulating covariance cov simplify treatment assume covariance factors inputs outputs vij univariate kernel positive matrix output covariances mercer theorem rasmussen williams kernel decomposed converging series eigenfunctions functions orthonormal relative measure precise choice irrelevant time property dual control approximate bayesian reinforcement learning precisely sense gaussian process regression written bayesian linear regression use suggestive somewhat abusive notation generative model defined fki lij matrix satisfying cholesky decomposition elements draws white gaussian process mercer theorem exists expectation sense notation allows writing nonparametric prior mean covariance infinite diagonal matrix diagonal elements matrix multiplication defined using notation tedious straightforward linear algebra derivation see appendix shows posterior number gaussian observations tractable gaussian process gram matrix consists parts appropriate size depending current time state transition matrix needed account effect measurement noise time jacobians posterior mean evaluates posterior covariance comprised klenske hennig formulation together expositions preceding sections defines nonparametric dual control algorithm gaussian process priors important stress posterior indeed tractable far depends gram matrix size posterior computed time despite state space approximation constant cost practical control applications continuously rising inference cost rarely acceptable thus necessary project belief onto finite representation replacing infinite sum finite one bound computational cost matrix inversion eqs projecting finite basis functions drawn kernel respect lebesgue measure approach recently popular elsewhere regression rahimi recht readers unaware line work short introduction bochner theorem stein rasmussen williams covariance function stationary continuous random process represented fourier transform positive finite measure measure density fourier dual means eigenfunctions kernel trigonometric functions stationary covariance functions like commonly used square exponential kernel exp approximated sine cosine basis functions sin cos frequencies feature functions sampled power spectrum process example kernel approximation shown fig increasing number features approximation chosen closely true covariance function needed keeping number features range still feasible within time constraints control algorithm dual control feedforward neural networks another extension parametric linear models section allow nonlinear parametrization dynamics function dual control approximate bayesian reinforcement learning full kernel prior kernel appximation posterior kernel figure prior left posterior middle kernel function right full kernel function top row approximate kernel bottom row thick lines represent mean thin lines show two standard deviations dashed lines samples shown distributions zkj zki wij xjk figure feedforward neural network sketch illustrate structure klenske hennig particularly interesting example structure multilayer perceptrons consider network logistic link function weights latent output layer weights input hidden units biases hidden units see fig neural networks used control quite regularly see nguyen widrow instead using backpropagation stochastic gradient descent applications neural networks rumelhart robbins monro ekf inference procedure used train weights well singhal possible ekf linearization also applied nonlinear link function logistic function speaking terms feature functions weight feature also shape steepness inferred limiting factor inference naturally number data points features parameters introduced data points necessary learn using state augmentation linearizing parameters step ekf inference neural network parameters allows apply relatively cheap inference also use dual control framework plan control signals accounting effect future observations subsequent change belief means adaptive dual controller described sec identify parts neural net relevant applying optimal control problem hand sec show experiment properties experiments series experiments tasks continuous state space highlights qualitative differences adaptive dual controller three controllers oracle controller access true parameters provides unattainable lower bound achievable performance certainty equivalent controller described sec controller minimizing sum cost bayesian exploration bonus beb see kolter beb sqrt diag sqrt diag scalar exploration weight additional cost term beb evaluated predicted parameter covariance prediction time chosen according order system effect current control signal shows belief parameters type controller sometimes also called dual control referred explicit dual control dual features obtained modified cost function filatov unbehauen every experiment repeated times different random seeds shared across controllers comparability systems presented simple setups primary point show qualitative differences controllers behavior experiments done different approximations preceding section show experimental feasibility dual control approximate bayesian reinforcement learning feature set used specific application part prior assumptions application large uncertainty requires flexible models take longer converge require exploration feature selection important since independent dual control framework broad topic beyond scope paper following experiments different feature sets used examples flexibility framework also model different structural knowledge problems hand simple scalar system control matches exact dual control well linear system sec known fig right compares cost functions various controllers approximately exact sampling solution available simple setup cost functions shifted irrelevant constant cost quadratic indifferent zero beb gives additional structure near zero encourages learning qualitatively similar dual cost global minimum almost location dual control approximates sampling solution much closer faced state cost control holds exploration suitable cart rail simple example dynamical system combined nonlinearly varying slope simple nonlinear system constructed dynamics prior beliefs true values parameters chosen superscripts denote vector elements nonlinear functions shifted logistic functions form chosen use setup testbed exploration problem actual system dynamics relatively irrelevant focus complication caused cost function reference tracked also shown plot fig dashed orange line state weighting klenske hennig cost state cost high state cost state cost figure top four density estimate trajectories first state top bottom optimal oracle control gray certainty equivalent control red bayesian exploration bonus blue approximate dual control green reference trajectory dashed orange bottom mean cost per time step shown bottom plot colors matching controllers noted control cost relatively low task thus first keep cart fixed starting position high precision first time steps followed loose period time steps cart moved one side back center side back high cost good exploration strategy setting act cautiously first time steps aggressively explore loose phase finally able control motion high precision inference model approximated kernel described sec use alternating sine cosine features distributed according power spectrum full kernel since true nonlinearity form approximation model lower bound controller represents perfectly learned still exact model fig shows density estimated state trajectories four different controllers lower bound controller top controls precisely times high cost nothing times zero cost controlling perfectly measurement state disturbances certainty equivalent controller second top never explores actively learns accidentally observations arising run since initial trajectory requires little action left bad model reference starts move time step exploration bonus controller second bottom continuously explores dual control approximate bayesian reinforcement learning way knowing loose phase ahead course strategy incurs higher cost initially dual controller bottom efficiently holds exploration reaches loose phase explores aggressively control distinguishes necessary unnecessary parameter exploration system including nonlinearities experiment although noise parameters reference trajectory state weighting much simpler though weighting allowing identification beginning penalizing deviations first state later time steps important note reference trajectory passes areas state space strong negligible good exploration thus ignore found reasoning future trajectories experiment learned model neural network form described sec use logistic features see two free parameters equally spaced locations true nonlinear features means possible learn perfect model case fig shows density estimated state trajectories four different controllers symmetry cost function feature functions beb decide relevant irrelevant choosing exploration direction stochastically thus sometimes reduces uncertainty help subsequent control controller ignores completely identifies early phases leading good control performance control maintains useful knowledge last experiment similar sec uses different set nonlinear functions namely shifted gaussian functions radial basis functions experiment model learned parametric linear regression according sec fundamental difference experimental setups model assumes parameter drift results growing uncertainty parameters time true parameters kept constant simplicity klenske hennig cost state cost state cost figure top four density estimate trajectories second state top bottom optimal oracle control gray certainty equivalent control red bayesian exploration bonus blue approximate dual control green reference trajectory dashed orange bottom mean cost per time step shown bottom plot colors matching controllers noted reference tracked passes nonlinear features stays one cost structure cost starting linear reference trajectory time instant fig shows parameter belief relevant state single run experiment time shows clearly beginning necessary parameter learned early beb controller learns accidentally beb controller also learns second parameter beginning even though knowledge lost time trajectory reaches zone second parameter beb controller tries lower growing uncertainty every visible drops state incurring high cost control completely ignores growing uncertainty reaching area thus preventing unnecessary exploration dual control approximate bayesian reinforcement learning figure parameter knowledge left middle state trajectory right different controllers top bottom certainty equivalent control red bayesian exploration bonus blue approximate dual control green true parameters black lines exp mean std oracle exp mean std exp mean std exp mean std table average standard deviation costs experiments runs quantitative comparison experiments aim emphasize qualitative strengths control simpler approximations desirable controllers deal flexible models many parameters many invariably superfluous reference table also shows quantitative results averages standard deviations cost runs controller controller shows good performance overall interestingly also low variance beb prone instabilities conclusion bayesian reinforcement learning dual control offers elegant answer explorationexploitation relative prior probabilistic beliefs intricate intractable structure requires approximations balance another kind computation performance work investigated old approximate framework control klenske hennig language reinforcement learning extended apply contemporary inference methods machine learning including approximate gaussian process regression networks result tractable approximation captures notions structured exploration like value waiting future exploration opportunities distinguishing relevant irrelevant model parameters dual control framework clearer form offers interesting directions research reinforcement learning including combination recent new developments learning planning following conceptual work main challenge development still comparably high tractable numerical load dual control particularly problems higher dimensionality dual control approximate bayesian reinforcement learning appendix nonparametric ekf form standard kalman filter found many textbooks therefore restated starting standard equations derive general formulation classic state augmentation weight vector gives expected result derivation formulation assuming result gaussian process framework identical certain circumstances wish transform formulation full gram matrix therefore prediction update step combined pretty straightforward adopting standard notation covariance time step jacobian drift measurement covariance done second time step beneficial introducing compact notation predictive covariance first using compact notation defining analogously write update application schur lemma gives klenske hennig assuming full state measurement compactness notation update already looks similar inference generalize result general form building gram matrix according fpf fqf individual parts appropriate size depending current time state transition matrix also needed shift initial covariance drift covariances time put together results fpf fqf compact notation achieved using obtain calculating mean prediction done analogously augmenting state instead tracking state covariance setting also dynamics function inferred system equations nonlinear system hxk dual control approximate bayesian reinforcement learning inference model done ekf augmented state adopt view see rasmussen williams augment state weight vector original replaced augmented choosing recovers original states augmented state vector obtain calculations similar gram matrix additional terms including feature functions prior point important note infinite inner product corresponds evaluation kernel means write gram matrix written save space total gram matrix fpf fqf fkf since inference compact numerically stable absorb gram matrix define inference done according covariance mean klenske hennig appendix gradients hessians dynamics functions neural network basis functions neural network dynamics function derivatives logistic gradient easily found hessian written parts using fourier basis functions fourier approximation dynamics function form sin cos gradient easily verified cos sin sin cos sin cos dual control approximate bayesian reinforcement learning hessian written parts using normalization sin cos cos sin odd even radial basis functions radial basis function features dynamics function exp gradient exp exp exp hessian written parts exp exp references badgwell qin rawlings wright nonlinear predictive control moving horizon estimationan introductory overview advances control pages springer aoki optimization stochastic systems academic press new york london audibert munos tradeoff using variance estimates bandits theoretical computer science stochastic dynamic programming caution probing ieee transactions automatic control klenske hennig tse dual effect certainty equivalence separation stochastic control ieee transactions automatic control tse caution probing value information control uncertain systems annals economic social measurement bellman adaptive control processes guided tour princeton university press bertsekas dynamic programming optimal control athena scientific edition chapelle empirical evaluation thompson sampling 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university boston rasmussen williams gaussian processes machine learning mit robbins monro stochastic approximation method annals mathematical statistics rumelhart hinton williams learning representations backpropagating errors nature bayesian filtering smoothing cambridge university press singhal training multilayer perceptrons extended kalman algorithm advances neural information processing systems nips pages srinivas krause kakade seeger gaussian process optimization bandit setting regret experimental design international conference machine learning icml stein interpolation spatial data theory kriging springer verlag sternby simple dual control problem analytical solution ieee transactions automatic control thompson likelihood one unknown probability exceeds another view two samples biometrika tse actively adaptive control linear systems random parameters via dual control approach ieee transactions automatic control klenske hennig tse meier iii adaptive dual control nonlinear 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optimal inference model selection william dennis jonathan apr department statistics university california berkeley department statistics california polytechnic state university department statistics stanford university april abstract perform inference model selection propose controlling selective type error error rate test given performed recover frequency properties among selected hypotheses analogous apply classical context proposal closely related data splitting similar intuitive justification powerful exploiting classical theory lehmann derive powerful unbiased selective tests confidence intervals inference exponential family models arbitrary selection procedures linear regression derive new selective generalize recent proposals inference model selection improve power new selective require knowledge error variance introduction typical statistical investigation thought consisting two stages selection analyst chooses statistical model data hand formulates testing estimation problems terms unknown aspects model inference analyst investigates chosen problems using data selected model informally selection stage determines questions ask inference stage answers questions statistical methods carry implicit assumption selection nonadaptive choices model use hypothesis test parameter estimate made seeing data adaptive selection also known colloquially data snooping violates assumption formally invalidating subsequent inference cases possible specify question prior collecting instance data governed known physical law however applications choice question least partially guided data example often perform exploratory analyses decide predictors interactions include regression model check whether assumptions test satisfied goal paper codify means inference valid presence adaptive selection propose methods achieve selective account properly adaptive model selection resulting inferences troubling frequency properties illustrate example correspondence addressed example file drawer effect suppose one scientific research groups make independent measurements quantities focus apparently large effects selecting say indices scientist wishes test significance level practitioners intuitively recognize nominal test rejects invalidated selection exactly invalid test probability falsely rejecting given still since simply tested time rather troubling feature error rate among hypotheses selected testing possibly much higher precise let number true null effects suppose long run fraction errors among true nulls test reject true false rejections true nulls selected true reject reject nominal test thus see probability false rejection conditional selection natural error criterion control presence selection example directly control level simply finding critical value solving case stringent nominal cutoff paper develop theory inference selection based controlling selective type error rate guiding principle answer must valid given question asked simplicity example regarded stylized model science imagine represents estimated effect size scientific study however large estimates ever caricature may far truth recently demonstrated franco compound problem may many reasonable methodologies choose even analyst decided roughly scientific question address gelman loken resulting selection bias error rate among published claims may high leading even speculation published research findings false ioannidis thus selection effects may partial explanation replicability crisis reported scientific community yong popular media johnson setting example studied extensively literature simultaneous selective inference several authors proposed adjusting selection means conditional inference pritchard zhong prentice construct selectionadjusted estimators intervals association studies genes pass fixed initial significance threshold based conditional gaussian likelihood cohen sackrowitz obtain unbiased estimates mean population whose sample mean largest conditioning ordering observed sample means sampson sill sill sampson apply idea obtain estimates drug adaptive clinical trial design hedges hedges propose methods adjust file drawer effect scientists publish significant results another framework selection adjustment proposed benjamini yekutieli consider problem constructing intervals number parameters selected viewing data letting denote number intervals among constructed define false rate fcr expected fraction max noncovering intervals controlling fcr level thus amounts coverage average among selected see section fcr control closely related selective error control criterion propose fact weinstein employ conditional inference construct intervals context example rosenblatt benjamini propose similar method finding correlated regions brain also view toward fcr control conditioning selection classical statistical inference notion inference selection exist analyst must specify model well hypothesis tested advance looking data classical test hypothesis model must control usual nominal type error rate reject subscript reminds probability computed assumption data generated model true misspecified guarantees rejection probability statistical practice unrealistic rule model selection altogether statisticians trained check models tweak diagnose problem purist even model checking suspect since leaves open possibility model change see data argue model hypothesis selected adaptively instead control selective type error rate reject selected one argue models hypotheses practically never truly fixed chosen randomly since based outcomes previous experiments random scientific process typically ignore random selection use classical tests control implicitly assuming randomness selecting independent data used inference case reject selected reject may seem pedantic point model selection random based previous experiments viewpoint justifies common prescription previous experiments dictate model possible split data independent imitate scientific process setting aside selection inference selection depends nominal test based value satisfy nominal test based also controls selective error generating selective procedures nominal ones called data splitting sample splitting idea dates back least far cox despite paucity literature topic common wisdom among practitioners example customary genetics use one cohort identify loci interest separate cohort confirm sladek wasserman roeder meinshausen discuss approaches inference data splitting owes much popularity transparent justification even nonexpert appreciate imagine observed first proceed analyze though model selection took place ahead equation guarantees temporal metaphor lead astray even describe actually collected data splitting elegantly solves problem controlling selective error cost reduces amount data available inference also reduces amount data available selection furthermore always possible split data independent parts case autocorrelated spatial time series data article propose directly controlling selective error rate conditioning event selected data splitting treat data though revealed stages first stage observe enough data resolve decision whether test treat data selected yet observed second stage commences intuition paragraph expressed formally terms filtration used selection used inference denotes generated random variable informally everything know data observing trivial representing complete ignorance selection event selected think time progressing left right stage one learn enough decide whether test advancing state knowledge begin stage two discover actual value advancing knowledge selection decision made end stage one everything revealed stage two fair game inference effect controlling type error conditional prevents appealing fact evidence even extremely surprising still reject unless surprised anew second stage sense conditioning random variable discards information carries parameter hypothesis interest contrast data splitting viewed conditioning instead advocate discarding little information possible reserving rest stage two frugality results efficient division information carried call data carving outline section formalize problem selective inference discuss general properties selective error control address key conceptual questions conditioning selection event effectively discards information used selection information left inference also see major advantage selective error control allows consider one model time designing tests intervals even priori many models consideration law random variable follows exponential family model event follows closely related exponential family model result selective inference dovetails naturally classical optimality theory lehmann section briefly reviews theory derives powerful unbiased selective tests arbitrary exponential family models arbitrary model selection procedures conditioning data necessary saps power tests data splitting yields inadmissible selective tests general conditions section gives general strategies computing rejection cutoffs tests prescribed section sections derive selective tests specific examples section focuses case linear regression generalizing recent proposals tibshirani lee others derive new powerful selective well selective require knowledge error variance several simulations section compare selective data splitting illustrate tradeoff using data initial stage reserving information second stage section compares contrasts selective inference multiple inference section concludes problem selective inference example regression lasso previous section motivated idea conditioning selection arguably familiar example selection variable selection linear regression regression observed data assumed generated multivariate normal distribution goal model mean linear function predictors obtain parsimonious model simply identifiable model researchers often use subset predictors subset leads different statistical model corresponding assumption denotes matrix consisting columns customary report tests coefficient model chosen way control selective error must condition selected case conditioning selected model many methods variable selection linear regression ranging aic minimization forward stepwise selection hastie consider one procedure particular based lasso mostly selective inference context lasso lee main motivation present work lasso tibshirani provides estimate solves argmin full matrix consisting predictors first term usual objective second term encourages many coefficients exactly zero property makes sense define model selected lasso set variables coefficients take possible values one subset notice form partition regions correspond model regions control selective error selecting particular must condition event landed partition lasso problem variables dimensions shown figure explicit characterization lasso partition found lee see also harris interactive visualization way lasso partitions sample space different selection procedure would partition sample space differently characterizations figure example lasso observations based distribution conditional landing highlighted region partitions forward stepwise selection marginal screening found loftus taylor lee taylor respectively imagine stage one loaded data software package computed remain otherwise ignorant value observed regions falls region chosen model construct tests selected variables example shown figure selected variables thus test two hypotheses notice careful always specify model along coefficient since coefficient variable necessarily consistent interpretation across different models regression coefficient summarizes effect variable adjusting variables model example effect salary genuinely different question effect salary adjusting years education questions meaningful fundamentally chosen model conditioned selection base tests precise location know yet conditionally gaussian follow exponential family result appeal classical theory lehmann construct tests confidence intervals natural parameters known otherwise concrete example mind develop general framework selective inference much broadly applicable explicitly allowing models hypotheses random necessary carefully define inferential goals first discuss selective inference context hypothesis testing closely related developments confidence intervals follow section use word effect informally refer regression coefficient recognizing regression establish causal claims selective hypothesis tests introduce notation use remainder article assume data lies measurable space unknown sampling distribution analyst task pose reasonable probability model family distributions believes contains carry inference based observation let denote question space inference problems might tackle hypothesis testing problem pair model null hypothesis mean submodel write model hypothesis corresponding without loss generality assume tested alternative hypothesis avoid measurability issues assume throughout countable although framework extended uncountable additional care section test variable selected regression model question space note slight abuse notation using interchangeably refer subset variable indices corresponding probability model model selective inference process two distinct stages selection collection possible questions analyst selects subset test based data inference analyst performs hypothesis test case simple regression example shown figure selected variables would consist hypotheses two variables model correctly specified model one contains true sampling distribution importantly expressly assume candidate models correctly specified analyst must choose without knowing could choose poorly case may formal guarantees behavior test performs stage two degree misspecification rule rather exception real statistical applications whether models specified adaptively analyst would position select probably wrong model using use model perform test new data collected confirmatory experiment see section discussion issue purposes hypothesis test function taking values representing probability rejecting cases value function either discrete variables randomization may necessary achieve exact level adjust selection testing condition event question asked describe selection event event among questions asked general selection events different questions disjoint regression example test model identify null hypothesis like corresponding subfamily null model remind error guarantees test necessarily extend beyond model designed convention take selected conditioning equivalent simply conditioning reflect idea hypothesis tested rejected note convention affect selective properties selective inference mainly interested properties test question conditional say controls selective type error level define selective power function countable relevant notice model hypothesis relevant defining selective level power test means designing valid concentrate one time even many mutually incompatible candidate models long controls selective error level given selection event global error also controlled false rejections true nulls selected provided denominator finite equation holds countable regardless dependence structure across different fact design tests one time makes much easier devise selective tests concrete examples take sections comparison familywise error rate selective error control neither weaker stronger control familywise error rate fwer probability rejecting true null hypothesis fwer although fwer usually considered conservative control guarantee scale easily across different researchers suppose example observation collected different scientific research team different university team publishing nominal test otherwise moving another project level single research group experiment fwer controlled major multiplicity problem consider discipline whole contrast selective error control scales naturally across multiple research groups requires coordination among groups research team discipline controls selective error rate experiments discipline whole achieve control type error rate among true selected null hypotheses would selection proposition error control suppose independently operating research groups scientific discipline shared countable question space research group carries selective tests collects data applies selection rule assume research group probability least carrying least one test true null common efi grows discipline whole achieves control frequentist error rate lim sup false rejections true nulls selected figure instead conditioning selection event question asked condition finer event value random variable call selection variable proof deferred appendix counterpart proposition popular error rates false discovery rate fdr benjamini hochberg familywise error rate fwer section discusses relationship selective error control common error rates multiple inference selective confidence intervals goal instead form confidence intervals parameter convenient think containing pairs model parameter analogy call set selective confidence set next result establishes selective confidence sets obtained inverting selective tests one would expect analogy classical case proposition duality selective tests confidence sets suppose form confidence interval event suppose also event form test let set always reject selective test selective confidence set proof selective probability conditioning discards information performing inference conditional random variable effectively disqualifies variable evidence hypothesis typically want condition little data possible stage two even selective inference procedures condition example data splitting viewed inference conditional part data used selection generally say selection variable variable whose level sets partition sample space finely informally think conditioning finer partition shown figure say controls selective type error respect level error rate less given formally taking coarsest possible selection variable recovers baseline selective type error definition selective confidence set may generalized way generalizing finer selection variables gives used selection used inference suggesting refine less data left inference indeed finer stringent requirement proposition monotonicity selective error suppose controls type error rate level finer selection variable also controls type error rate level coarser proof coarsest possible choice test controlling type error selection variable also controls selective error extreme improve trivial test proposition suggests typically sacrifice power move coarser finer selection variables even refining selection variable useful computational reasons example case lasso conditioning additionally signs nonzero selection event becomes convex region instead union disjoint convex regions lee another valid reason refine beyond strengthen inferential guarantees meaningful way example achieve achieve false rate fcr control choosing see section proposition data splitting corresponds setting every selection variable equal result data splitting use information remains conditioning see informally filtration used selection wasted used inference see section waste information means data splitting inadmissible fairly general conditions quantify amount leftover information terms fisher information remains conditional law given smooth parametric model decompose hessian conditional expectation selective confidence interval selective nominal information leftover fisher information observed leftover fisher information function little information conditional distribution since conditionally highly concentrated virtually information lost confidence intervals inverting umpu tests section interval essentially coincides nominal interval close wide interval reflects potentially severe selection bias figure univariate gaussian selection event leftover fisher information selection leftover information essentially missing information orchard find leftover intuitive descriptor missing context since information disposal taking expectations obtain thus average price conditioning price selection information carries cases loss may quite small simple example elucidates example consider selective inference univariate gaussian model conditioning selection event figure plots leftover information function little information conditional distribution whether conditionally highly concentrated contrast conditional law practically different marginal law virtually information lost conditioning figure shows confidence intervals result inverting tests described section interval essentially coincides nominal interval hardly selection bias real adjustment necessary contrast close potentially subject severe selection bias fact reflected confidence interval longer nominal interval centered value significantly less note necessarily every fact interesting counterexamples certain take conceptual questions pause address conceptual objections encountered explaining work objections easily expressed answered setting single selected model single selected hypothesis test confidence interval construct always singleton common theme every one conceptual objections follow equally good grounds objecting data splitting matter selecting model hypothesis based prior experiment whose outcome random thus good exercise ask would answer question asked data splitting likely answer applies equally well data carving model random framework inference based statistical model allowed chosen randomly based data common first reaction data generated according model model selected based data whole business circular nonsensical resolve conundrum note framework true sampling distribution selected sense entirely outside analyst control thing selected working model tool analyst uses carry inference may may include true thus sampling distribution comes first data model random selection models new trouble confuse would random selected via data splitting matter based prior experiment thoughtful skeptics may find reasons concern approaches believing statistical testing appropriate model based purely convincing theoretical considerations answer point view would rule scientific inquiries statistics ever used however comfortable choosing random model using data splitting previous experiment see special reason concerned choosing random model using data carving case means required random framework applies many interesting settings always parametric nonparametric model adaptively choose hypotheses test parameters estimate example clinical trial example section statistical model always choose null hypotheses test inspecting data true conditional confidence intervals weinstein selective proposed lee discussed rank verification methods proposed hung fithian selected model wrong writing topic selective inference might begun stating formal mathematical assumption sampling distribution belongs known model devised test behaves well might work well example choose apply sample whose observations highly correlated probability rejection may great deal larger nominal even mistake formal theory make inherently invalid test rather validity invalidity test defined respect behavior given application analyst must choose among many statistical methods knowing one designed work particular set parametric nonparametric assumptions particular model theory encompasses choice subsequent analysis would sensible assume analyst infallible always selects correct model typically candidate models correctly specified others analyst never know sure model selection procedure using data splitting data carving prior experiment always carries risk selecting wrong model cases type error guarantees force model correct course possibility misspecification restricted adaptive procedures like data carving data splitting selecting inappropriate model seeing data leaves better worse chosen inappropriate model seeing data alternative adaptive model selection infallible model selection model selection separate question robustness close sense may still want procedure behave predictably however even model gives reasonable approximation guarantee induced model reasonable since conditioning introduce new robustness problems example suppose test statistic tends distribution setting might comfortable modeling gaussian basis hypothesis testing case also true converges truncated gaussian law fixed approximation may much poorer intermediate values worse use increasing thresholds truncated gaussian approximation may never become reasonable understanding interaction selective inference asymptotic approximations area active ongoing study see tian taylor tibshirani tian taylor taylor tibshirani subsequent works discussing asymptotics without gaussian assumptions result marginal interpretation consider adaptive clinical trial select promising subgroup patients based preliminary analysis report confidence interval average treatment effect subgroup realizations data might decide return interval effect men age realizations might decide return interval effect hispanic women high blood pressure let denote selected subpopulation random region covariate space let denote true average treatment effect given subpopulation let denote confidence interval construct find confusion often occurs people attempt interpret marginal confidence interval treatment effect random subpopulation ran experiment selection procedure might choose completely different giving completely different meaning realizations data might produce disjoint realizations meant cover different true parameter values technically correct interpretation usually best avoided rather recommend thinking different confidence interval different fixed parameter nothing selection stage choose one construct leave intervals undefined words interval useful interpretation first stage complete fixed pointless try interpret answer even decide question ask true might asked different parameter data looked different token might performed entirely different experiment recent grant application funded neither contingencies source confusion experiments performed parameters selected irrelevant situation hand prior work selective inference article takes main inspiration recent ferment work problem inference linear regression models model selection lockhart derive asymptotic test whether nonzero fitted coefficients given knot lasso path contain true nonzero coefficients tibshirani provided exact version result extended lars path lee loftus taylor lee taylor used similar approaches derive exact tests lasso fixed value regularization parameter forward stepwise regression regression marginal screening respectively approaches derived assuming error variance known independent estimate available present work attempts unify approaches common theoretical framework generalizing classical optimality theory lehmann elucidate previously unexplored questions power also lets generalize results case unknown arbitrary exponential families arbitrary selection events since initial appearance work applied many settings see taylor tibshirani recent review works viewed selective inference multiple inference problem recent work vein found berk barber section argues inference model selection multiple inference distinct problems different scientific goals see benjamini discussion distinction empirical bayes approach estimation found efron also recently work inference linear regression models notably belloni belloni zhang zhang javanmard montanari van geer see dezeure review works focus approximate asymptotic inference fixed model many variables consider inference selecting smaller submodel focus inferential goals leeb prove certain impossibility results regarding estimating distribution estimators results apply framework statistical models use distributions test statistics known thus require estimation foregoing works frequentist work bayesian inference conditions entire data set conditioning first selection event typically operative effect posterior respectively marginal likelihood prior dawid yekutieli argues certain cases appropriate condition likelihood selection without changing prior reflect conditioning resulting posterior proportional credible intervals discussed yekutieli resemble confidence intervals proposed article discussion therein presents somewhat different perspective conditioning adjust selection though goals different theoretical framework respects similar conditional confidence framework kiefer inference made conditional estimate confidence decision made see also kiefer brownie kiefer brown berger olshen discussed error control given selection multiple comparison procedure first performed applied rejects large enough rejection thresholds simultaneous coverage second stage less conditional rejection stage one selective inference exponential families discussed section construct selective tests one time hypothesis pair conditional corresponding selection event ignoring models previously consideration candidate models hypotheses irrelevant satisfying reason suppress explicit dependence except necessary resolve ambiguity framework selective inference especially convenient corresponds multiparameter exponential family exp respect dominating measure conditional distribution given measurable another exponential family natural parameters sufficient statistics different carrier measure normalizing constant exp fact lets draw upon rich theory inference multiparameter exponential families conditional inference nuisance parameters classically conditional inference exponential families arises means inference presence nuisance parameters model model exponential family nuisance parameters follows exponential family sufficient statistics dimension respectively exp open assume corresponds parameter interest unknown nuisance parameter conditional law depends exp letting eliminate problem conditioning obtain family consider testing null hypothesis alternative say selective test selectively unbiased condition specializes usual definition unbiased test selection unbiasedness rules tests privilege alternatives detriment others tests alternative uniformly powerful unbiased umpu selective test one whose selective power uniformly highest among tests satisfying selectively unbiased confidence region one inverts selectively unbiased test confidence regions inverting umpu selective tests called uniformly accurate unbiased umau specialize usual definitions see lehmann romano brown thorough reviews rich literature testing exponential family models particular following classic result lehmann gives simple construction umpu tests exponential family models theorem lehmann model consider testing hypothesis level umpu test form chosen satisfy condition constrains power obtained differentiating power function setting derivative exponential family simply apply theorem conditional law obtain analogous construction selective setting corollary umpu selective tests model consider testing hypothesis selective level selection event umpu selective test form solve emphasize test defined merely umpu among selective tests condition rather umpu among selective tests see lehmann romano details cases may useful interpret conditionally example observed leads less powerful test worth keeping mind unbiasedness one way choose test completely ump one example another simple choice use test conditional law rejection region simply union rejection regions umpu tests choose different ways tests take form fact see next admissible tests form implies data splitting tests usually inadmissible conditioning admissibility data splitting selective test inadmissible selection event exists another selective test inequality strict least one main result section show tests based data splitting nearly always inadmissible let observation model suppose wish test assume tests functions sufficient statistic write abuse notation without loss generality test obtain new test function power function original therefore inadmissible original test apply following result matthes truax theorem matthes truax theorem let observation model suppose wish test let denote class tests form int convex set every exists notice holds equality completeness hence every admissible test almost surely equal test order apply result data splitting first introduce generic exponential family composed two independent data sets governed parameters model exponential family data splitting model independent random variables exp models satisfying model model would example cover case responses two linear regressions different design matrices regression coefficients selection event say test selection stage uses inference stage uses assume without loss generality test form next define cutoff gap largest acceptance rejection regions separated cushion width conditionally independent copy given sup note support depend thus neither tests example either cutoff interior supp positive probability randomized test discrete next prove main technical result section inadmissible unless determined within amount variability theorem let denote copy conditionally independent given let test model inadmissible proof construct conditionally independent copies assume form otherwise could admissible matthes truax must equivalent test form exist assumption exists occurs positive probability definition event also occurs positive probability since two events independent occurs positive probability next assume event occur otherwise could reparameterize natural parameter would sufficient statistics event occurs positive probability two ruling possibility typical case corollary suppose test model inadmissible unless function expected length interval length information leftover fisher information data splitting data carving data splitting data carving fisher information available inference expected confidence interval length figure contrast data splitting data carving example independently data splitting discards entirely data carving uses leftover information inference data carving also uses one data point inference since information left conditioning barely effects law data carving nearly two data points left example illustrate theorem consider bivariate version example condition selection event data splitting could construct confidence interval using namely interval valid use information available powerful alternative construct interval based law uses leftover information figure shows fisher information available test function fisher information data splitting exactly matter whereas optimal selective test information approaching increases figure shows expected confidence interval length equal tailed interval function data splitting interval roughly longer needs limit factor together plots tell consistent story selection event unlikely discarding first data set exacts unnecessary toll power procedure selective inference linear regression concrete example exponential family framework discussed section turn linear regression one important applications selective inference linear regression data arise multivariate normal distribution modeled avoid trivialities assume full column rank consideration depending whether assumed known unknown hypothesis tests coordinates generalize either case based coordinates ordinary least squares ols estimator penrose pseudoinverse particular convenient write remainder adjusting columns pxm denotes projection onto column space letting kpx test statistics respectively distributed henceforth suppress subscript superscript simply writing ambiguity optimal selective based test statistics compared different null distributions consider two distinct modeling frameworks section concerns inference restrictive selected linear model family distributions hold section concerns inference general saturated model assumes performs inference see tests powerful tests extra power comes price since inferences valid restrictive modeling assumptions section compares contrasts two approaches inference selected model suppressing superscript selected model form exp kyk known sufficient statistics inference based otherwise represents another sufficient statistic inference based decomposing pxm pxm see fixed affine transformation condition known equivalently base selective test marginally independent generically conditionally independent given null distribution generically depends unknown may observe kpxm kpx writing kpxm monotone function fixing thus test based appropriate conditional law note given neither unbiased recommend viewing serious estimate selective setting constructing selective straightforward general case described section natural parameter selected model rather testing equivalent testing testing correspond point null hypothesis however define bxj bxj natural parameter carry umpu selective based law inference saturated model even take linear model seriously still best linear predictor population design matrix arg min call least squares coefficients according point view sponds linear functional point view convenient parameters general saturated model leading meaningful inference even poor job selecting predictors particular berk adopt perspective way avoiding need consider multiple candidate statistical models several recent articles tackled problem exact selective inference linear regression specific selection procedures lee loftus taylor lee taylor works well berk assume error variance known estimate may obtained independent data target parameters saturated model selected model whereas saturated model may exist compared selected model saturated model additional nuisance parameters corresponding write saturated model exponential family form kyk exp natural paramaters unknown otherwise perform inference coefficient rewrite exp kyk known inference selection event based conditional law equivalently unknown must instead base inference unfortunately conditioning restrictive set kyk line intersected sphere consists two points equally likely hypothesis thus saturated model conditioning leaves insufficient information carry meaningful test saturated model selected model known choice whether carry test statistic saturated selected model words must choose either assume treat unknown nuisance parameter writing must choose whether condition saturated model selected model conditioning never increase power relative conditioning unless tests coincide lead inadmissible test per theorem case choice makes difference since mutually independent selective case however choice may major consequence lead different tests general conditionally independent given may play important role determining conditional distribution needlessly condition may lose great deal power whereas failing condition could lead astray large simple example elucidate contrast example suppose design matrix choose bestfitting model choose otherwise figure shows one realization process little larger choose yellow highlighted region chosen selection event selected model case since one column selectedmodel test based whereas test based second conditioning set union two rays plotted brown hypothesis realized quite large given giving contrast terribly large given leading conditional null distributions conditioning sets density saturated model selected model observed value conditioning set union quadrants plotted yellow saturatedmodel conditioning set union rays plotted brown conditional distributions hypothesis realized quite large given giving contrast large given giving figure contrast tests example fit model design matrix test based whereas test based approach especially testing goodness fit selected linear model case prefer test level rather reject high probability important variables selected fithian consider sequential testing path increasingly complex models selected method like lasso forward stepwise regression example illustrates especially large near ties much larger result selectedmodel test much powerful early steps path multiple strong variables compete enter model first details see fithian computations saw section inference exponential family requires knowing conditional law cases saturated model viewpoint conditional law determined fairly explicitly cases need resort monte carlo sampling section suggest general strategies gaussians saturated model discussed section previous papers lee loftus taylor lee taylor adopted saturated model viewpoint known case truncated univariate gaussian since gaussian random variable independent convex truncation interval endpoints represent maximal extent one move figure inference generic convex selection set conditioning yellow set largest get smallest get test statistic takes distribution standard gaussian random variable truncated interval result uniformly distributed direction height still remaining inside sup inf geometric intuition illustrated figure specifically polytope obtain expressions generalization regions straightforward instead truncating single interval truncate union intervals discussion points see lee monte carlo tests intervals generic setting may easy formula conditional law case several options inference using monte carlo methods obtain stream samples value carry hypothesis tests construct intervals done efficiently via rejection sampling example sample efficiently small otherwise specialized sampling approaches may required little abstractly consider constructing test based statistic distributed according exponential family exact monte carlo tests suppose addition given independent sequence reference distribution exact monte carlo test rejects observed value among largest barnard even samples available procedure level provided law exchangeable besag clifford propose ingenious procedure obtaining exchangeable sequence know run markov chain stationary distribution beginning take steps backward chain run independent chains steps forward beginning chain letting denote end state ith chain sequence exchangeable note test level using small values makes test random reducing power chain irreducible test converges deterministic test approximate monte carlo intervals reweighting samples use test denote empirical expectation integrable effect put exponential family empirical distribution manner efron see also besag monte carlo cutoff test test rejects smallest randomizes appropriately test bit involved similar principle solve appendix discuss solved efficiently fixed inverted obtain confidence interval monte carlo inference described computationally straightforward obtained generally could represent importance samples weights steps markov chain stationary distribution methods apply long still integrable numerical problems may arise solving far away reference parameter used sampling combining appropriately weighted samples several different reference values help keep effective sample size getting small references monte carlo inference see jockel forster mehta sampling gaussians affine quadratic constraints case gaussian several simplifications possible one many ways sample truncated multivariate gaussian distribution paper use gibbs sampling algorithms pakman paninski suggest another approach based hamiltonian monte carlo efficient sampling multivariate gaussian distributions constraints main algorithmic challenge gaussian selective tests proposed paper works cited use saturated model exclusively means require sampling many cases sampling problem may greatly facilitated refining selection variable use example lee propose conditioning variables selected lasso well signs fitted leading selection event consisting single polytope condition selected variables signs selection event union polytopes number variables selected model though polytopes might excluded conditioning refining selection variable never impairs selective validity procedure typically leads loss power however loss power may quite small example conditional law puts nearly mass realized polytope price power acceptable way obtain tractable test quantifying tradeoff computation power interesting topic work carrying selective necessary condition realized vector length adding quadratic equality constraint support deal sample instead ball project samples onto sphere using importance sampling scheme appendix gives details selective inference settings section describe tests two simple settings selective inference binomial problem tests involving scan statistic poisson process models generally address question selective inference generalized linear models selective clinical trial illustrate application approach simple setting discuss selective clinical trial binomial data experiment discussed similar adaptive design proposed sill sampson consider clinical trial candidate treatments heart disease give treatment patients corresponding placebo number patients treatment suffer heart attack trial ind binom log measures efficacy treatment likelihood exp exponential family sufficient statistics define let denote jth smallest order statistic observing data select best treatments construct confidence interval one odds ratio relative placebo ties select treatments could possibly select treatments simplicity assume treatments ones selected inference based conditional law selected law fixed remaining unknowns conditioning selection multinomial table control treatment heart attack heart attack margins fixed conditioning selection gives additional constraint side known conditioning rejecting conditionally extreme amounts selective fisher exact test aside constraint support distribution hypergeometric otherwise noncentral hypergeometric noncentrality parameter use family construct interval poisson scan statistic second simple example consider observing poisson process interval intensity possibly elevated unknown window poisson otherwise goal locate maximizing scan statistic test whether construct confidence interval assume always true example use likelihood ratio statistic proposed rivera walther density written exponential family form exp log exp event chosen carry inference respect note conditional uniform random sample condition event values uniform thus sample taking include uniformly random points rejecting samples selected window generalized linear models framework extends logistic regression poisson regression generalized linear model glm response design matrix since glm model may represented exponential family form exp result proceed case linear regression reduced model conditioning basing inference difficulty may arise logistic poisson regression due discreteness response distribution control variable continuous almost every realization configurations yield unique values case conditioning means conditioning information left inference best exact selective test trivial one contrast control variables discrete variables like gender ethnicity conditioning may constrain much approximately multivariate gaussian random variable promising approach may base inference asymptotic gaussian approximation taylor tibshirani simulation regression simple illustration compare selective inference linear regression lasso rows design matrix drawn equicorrelated multivariate gaussian distribution pairwise correlation variables columns normalized length simulate model entries set magnitude chosen data splitting half data yielded superset true variables roughly instances data splitting carving partitioned selection inference data sets containing data points respectively assume error variance known carry lasso lagrange parameter described negahban compare two inference procedures data splitting lasso use lasso select model use inference data carving lasso use lasso select model use whatever left inference data carving procedures use section addition condition signs active lasso coefficients procedure test proposed lee know theorem procedure strictly dominates procedure tradeoff procedures carven form selection event use lasso data points test statistic thus distinction conditionally independent lasso data points pscreen fdr power level algorithm table simulation results pscreen probability successfully selecting true variables power power conditional successful screening tests true variables data use selection better selected model quality cost power appears finding good tradeoff competing goals always outperforms predicted theorem pscreen fdr power level algorithm table simulation results misspecification errors drawn independently student conclusions identical table carven uses data selection therefore likely select superior model whereas procedure reserves power second stage let size model selected number noise variables included compare procedures respect aspects selection performance chance screening obtaining correct model pscreen expected number noise variables selected expected number true variables selected false discovery rate true variables selected max fdr conditional obtained correct model also compare aspects second stage performance probability correctly rejecting null one true variables power probability incorrectly rejecting null noise variable level results shown table bear intuition section procedure uses information first stage performs best terms model selection pays screening power probability screening power carving power splitting data carving data splitting data points used selection data points used selection probability successful screening power conditional screening probability successful screening times power conditional screening figure tradeoff power model selection increases data used first stage better chance successful screening picking true nonzero variables however increasing also leads reduced power second stage data splitting suffers much data carving though affected price lower power relative procedure clearly dominates expected increasing improves pscreen suffers drop power procedure seems strike better compromise figure shows tradeoff curve model selection success measured probability successful screening power conditional successful screening increases performance improves performance declines decline much slower data carving surprisingly much higher power respectively explain holding one two data points first stage improves power dramatically better understanding tradeoff interesting topic work finally check robustness data carving replace gaussian errors independent errors drawn student distribution five degrees freedom numbers barely change see table tian taylor rigorously analyze case errors conditioning device multiple inference point argued controlling selective type error goal right also serve device controlling traditional multiple inference goals section discuss two examples confidence intervals selected parameters control false rate fcr familywise error rate fwer suppose correspond parameters common fixed model interesting construct confidence adaptively designate number benjamini yekutieli propose controlling false interval rate fcr max number intervals constructed authors addressed inference selection proposing control fwer chance selected test incorrectly rejects null constructed confidence interval fails cover parameter example inference posi method berk constructs simultaneous confidence intervals parameters linear regression models ever consideration result matter choose model overall probability constructing interval controlled choosing appropriate selection variables control fcr fwer desired using intervals selective coverage proof generalizes extends result weinstein also use conditional control achieve fcr control specialized setting using similar proof also show using adaptive bonferroni rule adjusts test level based random number intervals actually constructed achieve fwer control proposition fcr fwer control via selective error control assume countable corresponding different parameter model let define enjoys coverage level given collection controls fcr level intervals max controls fwer enjoys coverage level given level proof let selective coverage hence repeat argument selective coverage obtain marginalizing bound gives result however converse proposition true fwer control general guarantee control relevant selective error rates example suppose construct interval effect red meat consumption heart disease probability effect statins heart disease otherwise selective error rates respectively overall fwer still controlled conservatism asking smoking compensate asking coffee perhaps readers primarily interested statins consistently misled readers primarily interested red meat consumption see unnecessarily conservative intervals averaging error rates across two questions two different interpretations seems inappropriate problematically different questions correspond different models example examine residuals decide poisson model model especially unintuitive focus error rates averaged across different choices model contrast different questions represent bag relatively anonymous priori undifferentiated hypotheses prioritizing research association study error rate like fdr likely better proxy scientific goals discussion selective inference concerns properties inference carried using procedure select questions ask recover frequency properties among answers selected questions would obtain classical setting follow guiding principle selective error control answer must valid given question asked happily living principle simple matter exponential family models including linear regression due rich classical theory optimal testing exponential family models even possibly selecting large menu diverse incompatible models still design tests one model time control selective error using test designed selected model generally pay price conditioning desirable condition little possible data carving dramatically improve data splitting using leftover information data set initially designated selection many challenges remain deriving cutoffs sample carving tests computationally difficult general addition entire development article takes model selection given reality choose work needed learn model procedure selection procedures lead favorable properties data sets research questions become complex less less hope specifying adequate statistical models ahead time key challenge complex research balance goal choosing realistic model goal inference chosen hope ideas article represent step right direction reproducibility git repository code generate figures file available first author website acknowledgements william fithian supported national science foundation vigre grant gerald lieberman fellowship dennis sun supported part stanford genome training program ric weiland graduate fellowship jonathan taylor supported part national science foundation grant air force office sponsored research grant would like thank stefan wager trevor hastie rob tibshirani brad efron yoav benjamini larry brown maxwell grazier sell subhabrata sen yuval benjamini helpful 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richard berk lawrence brown andreas buja kai zhang linda zhao valid inference annals statistics julian besag markov chain monte carlo statistical inference center statistics social sciences julian besag peter clifford generalized monte carlo significance tests biometrika lawrence brown contribution kiefer theory conditional confidence procedures annals statistics pages lawrence brown fundamentals statistical exponential families applications statistical decision theory lecture series pages brownie kiefer ideas conditional confidence simplest setting communications methods arthur cohen harold sackrowitz two stage conditionally unbiased estimators selected mean statistics probability letters cox note evaluation significance levels biometrika dawid selection paradoxes bayesian inference lecture series pages ruben dezeure peter lukas meier nicolai meinshausen inference confidence intervals hdi statistical science bradley efron tweedies formula selection bias journal american statistical association bradley efron robert tibshirani using specially designed exponential families density estimation annals statistics william fithian jonathan taylor robert tibshirani ryan tibshirani adaptive sequential model selection arxiv preprint jonathan forster john mcdonald peter smith monte carlo exact conditional tests logistic models journal royal statistical society series methodological pages annie franco neil malhotra gabor simonovits publication bias social sciences unlocking file drawer science andrew gelman eric loken garden forking paths multiple comparisons problem even fishing expedition research hypothesis posited ahead time downloaded january naftali harris visualizing lasso polytope geometry june url http trevor hastie robert tibshirani jerome friedman hastie friedman tibshirani elements statistical learning volume springer larry hedges estimation effect size nonrandom sampling effects censoring studies yielding statistically insignificant mean differences journal educational behavioral statistics larry hedges modeling publication selection effects statistical science pages kenneth hung william fithian rank verification exponential families arxiv preprint john ioannidis published research findings false plos medicine adel javanmard andrea montanari hypothesis testing regression gaussian random design model asymptotic theory ieee transactions information theory jockel finite sample properties asymptotic efficiency monte carlo tests annals statistics pages george johnson new truths one see new york times jack kiefer admissibility conditional confidence procedures annals statistics pages jack kiefer conditional confidence statements confidence estimators journal american statistical association jason lee jonathan taylor exact post model selection inference marginal screening advances neural information processing systems pages jason lee dennis sun yuekai sun jonathan taylor exact inference application lasso annals statistics hannes leeb benedikt model selection inference facts fiction econometric theory hannes leeb benedikt one estimate conditional distribution estimators annals statistics pages hannes leeb benedikt one estimate unconditional distribution estimators econometric theory lehmann joseph romano testing statistical hypotheses new york springer lehmann henry completeness similar regions unbiased estimation part indian journal statistics richard lockhart jonathan taylor ryan tibshirani robert tibshirani significance test lasso discussion annals statistics joshua loftus jonathan taylor significance test forward stepwise model selection arxiv preprint ted matthes donald truax tests composite hypotheses multivariate exponential family annals mathematical statistics pages cyrus mehta nitin patel pralay senchaudhuri efficient monte carlo methods conditional logistic regression journal american statistical association nicolai meinshausen lukas meier peter regression journal american statistical association sahand negahban pradeep ravikumar martin wainwright bin unified framework analysis decomposable regularizers statistical science november issn doi url http richard olshen conditional level ftest journal american statistical association terence orchard max woodbury missing information principle theory applications proceedings berkeley symposium mathematical statistics probability volume pages university california press berkeley ari pakman liam paninski exact hamiltonian monte carlo truncated multivariate gaussians journal computational graphical statistics camilo rivera guenther walther optimal detection jump intensity poisson process density likelihood ratio statistics scandinavian journal statistics rosenblatt yoav benjamini selective correlations voodoo neuroimage allan sampson michael sill design normal case biometrical journal michael sill allan sampson design binomial case computational statistics data analysis robert sladek ghislain rocheleau johan rung christian dina lishuang shen david serre 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models annals statistics larry wasserman kathryn roeder high dimensional variable selection annals statistics asaf weinstein william fithian yoav benjamini selection adjusted confidence intervals power determine sign journal american statistical association daniel yekutieli adjusted bayesian inference selected parameters journal royal statistical society series statistical methodology yong replication studies bad copy nature zhang stephanie zhang confidence intervals low dimensional parameters high dimensional linear models journal royal statistical society series statistical methodology hua zhong ross prentice estimators confidence intervals odds ratios association studies biostatistics sebastian jonathan pritchard overcoming winners curse estimating penetrance parameters data american journal human genetics proof proposition proof group let number true nulls selected let denote number false rejections znv znr need show lim znv design result znv znr every need show two sums far expectations var apply kolmogorov strong law large numbers independent sequence obtain eznr znr znr eznr eznr znv eznv znv znv eznv eznr words znr lim supn znv monte carlo tests confidence intervals details assume arises exponential family wish compute monte carlo umpu rejection region hypothesis let unif auxiliary randomization variable define dictionary ordering region implements rejection region test cutoffs boundary randomization parameters write correct cutoffs fixed decreasing increasing increasing let sequence random variables integrable would true valid sample importance sample come valid markov chain monte carlo algorithm defined analogously replaced importanceif satisfy weighted empirical versions pointwise monotonicity properties almost surely result almost sure convergence compacta sup max compact effect defining carry tests solving solve umpu tests exponential family approximating empirical measure specifically define inf set empty given lower cutoff define upper cutoff increasing obtain acceptance region possible function solve using binary search let denote rejection region obtained fact note paired analogous test whether right tail gives quick way carry test also allows quickly find upper lower confidence bounds approximating empirical family via binary search sampling selective details let denote set nonempty interior consider problem integrating integrable function uniform probability measure unit sphere dimension assuming intersection assume given sequence uniform samples unit ball let unif unif let use sequence importance samples weights since carry selective need sample pxm let pxm let fixed selection event let hyperplane intersected event would sample selective uniformly distributed assume resample uniformly sampling additional quadratic constraint sample ball radius intersected turn sample sphere via scheme outlined importance weight suffices turn sample selective conditioning set
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population synthesis via neighbor crossover kernel fujitsu laboratories kanagawa japan email recent development simulations brings need population synthesis task reconstructing entire population sampling survey limited size supplying initial conditions simulations begin paper presents new kernel density estimator task method analogue classical estimator employs novel techniques harness huge degree freedom required model nonlinearly correlated datasets crossover kernel neighbor restriction kernel construction set bagging kernels performance statistical estimator examined real synthetic datasets provide parameter selection rule method theory method works computational cost analysis demonstrate usefulness population synthesizer method applied household synthesis task urban ntroduction recent years increasing computational power enables conduct simulations highly public subjects urban planning transportation energy management disaster prevention welfare engineering carry simulations highly public subjects face difficulties collecting detailed survey data supply realistic initial conditions simulators example modern urban require disaggregate study area including every household residential place family structure car ownership income well person age gender job daily activities etc complete survey massive detailed information usually impracticable cost privacy reasons sampling survey available recover entire population obtained sample purpose iterative proportional fitting ipf extensions iterative proportional updating ipu widely used aforementioned areas however ipflike approach simply weights copies sample points synthesize population reproduce diversity original population might missed sampling survey moreover since approach accepts categorical variables numerical variables age income must roughly discretized usually two five one promising alternative statistical approach first estimates probability distribution sample points resamples population estimated distribution though easy model complicated distributions data annual income aug naoki hamada katsumi homma hiroyuki higuchi hideyuki kikuchi age fig age annual income washington state citizens american community survey note values high ages incomes survey order protect confidentiality survey respondents simulations figure shows scatter plot person age income real city commonly used attributes person agents simulations see even simplified example defies parametric models distribution part ages high income earners middle ages constitute long tail furthermore artifact lines arise due problem much harder data often consist variables variables nonlinearly correlated together parametric modeling formidable task even domain experts kernel density estimation shown possess excellent flexibility fit various complex distributions spaces fitting method select bandwidth parameters kernel functions minimize measure high dimensions fitting strategy becomes ineffective sample point requires different bandwidth matrix representing local correlation varies locations nonlinear manner due large number parameters model fitting procedure involves nonlinear optimization requires huge computational efforts limits applicable dimensionality shape data distribution recently new kind kernels density estimation crossover kernel proposed employs crossover operator genetic algorithm kernel function kernel density estimation crossover kernel require bandwidth parameters specify shape function instead uses subset sample points called kernel construction set kcs hence potential obviate costly fitting procedures quickly resample data unfortunately merely two studies choose kcss one appeared sakuma kobayashi original work proposed algorithm calculate sample weights describe probability sample point belongs kcs another kimura matsumura pointed method fit distributions matter weights optimized proposed optimizing choice kcss rather weights methods rely optimization thus involve heavy fitting procedures aim paper develop optimizationfree crossover kernel fast accurate resampling introducing neighbor restriction kcs choice show simple idea works surprisingly well problem certain theoretical background contributions summarized follows proposed neighbor restriction kcs choice showed method faster also accurate conventional gaussian kernels crossover kernel complicated datasets dimensions examined parameter sensitivity method gave rule thumb choosing neighborhood size kcs size without optimization found setting decreases generalization error experiments showed rationale viewpoint bagging demonstrated simulation accuracy urbansim enhanced method supplies initial households instead ipu roblem efinition otations rganization paper throughout paper following assumptions made sample size given taken unknown population population set points unknown continuous density going estimate say resample points recover population problem population synthesis discussed two different situations case unbiased sample sample uniformly drawn population goal thus estimate density identical subsequent three sections paper discuss case section iii reviews existing kernel density estimators crossover kernels section presents proposal properties section shows effectiveness proposed method comparing conventional kernel density estimators benchmark datasets mechanisms behind method discussed case biased sample marginal frequencies applications samples available often biased rather uniform case requires extra information bias introduced assume population binned frequencies given binning applied marginal distributions variables incomplete recover entire joint distribution thus case correct sampling bias frequencies extract variable correlation sample combining estimate entire joint density section tackle application type present bias correction technique incorporated resampling method developed section technique applied task generating initial population households urban iii onventional ethods focusing case section reviews existing kernel density estimators crossover kernels kernel density estimation kernel density estimators classified two versions sample point estimator balloon estimator resampling purpose estimate must density easy sample sample point estimator satisfies requirements kernel density easy sample balloon estimator usually density even kernel thus concentrate former following form kernel function specified bandwidth matrix symmetric matrix parameters estimating paper consider multivariate gaussian kernel det exp measure mean integrated squared error mise commonly decomposition given mise squared bias variance various measures based divergence hellinger distance used purpose case problem point section still arises case fixed matrix bandwidths bandwidth selectors using plugin developed however comes variable bandwidths existing techniques restrictive bmp estimator classical one uses scalar bandwidths determines constant multiple euclidean distance kth nearest sample point fixed asymptotically equivalent choosing abramson proposed square root law choosing regardless showed reduces bias accelerates rate convergence handle matrix bandwidths sain studied binning technique decreases number parameters sharing bandwidth matrix sample points bin simulation results clarified even moderate sample size unbiased crossvalidation tends yield small namely overfit bandwidth matrices since unbinned variable bandwidth matrix parameters situation much worse higher dimensions developing bandwidth selectors case currently open question another problem high dimensions relative contribution variance dominates bias mise sain pointed fixed scalar bandwidth ratio order bias order variance therefore dimensionality increases bias reduction techniques get less importance variance reduction techniques including bagging become core estimator crossover kernels genetic algorithm literature kita argued desirable behavior crossover operators suggested guideline preservation statistics tells crossover parents children mean covariance specifically suppose parents children realizations random variate respectively following satisfied sakuma kobayashi pointed parents realizations implicitly determine probability density function thus considered kernel function context set parents called kernel construction set kcs one chooses crossover satisfies gaussian distribution crossover coincides maximum likelihood estimator mle calculated kcs sakuma kobayashi also noticed undxm fulfills conditions kcs size set remaining question construct entire density estimate crossover kernels sakuma kobayashi used gaussian mixture model final estimate parameters optimized modified algorithm gaussian component tth step replaced average crossover kernels xil kcss xil chosen random probability proportional weights calculated step describe possibility ith component generates jth sample point superiority algorithm original one shown simulation studies kimura matsumura proposed direct way giving entire estimate chosen maximize log starting random choice kcss randomly remade one time improves loglikelihood showed high flexibility approach applying circularly distributed dataset earest eighbor rossover ernel seen bmp estimator fast restricted scalar bandwidth whereas crossover kernel slow able handle covariance matrix idea combine strengths estimators rex kernel mentioned section coincides gaussian mle kcs size order extend property arbitrary size employ rex kernel function rex generates resampling point kcs follows random number drawn univariate gaussian distribution following sakuma kobayashi derivation kernel function get probability density function namely rex kernel det exp regardless thus always gaussian mle calculated comparing rex kernel gaussian kernel see covariance matrix corresponding bandwidth matrix resampling gaussian kernel explicit mle costs time rex kernel resampler bypasses since random number generation relies cholesky decomposition requires time calculate time cholesky decomposition algorithm rex kernel sample proposed bmp fig example local correlation two typical proposed method shown solid ellipsoids corresponding bmp estimator drawn dashed circles sample points dots underlying density gray scale pdf tion process enables generate sample point time neighbor restriction kcs choice order adapt kcss data without optimization choose kcss neighbor sample point simple yet powerful trick realize crossover kernel analogue bmp estimator shown appendix covariance matrix points sample point asymptotically form sufficiently large expect thus first term overwhelms second term means approximates scalar bandwidth bmp bandwidth asymptotic behavior strength weakness one hand method inherits rich asymptotics bmp estimator including normality consistency leading term rate convergence etc hand matrix extension bandwidth gives advantage bmp estimator asymptopia second term influential nonlinear correlation presents notice function second term captures local correlation around taken account bmp estimator easy check matrix always rank one unless gradient vanishes unique nonzero eigenvalue equals corresponding eigenvector parallel dividing thought relativization effect thus subtracting matrix bmp bandwidth makes bandwidth shrink gradient direction figure illustrates effect captures local correlation example gradient everywhere normal input sample training data neighborhood size kcs size population size output synthetic population calculate say end repeat draw random draw random generate return circumference circular density becomes large near circumference thus bandwidth shrinkage occur normal directions tangent directions respect density manifold resulting better fit shown fig accordingly matrix extension bandwidth gain improvement finite sample size another advantage method bmp estimator finite sample size bmp bandwidth depends kth nearest sample point points ignored bandwidth determined points make bandwidth computation stable bagging kernels even mle stabler bmp distance severe variability samples still fluctuate resulting kernel order exploit sample information variance reduction employ bagging kernels kcs subset change kcs every time resampling point generated resulting entire estimate becomes follows denotes expectation possible choices kcs size course compute value since mixture many gaussians easily draw sample simply choosing points random passing end section presenting resampling algorithm via rex kernel algorithm xperiments order evaluate proposed method conduct numerical experiments benchmark datasets population reproducibility parameter sensitivity bagging effect cpu time studied table name pums person swiss roll road network letter recognition escription datasets dimension sample size url methods parameter settings rex kernel algorithm used proposed method parameters set follows neighborhood size kcs size rex kernel crossover kernel combined model rex kernel parameters set follows number kcss kcs size optimization kcs choice ran consecutive iterations increase bmp gaussian kernel used see performance progress owing bandwidth matrization bagging parameters set follows neighborhood size constant multiplier fixed gaussian kernel used conventional parameter bandwidth set datasets examine effect dimensionality sample size four datasets shown table used every dataset affinely transformed whitening mean covariance zero vector identity matrix respectively datasets detailed descriptions available online consult references url column table criterion order evaluate accuracy population reproduction measured hellinger distance test data synthesized population dataset evenly divided used inverted crossvalidation icv one subset used training data remaining subsets constituted test data since compute value rex kernel density stated section binned version hellinger distance used reconstructed population set test data performance measure written set bins resp subset resp whose elements fall bin virtually exactly identical using fixed matrix bandwidth datasets normalized affine transformation datasets divided remainders randomly dropped set cardinality holds method generates better accuracy method generates setting icv done average standard deviation hellinger distances calculated results table shows hellinger distances test data points population points synthesized method training data hellinger distances training data also displayed baseline note values attainable simply copying bootstrapping training data every method much better reproduced population baseline datasets except rex kernel road network sense kernel density estimation less generally preferable copying bootstrapping approaches purpose population synthesis especially sample small complex among others knn rex kernel gained significant improvements cases visual comparison provide scatter plots synthetic populations appendix surprising fact found comparison standard deviations rex kernel standard deviations less rex kernel low three cases remember model matrix bandwidth extension bmp model uses lot kernels model thus seems natural expect standard deviations greatest values paradoxical result implies success variance reduction techniques holds good even letter recognition small sample size case discussions parameter sensitivity figure gives survey results rex kernel filled area plot indicates parameters whose results surpass best performance fixed gaussian kernel every filled area horizontally long part means performance rex kernel insensitive neighborhood size observation consistent sensitivity analysis bmp estimator studied breiman figure also clarifies choice kcs size critical performance practitioners find suitable pums person optimal value agrees common practice genetic algorithm search space dimensions crossover use parents swiss roll road network letter recognition optimal respectively looks inconsistent intrinsic dimensions considered rule found swiss roll spiraling band distribution road network data observed along roads intrinsic dimension therefore dimensionality letter recognition reduced nonlinear dimensionality reduction methods remind practice multiobjective table ellinger distance average standard deviation fold icv best parameter settings row bolded average smaller one statistical significance elch test rex kernel dataset dimensions training points pums person swiss roll road network letter recognition rex kernel bmp gaussian kernel pums person fixed gaussian kernel training data baseline swiss roll kcs size hellinger distance neighborhood size road network neighborhood size letter recognition kcs size hellinger distance kcs size neighborhood size hellinger distance kcs size hellinger distance neighborhood size fig parameter hellinger distance rex kernel hellinger distance averaged icv legend means follows hellinger distances filled squares better best tuned gaussian kernel hellinger distance parameter settings achieve better distances legend read similarly genetic algorithm pareto solution set forms dimensional submanifold search space using parents exhibits better performance consequence suggest parameter selection rule determines following steps choose identify intrinsic dimensionality dataset priori knowledge dimensionality reduction set near possible effect bagging bagging different breiman original proposal points given sample size breiman bagging makes particular replicates size random sampling replacement bagging aims implicitly achieve similar effect changing kcs every time corresponds taking possible replicates size random sampling without replacement despite differences fig shows get better results users set much greater case many possibilities kcs choice arise averaging effect bagging would enhanced therefore gain much variance reduction also decrease hellinger algorithm rex kernel bias correction distance computational cost analyze time complexity rex kernel examine algorithm lines compute points dimension implemented naive time algorithm calculates euclidean distances among points lines compute points generated time therefore overall complexity dlm similar way also get complexity bmp estimator difference second term related resampling usually small intrinsic dimensionality plus one thus additional computation cost extensions significant cases actual computation time shown table iii measured running single threaded program server machine intel xeon ghz ram method time includes reading dataset file computing density synthesizing population writing file one particular parameter setting excludes selecting parameters calculating hellinger distance table shows every dataset time knn rex kernel comparable bmp gaussian kernel result supports theoretical comparison time complexities method approximately four times faster rex kernel cases implies kcs choice restriction certain advantage optimization terms speed table also shows fixed gaussian kernel much faster method however result excluding time bandwidth selection practice method parameter selection rule presented section enable find good parameter setting within several parameter examinations methods need kind grid search typically surveying tens hundreds parameter settings fairer comparison time parameter selection taken consideration method would fastest bottleneck method bmp estimator computation complexities imply fact road network letter recognition consumes overall time one would like accelerate variety nearest neighbor search techniques available pplication rban icro imulation last section seen kernel density estimation generally enjoys significant improvements bootstrapping approaches given unbiased sample case proceed case biased sample marginal frequencies interest section whether yet beneficial simulations urbansim urbansim one advanced urban microsimulators simulates location choice business units households development pricing land real estate govermental policies involved study area investigate change years parcel level several applications cities honolulu hawaii eugenespringfield oregon detroit michigan salt lake city utah input pums household microdata household marginal frequencies neighborhood size kcs size number households output synthetic households calculate say end repeat find vacant bin argmaxb draw random else generate sampling uniform distribution calculate end draw random generate remove point random end return seattle washington europe paris france brussels belgium zurich switzerland urbansim requires data describe detailed attributes every household study area part initial condition simulation united states census bureau publishes annual survey results sample households living area public use microdata sample pums marginal distribution attribute households published census summary files combining one generate households run urbansim popgen supply households urbansim popgen available software implements iterative proportional updating ipu weights sample point synthesize population weights determined population marginal distributions match given marginals rex kernel bias correction apply rex kernel task need little modification outcome matches marginals algorithm version algorithm denotes frequency bin denotes subset whose elements bin since urbansim requires integer attributes round attributes real numbers generated algorithm experimental settings used pums sample households seattle extracted six attributes age head annual income number people number kids race number workers popgen ipu method generated households number households seattle table iii cpu time seconds average standard deviation fold icv old fonts mean table dataset dimensions training points pums person swiss roll road network letter recognition rex kernel table ellinger distance synthesized households pums average standard deviation trials old fonts mean table year urbansim urbansim rex kernel rex kernel popgen ipu households generated method ran urbansim evaluated hellinger distance simulated households actual ones pums popgen used baseline note settings involve person attribute used urbansim kept settings initial households default values demo project file seattle parcel used method according parameter selection rule section results discussions result shown table simulation rex kernel households closer pums sample ipu tendency remains simulation resulting better prediction guess diversity household attributes better reproduced method ipu simply copies given sample synthesize population never includes households exist city present pums sample contrast rex kernel estimates density underlying population resamples probabilistic manner generate households present given sample vii onclusions paper proposed neighbor crossover kernel population synthesis showed method outperforms accuracy speed conventional fixed kernel variable kernel crossover kernel gave parameter selection rule rationale method demonstrated usefulness method household synthesis task urban simulation future work plan apply proposed method tasks reveal generality method especially oversampling imbalanced data would fruitful application another direction interchange knowledge kernel density estimation evaluation sampling bias pums corrected using pums weights bmp gaussian kernel fixed gaussian kernel genetic algorithm theories developed former community may help understand genetic algorithms work algorithms developed latter community provide calculate estimators theoretically complex derive ppendix erivation quation loftsgaarden quesenberry showed consistency analysis density estimator chosen sequence positive integers ball tends contain arbitrarily many points volume converges zero probability increases analysis motivated result however simplify situation evaluate analytic covariance matrix instead sample one random variate ball whose radius deterministically goes zero thus rigorous proof still gives insight asymptotic behavior sample covariance points rex kernel assume density continuously differentiable value point consideration consider taylor series approximation density around point converges since covariance matrix translation invariant fix without loss generality integration radius centered origin written cos sin cos sin cos sin cos sin sin calculate integral useful formulas definite integrals trigonometric functions nonnegative integers even sin cos odd even sinp cosq otherwise beta function gamma function satisfies integer max using formulas property odd function get following equations ydy suppose random variate density mean ydy ydy ydy ydy mean finally covariance volume unit ball matrix tensor let consider density restricted ydy ydy result implies rex kernel generally generic crossover kernels crossovers compliant kita preservation statistics order bandwidth bmp estimator especially choose kcs size rex kernel constant multiplier bmp estimator asymptotically coincides consequently rex kernel considered matrix bandwidth extension bmp estimator ppendix isual omparison figure shows distribution synthetic populations pums person plots intuitively support correctness numerical evaluation presented table rex kernel annual income annual income age age bmp gaussian kernel annual income annual income fixed gaussian kernel age age training data annual income annual income rex kernel test data age age fig synthetic populations training data test data pums person eferences waddell urbansim modeling urban development land use transportation environmental planning journal american planning association vol milkovits huang antoniou lopes dynamit next generation dynamic traffic assignment system proceedings second international conference advances system simulation simul vytelingum ramchurn voice rogers jennings trading agents smart electricity grid proceedings international conference autonomous agents multiagent systems aamas massaguer balasubramanian mehrotra venkatasubramanian simulation disaster response workshop agent technology disaster management atdm autonomous agents multiagent systems aamas endriss maudet welfare engineering multiagent systems engineering societies agents world ser lecture notes computer science lncs springer berlin heidelberg vol deming stephan least squares adjustment sampled frequency table expected marginal totals known annals mathematical statistics vol konduri pendyala sana waddell methodology match distributions household person attributes generation synthetic populations annual meeting transportation research board public use microdata samples pums online available http sain multivariate locally adaptive density estimation computational statistics data analysis vol sakuma kobayashi crossover role kernel density estimation proceedings annual conference genetic evolutionary computation gecco new york usa acm kimura matsumura density estimation using crossover kernels application genetic algorithm proceedings ieee congress evolutionary computation cec breiman bagging predictors machine learning vol jones variable kernel density estimates variable kernel density estimates australian journal statistics vol terrell scott variable kernel density estimation ann vol simonoff smoothing methods statistics ser springer series statistics springer duong hazelton bandwidth matrices multivariate kernel density estimation scandinavian journal statistics vol duong multivariate bandwidth selection unconstrained pilot bandwidth matrices test vol breiman meisel purcell variable kernel estimates multivariate densities technometrics vol abramson bandwidth variation kernel square root law annals statistics vol kita ono kobayashi extension unimodal normal distribution crossover genetic algorithms proceedings congress evolutionary computation cec vol kobayashi frontiers genetic algorithms journal japanese society artificial intelligence vol silverman density estimation statistics data analysis ser chapman monographs statistics applied probability london taylor francis vol fukunaga introduction statistical pattern recognition ser electrical science new york usa academic press gerber cems conditional expectation manifolds package version online available http bache lichman uci machine learning repository online available http bhattacharyya measure divergence two statistical populations defined probability distributions bulletin calcutta mathematical society vol zhang wang manifold learning applications recognition intelligent multimedia processing soft computing ser studies fuzziness soft computing tan yap wang eds springer berlin heidelberg vol hamada sakuma kobayashi ono functionalspecialization genetic algorithm fsmoga parallel problem solving nature ppsn ser lecture notes computer science lncs berlin heidelberg vol konduri popgen population generator online available http loftsgaarden quesenberry nonparametric estimate multivariate density function annals mathematical statistics vol
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towards reversible francesco tiezzi nobuko yoshida imt institute advanced studies lucca italy imperial college london work incorporate reversibility structured programming allow parties session automatically undo rollback fashion effect previously executed interactions permits taking different computation paths along session well reverting whole session starting new one aim define theoretical basis examining interplay concurrent systems reversible computation interaction thus enrich variant memory devices dedicated keep track computation history sessions order reverse discuss initial investigation concerning definition session type discipline proposed reversible calculus practical advantages static verification safe composition distributed software performing reversible computations introduction reversible computing aims providing computational model besides standard forward executions also permits backward execution steps order undo effect previously performed forward computations reversibility key ingredient different application domains since many years recently also design reliable concurrent systems permits understanding existing patterns programming reliable systems compensations checkpointing transactions possibly improving developing new ones promising line research topic advocates reversible variants process calculi ccs formalisms studying reversibility mechanisms concurrent systems work incorporates reversibility variant equipped session primitives supporting structured programming binary session consists series reciprocal interactions two parties possibly branching recursion interactions session performed via dedicated private channel generated initiating session session primitives come together session type discipline offering simple static checking framework guarantee correctness communication patterns practically combining reversibility sessions paves way development distributed software intrinsically capable performing reversible computations way without coding effort application programmer interaction among session parties relaxed computation automatically back thus allowing take different paths current one satisfactory application example used paper illustrating approach consider simple scenario involving client multiple providers offering service video streaming client connects provider request given service specifying title movie video quality provider replies quote determined according requested quality service servers status current load available bandwidth work partially supported cost action betty project ascens italian miur prin project cina alastair donaldson vasco vasconcelos eds proceedings workshop programming language approaches concurrency software places eptcs tiezzi yoshida work licensed creative commons attribution license towards reversible sessions client either accept negotiate reject quote problem occurs interaction client provider computation reverted order allow client automatically start new session possibly another provider proposed reversible calculus relies memories store information interactions effects system otherwise would lost forward computations data used enable backward computations revert effects corresponding forward ones memory devoted record data concerning single event correspond taking place communication action choice thread forking memories connected order keep track computation history using unique thread identifiers links like formalisms reversible computing concurrent settings forward computations undone fashion backtracking necessarily follow exact order forward computations reverse independent actions undone different order resulting formalism offers theoretical basis examining interplay reversible computations structured interactions notice reversibility enables session parties partially undo interactions performed along current session also automatically undo whole session restart possibly involving different parties advantage reversible approach behaviour realised without explicitly implementing loops hand session type discipline affects reversibility forces concurrent interactions follow structured communication patterns fact linearizing behaviours sessions reduces effect causal consistency concurrent interactions along session forbidden hence rollback along session follows single path however interactions along different sessions still concurrent therefore reverted usual fashion notably interesting issues concerning reversibility session types still open questions especially concerns validity reversible setting standard properties progress enforcement possibly new properties reversibility ongoing session history irreversible closure sessions related work review closely related works concern definition reversible process calculi refer detailed review reversible calculi reversible ccs rccs first proposal reversible calculus subsequent works drew inspiration currently running thread associated individual memory stack keeping track past actions well forks synchronisations information pushed memory stacks upon forward transition used memories also serve naming scheme yield unique identifiers threads process divides two inherits father memory together fork number indicating two sons thread drawback approach parallel operator satisfy usual structural congruence rules commutativity associativity nil process neutral element another reversible variant ccs mainly aims formalising biological systems like rccs relies memory stacks storing data needed backtracking also includes events corresponding unfolding process definitions differently rccs specific identifiers used label threads different approach used dealing forking ccs communication keys ccsk reversible process calculus obtained applying general procedure produce reversible calculi relevant aspect approach rely memories supporting backtracking idea maintain structure processes fixed tiezzi yoshida throughout computations thus avoiding consume guards alternative choices properly revert synchronisations two threads agree key uniquely identifying communication reversible variant borrows rccs use memories keeping track past actions although stacks syntactically associated threads parallel terms one dedicated single communication connection memories threads kept resorting identifiers resemble ccsk keys fork handling based structured tags used connect identifier thread identifiers approach reversibility applied distributed language another reversible variant similarly rccs calculus relies memory stacks recording communication events forking differently considers standard without choice replication host calculus semantics defined terms labelled transition relation finally reversible structures simple computational calculus modelling chemical systems reversible structures exploit memories maintains structure terms uses special symbol indicate next forward backward operations term perform work mainly take inspiration approach fact approaches based ccs directly applied calculus moreover approach preferable one former proposes reduction semantics interested latter proposes labelled semantics would complicate theoretical framework order properly deal scope extrusion reversible section introduce reversible extension enriched primitives managing binary sessions structured interactions two parties call reversible formalism due lack space technical details semantics results omitted refer interested reader complete account approach keep track computation history follows tag processes unique identifiers tagged processes called threads use memories store information needed reverse single forward reduction thus history reduction sequence stored number small memories connected using tags links way terms perform besides forward computations denoted also backward computations denoted undo effect former ones fashion processes expressions given grammars figure synchronisation shared channel processes initiates session along fresh session channel channel consists pair dual endpoints denoted one dedicated one party exchange values endpoints replace variables means substitution application order used respectively later communications primitives denote output input via session endpoints identified respectively communication primitives realise standard synchronous message passing messages result expressions evaluation may contain endpoints delegation constructs denote label selection branching pairwise distinct via respectively interaction primitives combined conditional choice parallel composition restriction recursion inaction towards reversible sessions shared channels labels channels session channels process variables names session endpoints tags session ids variables shared ids processes else expressions values true false processes nil memories khei figure syntax processes built upon processes labelling tags uniquely identify threads uniqueness tags ensured using restriction operator considering reachable terms def moreover extends three kinds memories action memory stores action event together tag active party action tag passive party tags new threads activated corresponding reduction action event records information necessary revert kind interactions either session initiation communication along established session khei branch selection choice memory stores choice event together tag conditional choice new activated thread event records evaluated expression processes respectively fork memory stores tag splitting thread form tags new activated threads memories analogous connectors threads memories composed parallel composition restriction operators processes allowed syntax semantically meaningful general term history stored memories may consistent due use tags broken connections continuation tags within memories corresponding threads example given choice memory broken connection thread tagged exists process memory form exists thus ensure history consistency consider reachable processes processes obtained means forward backward reductions processes unique tags memory def reachable processes set reachable processes closure see set terms whose threads distinct tags generated nil semantics operational semantics given terms reduction relation given union forward backward reduction relations report way examples forward backward rules session initiation require fresh forward rule tiezzi yoshida two parallel threads synchronise establish new session two fresh tags created uniquely identify continuations moreover relevant information stored action memory tag initiator thread executing prefix form tag thread executing dual action tags continuations shared channel used synchronisation replaced variables generated session channel processes substitutions applied information exploited backward rule revert reduction particular corresponding backward reduction triggered coexistence memory described two threads tagged within scope session channel tags generated forward reduction fact removed backward one considering reachable processes due tag uniqueness processes coincide indeed registered memory latter processes tagged forward reduction therefore fact two threads tagged parallel memory ensures actions possibly executed two continuations activated forward computation undone hence safely undone forward computation multiple providers scenario scenario involving client two providers informally introduced section rendered pclient client process pclient alogin hsrv reqi yquote accept yquote lacc pacc else negotiate yquote lneg pneg else lre pprovider alogin zreq hquotei zreq lacc qacc lneg qneg lre client contacts first provider accepts proposed quote system evolves pacc qacc srv memories keep track computation history problem occurs subsequent interactions computation reverted allow client start new session possibly another provider pclient properties show enjoys standard properties reversible calculi first demonstrate conservative extension fact reversible calculi decoration host calculus decoration erased means forgetful map mapping terms ones removing memories tag annotations tag restrictions following lemmas show forward reduction process corresponds reduction corresponding process vice versa lemma let two processes lemma let two processes process exists process show backward reductions inverse forward ones vice versa towards reversible sessions lemma loop lemma let two reachable processes conclude causal consistency result stating two sequences reductions called traces ranged initial state coinitial equivalent standard notion causal equivalence lead final state cofinal thus case rollback initial state reversing two traces theorem let coinitial traces cofinal discussion type discipline question answered defining static type discipline reversible calculus type check processes stored memories question arises fact able determine process case answer yes otherwise typability would preserved reduction subject reduction would satisfied indeed easy define process see containing memory even consistent triggers backward reduction leading untypable term type system defined host calculus one could wonder possible type processes way separately type checking term resulting application single memory using type system memory would check term triggered forward reduction generating memory general approach work term stored memory type checked isolation without taking account context example consider memory corresponding communication along session typable typing int int parallel term resulting corresponding backward reduction typable typings composable indeed defined memory context considered extending type system rules permits typing processes stored memories ignoring tag annotations tag restrictions see definition type system way type checking typings memories threads must composed means rule parallel composition thus process mentioned rightly untypable type system properly works simplified setting permits avoiding deal dependencies among memories threads outside memories could cause unwanted conflicts type checking specifically consider class processes extending def obtained means forward backward reductions processes unique tags memory session initialised conditional choices recursions delegation characteristic processes memory inside process exists within process ancestor memory corresponding initialisation considered session type system checks latter kind memories significantly simplifies theory coming back multiple providers scenario verify initial process welltyped particular channel alogin typed shared channel type request quote lacc lneg lre end sorts request quote used type requests quotes respectively let consider scenario client wills concurrently submit two different requests provider would concurrently serve consider particular following specification client alogin hsrv req hsrv req tiezzi yoshida new specification clearly due use parallel threads within session permits avoiding mixing messages related different requests wrongly delivering order properly concurrently submit separate requests client must instantiate separate sessions provider one request concluding remarks bring benefits reversible computing structured programming defined theoretical framework based used formal basis studying interplay reversibility structured interaction type discipline still subject study fact type system mentioned section completely satisfactory use limited restricted class processes consider broader class appropriate static type checking approach memories devised memory would check term composed threads stored memory context composed threads generated execution memory threads concerning reversible calculus plan investigate definition syntactic characterisation consistent terms statically enforces history consistency memories worth noticing calculus fully reversible backward computations always enabled full reversibility provides theoretical foundations studying reversibility suitable practical use line plan enrich language mechanisms control reversibility moreover intend enrich framework irreversible action committing closure sessions way computation would backward forward allowing parties try different interactions session successfully completed instance process pacc example could terminate performing irreversible action commit synchronise action commit qacc differently sessions terminated session terminated commit synchronisation unbacktrackable irreversibility due fact backward rule defined revert interaction type theory tailored properly deal kind session closure goals intend apply proposed approach formalisms consider asynchronous sessions multiparty sessions moreover plan investigate implementation issues may arise incorporating approach standard programming languages particular case distributed setting references luca cardelli cosimo laneve reversible structures cmsb acm cristescu krivine varacca compositional semantics reversible lics ieee vincent danos jean krivine reversible communicating systems concur lncs springer vincent danos jean krivine formal molecular biology done electr notes theor comput sci giachino lanese mezzina tiezzi reversibility language technical report http towards reversible sessions lanese mezzina schmitt stefani controlling reversibility concur lncs springer lanese mezzina stefani reversing concur lncs springer iain phillips irek ulidowski reversing algebraic process calculi log algebr program francesco tiezzi nobuko yoshida towards reversible sessions technical report http yoshida vasconcelos language primitives type discipline structured programming revisited two systems session communication electr notes theor comput sci
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jan equation left nilpotent left braces ferran tatiana agata smoktunowicz abstract study involutive solutions equation call simply solutions show structure group finite solution engel group known structure group finite multipermutation solution group thus result gives rich source examples braided groups left braces groups engel groups also show finite solution equation embedded convenient way finite brace finite braided group left brace explore close relation multipermutation level solution associated radical chain introduced rump introduction braces introduced rump study involutive settheoretic solutions equation recall left brace set two operations abelian group group every right braces defined similarly changing property brace left brace also right brace left brace one defines another operation rule known brace jacobson radical ring conversely jacobson radical ring one date january mathematics subject classification primary key words phrases brace equation nilpotent braces smoktunowicz defines new operation called adjoint group radical ring brace hence study braces equivalent study jacobson radical rings general operation left brace associative left distributive respect sum let left brace let map defined known automorphism additive group left brace map aut defined morphism groups kernel morphism called socle soc fact socle left brace ideal normal subgroup multiplicative group invariant maps particular soc also subgroup additive group note soc soc therefore quotient multiplicative group soc also quotient additive group soc left brace left brace quotient modulo ideal soc let set recall map solution equation maps idx idx write map every maps bijective involutive idx convention solution ybe shortly solution mean nondegenerate involutive solution equation let solution ybe structure group group presentation authors call group nilpotent braces equation follows results naturally embedded one define sum free abelian group basis moreover left brace see say canonical left brace structure known group acts set left right moreover assignments extend group homomorphism symx image homomorphism subgroup symx called permutation group denoted known hlx group epimorphism kernel ker soc sets thus inherits structure left brace via natural isomorphism groups say canonical structure left brace moreover soc symmetric groups involutive braided groups left braces paper prove general results braces apply study close relations properties solutions associated left braces spirit results results section motivated result assures structure group solution yang baxter brace let left brace usual positive integer denote denote appears times lemma let left brace whose additive group assume integer occurs times equation assume moreover lan integer equivalently proof note let positive integer let proved induction smoktunowicz since lan thus hence nek positive integer suppose let smallest positive integer nen since contradiction definition let group following notation denote element recall group engel group exists positive integer occurs times theorem let left brace additive group torsionfree soc multiplicative group left brace engel group trivial brace proof let soc note hence nilpotent braces equation hence therefore let since soc soc ideal soc since engel group exists positive integer occurs times hence appears times torsion free hence occurs times lemma therefore equivalently trivial left brace call left brace left nilpotent chain introduced rump consequence lemma theorem following two results theorem let finite solution ybe assume positive integer equality holds occurs times occurs equality trivial solution particular left nilpotent left brace trivial solution proof since soc subgroup symmetric group symx finite set soc hence lemma particular trivial solution known ordered group abelian see example section also known nilpotent group ordered see lemma recall finite solution ybe solvable group see therefore nilpotent abelian case canonical left brace structure trivial trivial solution following related result theorem let finite solution ybe structure group engel group trivial solution smoktunowicz proof consequence theorem right nilpotent left braces etingof schedler soloviev introduced retract solution given solution ybe let solution ybe retract relation set respect defined retraction ret denotes define ret retk ret solution ybe called multipermutation solution level smallest nonnegative integer solution retm cardinality case write mpl let left brace mean chain ideals introduced rump say right nilpotent exists positive integer recall left brace map defined solution ybe solution ybe associated left brace see proposition let nonzero left brace let associated solution ybe multipermutation level proof note soc every first shall prove implication mpl use induction mpl suppose mpl therefore equivalent follows nonzero left brace hence gives base induction suppose condition mpl implies assume mpl retraction ret multipermutation level moreover isomorphism left braces equivalently isomorphism braided groups soc ret isomorphic solution ybe associated soc hence inductive nilpotent braces equation assumption soc soc implies soc subset soc therefore prove inverse implication mpl base induction clear assume implication true suppose left brace recall therefore soc hand imply soc inductive assumption mpl ret therefore mpl proves proposition embedding solutions groups finite braces finite rings section show finite solution ybe embedded explicit way finite left brace recall shown canonical correspondence left braces symmetric groups sense takeuchi therefore proposition also shows explicitly embed finite solution ybe finite symmetric group proposition let finite solution ybe finite left brace generates additive group moreover subsolution solution associated canonically left brace restriction proof let structure group finite solution know additive group left brace associated free abelian basis since finite soc say soc consider set claim ideal brace normal subgroup multiplicative group invariant respect left actions elements clear additive subgroup soc one subgroup let hng thus normal subgroup also invariant left action elements therefore ideal left brace difficult smoktunowicz show brace quotient finite left brace order cardinality observe also two elements one since additive group free abelian basis restriction natural map set injective proposition proved conference porto cesareo amberg mentioned collaborators first became interested jacobson radical rings gave way construct examples groups later found ways constructing examples triply factorized groups also used define braces see theorem interesting results triply factorized groups found triply factorized groups example useful investigating structure normal subgroups group product two subgroups several authors investigated connections triply factorized groups nearrings might interesting investigate connections nearrings braces would like pose related open question question investigate whether relation nearrings solutions ybe multiplicative group brace also called adjoint group brace observe corollary asserts every finite solvable group subgroup adjoint group left brace also make following simple remark follows lemma corollary remark related corollary lemma every finite nilpotent group subgroup adjoint group finite nilpotent ring let prime lemma every finite isomorphic subgroup adjoint group finite nilpotent ring cardinality power let finite nilpotent group let distinct prime divisors order let sylow isomorphic subgroup adjoint group finite nilpotent clear isomorphic subgroup ring since adjoint group finite nilpotent ring ring adjoint semigroup defined following technique proof lemma get following result nilpotent braces equation proposition let group let ring unit isomorphic subgroup adjoint semigroup group ring proof let map defined clearly injective let therefore injective homomorphism semigroups adjoint semigroup acknowledgments research first author partially supported grants dgi miciin mineco research second author partially supported grant computational combinatorial methods algebra applications bulgarian national science fund abdus salam international centre theoretical physics ictp trieste institute mathematics bonn research third author supported erc grant references amberg triply factorized groups groups andrews vol london mathematical society lecture note series cambridge university press franciosi giovanni triply factorized groups communications algebra volume issue amberg kazarin nilpotent factorized proceedings groups andrews london math society lecture note series vol cambridge univ press amberg sysak radical rings products groups groups andrews bath edited campbell ferran eric jespers jan braces equation ferran eric jespers jan braces equation communications mathematical physics april volume issue ferran eric jespers involutive groups trans amer math soc etingof schedler soloviev solutions quantum yangbaxter equation duke math tatiana solutions equation braces symmetric groups van den bergh semigroups algebra smoktunowicz peter hubert nearrings construction triply factorized groups doctoral dissertation johannes mainz eric jespers jan okninski binomial semigroups algebra passman algebraic structure group rings wiley new york robinson course theory groups gtm vol wolfgang rump braces radical rings quantum equation journal algebra volume issue january pages sysak products groups local nearrings note mat suppl takeuchi survey mached pairs groups elementary approach theory banach center publ ferran departament universitat barcelona bellaterra barcelona spain tatiana american university bulgaria blagoevgrad institute mathematics informatics bulgarian academy sciences sofia bulgaria agata smoktunowicz school mathematics university edinburgh james clerk maxwell building kings buildings mayfield road edinburgh cedo tatyana
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constructing routes personalized submodular maximization poi features hongwei liang wang school computing science simon fraser university canada consider practical route problem given collection points interest pois rated features traveling costs pois user wants find routes source destination maximally match needs feature preferences completed within travel cost budget one challenge dealing personalized diversity requirement user different quantity number pois specified feature variety coverage specified features another challenge large scale poi network great many alternative routes search model personalized diversity requirement whole class submodular functions present optimal solution route problem index structure retrieving relevant pois feature route spaces various strategies pruning search space using user preferences constraints also present heuristic solutions evaluate solutions real life poi network data ntroduction advancement mobile devices triggered revolution location based services one emerging thread route planning applications trip recommendation ride sharing delivery majority current route planning systems focuses cost paths explores points interest pois popular geographically close works recommend routes based learnt user preferences historical data popular travel planner google trips suggests day plans traversing famous places user selected pois able respond user detailed requirement trip containing museum park relatively understudied problem many practical applications user wants suggested small number routes best meet requirements satisfying certain constraints let consider several applications scenario new visitor rome wants well spend remaining hours taking flight back home wishes recommended trip starting hotel ending airport allows visit museums souvenir shops eat good italian restaurants necessarily order values variety number places visited example route consisting one museum one shop one italian restaurant preferred route consisting two museums two shops scenario home seeker wants plan route see open houses many features bedrooms close schools within budget scenario driver wants suggested sequence pickups route feature vector park museum food oct hongweil wangk fig sample poi map nodes represents poi poi features park museum food feature numeric rating range indicated vector aside poi edge weight indicating cost traveling edge traveling time previous pickups next optimize certain objective integrates factors factors profit driving conditions allow dinner time route problem examples illustrate common structures requirements various emerging route planning applications first database poi map containing collection pois connected edges weighted traveling cost pois poi location set features museum bedroom feature numeric binary rating features ratings poi ubiquitous easily accessible real world instance foursquare encourages users rate venue numeric score features cleanliness write text tips features ratings also extracted user provided text reviews second user wants find routes query several parameters pair origin destination route weight vector specifying personalized weight feature budget constraint total travel cost route addition user personalized route diversity requirement terms quantity variety pois route scenarios modeled carefully chosen aggregation function feature scores pois route match degree route defined gain function gain routes highest gain satisfying origin destination constraints cost budget challenges one challenge modeling route problem choosing general aggregation function model range personalized route diversity requirement intuitively sum aggregation returns sum scores pois input route whereas max aggregation returns max score pois input route choices present two extremes quantity variety model consider poi map figure query specified park museum food sum aggregation prefers route though route fails cover specified feature park max aggregation distinguish two routes maximum score museum park respectively though latter may preferred user owing additional second challenge large search space route problem even single feature route problem subsumes orienteering problem heuristic solutions could speed search provide guarantee optimality complexity route problem practically compounded feature space route space makes quite meaningful develop efficient exact algorithms provide optimal results reasonable responding time key push user requirements search prune unpromising routes speed shortest distance computation precomputed index structures furthermore approach work wide range aggregation function model personalized route diversity requirement contributions variety minded user important cover many specified features possible desirable diminish incremental value additional poi repeating feature diminishing marginal utility properties naturally modeled submodular set function submodularity widely used many real world problems knowledge used model route diversity requirement problem contributions follows formulate route problem personalized route diversity requirement submodular aggregation function query parameter show function fulfils submodularity personalized route diversity requirement specified law parameter section iii propose novel structure index pois feature scores travel cost component eliminates irrelevant pois expensive access travel cost computation query time section propose optimal solution pacer topk route problem novelty design various strategies reducing computation search space work submodular function including route enumeration strategy facilitates reuse computed results pruning strategy eliminates dominated routes upper bound pruning strategy section present two heuristic algorithms far smaller search spaces good utility section evaluate algorithms analytically empirically suggests pacer significantly faster algorithms section vii elated orks orienteering problem finds path limited length visits vertices maximizes total points collected vertices path arc orienteering problem aop associates utility edges instead nodes route problem generalization modeling node poi features ratings modeling personalized user preference diversity requirement problem also generalizes aop edge utility modeled inserting dummy poi edge however clear solution aop solve problem next poi travel package recommendation works either recommend poi visited next recommend set pois quite different aim find route sequence featured pois trajectory search works category require database trajectories collected either retrieve existing segments trajectories match certain similarity query route based segments trajectories problem assumes poi map instead trajectory database route construction sequential location body works suggests poi sequence location path user learns historical travel behaviors recommends travel routes sequentially predicting next location using markov model approach works existing users historical data interactively plans route based user feedback selection approach specifies desired routes user query without requiring historical data interactive feedbacks user several works recommend route maximizing certain user satisfaction assumes poi single type searches route pois following order types allows user specify minimum number poi types instead exact types route uses user historical photos estimate personalized ratings stay time pois estimates user preferences none works considers feature preferences queries like finds route maximizes number poi desired keywords given distance threshold constructs optimal route covering categories locations since coverage containing keyword associated locations visiting one park considered good visiting two parks neither addresses personalized diversity requirement users perhaps related work adopts keyword coverage function measure degree set query keywords covered route similar however pruning strategies designed specifically specific keyword coverage function thus work address personalized route diversity requirement different submodular function may required user pruning strategies depend submodularity function specific form thus approach supports personalized route diversity requirement finally produces single route empirical performance surpass method much iii reliminary formally define problem studied paper table summarizes notations frequently used throughout paper variables vectors matrices problem statement definition poi map poi map connected graph set poi nodes set edges nodes set features pois denotes matrix rating feature poi poi associated staying cost edge travel cost choices depend applications time expenses cost definition routes route path origin destination sequence pois except possibly denotes set pois denotes least traveling cost next visited necessarily adjacent poi map cost defined cost route contains pois user actually visits consuming staying time route path least traveling cost poi path serves intermediate node visited user staying times either considered ignored depending user choice latter case modeled setting minimum user origin destination route necessarily distinct budget cost route addition user may want pois certain features specified vector weight feature user also specify filtering table nomenclature notation gain topk interpretation poi map node set edge set feature set pois staying cost poi matrix rating feature poi traveling cost edge least traveling cost poi route included poi set user query parameters source destination location travel cost budget feature preference vector filtering vector feature ratings feature aggregation functions poi candidates set retrieved size filtered gain route given query found routes vector set less denotes filtering finally user may specify route diversity requirement feature aggregation function vector feature returns aggregated rating feature pois definition query gain user query route valid starts ends cost gain gain note specification required specification optional provided user default choices used gain set function routes differ order pois gain order pois affects cost among routes poi set consider route smallest cost definition route problem given query integer find construct valid routes different poi sets highest gain ties ranked cost routes denoted topk rest paper use gain gain submodular function address personalized route diversity requirement consider submodular model diminishing marginal utility pois feature added route set function submodular every monotone every next theorem follows fact gain nonnegative linear combination theorem nonnegative monotone submodular gain coeff rank fig power law coefficient rank varied aim provide general solution route problem whole class nonnegative monotone submodular model various personalized route diversity requirement illustrate modeling power example consider defined power law function returns rank poi rating feature among pois largest value ranks first power law exponent feature increases note existing pois change new poi added route incorrect compute new simply adding marginal brought existing value ease presentation drop subscript use rest paper figure shows varies rank increases different note sum aggregation max aggregation special cases general larger means lower ranked pois faster diminishing factor feature ratings diminishing incremental value hence eqn supports spectrum diversity requirement ranging sum aggregation max aggregation setting theorem defined eqn nonnegative monotone submodular proof nonnegativity monotonicity eqn straightforward show submodularity let set pois contained two routes pois arranged descending order let let suffices show proof straightforward thus assume also assume contains exactly one poi say general case containing pois proved repeating argument assumed case times besides assume feature otherwise always true given feature rank existing poi reduced inserting thus consists two parts increment brought decrement existing pois caused reduced ranks includes one extra poi compared rank following two cases case ranks must ranked lower case increment brought decrement larger owing extra hence case ranks different must ranked ahead pois ranked lower suppose rank insertion ranks pois ranked higher remain unchanged rank poi ranked lower falls one thus let resp denote poi whose rank feature resp increments rank similarly increments rank get analogy get subsequent ranks hence rest paper assume nonnegative monotone submodular gain theorem route problem subsumes two problems submodular maximization problem orienteering problem thus poi map feature space path space must carefully indexed novel search strategies necessary prune unpromising much possible first consider indexing strategy section consider search strategies section overview ndexing algorithm offline component online component processing query offline component builds indices based poi map speeding poi selection travel cost computation online component responds user query poi candidates retrieval retrieves relevant routes finding searches routes using section explain offline component poi candidates retrieval online component routes finding presented later sections offline indexing poi map data stored disk answer user queries rapidly low access speed travel cost computation build two indices stored disk inverted index mapping feature list pois rating entry indicates feature rating poi sorted descending order sample poi map index feature index park museum food fig built poi map figure helps retrieving pois related features specified query least traveling cost arbitrary two pois frequently required online component compute efficiently employ labeling shortest distance querying weighted graphs shows scalable results finding labels unweighted weighted graphs index built using labels generated algorithm undirected graph one list pivot labels node label contains pivot node traveling cost denotes list labels sorted ascending order according computed query specific index hiq query specific feature index fiq park museum fig retrieve poi candidates given query retrieve subindex fiq subindex hiq inding ptimal routes present second part online component algorithm finding routes based min extracted poi candidate set fiq hiq algorithm design two goals mind prune unthe poi map figure shown promising routes aggressively possible preserving figure example compute search optimality answers ensure pruning common pivot nodes pivot label lists strategies applicable whole class nonnegative find pivot node minimizes traveling monotone submodular aggregation functions procost pose novel algorithm called prefix based compact states case directed graph poi two growth pacer incorporate idea memoization lists labels viin ending node variation dynamic programming fuse costhi viout starting node eqn becomes based pruning strategy pruning strategy min unified way first introduce several terminologies viout vjin route associated several variables gain ending poi end cost online poi candidates retrieval visited consume stay time every given query first thing retrieve poi set applied poi sequence candidates likely used routes finding open route starts visits several pois part particular pois contain feature closed route starts ends preference vector pass threshold initial open route includes open route never used ones visited feasible closed form satisfies cost way source destination within following discussion denotes either open route budget implemented retrieving query specific closed route open route ending poi fiq hiq extended longer open route poi figure illustrates retrieval works query variables updated gain weights park museum food end power low exponent eqn elements cost cost vector value features fiq retrieved using next present enumeration pruning strategies directly locates lists user preferred followed algorithm complexity analysis features park museum used cut lower rated pois sorted lists indicated red scissors compact state growth memoization contains remaining pois compact states gain depend poi hiq formed retrieving set route independent pois lists poi also ordered therefore group open routes sharing used cut sorted lists indicated red scissors compact state denoted let denote also check whether poi current actually reachable list open routes poi set associated checking visit cost following fields remove remove list hiq gain gain routes grouped indicated blue shading final poi candidates end cost typically much less original fiq hiq retrieved kept memory information cached hash map key pruning strategy abc abd acd bcd abc abd abcd acd bcd fig compact state enumeration tree pacer number aside node indicates order enumeration assume pois arranged lexicographical order poi ids compact states enumerated subsets represented nodes tree included every compact state omit figure shows compact state enumeration tree excluding capital letter represents poi node represents compact state define set pois precede order poi set prefix poi compact states generated specific manner longer open routes extended earlier computed shorter ones initially root labeled empty set child node current node generated appending poi precedes pois front child nodes arranged order example node abc generated child node node appending front precedes routes generated extending cached routes compact states node generate open route selecting routes append end compute gain cost based accessed information hash map kept feasible example generate open routes node abc access cached open routes nodes append missing poi ending poi represents open routes ending poi first two pois order note need materialize entire tree memory materialize current expanded branch tree closed route used update routes topk empty compact state kept compact state expandable stops enumeration yield final topk section give detailed algorithm note include open routes enumerating routes expensive present two strategies prune unpromising routes cost dominance pruning consider two feasible open routes say dominates identical poi set ending poi cost thus produce extended route also produce extended route using extension latter cost larger former note poi set also ending poi hence compact state generating given select open route feasible least cost thus dominates routes reduces open routes dominating open routes compact state one without affecting optimality example node abc represents one open route call strategy cost dominance pruning emphasize subtree pruning example node pruned open routes starting node node never considered though prunes dominated open routes many remaining dominating open routes may lead topk closed routes due small gain next strategy prunes unpromising dominating routes pruning strategy extend dominating open route step step using remaining budget cost closed route gain gain pois used extension step reachable current end therefore chosen set unvisited poi tend computed help hiq marginal gain concatenating existing gain gain let denote highest gain ranks lower current top routes topk promising open routes extended pruned motivates next pruning strategy marginal gain upper bound pruning finding hard finding optimal route scratch seek estimate upper bound marginal gain gain less gain current topk promising thus extensions pruned without affecting optimality call marginal gain upper bound pruning routes enumerated gain current topk increases pruning becomes powerful challenge hard estimate cost extended part without knowing order pois independent order pois ignore order estimate route cost set cost sum cost poi remaining route larger actual cost included remaining route define min min cost cost since order pois ignored easy verify min ensures property end destination cost end min tend min end set cost approach route cost replacing respectively choosing finding min min incur computation cost remaining task estimating becomes optimizing following problem max defined eqn end note include ended let denote maximum end larger actual costs thus using never loses optimality solve eqn first show properties theorem assume nonnegative monotone submodular given open route extended part route nonnegative monotone submodular polynomially computable function proof show submodular proof properties straightforward according set function submodular disjoint sets residual function defined also submodular since gain submodular theorem since disjoint gain gain residual thus submodular apparently eqn submodular maximization problem unfortunately also polynomial time exact algorithm unless thus switch estimate upper bound showed exists polynomial time greedy algorithm presented obtain approximate solution problem approximation ratio upper bound achieved however greedy algorithm runs defined eqn simplified version algorithm runs achieve approximation ratio mentioned offline bounds stated advance running actual algorithm compared offline bounds authors showed use submodularity acquire tight online bound arbitrary given solution obtained using algorithm problem maximizing submodular set function subject cost constraint empirically show bound much tighter bound applying problem eqn following theorem theorem set poi let let let sequence pois decreasing order let let theorem measures far given set optimum purpose estimating upper bound let hence computed without running greedy algorithm also empirically proved online bound eqn outperforms offline bounds tightness computational cost thus finally choose online bound algorithm algorithm pacer recursive funcion required fiq hiq compute gain cost parameters compact state set pois extending output priority queue topk forall poi set order compute gain forall poi dominating route cost minimum cost compute using eqn gain gain topk insert route updatetopk topk pacer prefix pacer algorithm incorporates compact state enumeration pruning strategies global input query fiq hiq used compute gain cost eqn integer output priority queue storing topk results pacer recursively enumerates subtree current compact state poi set available extending initial call pacer included explained section line extends set order creating child node computing gain lines generate dominating promising open routes specifically selected ending poi rest pois previously computed compact state finds dominating route line corresponds considered feasible applied check sum gain estimated upper bound marginal gain less gain route topk lines inserted finalized closed route open route least cost chosen update topk line information new compact state eqn added hash map last extended recursively pois prefix line let summarize good properties pacer follows properties pacer first pacer relies nonnegative monotone submodular independent form important dealing personalized diversity requirement second pacer enumerates open routes compact states depthfirst order enables constructing open routes incrementally memoization third compact state represents dominating feasible open routes instead routes fourth dominating feasible open routes estimated maximum achieved gain less current topk kept pruning tightened closed routes enumerated complexity analysis one reasonable measure computational complexity number routes examined two main factors impact number routes examined size poi candidate set maximum length routes examined excluding maximum let denote parameters respectively analyze pacer relatively search approximation solution pacer compact states level enumeration tree compute routes containing pois thanks compact states level represents routes computed thus pascal rule number routes examined therefore computation cost pacer also enabled prunes percent routes examined pacer computation cost pacer algorithm algorithm based expansion examines routes times pacer general next poi visited route immediate neighbor previous one approximation algorithm proposed quasipolynomial time approximation algorithm orienteering problem modified solve problem produces single route approximation ratio computation cost log improved version reduces cost log log estimated upper bound optimal gain however costs remain expensive many discrete values example minutes discrete values single decimal point precision computation cost improved version log noted took seconds small graph nodes comparison computation cost pacer given eqn cost reduced pacer finds optimal routes whereas finds single approximate solution experimentally compare pacer euristic olutions although pacer employs carefully designed enumeration pruning strategies remains expensive constraints loose large cost budget large poi candidate set need either approximation heuristic algorithms limit search space extreme cases arise approximation algorithm mentioned shown practically scalable therefore section design two heuristics state collapse heuristic cost dominance pruning pacer keeps open routes compact state representing set pois excluding aggressive pruning keep single open compact state one least cost heuristic route likely visits pois clearly heuristic longer guarantees optimality produce routes denote heuristic algorithm stands state collapsing let defined section number examined routes number nodes enumeration tree still computation cost around pacer eqn enabled pacersc computation cost reduces greedy algorithm computation complexity remains exponential route length next greedy algorithm runs polynomial time starts initial route iteratively inserts unvisited poi current route maximize marginal ratio gain gain denotes set pois current route inserts two adjacent pois current route total cost resulting route minimized term constrains selected pois far away two end points expansion process repeated budget used algorithm produces single route examines routes insertion consider unvisited pois vii xperimental valuation evaluated proposed algorithms two real life data ubuntu lts machine intel core cpu ghz ram algorithms implemented experimental setup datasets singapore denotes foursquare data collected singapore austin denotes gowalla data collected austin previously used singapore locations users austin locations users suggested built edge two locations visited date user poi map singapore locations austin locations locations connected edges ignored filled edge costs querying traveling time minute using google maps api driving mode average singapore minutes austin minutes staying time generated following gaussian distribution mean hours standard deviation hours costs minute extracted unique features respectively singapore austin based user mentioned features suggested rating feature poi calculated min number poi containing feature set pois containing maximump feature rating set data sets average number calculation scales middle value ratio poi count average count algorithms compared following algorithms method mentioned section optimal algorithm section cost dominance pruning enabled enables marginal gain upper bound pruning state collapse algorithm section greedy algorithm section approximation algorithm proposed see section algorithm proposed since works specific keyword coverage function compared section adapt submodular aggregation function coverage function fair comparison algorithms use extracted subindices section reduce access search space note produce exact solution produce greedy approximate solution queries query six parameters concreteness choose eqn controlling diversity pois desired route assume features singapore set singapore zoo nanyang technological university austin set austin four seasons hotel austin data set generated weight vectors model feature preferences users follows first draw number features uniform distribution draw features following probability distributions defined eqn selected feature let set selected features finally consider hours default settings bold face setting generated queries using vectors hours therefore specifies budget mins first study effect indexing section evaluate performance proposed algorithms section finally compare algorithm section effect indexing poi candidate retrieval section extracts two poi candidate set relevant query search space largely determined size denoted table compares average queries setting fixed default settings compared number pois original data singapore austin significantly reduced poi candidate set paves way subsequent efficient online route search table average size poi candidate set singapore austin optimal optimal runtime runtime routes routes gain fig experimental results singapore run time search space routes logarithmic scale labels beside data points indicate ratio queries succesfully responded algorithm parameter setting label query fail data point bar drawn half fail respond queries small performance study evaluate algorithms various settings chosen ranges mentioned particular evaluate gain cpu runtime search space number examined open routes processing query report average queries vectors setting produce routes thus first set order compare algorithms impact presented section figure report experiments singapore austin respectively row corresponds various settings one fixing two default settings optimal denotes optimal gain terminated algorithm given query runs hour runs memory used label beside data point indicate percentage finished queries half queries terminated data point shown impact budget affects length routes number pois included figure shows runtime search space gain worst consistent analysis section shows suffers high complexity many discrete values example gain runtime runtime gain routes time budget hour routes time budget hour gain runtime time budget hour gain routes time budget hour runtime sec gain routes routes runtime sec runtime time budget hour gain routes time budget hour gain gain routes routes runtime sec runtime sec runtime sec gain gain routes runtime sec routes gain fig experimental results austin setting discrete values minute majority queries finish second worst efficiency drops dramatically increases since number open routes becomes huger processing time memory consuming search space two orders magnitude smaller thanks compact state enumeration strategy cost dominance pruning best among exact algorithms comparing one order magnitude speedup runtime two orders magnitude reduction search space clearly demonstrates additional pruning power gain based upper bound pruning trades optimality efficiency surprisingly shown figure performs quite well gain close optimal always finishes less seconds singapore achieved gain worse optimal whereas austin difference small source destination singapore relatively remote central city greedily select pois far away thus many pois possibly higher feature ratings located central city less likely chosen results larger gap gain singapore contrast austin downtown area situation avoided cases impact filtering threshold figure show performance versus threshold feature becomes larger size poi candidate set reduced algorithms run faster majority experiments finish esplanade park tiong shian house peach garden diversity requirement bukit timah nature reserve tampines eco green bedok reservoir park hilton singapore onal museum fort canning park hilton singapore national museum fort canning park bedok reservoir park tampines eco green bukittimah nature reserve hilton singapore gain cost mins without diversity requirement fig two routes found singapore query hilton singapore represent park museum chinese restaurant results shown study suggests reasonable value reduces searching cost greatly little loss quality found routes impact diversity parameter figure show results various settings represent user diversity requirements route slightly affected increases marginal return diminishes faster behaves towards max aggregation case becomes less effective eqn becomes sum aggregation value gain difference optimal reach maximum emphasize make sense compare gain value different settings fact different setting values things differently figure evaluates effectiveness power law function eqn modeling personalized route diversity requirement run two queries singapore one specifies diversity requirement one specifies usual sum aggregation query parameters figures show best routes found query pois route labeled sequentially red dots represent source destination route covers specified features two pois feature maximizing total gain route four parks five pois due higher weight park although gain value second route higher first route less preferred user values diversity fact second route gain value evaluated using impact vary range fixing default values run algorithms except datasets influences pruning performance unchanged routes time budget hour runtime singapore routes singapore time budget hour routes fort canning park singapore art museum runtime sec onal museum hilton singapore national museum fort canning park singapore art museum esplanade park peach garden tiong shian eating house hilton singapore gain cost mins runtime sec hilton singapore time budget hour runtime austin time budget hour routes austin fig logarithmic scale change limited less slower less slower small gain best route usually far away best route thus marginal gain upper bound pruning seriously influenced omit figures varying due limited space discussion shown fact exact algorithm pacer practically runs much faster approximation algorithm theoretical approximation guarantee therefore real applications budget large prefer pacer returns best solutions reasonable responding time relatively large switch collapse heuristic version pacer sacrifices optimality little bit runs much faster large switch fastest greedy algorithm usually return acceptable solution comparison works keyword coverage function finds single route range set number poi feature average case single poi yields maximum value feature pois considered compare algorithm set eqn speed fair comparison also equip indices poi candidate retrieval note maximum budget kilometers efficiency study minutes google maps driving mode much smaller settings hours figure shows comparison modified datasets report gain omitted exact algorithms outperforms especially large several queries austin even failed hours although pruning strategy specifically keyword coverage function search strategy bottleneck besides gainbased pruning based greedy algorithm presented bound looser online bound fact experiments showed times faster algorithm contrast algorithm enables dynamic programming based search method multiple pruning strategies making search space much smaller algorithm shown section viii onclusion xtensions considered personalized route search problem large scale poi maps combination search feature space spatial space path space make problem computationally hard personalized route diversity requirement demands solution works reasonable route diversity specification presented exact solution address challenges multiple pruning strategies applicable whole class diversity requirements corresponding class submodular functions analytical empirical evaluation suggested solutions significantly faster algorithms also presented heuristic solutions introduce several possible extensions work feature order constraints user may certain order preference features route least one poi feature food visited pois shopping kind feature partial orders represented topological sorting open route violates order constraint violation removed appending pois end thus safe prune routes extension open route order constraints algorithms presented sections easily adopted prune search space additional constraint feature combination requirements following limited features combination requirements may interesting real application two pois feature visited consecutively either feature visited none visited iii exactly one visited constraints less routes satisfied thus extra pruning introduced eferences junglas watson services 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swish sharing jan fabrizio nov web media group university amsterdam netherlands department philosophy linguistics theory science university gothenburg sweden dipartimento matematica informatica university ferrara italy abstract recently see new type interfaces programmers based web technology example jsfiddle ipython notebook web technology enables solutions embedding tutorial web pages attractive rendering results cooperative development etc article describes swish web prolog public website exposes swiprolog using swish used run small prolog programs demonstration experimentation education connected swish cliopatria semantic web toolkit allows collaborative development programs queries related dataset well performing maintenance tasks running server embedded swish learn prolog online prolog book introduction web technology emerged state becomes useable implementing programming development environments major modern browsers implement javascript mature components available code editors framework styling widgets vizualization libraries using web technology rather traditional gui based technology microsoft visual etc various advantages network transparent allows controlling cloud hosted applications well prolog processes running headless devices web technology provides great infrastructure mashups pages integrate material several sources example embedding prolog tutorial pages embedding prolog queries modified documents describe data collections https http http http https https http swish sharing provide technology swi prolog swish consists javascript client browser code number prolog libraries realise server prolog application client code consists series plugins deal editing source code managing shared source repository entering queries rendering answers produced prolog libraries serve overall web application implement source store support editor predicate documentation templates results etc executing prolog queries swish relies pengines prolog engines pengine prolog engine controlled similarly prolog running terminal using http requests swish infrastructure originally developed prolog version jsfiddle later reimplemented modular jquery based infrastructure aiming collaborative exploration data hosted sql sparql server described section article organised follows section describes related work case systems inspired section describes architecture components swish section describe four applications current system conclude future work conclusions related work aware initiatives aim developing rich development environment prolog compare swish traditional editor gui based development environments prolog environments provide new opportunities pose new challenges instead discuss three applications served inspiration swish ipython stated initial inspiration swish jsfiddle unlike jsfiddle though prolog executed server rather browser interface statistical package although web application based webkit framework uses web based technology background came picture project provided grant developing swish toolkit analysis relational sql data interface similar layout swish providing source window console output plane typically shows results tables charts ipython notebook allows mixing markdown html text python sources rendered notebook shows text sources possible results form numbers tables charts swish embodies ideas behind jsfiddle embedding swish documents demonstrated section interactive editing documents embed swish discussed future work section https https http http ipython notebook work basis authentication either application command may executed swish operate public service granting access queries authenticated service run arbitrary queries example maintenance purposes see section swish application swish consists two parts client side running browser implemented series jquery plugins using bootstrap styling package management server side completely implemented builds top http server libraries pengines library ide support libraries provide data auto completion documentation highlighting following sections describe swish terms interface components discuss requirements user aspects client code supporting server functionality component first provide screendump illustrates main components figure answer pane runner render chessboard hover shows details query pane fig screendump swish left pane shows source code pane shows query runner exploits current selected answer renderer buttons continue first answer pane provides query editor access example queries stored source query history apply solution modifiers result presentation run button start query http code editor proper editor important component usable programming environment editor must support language including syntax highlighting auto indentation code completion based templates already existing code highlighting errors warning compiler editor used editing source code editing queries prolog difficult language support code editors due lack reserved keywords word starts embedded comment string word prolog refer predicate also predicate different arity constant etc another example arithmetic expression pair used next lack keywords ability extend syntax using new operators complicates implementation syntax support editing emacs oriented editor resolves problem closely integrating prolog editor typing current term clause directive parsed analysed context current file file imports keystroke term valid syntax tokens coloured according syntactic role well relation remainder program example call predicate coloured red call imported predicate blue call locally defined predicate black libraries implement analysis decoupled editor support source colouring documentation system pldoc two dominant open source actively maintained code editors available ace codemirror started swish ace basic prolog mode codemirror none nevertheless opted codemirror highlighting based raw javascript code rather regular expression based template language used ace low level implementation allows novel highlighting implementation highlighter consists javascript implemented prolog tokeniser tokenizing prolog sufficient colour comments quoted material strings quoted atoms variables constants atoms numbers also sufficient support smart indentation discussed sufficient highlighting role played atoms compound provide semantic highlighting illustrated figure forwarding changes content editor server server maintains mirror editor content asking server produce list semantically enriched tokens source tokens returned inner list represents tokens source term clause directive grouping tokens per source term allows incremental update yet implemented well see example fragment enriched token list may look like functor undefined goal variable singleton javascript tokeniser matches tokens http libraries yet used prodt additional complication formed codemirror highlighting support consequence must decide colour asserta know arity term called locally defined recursive fig semantic highlighter classifies addition syntactic category comment variable terms define call predicates based source code list basic type functor variable matches uses enrichment information singleton decide style basic token type match highlights token using basic syntactical category schedules request server new list enriched tokens request sent user pauses typing seconds request accompanied full source small list changes since last request source large waiting enriched tokens javascript highlighting code heuristically tries either uses uncertain results falls back plain tokens checks whether single added deleted modified token gets token stream fails tries next clause directive codemirror hover plugin used show basic information tokens pointer hovers goals includes goal defined iso swiprolog library locally documentation summary information available information requested server codemirror template plugin configured templates atom length extracted manual pldoc documentation imported libraries plugin shows menu applicable predicates templates finally user uses run button execute query program compiled compiler generates errors warnings inserted notes source code source code query management jsfiddle formed initial inspiration swish swish facility save program current version swish explicitly targets cooperative development prolog programs queries related dataset see section triggered implementation organised storage facility storage module implemented prolog inspired git file versioned independently rather maintaining version hierarchy files files referenced content using git compatible hash name name considered version head refers latest version name file save load interface provides following operations saving file anonymously produces randomly generated url similar jsfiddle saving file name saving new version interface shows available versions modifications forking file new name history remains linked original prolog source files include sources server using include filename including latest version include hash include specific version prolog source files embed example queries using structured comments sequence matching full stop token added examples menu query panel see figure comment illustrates call embedded source window examples append one two three list query editor query editor based jquery plugin realises code editor thus profits syntax highlighting template insertion hovering plugins addition provides three popup menus examples menu filled structured comments described examples menu shown figure history menu provides earlier executed queries solutions menu embeds existing query alter result currently provided operations aggregate count order distinct limit time debug trace figure shows menu used count solutions goal fig solutions menu used count results order filter duplicates etc upper runner shows answers query table running query runners answer pane answer pane placeholder runners runner represents query answer pane provides menu operations runners inside provided actions collapse expand stop clear query may completed running waiting user input swish manage multiple active queries time application defined maximum default runner provides set commands control specific query execution runner provides abort button query evaluation completes answer answers may available runner allows asking next results stop query addition runner shows text input field application wants read input may show debugger interaction buttons tracer used see section runner render answers two modes classical prolog mode table similar many database interfaces provide table mode particularly appealing querying datasets see figure former suitable rendering small amounts complex answers chessboard position figure default prolog terms rendered structured html objects rendered text prolog predicate server provide rendering libraries render prolog terms using dedicated html figure chess renderer loaded due use rendering chess directive chess renderer translates list length holding integers range chessboard queens addition chess rendering library swish provides rendering libraries sudoku puzzles parse trees tables cliopatria version adds renderer rdf resources renders resource compactly provides hyperlink obtaining details term rendered multiple ways interface provides hover menu select alternatives figure illustrates functionality render facility similar prolog hook allows changing result printed terms specific shape however exploit full potential html svg interface allow switching selected rendering rendering library module must define grammar rule term calling produce html prolog input term list variable bindings name variable user provided options current version new rendering modules must loaded swish server created swish user fig chess render library list integers interpreted queens chessboard user select rendering prolog term see actual term server side execution query execution query supported pengines library pengines library allows creating prolog engine represented prolog thread optionally pengine handed prolog program loaded pengine workspace program space workspace temporary module disposed pengine terminates pengine may asked questions http queries similar traditional prolog user interacting prolog running swish user hits run button content source pane used create new pengine subsequently content query pane sent one query executed execution query verified safe unless sandboxing disabled see section sandbox component discussed http https pengines execute multiple queries use facility fresh pengine starts predictable state standard operators empty dynamic database pengine default working module may code swish uses facility redefine prolog predicates etc cliopatria version section also preloads rdf libraries users run queries rdf database without explicitly importing required libraries sandboxing queries prolog environment contains global state form loaded modules defined operators dynamic predicates etc prolog exposes rich potentially dangerous interface operating system anonymous services want query start well defined state must ensure execution query make unwanted changes hosting computer leaks sensitive information education purposes data analysis one write meaningful programs without making permanent changes server server filesystem sandbox library comes sandbox library active loading source refuses add clauses modules pengine workspace accepts restricted set directives also aimed keeping changes local workspace prior execution sandbox unfolds query compares reachable goals whitelist whitelist contains free prolog predicates safe allows using dynamic database provided head affected predicate thus referenced predicate lives temporary program space pengine body safe allow calls module goal private predicates provide access predicates current current avoid leaking sensitive information sandbox test fails one conditions discovers goal deduce called code traditional example read call goal encountered signals instantiation error together trace explains insufficiently instantiated goal reached note deal normal predicates specified example following goal accepted safe maplist plus discovers goal whitelisted case signals permission error accompanied trace explains goal reached note pure prolog predicates unfolded also concerns predicates libraries belonging set built predicates discovers goal call predicate public normally tradition quintus prolog allows allowing swish would imply data kept secret limitation libraries keep data local dynamic predicates remain invisible users debugging swish debugger based traditional debugging model prolog figure shows tracer action lists example source debugger triggered line set clicking code editor debugging interaction deliberately kept simple similar traditional programming environments retry button added commonly seen step step step highlighting unique feature prolog goal fig debugging applications swish portability swish client libraries based mature well maintained javascript libraries client runs modern major browsers css javascript support frequently tested firefox chrome safari internet explorer server code basically many required libraries features shared least one prolog implementation none capable support full range summarise main problems scale involved prolog libraries demands closely compatible prolog module system probably sicstus yap used without significant rewrites http server libraries heavily based code interacts swiprolog foreign language interface prolog streams yap copied libraries capable run old version libraries pengines library depends http library interface thread api also provided yap xsb sandbox library section assumes whitelisted predicates indeed safe requires robust handling invalid calls resource overflows prolog systems satisfy requirement sicstus prolog would candidate sicstus support semantic syntax highlighting depends detailed source layout information provided read support term layout extended version quintus prolog term layout functionality significant parts code rely version extensions notably dict string types facilitate natural mapping prolog json data list clear fully functional port swish another prolog system immediately feasible yap probably comes closest still requires significant amount work realistic scenario though setup provides web interface development tools another language even necessarily prolog provides query solving interface two based interprocess communication target system robust safe capable supporting threads linking target system process using interface communication applications section describe evaluate four publicly available swish applications services regularly updated run latest version swish swiprolog swish runs plain publicly accessible copy swish allows running sandboxed see section prolog programs server collected programs september june month may executed prolog queries web site regularly used users prolog irc channel discuss programming solutions active use http steve matuszek umbc via thank much fantastic resource used teaching prolog semester really helped tighten loop students spent zero time tool installation overhead time understanding concepts even turn assignments via swish test queries examples cliopatria cliopatria semantic web rdf framework consists rdf triple store sparql server web frontend manage server explore data rdf store cliopatria extended using cpacks cliopatria pack plugin swish available cliopatria makes prolog shell available querying well maintenance tasks without login user run free queries rdf data stored cliopatria rdf database login administrative rights sandbox limitations lifted prolog shell used perform maintenance tasks rdf data data transformation cleanup well program maintenance tasks reloading modified source files swish used talk europe creative explore data speeches european although still immature users appreciated ability define efficient expressive queries provided sparql query interface ability save share programs perform interesting tasks data frequently used particular seek help fixing queries learn prolog learn prolog online version prolog book patrick blackburn johan bos kristina striegnitz established proof concept embeds swish online course realised prolog hosted proxy fetches pages main site serves enhanced pages user proxy identifies classifies code fragments terms source code queries dependencies next adds button source fragments pressed replaces html pre element iframe running swish filled source example queries added examples menu figure queries executed http program queries transferred using following http parameters code source code background source code loaded pengine visible editor examples queries appear examples menu initial query proxy server served pages may cplint swish cplint web application based swish reasoning probabilistic logic programs distribution semantics prolog source window used write logic program annotated disjunction query form ground atom answered returning probability true program computation probability done cplint system server using http http http http http http fig screendump learn prolog opened swish instance shows collected source well example queries subsequent text classified relating source embedded swish provides functionality available swish user presses close button swish removed original code pengines input program translated internal representation using source source transformation swish modified javascript code runner source code prepended code loading cplint library enabling source source transformation query wrapped call inference predicate call variable argument prob hold probability shown user answer pane trill swish trill probabilistic owl reasoner reuses swish swish cliopatria described section cliopatria cpack prolog source window replaced editor window used upload owl ontology query editor used pose prolog queries owl http ontology probability queries computed using trill reasoner prolog adopts distribution semantics probabilistic description logics also trill swish modify javascript code runner source code sent pengine obtained adding prolog code parsing string calling parsing predicate wrapping query performs syntactic checks misspellings future work although definitely usable current state swish work progress confident basic component selection organisation server client code stable work needed improve current system notably semantic highlighting yet perfect often fails degrade gradually server side annotation match client tokens perfectly pengine sandbox protection often restrictive several security flaws reported fixed already probably always advised run public services operating system sandbox current server suffers memory leaks stability problems although situation improved significantly main demo server needs restarted foresee several extensions swish improve current applications enable new opportunities deploying swish cliopatria authorised usage swish shows potential controlling servers embedded prolog engines addition small temporary prolog programs would like able edit existing create new prolog modules well pages languages javascript html css full editing capabilities would allow shared development server software without shell access server swish enabled software running editing enhance sandboxed swish application providing input output documents compare trill section using document input plan provide format specifically writing tutorials well dataset analysis documents first look like learn prolog example discussed section second envision document embedded code query fragments query fragments produce tables charts similar ipython notebook turn http reliable scalable resource examine possibility schools instantiate private version preloaded course material assignments restart server small consequences active users open queries killed source code mirror lost automatically recovered client asks new set semantically enriched tokens conclusion article presented swish sharing swish provides webenabled interface prolog based ideas jsfiddle ipython notebook consists javascript client side server side based swiprolog http pengines prolog engines libraries swish deployed many settings education data analysis server development maintenance swish whole tied languages even limited prolog could controlled swish made available open source downloaded acknowledgements development swish supported dutch national program references patrick blackburn johan bos kristina striegnitz learn prolog volume college publications christopher gandrud reproducible research studio crc press lager jan wielemaker pengines web logic programming made easy tplp fabrizio riguzzi terrance swift efficient inference probabilistic logic programming distribution semantics theory pract log special issue annual gulp conference march cyrille rossant learning ipython interactive computing data visualization packt publishing ltd jan wielemaker anjo anjewierden pldoc wiki style literate programming prolog patricia hill wim vanhoof editors proceedings workshop methods programming environments pages jan wielemaker zhisheng huang lourens van der meij web tplp riccardo zese elena bellodi evelina lamma fabrizio riguzzi fabiano aguiari semantics inference probabilistic description logics uncertainty reasoning semantic web iii volume lncs pages springer https
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efficient pose estimation provable guarantees nov shaofei wang beijing technologies beijing china konrad paul kording university pennsylvania philadelphia koerding julian yarkony experian data labs san diego abstract bounds pose estimation mppe natural images key meaningful use visual data many fields including movement science security rehabilitation paper tackle mppe approach starting candidate detections body parts convolutional neural network cnn grouping people formulate grouping body part detections people set packing mwsp problem set potential people power set body part detections model quality hypothesis person set mwsp augmented markov random field variables correspond correspond power set detections part introduction study problem pose estimation mppe model problem selecting subset proposals people supported image evidence prior model formulation mppe corresponds minimum weight set packing mwsp problem elements correspond detections body parts sets referred poses correspond subsets body parts detections poses associated real valued costs based occurrence cooccurrence probabilities detections pose defined deep neural network augmented tree structured model respectively fully specifies mppe optimization problem since set poses grows exponentially number detections employ implicit column generation icg strategy inference mwsp exploit augmented tree structure cost pose frame pricing dynamic program variables correspond body parts state given variable corresponds subset detections part since variable enormous power set detections given part introduce tool operations research called nested benders decomposition nbd avoids considering vector product adjacent variables tree nbd used variety applications including agricultural planning factory production planning stock portfolio optimization nbd formulation guaranteed achieve exact inference pricing problem practice orders magnitude faster naive dynamic programming nbd exploits fact pricing problems describe novel algorithm combines efficiency provable bounds mwsp problem employ implicit column generation strategy pricing problem formulated dynamic program efficiently solve dynamic program exploit problem structure utilizing nested bender decomposition nbd exact inference strategy speed recycling bender rows calls pricing problem test approach dataset showing approach obtains comparable results algorithm joint node labeling grouping problems nbd achieves considerable relative naive dynamic programming approach typical algorithms solve joint node labeling grouping problems use heuristics thus obtain proofs optimality approach contrast proves percent problem instances find globally optimal solution otherwise provide indicates part associated detection detection associated single part prior mwsp optimization short hand use denote set detections associated part use denote set possible poses index associate members detections using index pdp indicates detection pose allows formulate mppe search low cost sets model human poses augmented tree structure set parts described using matrix index part child part augmented tree use fourteen body parts head neck shoulder elbow wrist hip knee ankle standardized mpii dataset form augmented tree parts defined typical tree structure parts excluding neck followed connecting neck thirteen parts model design based observation real images people necks rarely occluded thus connections neck body parts handle cases body parts occluded keeping model relatively simple short hand use denote root tree part neck selected arbitrarily cost pose defined using terms index respectively refer terms unary pairwise respectively use indicate cost including detection pose similarly use indicate cost including detections common pose augmented tree structure respected regards costs thus child model prior number poses image using constant offset added cost pose define mapping poses costs using index cost associated pose defined formally pdp similar across iterations icg hot starting optimization given iteration benders rows produced previous iterations combination icg nbd promises efficient provably optimal solutions without enumerate vector product related work work closely related integer linear programming ilp formulation refer shorthand models problem mppe partitioning detections fourteen body parts plus false positive clustering detections poses clustering done according correlation clustering criteria costs parameterized part associated detection formulation notable models suppression allowing poses associated multiple detections given body part however optimization problem often hard solve thus attacked heuristic methods motivated difficulty inference work introduces alternative model called two tier formulation ttf ttf truncates model way achieve fast inference using icg ttf model icg inference outperforms regards finding parts ankle wrist also provides marginal improvement overall accuracy furthermore ttf provides additional capacities beyond prior number people image inference ttf model attacked using benders decomposition though inference demonstrated fast icg strategy ttf model works well practice optimize full cost privileges single exemplar detection body part model costs pose ignoring costs detections furthermore ttf requires detections associated parts advance optimization outline paper structured follows section introduce novel mwsp formulation mppe address inference icg section introduce nbd approach solving pricing problem icg section present experiments validation set finally conclude section provide additional derivations appendix thus fully defined cost single pose next section formulate mwsp problem using costs poses relaxation formulation mwsp formulation mppe frame search lowest cost set nonoverlapping poses integer linear program ilp use define selection poses pose selected write constraint selected poses overlap formulate mppe mwsp use denote sets human body parts sets body part detections index respectively describe surjection detections parts using express ilp along corresponding linear program relaxations follows using lagrange multipliers index min min max algorithm implicit column generation algorithm repeat max pdp dneck min pdp pdp end end appendix study bounds applied accelerate inference without loosening relaxation inference via implicit column generation given grows exponentially number detections explicitly consider mwsp optimization thus construct sufficient subset using icg solve exactly dual problem icg consists two alternating optimizations referred restricted master problem rmp pricing problem respectively figure implicit column generation procedure iteratively solve rmp followed pricing pricing compute one pose associated member power set neck detections add pose nascent set corresponds violated constraint dual terminate pose corresponds violated dual constraint rmp solve dual optimization set providing dual variables write dual optimization rmp max pdp write icg alg consider pricing problem dynamic program section termination alg solve mwsp using ilp solver practice find relaxation provides integral solution termination cases needed could employ branch price tighten bound odd set inequalities preserving structure pricing problem pricing using identify subset violated constraints corresponding members add subset includes violated constraint violated constraints exist icg terminates slack dual constraint corresponding given primal variable referred reduced cost thus pricing identifies lowest reduced cost terms primal anytime lower bounds compute anytime lower bound objective adding term based columns produced objective rmp recall detection assigned one body part thus rely proof mwsp bounded rmp objective plus lowest reduced cost times cardinality set elements negative reduced cost term exists compute lower bound given provided rmp follows min min pdp pricing step iterate power set neck detections compute lowest reduced cost pose containing exactly neck detections index power set neck detections use indicate neck detections exactly write pricing arbitrary subset neck detections min pdp observe minimization computed time pricing alg note conditioned specific set neck detections pairwise costs neck detections detections added unary costs detections thus structure becomes typical tree structure exact inference done via dynamic programming pricing using naive dynamic program observe corresponds computing maximum posteriori probability map markov random field mrf bijection parts except neck variables mrf similarly variable mrf bijection state space variable power set detections associated part mrf tree structured hence amenable exact inference via dynamic programming consider follows use denote state space variable describe using index respectively use sds indicate detection included configuration use refer value lowest cost solution rooted conditioned parent taking state cost includes pairwise interaction terms members define formally using helper terms existence decomposition known result stochastic programming literature notice minimization require either configuration children messages children hence state determined independently variables similarly variable determined independently children given state parent sets small cardinality solving easy thus construct sufficient subset denoted root using row generation cutting planes refer collection nascent sets given nascent sets construct lower bound denoted defined using helper function minr minr sds sds minr max sds root variable associated terms since parent associated terms overview constructing consider construction small sufficient sets outline remainder section follows section produce bounds map mrf identical termination nbd upper bound accompanied configuration cost equal upper bound section compute gap upper lower bounds introduced variable tree select variable associated largest increase gap section add benders row next section increase terms hence tightening relaxation without generating new rows section combine steps produce complete nbd inference approach finally provide implementation details section computing expensive since need search possible combinations state spaces two adjacent variables number possible states variable experiments nested benders decomposition alternative naive dynamic programming consider nbd alternative naive dynamic programming avoids expensive computation express using sum convex functions constructed maximum unique set affine functions called benders rows specifically part root denote corresponding set benders rows index describe using parent index given use indicate value associated offset using provide alternative description using helper term defined producing configuration corresponding bounds using nested benders decomposition section produce configuration mrf proceeding root leaves selecting state given variable given state parent describe configuration produced using use refer state parent process producing defined min arg min arg minr min cost configuration upper bound map associated lower bound map sdr max computing produce new benders row denoted defined follows sdx solving optimization dual add tiny negative bias objective corresponding terms ensures corresponding terms increase beyond needed produce optimal dual solution stabilizes optimization additional small may understood intuitively ensuring biases corresponds marginal cost using solution appendix solve dual efficiently reducing equivalent variables far fewer constraints section identify variable tree associated largest increase gap upper lower bounds given configuration produced section gap upper lower bounds introduced difference upper lower bounds variable minus corresponding gaps children use defined denote cost rooted corresponding lower bound gap introduced respectively max computing gap introduced variable tree sdr generating benders rows section identify violated benders row denoted given given parent takes sets formulate small scale linear program described rows rapidly updating leaving fixed provide complementary mechanism generating new benders rows mechanism motivated observation may increase never decrease benders rows added descendants given variable mechanism takes existing benders rows sets corresponding term use decision variable indicate state associated variable use decision variable index introduce dual variables lie minimum feasible value thus tightening corresponding constraint leaving fixed task faster generating benders rows via method section observe given satisfying always exists feasible setting given fixed select smallest feasible value follows minr algorithm overall algorithm nested benders given initial repeat step update terms leaves children root end end step compute configuration bounds arg children leaves arg end pose corresponding pdp step select variable add benders row leaves children sdx end arg step generate new benders row generate solving dual given parent taking pstate complete nested benders decomposition algorithm nbd approach iterates following steps step proceeding leaves children root set terms minimum feasible value given fixed done first iteration nbd empty step two proceed root leaves select state variable conditioned parent one benders rows associated children produces upper lower bounds associated variable tree step three select variable corresponding largest increase gap upper lower bounds tree step four add new benders row repeat procedure additional benders rows need added point configuration produced step two guaranteed global optima formalize procedure alg end implementation details section provide implementation details regards alg accelerating step one alg observe terms associated decreased new benders row added one descendants thus executing alg update terms associated ancestors variable benders row set augmented observe need update terms associated leaves first iteration nbd given call icg accelerating step two alg recall choice state given variable function benders rows associated children configuration parent parent iteration nbd first consider previous configuration mrf produced step two alg update state variable state parent changed ancestor variable benders row set augmented limiting number neck detections pose found best results regards timing modeling occur require pose include exactly one neck detection limiting state space variable limit number states given variable given user defined parameter value construct set follows begin state corresponding zero figure timing comparison achieved nbd accumulated running time problem instances nbd respectively factor nbd relative function computation time spent pricing note general factor grows problem gets harder constant bias set hand regularize number people solution compare solutions found nbd step icg problem instances optimization steps nbd obtains exactly solutions tie costs comparing total time spent nbd across problem instances found nbd faster faster extreme problem instances comparison accumulated running time used nbd instances shown fig observe factor speed provided nbd increases function computation time regards cost observe integer solution produced identical value problem instances thus certifying optimal integer solution produced instances relaxation fails produce integer results gaps objectives integer solutions within objectives sake completeness also report mppe accuracy terms average precisions aps compare solver uses primal heuristics note cost formulation differs allows pose associated multiple neck detections none model requires pose must exactly one neck detection maps detections parts prior icg also model includes prior number poses modeled shown table achieve equivalent results note although algorithm run fast code implemented pure matlab benefit using commercial solvers parallelizing pricing routines importantly formulation provides tions included add group states corresponding one detection included add group states corresponding two detections included etc adding group would state space exceed states variable add group terminate caching integrals accelerate alg storing value repeatedly used integrals change value course optimization time new produced store thus similarly store initializing initialize benders rows first round pricing icg thus initial state variable nbd ignores existence children corresponding initial lower bound timing observation experimentally observe total time consumed steps nbd ordered greatest least note step solving second least time consuming step nbd selecting root observe alg requires solving lps step four variables except root number constraints part exponential size avoid solving largest selecting root part associated detections alternatively could select root balanced tree goal leveraging parallel processing experiments evaluate approach naive dynamic programming based formulation validation set consists images unary pairwise costs trained using code part head shoulder elbow wrist hip knee ankle map ubody map time table display average precision approach versus running times measured intel cpu figure example output system references bounds cases certificates optimality andriluka pishchulin gehler schiele human pose estimation new benchmark state art analysis proc cvpr baldi sadowski whiteson searching exotic particles physics deep learning nature communications conclusion described pose estimation set packing mwsp problem address using implicit column generation solve corresponding pricing problem using novel nested benders decomposition nbd approach reuses bender rows calls nbd cases find provably optimal solutions practically important domains knowledge certainty matters interventions rehabilitation procedure solving pricing problem vastly outperforms baseline dynamic programming approach expect nbd find many applications machine learning computer vision especially solving dynamic programs large state spaces individual variables example could formulate tracking mwsp pricing using nbd bansal blum chawla correlation clustering journal machine learning pages barnhart johnson nemhauser savelsbergh vance column generation solving huge integer programs operations research bellman dynamic programming dover publications incorporated ben amor desrosiers carvalho inequalities stabilized column generation operations research benders partitioning procedures solving mixedvariables programming problems numerische mathematik yang ramanan articulated pose estimation flexible proc cvpr birge decomposition partitioning methods multistage stochastic linear programs operations research desrosiers primer column generation column generation pages springer felzenszwalb mcallester ramanan discriminatively trained multiscale deformable part model proc cvpr felzenszwalb girshick mcallester ramanan object detection discriminatively trained partbased models ieee transactions pattern analysis machine intelligence geoffrion graves multicommodity distribution system design benders decomposition management science gilmore gomory linear programming approach problem operations research volume gilmore gomory multistage cutting stock problems two dimensions operations research insafutdinov pishchulin andres andriluka schiele deepercut deeper stronger faster multiperson pose estimation model corr karp reducibility among combinatorial problems ibm research symposia series pages plenum press new york krizhevsky sutskever hinton imagenet classification deep convolutional neural networks proc nips levinkov uhrig tang omran insafutdinov kirillov rother brox schiele andres joint graph decomposition node labeling problem algorithms applications proc cvpr pishchulin insafutdinov tang andres andriluka gehler schiele deepcut joint subset partition labeling multi person pose estimation proc cvpr rumelhart hinton williams parallel distributed processing explorations microstructure cognition vol chapter learning internal representations error propagation pages mit press cambridge usa tang andres andriluka schiele subgraph decomposition tracking cvpr wang kording yarkony exploiting skeletal structure computer vision annotation benders decomposition arxiv preprint wang wolf fowlkes yarkony tracking objects higher order interactions via delayed column generation artificial intelligence statistics pages wang zhang ihler yarkony pose estimation via column generation arxiv preprint yan dang sun deep image scaling image recognition arxiv preprint yarkony fowlkes planar ultrametrics image segmentation proc nips appendix dual optimal inequalities dominates nbd computation achieve make following observations primal form section provide upper bounds lagrange multipliers called dual optimal inequalities doi computed prior icg use doi decreases search space icg needs explore thus decreases number iterations pricing required observe given iteration icg optimal solution primal relaxation need lie span limited producing primal solution useful allow values exceed one troduce slack vector indexed tracks presence detections included prevents contributing objective corresponding contribution negative offset cost detection cost least compensates likely overcompensates observe removal detection pose removes cost associated pose detection term define upper bound increase cost pose given removed express compactly introduce following terms max min min define follows min given constraint inactive given constraint inactive given constraint inactive observe following true pair since associated nonnegative term objective observe following max expanded mwsp objective dual relaxation given min pdp using observations rewrite optimization max observe term associated objective bound zero thus value primal therefor observe terms bound zero given objective value zero constraint common value using merge terms across follows using index helper term observe dual relaxation bounds bounds called doi cases pose required include neck detection use bound neck detections removal neck makes pose invalid therefore ignore term computing neck set ensure doi active termination icg offset small positive constant appendix deriving compressed benders row generation section compress dual form accelerate inference fact compressing observe optimization longer add tiny magnitude negative objective ensure smallest valued solution produced ensures corresponding terms increase beyond needed produce dual feasible solution stabilizes optimization use express max remove constraints corresponding members constraints form without altering solution observe slack constraints decrease increases thus determine constraints form need consider set zero vector constraints violated violated setting ignored solving practice find proportion constraints remove large thus achieve considerable time savings solving
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optimal longest paths dynamic kai christian karlsruhe institute technology karlsruhe germany fieger university vienna vienna austria feb abstract propose optimal algorithm solving longest path problem undirected weighted graphs using graph partitioning dynamic programming obtain algorithm significantly faster methods enables solve instances previously unsolved acm subject classification graph theory keywords phrases longest path graph partitioning digital object identifier introduction longest path problem find simple path maximum length two given vertices graph length defined number edges total weight edges path problem known several applications designing circuit boards project planning information retrieval patrolling algorithms multiple robots graphs example designing circuit boards length difference wires kept small longest path problem manifests length shorter wires supposed increased another example application project problem used determine least amount time project could completed organize rest paper follows introducing basic concepts related work section present main contribution section optimal algorithm longest path problem undirected graphs main ingredient algorithm dynamic programming technique based hierarchical partitions graph summary extensive experiments done tune algorithm evaluate performance presented section includes study algorithm performance respect different partitioning strategies order find good balance time spent partitioning overall runtime algorithm also compare algorithm optimal algorithms presented recent literature experiments show new algorithm solves significantly instances also faster optimal algorithms instances finally conclude section work partially supported dfg grants tomas balyo kai fieger christian schulz licensed creative commons license international symposium experimental algorithms sea leibniz international proceedings informatics schloss dagstuhl informatik dagstuhl publishing germany optimal longest paths dynamic programming preliminaries definitions following consider undirected graph edge weights extend sets denotes neighbors subgraph graph whose vertex edge set subsets another graph call subgraph induced every possible edge subset graph vertices called clique graph contains edge every two distinct vertex subset matching subset edges graph two edges vertices common sequence vertices pair consecutive vertices connected edge called path say source target path called simple contain vertex length path defined sum edge weights graph unweighted edge weights assumed one given graph well two vertices longest path problem find longest simple path another version longest path problem find longest simple path two vertices however problem solved introducing two vertices connecting vertices graph edges weight zero running algorithms tackling problem modified instance partition graph division blocks vertices balancing constraint demands lmax imbalance parameter objective typically minimize total cut eij eij related work paper based bachelor thesis kai fieger previous work stern mainly focuses possibility applying algorithms usually used solve shortest path problem longest path problem stern make clear difficult compared several algorithms presented frequently used solve minimization search problem modified order able solve search algorithms put three categories heuristic uninformed suboptimal algorithms first two categories yields optimal solutions problem relevant category paper heuristic searches heuristic provide extra information graph type graph heuristic searches require heuristic function estimates remaining length solution given vertex graph give important information helping prune search space stern show heuristic searches used efficiently longest path problem examples algorithms category dfbnb another category represents uninformed searches require information already given definition problem examples category dijksra algorithm dfbnb without heuristic modifying algorithms fit basically leads brute force algorithms means still look every possible path search space hence uninformed search strategies beneficial last category suboptimal searches authors looked large number algorithms find approximations longest path however relevant paper since present optimal algorithm aware recent work related optimal solving balyo karlsruhe high quality partitioning within work use open source multilevel graph partitioning framework kahip karlsruhe high quality partitioning precisely use distributed evolutionary algorithm kaffpae contained therein create partitions graphs better suited dynamic programming approach algorithms kahip able improve best known partitioning results walshaw benchmark many inputs using short amount time create partitions refer reader details longest path dynamic programming introduce main contribution paper new algorithm tackle longest path problem based principles dynamic programming hence algorithm called longest path dynamic programming lpdp algorithm solves longest path problem weighted undirected graphs start section introducing main approach includes preprocessing graph well combining paths end section show improve approach using hierarchical partitions input instance reducing number auxiliary nodes reduces size search space basic approach simple way solve longest path problem exhaustive search regular search dfs starts root vertex default vertex two states marked unmarked initially vertices unmarked except root marked dfs calls recursively unmarked vertex reachable edge current vertex parent vertices vertex marked done backtracks parent search finished dfs backtracks root vertex exhaustive dfs dfs unmarks vertex upon backtracking way every simple path graph starting root vertex explored problem solved exhaustive dfs using start vertex root search length current path stored compared previous best solution time target vertex reached current length greater best solution updated accordingly search done path maximum length found main idea lpdp partition graph multiple blocks run search similar exhaustive dfs block preprocessing step afterwards results combined single longest path partitioning preprocessing explain preprocessing routine first partition graph predefined number blocks modify input instance partitioning done using graph partitioning algorithm kahip replace every cut edge edge running two blocks introducing two new vertices edges one edges retains weight original edge edge weight set zero additionally insert two new vertices one connected start vertex one target vertex cases use edge weight zero newly generated vertices called throughout rest section next compute subgraphs set vertices block sea optimal longest paths dynamic programming well connected edges run vertices see figure example observe following property longest simple path function entry exit points block partition means longest simple path enter exit block inserted block contain every time path enters block also leave connecting pairs two sets interested subset block equivalent matchings exist consisting block words endpoint appear subset pairs pairs connected simple paths preprocessing algorithm computes longest connections block set modified version exhaustive dfs executed respective subgraph whose descriptions follows modified exhaustive dfs executed multiple times time using different root search algorithm divides current search path different path segments traverses vertices graph usual exception first segment starts root segment completed different reached happens algorithm starts new segment jumping continues traversing graph way segment starts ends startand endpoints segments resemble pairs best current result possible element stored updated necessary every time path segment completed avoid unnecessary traversal due symmetry graph path segment allowed end vertex higher start vertex additionally path segment start vertex higher starting vertices current search path combining paths performed preprocessing find longest simple path auxiliary graph contains edges graph two connected edge belong graph note every block original partition represented clique auxiliary graph additionally edges get introduced edges replaced previous graph replaced edges represent connections blocks order solve longest path problem use another modified version exhaustive dfs starts vertex representing start vertex algorithm creates set every block search path edge part block edge part search path pair element pair represents connection corresponding simple path simple paths pairs intersect best possibility combined length paths already calculated preprocessing phase following order receive valid pairs trying append new edges current search path first every block solution exist looked since best possible solutions calculated preprocessing step second search graph balyo figure example illustrating basic idea lpdp algorithm upper left corner graph partitioned three blocks indicated colors green yellow blue starting node green block target blue block border nodes named upper right corner see three graphs created removing edges connecting blocks adding auxiliary nodes indicated small circles finally bottom graphs used second stage algorithm combining paths left side see simple version right side contains improved version fever auxiliary nodes vertices alternating pattern edges part block edges connect two blocks otherwise would possible means two paths would intersect every time vertex representing original found paths every looked weight summed end different highest combined weight returned weight longest simple path actual longest simple path constructed looking paths given connections improvement hierarchical partitioning preprocessing computations done fast even parallel auxiliary graph searched variant naive approach avoid applying principles used accelerate exhaustive dfs recursively main idea combine groups blocks paths calculated blocks single new coarser block assigning vertex block split edges connect different blocks note cliques representing previous blocks stay intact representing start target vertex also connected new computation sea optimal longest paths dynamic programming combination paths done using modified exhaustive dfs new block way actually recursive call algorithm sense partition original graph hierarchical blocks subsequently combined step step larger ones one left longest path calculated improvement reducing number auxiliary vertices defined every cut edge replaced two edges let set connected vertex ignore possibility two elements connecting see path segment current block connecting also valid path replace leads improved algorithm single every vertex hence contrast previous formulation weight cut edge get divided among replacing edges instead edge gets introduced auxiliary graph combining paths retains weight weights set zero note variant algorithm every edge replaced meaning turn auxiliary graph traversed since edges different blocks weights associated obtain correct result sum weight weighted edges search path add length currently induced path additionally single vertex connected multiple different blocks still search possible valid paths allow multiple consecutive edges search path connects different blocks compared purely alternating pattern entering leaving vertex two edges corresponds connection mentioned represents usage single vertex block follows search path induces set block also set excluded vertices hence find best possibility connect pairs without using vertices preprocessing calculate possibilities choose best one looking unused vertices current solution get best possibility note vertex would simply correspond additional pair previous formulation algorithm example improved auxiliary graph see figure experimental evaluation methodology implemented algorithm described using compiled using gcc full optimizations turned flag implementation freely available karlsruhe longest paths package kalp gnu gpl license use multiple implementations provided stein comparison exhaustive dfs naive approach well algorithm dfbnb solver run algorithm input pair time limit one hour experiments run machine equipped two processors ghz cores ram present multiple kinds data first foremost use cactus plots number problems plotted running time plot shows runtimes balyo achieved algorithm problem running times sorted ascending order algorithm point curve means xth fastest solved problem solved seconds problems solved within time limit shown addition utilize tables reporting number solved problems well scatter plots compare running times two different solvers plotting points instance benchmark problems mainly use instances similar ones used previous work stein based mazes grids well road network new york additionally use subgraphs word association graph graph describes results experiment free word association performed participants vertices correspond words arcs represent pair first set instances generated using mazes grids square fields given start target field one move adjacent fields horizontally vertically field obstacle goal find longest simple path start field target field represent grids graphs every free field insert vertex add edge weight two vertices whose fields horizontally vertically adjacent generate grids stein top left bottom right field start target fields random fields grid consecutively made obstacles figure example maze certain percentage fields obstacles longest path filled afterwards path start target searched make sure solution longest path problem exists example maze shown figure sizes used mazes range second third set instances subgraphs road network new york well subgraphs word association graph respectively subgraph extracted follows start search random vertex network stop certain number vertices reached vertices touched search induce instance one touched vertices randomly chosen experimental results compare dfbnb exhdfs presented stein algorithm lpdp using two configurations configurations differ amount time spent partitioning input instance use either configuration kaffpa lpdpe good trade solution quality running time strong configuration kaffpae aims better partitions investing time partitioning lpdps latter case set amount block imbalance note lpdps spends much time graph partitioning phase algorithm ldpde results reporting running time paper include time spent partitioning figures table summarize results experiments apparent cactus plots figure configurations lpdp significantly outperform sea optimal longest paths dynamic programming time seconds lpdpe lpdps dfbnb exhdfs random grid maze problems time seconds lpdpe lpdps dfbnb exhdfs road network problems time seconds lpdpe lpdps dfbnb exhdfs word association problems figure cactus plots three kinds benchmark problems comparing previous algorithms lpdp three different partitioning configurations running times include time spent partitioning lpdp variants balyo lpdpe dfbnb lpdpe exhdfs lpdpe lpdps dfbnb lpdps exhdfs lpdps speedups problems figure speedup lpdpe lpdps relation previous algorithms problems solved within time limit five tested algorithms time lpdps sec time lpdpe sec figure scatter plot comparing running times lpdpe lpdps entire benchmark set points green line represent instances lpdpe lpdps faster points right top blue line represent instances solved within time limit one hour lpdpe lpdps sea optimal longest paths dynamic programming solver dfbnb exhaustive dfs lpdpe lpdps grid mazes number solved instances road network word assoc total table number instances solved within time limit one hour tested solver configurations collection benchmark problems total previous algorithms kind tested benchmark except easy problems problems typically solved seconds algorithms cases time algorithm spent partitioning phase moreover lpdp algorithms solve significantly problems seen cactus plots well table problem instances solved five solvers within time limit figure provide speedup lpdpe lpdps three original algorithms instances speedup actually data know happens easy problems solvable within couple seconds solver slowdown easy instances due overhead caused partitioning nevertheless average speedups still lpdpe dfbnb exhdfs respectively lpdps dfbnb exhdfs respectively differences running time highest grid maze instances lowest word association graph problems believe due structure graphs particular well partitioned loosely connected subgraphs algorithm excels problems successfully partitioned competitive kinds graphs evaluating algorithm different partitioning configurations see spending extra time partitioning get better solutions pays particular lpdps able solve instances especially instance appears hard worth invest time partitioning additionally depends well graphs partitioned highest grid mazes smallest word association looking scatter plot figure see lpdpe faster instances significantly unsolved instances nevertheless three instances solved ldpde lpdps three instances come different benchmark set shows spending effort partitioning necessarily increase number solved instances conclusion presented optimal algorithm longest path problem undirected graphs based dynamic programming experiments show new algorithm faster nontrivial problems previous optimal algorithms solve significantly benchmark instances time limit per instance given important future work includes parallelization algorithm improve solver speed even balyo references balyo fieger schulz kalp karlsruhe longest paths homepage http brucker scheduling algorithms new york secaucus usa demetrescu goldberg johnson shortest path problem ninth dimacs implementation challenge volume american mathematical url http fieger using graph partitioning accelerate longest path search bachelor thesis karlsruhe institute technology garey johnson computers intractability guide theory freeman new york usa university milano laboratory web algorithms datasets url http ozdal wong routing algorithm highperformance printed circuit boards ieee transactions design integrated circuits systems ozdal wong algorithmic study bus routing highspeed boards ieee transactions design integrated circuits systems portugal rocha msp algorithm patrolling based territory allocation using balanced graph partitioning proceedings acm symposium applied computing sac pages new york usa acm sanders schulz engineering multilevel graph partitioning algorithms proceedings european symposium algorithms volume lncs pages springer sanders schulz distributed evolutionary graph partitioning proceedings workshop algorithm engineering experimentation alenex pages sanders schulz think locally act globally highly balanced graph partitioning proceedings international symposium experimental algorithms sea volume lncs pages springer soper walshaw cross combined evolutionary search multilevel optimisation approach journal global optimization stern kiesel puzis feller ruml max min solving maximization problems heuristic search proceedings seventh annual symposium combinatorial search socs wong lau king information retrieval networks using genetic algorithm special interest tracks posters international conference world wide web www pages new york usa acm sea
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analysis models coupled dynamics may sander garth richard andrew abstract article considers iterative solution finite element discretisation magma dynamics equations simplified form magma dynamics equations share features stokes equations therefore formulate analyse numerically test elman silvester block preconditioner magma dynamics prove analytically demonstrate numerically optimality preconditioner presented analysis highlights dependence preconditioner parameters magma dynamics equations affect convergence iterative linear solvers analysis verified range threedimensional numerical examples unstructured grids simple illustrative problems large problems subduction geometries computer code reproduce numerical examples freely available supporting material key words magma dynamics mantle dynamics finite element method preconditioners ams subject classifications introduction mantle earth extends bottom crust top iron core mantle rock composed silicate minerals behaves elastic solid time scale seismic waves geological time mantle convects high rayleigh number creeping viscous fluid convective flow hidden engine plate tectonics giving rise plate boundaries ridges divergent subduction zones convergent plate boundaries host vast majority terrestrial volcanism volcanoes fed magma extracted partial melting mantle rock occurs typically depths less partially molten regions mantle interest geoscientists role tectonic volcanism chemical evolution earth depth regions makes inaccessible direct observation hence studies dynamics typically involved numerical simulation simulations often based system partial differential equations derived mckenzie since elaborated generalised authors equations describe two interpenetrating fluids different density vastly different viscosity solid molten rock mantle magma grains rock form viscously deformable permeable matrix magma percolate captured theory coupling stokes equations mantle darcy law magma although phase independently incompressible mixture allows divergence convergence solid matrix locally increasing decreasing volume fraction magma process modulated compaction mathematical institute university oxford andrew wiles building radcliffe observatory quarter woodstock road oxford united kingdom department earth sciences university oxford south parks road oxford united kingdom department engineering university cambridge trumpington street cambridge united kingdom department earth sciences university oxford south parks road oxford united kingdom mathematical institute university oxford andrew wiles building radcliffe observatory quarter woodstock road oxford united kingdom rhebergen wells katz wathen viscosity gives rise much interesting behaviour associated coupled dynamics governing equations solved variety contexts idealised studies localisation wave behaviour applied studies boundaries especially ridges studies employed finite volume techniques regular cartesian grids unlike ridges subduction zones plate geometry awkward cartesian grids however conveniently meshed triangles tetrahedra also focus resolution needed finite element simulations pure mantle convection subduction zones common literature remains challenge model dynamics subduction even though area active research models require highly refined computational meshes resulting large systems algebraic equations solve systems efficiently iterative solvers together effective preconditioning techniques necessary although governing equations similar stokes flow prior analysis discretisation numerical solution finite element method computationally expensive step modelling partially molten mantle typically solution problem velocity solid matrix address bottleneck context large unstructured grids finite element discretisations describe analyse test preconditioner algebraic system resulting simplified mckenzie equations system equations similar stokes problem preconditioner proven optimal iteration count iterative method independent size algebraic system variety discretisations stokes equations see also key lies finding suitable approximation schur complement block matrix resulting finite element discretisation follow approach prove demonstrate numerically optimality preconditioner coupled dynamics problems analysis numerical examples highlight issues specific dynamics simulations regarding impact model parameters solver performance best knowledge together work katz takei present first three dimensional computations simplified mckenzie equations first analysis preconditioner problem work incorporate analysis subduction examples software implementation analysis confirmed numerical examples range illustrative cases large representative models subduction zones solved using parallel computers computer code reproduce presented examples parallelised freely available lesser gnu public license lgpl part supporting material proposed preconditioning strategies implemented using libraries fenics project petsc fenics framework provides high degree mathematical abstraction permits proposed methods implemented quickly compactly efficiently close correspondence mathematical presentation paper computer implementation supporting material outline article follows section introduce simplified mckenzie equations coupled dynamics followed finite element method equations section preconditioner analysis conducted section construction discussed section numerical simulations analysis preconditioners coupled dynamics section verify analysis conclusions drawn section partially molten magma dynamics let bounded domain mckenzie model reads porosity matrix velocity strain rate tensor permeability melt viscosity shear bulk viscosity matrix respectively constant acceleration due gravity unit vector melt pressure constant melt matrix densities respectively density assume constants function magma fluid velocity obtained useful decompose melt pressure dynamic pressure lithostatic pressure equations may written constitutive relations given characteristic porosity characteristic permeability dimensionless constant ratio matrix bulk shear viscosity using primed variables velocity scaling given length scale dropping prime notation mckenzie equations form given rhebergen wells katz wathen compaction length defined becomes solving mckenzie model numerically simulations usually decoupled porosity updated velocity pressure determined solving iteration used better capture coupling expensive part procedure solving work study optimal solver equations given porosity field remark alternative decoupling use composable linear solver full system see brown case optimal solver may used preconditioner part composable linear solver rest paper replace constant furthermore replace spatially variable function independent obtain problem coupled dynamics problems may range approximately reason assume paper also bound setting note everywhere compaction stress vanishes velocity field divergence free reduces stokes equations generally case finite element formulation discussed following section boundary domain impose given boundary data satisfying compatibility condition finite element formulation section assume without loss generality homogeneous boundary conditions analysis preconditioners coupled dynamics let triangulation associated finite element spaces finite element weak formulation given find proposition exists proof proposition follows see ref application korn inequality consider finite elements stable degenerate limit coercive see proposition exists constant independent max kqh particular use finite elements simplices note setting stokes equations recovered generally case discrete weak formulation obtaining stokes limit finite element setting requires property divergence functions lie pressure space case finite elements discrete system written block matrix form rnu rnp respectively vectors discrete velocity pressure variables respect appropriate bases space satisfies zero mean pressure condition rhebergen wells katz wathen later convenience define negative pressure schur complement scalar pressure mass matrix kqh hqq rnp vector coefficients associated pressure basis denotes standard euclidean scalar product differences matrix formulation equations stokes equations lie matrices case dynamics includes discretisation compaction stresses term weighted factor terms known problematic context multigrid methods modes associated lowest eigenvalues well represented coarse grid number investigations issue div finite element problems second matrix differs stokes discretisation sufficiently large term provides pressure stabilisation elements would otherwise unstable stokes problem optimal block diagonal preconditioners model dynamics subduction efficient iterative solvers together preconditioning techniques needed solve resulting algebraic systems equations goal section introduce prove optimality class block diagonal preconditioners prove optimality block preconditioner mckenzie problem first present number supporting results proposition bilinear form satisfies proof follows directly lemma matrices given pressure schur complement pressure mass matrix stable formulation satisfying following bounds hold hsq given analysis preconditioners coupled dynamics min constant constant proof since symmetric positive definite definition hsq hck sup hck hav definition matrices follows hsq sup using inequality kqh hence kqh combining hsq kqh hqq hck proves upper bound next determine lower bound using condition max max kqh rhebergen wells katz wathen leads hsq hqq hck using proposition inequality hck kqh hck hqq combining hqq hck hsq setting case otherwise setting hsq min deduced discretisation stokes equations shown pressure spectrally equivalent schur complement recovered lemma everywhere lemma matrices pressure mass matrix condition satisfied hav rnu constant proof lemma symmetry positive inserting defining denoting maximum eigenvalue associated eigenvector since symmetric follows hence sides analysis preconditioners coupled dynamics letting becomes follows rnu taking rnu lemma follows consider diagonal block preconditioners form rnu rnp assume symmetric satisfy hav rnu independent may depend model parameters discrete system indefinite hence positive negative eigenvalues speed convergence minres krylov method preconditioned system depends tightly positive negative eigenvalues generalised eigenvalue problem clustered section aim develop bounds eigenvalues independent mesh parameter theorem let constants lemma matrices given pressure schur complement pressure mass matrix satisfy eigenvalues satisfy eigenvalues satisfy rhebergen wells katz wathen proof lemmas provide bounds hsq hav rnu using bounds together bounds given result follows directly following proof theorem elman generally pestana wathen main result section theorem states eigenvalues generalised eigenvalue problem independent problem size theorem see constants independent problem size independent tells find spectrally equivalent respectively iterative method preconditioner optimal interval shows dependence eigenvalues upper lower bounds positive eigenvalues well behaved lower bound negative eigenvalues upper bound negative eigenvalues tends zero case rate convergence iterative method may slow see time preconditioner construction implementation proposed preconditioner requires provision symmetric positive definite matrices satisfy obvious candidates direct solver used compute action use small problems following section study performance block preconditioning application direct solver practical however large case advocate use multigrid approximations inverse provide general guidance first reproduce following lemma elman lemma lemma solution system iteration error satisfies hav proof see elman proof lemma lemma implies solver optimal satisfy therefore candidate likewise obvious candidates multigrid preconditioners applied respectively however show example section increases therefore compaction stresses term become important multigrid becomes less effective preconditioner effective treatment large case subject ongoing investigations analysis preconditioners coupled dynamics numerical simulations section verify analysis results numerical examples test cases use finite elements simplices numerical examples deliberately address points practical interest spatial variations parameter wide range values large problem sizes unstructured grids subduction geometries consider two preconditioners first take apply direct solver compute action inverses preconditioner referred preconditioner second use aamg amg use amg denote use algebraic multigrid approximate inverse preconditioner referred amg preconditioner preconditioner introduced reference preconditioner amg preconditioner compared preconditioner suitable large scale problems note never construct inverse use action inverse tests use minres method solver terminated relative true residual reached multigrid approximations smoothed aggregation algebraic multigrid used via library multigrid approximations classical algebraic multigrid used via library boomeramg unless otherwise stated use multigrid two applications chebyshev jacobi smoothing level pre post case smoothed aggregation symmetric classical algebraic multigrid computer code developed using finite element library dolfin block preconditioner support petsc construct preconditioners computer code reproduce examples freely available supporting material verification optimality test case verify optimality block preconditioned minres scheme observing convergence solver varying solve unit square domain using regular mesh triangular cells permeability consider tanh tanh tanh tanh ignore body forces add source term right hand side dirichlet boundary condition source term constructed exact solution pressure velocity cos cos sin sin cos cos table shows number iterations minres method required converge using amg preconditioners varying clearly see preconditioner optimal iteration count independent problem size predicted analysis see theorem using amg preconditioner slight dependence problem size results table indicate preconditioner rhebergen wells katz wathen table number iterations amg preconditioned minres unit square test different levels mesh refinement different values number denoted case four applications chebyshev smoother one symmetric iteration application used amg amg amg amg ltx lbx fig description wedge geometry subduction zone uniform respect theorem indicates possible dependence constant however sufficiently small sufficiently large dependence becomes negligible small impact iteration count amg preconditioner hand shows strong dependence issue multigrid solvers discussed section manifest table observed tests effectiveness multigrid preconditioned solver operator deteriorates increasing manifest increasing increasing results case large spatial variations permeability presented tables cases respectively dependence iteration count permeability observed smaller larger iteration counts amg preconditioners also observe given little influence iteration count comparing results tables see preconditioner shows dependence amg preconditioner iteration count increases increases magma dynamics problem two dimensions test case solve domain depicted figure using unstructured meshes triangular cells take ltx set permeability tanh porosity analysis preconditioners coupled dynamics table number iterations reach relative tolerance using preconditioned minres unit square test varying levels mesh refinement varying pairs number degrees freedom denoted amg amg amg amg amg amg amg amg amg amg amg amg amg amg amg amg consider two test cases geometry first test problem denote analytical corner flow test problem second test lem problems prescribe following conditions uslab analytic corner flow analytical corner flow problem prescribe ucorner analytic expression section velocity components given cos sin sin cos arctan sin sin cos sin cos sin sin cos sin rhebergen wells katz wathen table number iterations reach relative tolerance using preconditioned minres unit square test varying levels mesh refinement varying pairs number degrees freedom denoted amg amg amg amg amg amg amg amg amg amg amg amg amg amg amg amg angle figure show computed streamlines magma matrix velocity fields problem table presents number solver iterations amg preconditioners different values observe similar behaviour saw test section preconditioner optimal uniform amg preconditioner shows slight dependence problem size increased iteration count grows problem problem instead prescribing ucorner prescribe boundary condition figure shows computed streamlines magma matrix velocity fields problem solver iteration counts problem different levels mesh refinement different values presented table analytic corner flow problem section preconditioner optimal uniform expected using preconditioner solver uniform analysis preconditioners coupled dynamics fig streamlines magma light matrix dark velocity fields wedge subduction zone using corner flow boundary condition solution computed mesh elements table number iterations required corner flow problem using amg preconditioned minres different levels mesh refinement varying case four applications chebyshev smoother one symmetric iteration application used amg amg amg respect magma dynamics problem three dimensions final case test solver problem geometrically representative subduction zone solve domain depicted figure set ltx lbx use unstructured meshes tetrahedral cells set permeability tanh porosity boundary conditions prescribe uslab figure show computed vector plots matrix magma velocities table shows number iterations needed amg preconditioned minres method wedge problem preconditioned solver practical problem using reasonable mesh resolutions cases computed parallel using processes computed examples span range problem sizes relatively small changes iteration count observed changes number degrees freedom becomes larger iteration count conclusions work introduced analysed optimal preconditioner finite element discretisation simplified mckenzie equations dynamics analysis preconditioner showed schur rhebergen wells katz wathen fig streamlines magma light matrix dark velocity fields wedge subduction zone using stress boundary conditions solution computed mesh elements table number iterations reach relative tolerance using amg preconditioned minres different values test case four applications chebyshev smoother one symmetric iteration application used amg amg amg plement block matrix arising finite element discretisation simplified mckenzie equations may approximated pressure mass matrix plus permeability matrix analysis verified numerical simulations unit square wedge flow problems inspired subduction zones computations used finite elements stable degenerate limit vanishing permeability numerical tests demonstrated optimality solver observed multigrid version preconditioner uniform respect ratio increased iteration count solver increases observe similar behaviour increases analysis testing optimal block preconditioning method dynamics presented work lays basis creating efficient optimal simulation tools ultimately put use study genesis transport magma subduction zones optimality demonstrated open questions remain regarding uniformity respect model parameters acknowledgements thank alisic rudge many discussions held related paper also thank reviewers knepley spiegelman wilson one remained anonymous whose comments helped improve paper authors acknowledge support natural environment research council grants katz furthermore analysis preconditioners coupled dynamics ltx lbx fig description wedge subduction zone matrix velocity magma velocity matrix velocity magma velocity fig vector plots magma matrix velocities wedge threedimensional subduction zone using boundary conditions grateful support leverhulme trust references aharonov whitehead kelemen spiegelman channeling instability upwelling melt mantle geophys logg rognes wells unified form language language weak formulations partial differential equations acm trans math software rhebergen wells katz wathen table number iterations required amg preconditioned minres subduction model different levels mesh refinement different values number degrees freedom denoted case four applications chebyshev smoother one symmetric gaussseidel iteration application used tests run using mpi processes arnold falk winther preconditioning div applications math arnold falk winther multigrid div curl numer balay gropp mcinnes smith efficient management parallelism object oriented numerical software libraries arge bruaset langtangen editors modern software tools scientific computing pages press balay brown buschelman eijkhout gropp kaushik knepley mcinnes smith zhang petsc users manual technical report revision argonne national laboratory balay brown buschelman gropp kaushik knepley mcinnes smith zhang petsc web page url http barcilon richter nonlinear waves compacting media fluid batchelor introduction fluid dynamics cambridge university press new york bercovici ricard energetics model lithospheric damage shear localization formation geophys brezzi fortin mixed hybrid finite element methods springerverlag new york brown knepley may mcinnes smith composable linear solvers multiphysics international symposium parallel distributed computing ispdc pages url http elman silvester wathen finite elements fast iterative solvers numerical mathematics scientific computation oxford university press gee siefert tuminaro sala smoothed aggregation users guide sandia national laboratories tech ghods melt migration beneath ridges geophys analysis preconditioners coupled dynamics grinevich olshanskii iterative method problem variable viscosity siam sci henson yang boomeramg parallel algebraic multigrid solver preconditioner appl numer katz magma dynamics enthalpy method benchmark solutions magmatic focusing ridges katz takei consequences viscous anisotropy deforming aggregate part numerical solutions full equations fluid katz spiegelman holtzman dynamics melt shear localization partially molten aggregates nature katz knepley smith spiegelman coon numerical simulation geodynamic processes portable extensible toolkit scientific computation phys earth planet keller may kaus numerical modelling magma dynamics coupled tectonic deformation lithosphere crust geophys kolev vassilevski parallel auxiliary space amg solver div problems siam sci logg wells dolfin automated finite element computing acm trans math software logg mardal wells editors automated solution differential equations finite element method volume lecture notes computational science engineering springer may moresi preconditioned iterative methods stokes flow problems arising computational geodynamics phys earth planet mckenzie generation compaction partially molten rock wells optimisations quadrature representations finite element tensors automated code generation acm trans math software pestana wathen natural preconditioners saddle point systems url http rhebergen katz wells wathen supporting material url https schubert turcotte olson mantle convection earth planets cambridge university press silvester wathen fast iterative solution stabilised stokes systems part using general block preconditioners siam numer simpson spiegelman weinstein multiscale model partial melts effective equations geophys simpson spiegelman weinstein multiscale model partial melts numerical results geophys spiegelman flow deformable part simple analysis fluid spiegelman flow deformable porous media part numerical relationship shock waves solitary waves fluid rhebergen wells katz wathen takei katz consequences viscous anisotropy deforming aggregate part governing equations linearised analysis fluid van keken currie king behn cagnioncle katz lin parmentier spiegelman wang community benchmark subduction zone modeling phys earth planet wilson spiegelman van keken terraferma transparent finite element rapid model assembler problems earth sciences url https
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bounded degree cosystolic expanders every dimension shai evra tali kaufman jan january abstract work present new local global criterion proving form high dimensional expansion term cosystolic expansion applying criterion ramanujan complexes yields every dimension infinite family bounded degree complexes topological overlapping property answer affirmatively open question raised gromov introduction expander graphs central objects study computer science mathematics past decades informally expanders sparse highly connected graphs numerous applications see hlw references therein recent years new theory high dimensional expanders emerged pioneered works linm gromov gro recent survey see gromov suggested two essentially equivalent generalizations notion graph expansion higher dimensions even though works completely different research directions gromov question gro gromov studied complexity embedding simplicial complexes euclidean spaces specifically considered following topological overlapping property definition simplicial complex overlapping property every continuous map exists point whose preimage intersects least family simplicial complexes said topological overlapping property exists member family overlapping property gromov proved remarkable result fixed family complete ddimensional complexes topological overlapping property result striking generalization classical results convex combinatorics due hebrew university israel email research supported part erc university israel email kaufmant research supported part irg erc bsf barany bar gromov proceeded give examples families complexes posses topological overlapping property spherical buildings random complexes however examples provided unbounded degree number faces incident single vertex grows number vertices complex naturally led gromov following question question gromov infinite family bounded degree simplicial complexes topological overlapping property arbitrary recent work kkl kaufman kazhdan lubotzky able give affirmative answer gromov question dimension two showed exists infinite family bounded degree complexes topological overlapping property see kkl theorem however proof method inherently suited dimension two work give complete answer gromov question theorem every exist infinite family bounded degree complexes topological overlapping property immediate application theorem improvement result gromov guth theorem gives high dimensional generalization classical result kolmogorov barzdin corollary every exists infinite family ddimensional bounded degree complexes satisfies following embedding images two simplices distance least volume image least criterion topological overlapping property let start addressing question one prove certain complex posses topological overlapping property gro gromov showed topological overlapping property implied certain notion expansion wish define first need notations definition let simplicial complex define collection faces space map ker space space norm position present first notion high dimensional expansion definition coboundary expanders let simplicial complex let say expander every expb min notion coboundary expansion first appeared implicitly work linm whose motivation generalize higher dimensional complexes phase transition phenomenon connectivity random graphs model later gromov gro came across essentially notion expansion studying fiberwise complexetiy showing coboundary expansion implies topological overlapping property unfortunately date bounded degree coboundary expanders known exists one use gromov result answer question problem coboundary expansion requires vanishing complex property strong even certain ramanujan complexes known satisfy see kkl order bypass problem define weaker notion expansion call cosystolic expansion notion strictly weaker coboundary expansion since allows existence cocycles coboundaries long sufficiently large definition cosystolic expanders let simplicial complex let say expander every expkz min systk min kzk dkw authors strengthen aforementioned result gromov proving cosystolic expansion implies topological overlapping property theorem gromov topological overlapping criterion dkw every exist expander overlapping property hence gromov topological overlapping criterion order prove theorem one needs prove existence infinite family bounded degree cosystolic expanders criterion cosystolic expansion first infinite family bounded degree cosystolic expanders constructed kkl shown skeletons see ramanujan complexes sufficiently large degree cosystolic expanders work intention generalize results kkl dimensions however proof method kkl specifically designed case indeed order apply proof dimension three higher one need assume graph whose vertex set set edges complex edge set set pairs edges form triangle complex excellent expander graph turns condition strong hold ramanujan complex main novelty work present new method transforming expansion lower dimension high dimension complex allow prove main result local global criterion cosystolic expansion dimensions state criterion need define notions skeletons links skeleton expander complex definition let simplicial complex skeleton denoted complex obtained deleting faces dimension greater link face denoted complex obtained picking faces contains removing faces link called proper link complex said expander satisfy following graph expansion property kak edges vertices ready state local global criterion cosystolic expansion theorem main theorem exists complex satisfying degree proper links expanders proper links expanders expander seems natural compare local global expansion result garland gar however philosophically two results unrelated garland work moreover proof method results quite different ramanujan complexes well known ramanujan complexes see constructed fixed residue field bounded degree high dimensional complexes excellent skeleton expanders follows directly ramanujan property proper links spherical buildings admitting strongly transitive action follows work gromov gro see also lmm coboundary expanders therefore left prove links spherical buildings also good skeleton expanders theorem exist prime power spherical buildings admitting strongly transitive action expanders conclusion ramanujan complexes sufficiently large fixed degree satisfies requirements theorem together gromov topological overlapping criterion theorem implies following corollary regarding expansion ramanujan complexes corollary exists prime power exists skeleton ramanujan complex constant definition skeleton expansion unnatural implies notion strictly weaker graph expansion see accurate notion skeleton expansion however notion skeleton expansion sufficient use completion paper izhar oppenheim pointed similar result theorem could potentially deduced work opp special case theorem proof exploits geometric structure spherical buildings allows get proof significantly shorter expander overlapping property finally combining explicit construction ramanujan complexes corollary get explicit construction bounded degree cosystolic expanders topological overlapping property every dimension theorem interesting note random construction bounded degree high dimensional complexes topological overlapping property currently known organization paper section review basic definition related simplicial complexes norms links skeletons high dimensional expansions section prove local global criterion cosystolic expansion theorem section prove one sided mixing lemma regular complexes section review definition spherical buildings show good skeleton expanders finally section prove ramanujan complexes satisfies conditions theorem hence skeletons therefore examples bounded degree complexes topological overlap property acknowledgments authors grateful david kazhdan alex lubotzky useful discussions advices ron rosenthal many valuable improvements paper first author would like especially thank alex lubotzky introducing teaching encouraging work problem without work would possible also thank erc bsf support work part thesis first author hebrew university jerusalem israel preliminaries complexes expansions section present basic definitions properties simplicial complexes norms well notions expansions complexes throughout paper shall use following notations regarding simplicial complexes simplicial complex set vertices family subsets closed inclusions note always face complex call elements faces simplices dimension simplex defined dim dimension entire complex defined maximal dimension simplex dim maxf dim complex said pure maximal faces dimension convenience sake mean face complex always mean finite pure simplicial complex let denote collection space usual denote map ker spaces respectively recall hence since working indicator function subset vice versa subset defines supp identify subset following definition norm define following norm space cochains kak note kak kak kak kbk equality disjoint definition container let let define following lemma let let kak kak proof first note one hand kak hand since contains belong hence previous calculation get kak finishes proof links next introduce notion link complex combinatorial analogue unit sphere complex thus links serve local views complex definition link let let link defined following denote space norm associated complex map let observe following cases degenerate links link unique empty set entire complex link collection isolated vertices order avoid degenerate cases proper link mean link face dimension dim definition localization lifting let let define following maps original complex link complex first map called localization takes cochain original complex restrict faces contains delete producing cochain link concretely second map called lifting takes cochain link complex adds face producing cochain original complex concretely let mention immediate observations localization lifting operators let following holds following three lemmas describes relations localization lifting operators global local norms complex norms original complex links lemma let let proof denote weight function original complex weight function link language links weight norm interpreted similarly since norm cochain define extending linearly weight function suffice show claim singleton finishes proof lemma let let proof kak follows immediately lemma first note every face contains combining observation equation get kak using assumption kak imply finally applying fact last inequality together equation yields finishes proof lemma let let kak proof definition norm fubini theorem kak skeletons define notions skeletons complex basically get forget higher dimensional faces given complex definition skeleton let let defined following dim example complex simply underlying graph note spaces complex operators complex also hand norms complex might quite different however complex bounded degree norms proportional lemma let let degree dimension kakx kakx kakx proof denote weight function original complex weight function skeleton since norm cochain define extending linearly weight function suffice show claim singleton suffice prove next note since complex pure degree dimension similarly since link pure degree dimension get finishes proof next define property graph expansion complex called skeleton expansion says complex satisfies form weak mixing behavior definition skeleton expander let let call expander kak edges vertices section prove one sided mixing lemma skeleton regular complex gives criterion skeleton expansion terms non trivial eigenvalues high dimensional expansion present different notions concerning high dimensional expansion simplicial complexes arose works linm gromov gro definition coboundary cocycle expansion let define cobonudary expansion parameter expkb min define cocycle expansion parameter expkz min let spell expansion parameters says special case graphs graph case coboundaries cocycles unions connected components graph cobonudary expansion parameter equal cheeger constant entire graph normalization cocycle expansion parameter equal minimum cheeger constants connected component graph large cocycle expansion parameter imply connected component graph good expander however graph disconnected particular expander let recall definition coboundary expanders introduction definition coboundary expander said expander expkb notion coboundary expansion first originate work linm connection vanishing cohomology actual term coboundary expansion later came recall cohomology quotient space simple exercise see following equivalences holds expkb furthermore cohomology trivial hence expansion parameters equivalent expkb expkz get following equivalent characterization coboundary expansion expkb expkz noted gromov see also dkw notion vanishing cohomology strong application since existence cocycle coboundary acceptable long small definition cosystoles comes play definition cosystoles let define minimal size systk min kzk finally recall definition cosystolic expanders introduction definition cosystolic expander said expander expkz systk remark recent years notion cosystoles close cousin systoles found applications field quantum error correcting codes see eot good lower bounds systoles cosystoles give rise quantum codes good parameters minimal cochains let introduce following notions minimal locally minimal cochains need later prove local global cosystolic expansion criterion definition mininmal locally minimal cochain said minimal kak min cochain said locally minimal localization minimal cochain link note expander minimal satisfies kak lemma minimal cochain locally minimal cochain proof let let lemma part get minimality equation kak proves locally minimal lemma minimal cochain also minimal cochain proof first note since disjoint kak next note since sum two cochains equal symmetric difference cochain second last step equality follows fact hence kak coboundry minimality get kak kak kak last inequality finishes proof cosystolic expansion criterion section prove following local global cosystolic expansion criterion following result slightly stronger theorem introduction theorem exists satisfying complex degree link expander link expander expkz systr particular expander order prove theorem follow kkl strategy noticed following isoperimetric inequality small cochains imply cosystolic expansion theorem exists satisfying link expander link expander locally minimal kak kak remark note theorem unlike theorem require bounded degree assumption simple fact makes theorem useful also unbounded degree complexes shown llr proved exists coboundary expanders bounded degree dimension every contained bounded number work llr generalization lubm remark constants theorem min constants theorem note made attempt optimize constants work sketch proof theorem diving proof theorem require define new technical notions would like provide proof sketch fix locally minimal cochain define notion fat faces respect complex essentially says face fat localization respect face large consider coboundaries localization respect fat faces note local coboundaries large assumption links cobundary expanders main idea proof local coboundary either actual coboundary original complex contains fat face smaller dimension last claim content proposition term essentially stands elements contains fat stands error term error term negligible due skeleton expansion assumption proposition iterating proposition get either large fat unique fat precisely large finishes proof fat machinery order prove isoperimetric inequality theorem first construct fatmachinery allow move calculations higher lower dimensions complex definition fat faces fix cochain define inductively fat faces call elements fat faces fatness constant following lemma shows sizes cochains fat faces bounded size original cochain constant lemma let kak proof fat applying lemma cochain get hence combining lemma hence iterating equation get kak kak finishes proof lemma get following consequence says small cochain unique face simple fact serve finishing argument proof theorem corollary let kak unique empty set fat face proof note empty set following interesting property local view everything lemma assumption size get kak hence definition empty set fat face next define cochain degenerate faces intuitively one think one trying move higher dimension lower dimension definition degenerate faces fix cochain pair two equal sized fat faces whose intersection face face said degenerate contains define cochain degenerate next define notion fat ladder definition fat ladders fix cochain fat define siting define next lemma essentially says inside fat ladder fat either deeper ladder contain dead end makes ambient face degenerate lemma let let either proof definition exists fat define fat since removing repetitions needed get otherwise may assume maximal fat since otherwise fat get hence proof theorem following proposition heart fat machinery allow move calculations higher lower dimensions proposition let let link expander locally minimal proof let evaluate following expression one hand since locally minimal minimal lemma also minimal assumption proper links expanders get note hence equation combining equation lemma part fact element contains faces get hand one following three possibilities must occur belongs contains case belongs contains case since must least one lemma get belongs contain case like since must least one lemma get either contains hence definition conclusion get combining equation together lemma yields krk finishes proof following proposition gives effective bound cochain degenerate faces terms skeleton expansion fatness constant let proposition let link expander kak proof denote contains least two different let lemma definition faces get lemma get skeleton expansion get lemma multiply sides get next definition fat faces summing last equality follows lemma applying lemma get combining equations together get kak kak kak finishes proof finally able prove isoperimetric inequality small cochains theorem proof theorem define following constants note definition kak corollary empty set non fat hence therefore constant get kak applying proposition equation constant get note kak combining proposition get kak kak finishes proof proof theorem fact coisoperimetric inequality small cochains theorem implies cosystolic expansion theorem first shown kkl sake add argument proposition kkl proposition let define let exists satisfies cochain locally minimal kak kak proof first note integer prove claim induction base case empty claim holds empty assume claim holds cochains kak locally minimal claim holds empty otherwise exist denote equation lemma part get kak since natural numbers induction assumption exist locally minimal cochain kak kak since get kck kak finishes proof hence taking noting kck using proposition isoperimetric inequality small cochains theorem able prove cosystolic expansion criterion theorem proof theorem let constants theorem let define min begin proving cocycle expansion let first note since kak cochain kak let assume let proposition apply cochain locally minimal cochain theorem fact get hence proposition part get min gives cocycle expansion expkz next prove cosystolic bound let cocycle exists nothing prove proposition let locally minimal kak note since also theorem fact cocycle get contradiction since kak gives cosystolic bound systk skeleton mixing lemma purpose section prove one sided mixing lemma regular complex see giving spectral criterion remark note mixing behavior spectral gaps random walks high dimensional complexes subjects already intensively studied several works see egl opp par prt ros mixing lemma present much simpler ones appearing mentioned works actually result graph rather complexes definition regular complex said regular satisfies exist partition vertices thereqexist kij contained exactly kij faces example simplest case dimension graph regular complex bipartite biregular graph remark note regular complex links skeletons next define second eigenvalue regular complex definition eigenvalue let regular define induced bipartite graph denote normalized second largest eigenvalue define normalized largest eigenvalue max let state one sided mixing lemma regular complex proposition let regular complex let normalized largest eigenvalue kak kbk kak kbk edges vertices note case mixing lemma already known lemma egl corollary let bipartite biregular graph let normalized second largest eigenvalue bipartite mixing lemma imply general skeleton mixing lemma proof proposition first note since regular complex let partition kak particular kak kbk restating lemma terms complex norm get kak kbk kak kbk kak kbk kak kbk let denote since partite hence similarly since kai kbj kai kbj kai kbj kai kbj kak kbk next note numbers one max applying kai kbj get kai kbj kai kbj kak kbk finishes proof particular since get following corollary let regular expander spherical buildings object section introduce notion spherical buildings show good skeleton expanders definition spherical buildings give definition spherical buildings list properties shall use buildings refer defining building let first define notion chamber complex definition chamber complex simplicial complex said chamber complex pure maximal faces two maximal faces sequence intersection chamber complexes call chamber call panel call sequence gallery chamber complex said thin panel contained exactly chambers said panel contained exactly chambers thick chamber complex mean one way define building follows equivalence common definition see theorem definition building building thick chamber complex together family subcomplexes called apartments satisfy following axioms apartment thin chamber complex two faces complex contained common apartment two apartments isomorphism fixes intersection building said respectively let note building chamber complex satisfy axioms building apartment also remark throughout paper concerns buildings simplicial complexes however mentioned buildings also complexes allowing chamber complexes let present example spherical building example let prime power denote fdq consider simplicial complex whose vertices proper subspaces faces flags subspaces spherical building moreover group gld acts strongly transitive way see next wish list basic properties spherical buildings complete proofs refer lemma define max apartment spherical building size proof theorem apartment spherical building spherical coxeter complex spherical coxeter complexes completely classified taken maximal size possible complexes corollary define max spherical building size proof let chamber building since building get chambers gallery distance iterating fact get chambers gallery distance since chamber building contained apartment containing since distance two chambers inside apartment size apartment lemma claim proven remark bound size spherical building mean tight since trying optimize constants work definition type function complex said admit function vertices function setting lemma proposition let building admits function vertices lemma proposition let building let face link also building lemma proposition let building let apartment gallery convex two chambers gallery building minimal length among possible galleries gallery sits completely inside apartment lemma proposition let spherical building let chamber apartment containing unique chamber denoted maximal gallery distance next define notion building posses many symmetries definition strongly transitive action building said posses strongly transitive action exist group automorphisms building aut preserves function building defined lemma two pairs chamber apartment containing chamber exists lemma let building group acts strongly transitively regular complex defined proof lemma exists let two type sets let two faces choosing two chambers contains respectively second property strong transitivity exists also first property strong transitivity preserves types hence since automorphism faces containing mapped bijectively faces containing particular cardinality proving building regular throughout section shall make use following notion group theory definition stabilizer let simplicial complex aut group automorphisms define stabilizer following subgroup stabg lemma let building group acts strongly transitively face apartment containing every passes apartment exists proof let maximal face containing let maximal face containing second axiom building exists apartment contains exists one hand preserves type function hence stabg hand particular needed lemma let building posses strongly transitive action let link also posses strongly transitive action proof let aut group acts strongly transitive define stabg stabilizer admits action link aut action since action pair chamber apartment containing inside lifted pair chamber apartment containing inside let two pairs pair contains chamber apartment containing chamber inside lifting pairs since pairs contains hence finishes proof expansion spherical buildings first observed gromov gro spherical buildings coboundary expanders see lmm simplified proof theorem exist spherical building expander purpose subsection prove spherical buildings sufficiently large thickness good skeleton expanders theorem let spherical building posses strongly transitively action expander max strategy proving theorem follows first define property bipartite graphs show graphs good bound second largest eigenvalue second show graphs spherical building posses strongly transitively action satisfy property finally applying corollary get skeleton expansion definition symmetric convex graph let bipartite graph let aut group acts graph automorphisms let vertex denote stabg stabilizer say graph number unique vertices maximal distance unique vertices maximal distance vertex maximal distance number neighbors dist dist proposition let bipartite connected graph normalized second largest eigenvalue bounded max proof let end adjacency operator let spec set eigenvalues let recall basic facts see egl since undirected graph operator hence spec since bipartite finally since note eigenvectors eigenvalues every vertex eigenvector exist fix vertex let stabg stabilizer define following follows vertices vertices number edges equal number edges vertex set vertices note independent choice finally let end adjacency operator let spec set eigenvalues definition get note since directed priori reason eigenvalues real however eigenvalue eigenvector defining get hence eigenvector eigenvalue particular spec spec hand spec pwith eigenvector defining get hence eigenvector eigenvalue particular spec applying trace formula get finally let use properties graph property number vertices property two parts unique vertex maximal distance since hence directed starting ending property vertex number directed starting ending max since corresponds following vertex either dist dist case possible note dist dist dist dist case possible get directed starting ending combining get max noting finishes proof following proposition shows bipartite graphs spherical buildings constant depends dimension building thickness proposition let spherical building posses strongly transitive action let let induced bipartite biregular graph graph regularity degrees least max proof let vertex building stabg let fixed apartment contains first let prove claim regularity degrees lemma regular complex hence bipartite biregular graph assume contained panel contain vertex type except contained chambers contains unique vertex neighbour graph course reasoning apply replacing lemma number bounded size apartment building lemma bounded let chamber contains lemma unique chamber maximal gallery distance let unique edge type inside since gallery distance coarser graph distance gallery path contains graph path two farthest vertices graph metric type respectively inside apartment hand since collection automorphisms hence preserves distances lemma unique chamber maximal gallery distance apartment get two apartments unique maximal distance vertices similarly unique maximal distance vertices since maximal distance neighbour dist dist axiom building let apartment contains edge let vertices neighbours satisfy dist dist minimal path passes also pass hence since gallery distance coarser graph distance minimal gallery building maximal face containing maximal face containing also pass maximal face containing apartment containing lemma minimal galleries lies particular lies last inequality lemma finally combining propositions corollary able prove spherical buildings good skeleton expanders theorem proof theorem proposition type induced bipartite graph building proposition normalized largest trivial eigenvalue finally applying corollary get building expander ramanujan complexes ramanujan complexes defined explicitly constructed finite quotients buildings type exhibits excellent spectral properties ramanujan complexes refer readers survey object section show ramanujan complexes sufficiently large degree satisfies requirement theorem namely theorem let ramanujan complex complex degree corollary proper link coboundry expander theorem proper link expander theorem expander proving theorem shall need following lemma lemma proper link ramanujan complex spherical building posses strongly transitive action proof lemma first since ramanujan complex quotient affine building type link ramanujan complex link covering affine building second since building posses strongly transitive action lemmas link also building posses strongly transitive action finally since affine building locally finite proper link finite hence spherical building proof theorem first three claims follows lemma together corollary theorem theorem left show excellent skeleton expander using notation section adjacency operator type induced graph hecke operator restricted graph hence remark max max combining corollary proves expander consequentially combining theorem together theorems give following high dimensional expansion properties ramanujan complexes sufficiently large degree corollary introduction together coisoperimetric inequality small cochains corollary exists skeleton dimensional ramanujan complex locally minimal cochain kak kak expander posses overlapping property finally theorem get prime power exist infinite family ramanujan complexes moreover ramanujan complexes constructed explicitly hence applying corollary ramanujan complexes constructed get affirmative answer gromov question introduction theorem mentioned introduction immediate application theorem get corollary corollary exists infinite family bounded degree complexes satisfies following embedding distance images two non adjacent simplices least volume image least proof corollary combine theorem proposition original result theorem gave slightly weaker result exists infinite family complexes whose degree might depends lower bound volume reader referred learn work kolmogorov barzdin generalization higher dimensions close work following concluding remarks remark theorem corollary holds finite quotient building large degree ramanujan complexes prove generalization one needs prove underlying graph finite quotient building large degree good expander graph done using explicit property get explicit bounds second eigenvalue graph remark note ramanujan complexes sufficiently large degree able prove cosystolic expanders following conjecture suspect ramanujan complexes fact finite quotients buildings cosystolic expanders contrast kkl proposition shown ramanujan complexes sufficiently large degree coboundary expanders cosystolic expansion best one could hope general references abramenko brown buildings theory applications graduate texts mathematics springer bar barany generalization carathodorys theorem discrete mathematics boros furedi number triangles covering center geometriae dedicata barzdin kolmogorov realization nets space probl cybernet see also selected works kolmogorov kluwer academic publishers dotterrer kahle coboundary expanders journal topology analysis dkw dotterrer kaufman wagner expansion topological overlap international proceedings informatics vol schloss fuer informatik egl evra golubev lubotzky mixing properties chromatic number ramanujan complexes international mathematics research notices eot eldar ozols thompson upper bounds systoles highdimensional expanders using quantum codes arxiv preprint fglnp fox gromov lafforgue naor pach overlap properties geometric expanders journal die reine und angewandte mathematik gar garland curvature cohomology discrete subgroups groups annals mathematics gro gromov singularities expanders topology maps part combinatorics topology via algebraic isoperimetry geometric functional analysis gromov guth generalizations kolmogorovbarzdin embedding estimates duke mathematical journal guth lubotzky quantum error correcting codes arithmetic hyperbolic manifolds journal mathematical physics gundert szedlk higher dimensional discrete cheeger inequalities proceedings thirtieth annual symposium computational geometry acm gundert wagner laplacians random complexes proceedings annual symposium computational geometry acm hlw hoory linial wigderson expander graphs applications bulletin american mathematical society kkl kaufman kazhdan lubotzky isoperimetric inequalities ramanujan complexes topological expanders geometric functional analysis kaufman mass high dimensional combinatorial random walks colorful expansion arxiv preprint kaufman mass walking edge cosystolic expansion arxiv preprint knowles rosenthal eigenvalue confinement spectral gap random simplicial complexes arxiv preprint lubotzky expander graphs pure applied mathematics bulletin american mathematical society lubotzky ramanujan complexes high dimensional expanders japanese journal mathematics linm linial meshulam homological connectivity random combinatorica lubm lubotzky meshulam random latin squares expanders advances mathematics lmm lubotzky meshulam mozes expansion complexes groups geometry dynamics israel lubotzky samuels vishne ramanujan complexses type journal mathematics llr lubotzky samuels vishne explicit construction ramanujan comed european journal combinatorics plexes type lubotzky luria rosenthal random steiner systems bounded degree coboundary expanders every dimension arxiv preprint meshulam wallach homological connectivity random complexes random structures algorithms uniform pointwise bounds matrix coefficients unitary representations applications kazhdan constants duke mathematical journal opp oppenheim isoperimetric inequalities local spectral expanders topological expanders arxiv preprint par parzanchevski mixing expanders arxiv preprint prt parzanchevski rosenthal tessler isoperimetric inequalities simplicial complexes combinatorica parzanchevski rosenthal simplicial complexes spectrum homology random walks random structures algorithms pin pinsker complexity concentrator international teletraffic conference stockholm ros rosenthal simplicial branching random walks applications arxiv preprint
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condition query active learning linear families nov xue xchen university texas austin eric price ecprice university texas austin november abstract consider problem learning function samples noise simplest agnostic learning setting number samples required robust estimation depends condition number arbitrarily large show improve dependence two natural extensions setting query access setting estimate function arbitrary points active learning setting get large number unlabeled points choose small subset label linear spaces functions family polynomials eliminates dependence condition number technique also yield improvements nonlinear spaces demonstrate family signals continuous frequencies supported nsf grant simons investigator award david zuckerman part work done author visiting simons institute theory computing introduction common task many fields estimate signal noisy observations problem takes many forms depending measurement model signal structure desired approximation norms classical solution problem empirical risk minimization erm observations loss function outputs arg min family signals learned important special case linear vector space set polynomials case solving referred ordinary least squares ols regression consider noise setting drawn distribution signal actually lies independent mean zero random variables case would like recover satisfies kfe probability body paper also consider noise independent mean zero desired guarantee weaker sample complexity required achieve guarantee depends function family sample distribution functions high variance positions rarely sampled sample complexity quite high one way bound measure condition number sup sup bounded condition number lets apply chernoff bound fixed function empirical norm approximate actual norm probability log samples dimension vector space classical results chapter used show bound sample complexity ols guarantee applying matrix chernoff improved log condition number always least much larger natural settings example univariate degree polynomials sampled uniformly chebyshev polynomials contain mass size region even worse adversarial distributions drive arbitrarily large unfortunately show section sample complexity required achieve random samples really log even via algorithms ols work show avoid dependence condition number two alternative models access signal first model may query wherever want second model active learning get fairly large number unlabeled points may choose small set see labels query access query access model may freely choose points see also know distribution used measure distances one may think uniform interval query access model one typically used function estimation problems sparse fourier transforms even though set functions sparse fourier transform form linear space techniques implications problem shall discuss algorithm query access model course simulate one sampling model choosing getting log sample complexity biasing choices towards points high engaging importance improve sample complexity particular sample points distribution sup output minimizes appropriately weighted version corollary show algorithm achieves log samples also show theorem least term necessary consider arbitrary noise arbitrary function case since may lie one hope converge number samples grows infinity instead guarantee kfe kgkd log sample complexity active learning active learning model semisupervised learning appropriate situations unlabeled data common labeling data requires expensive human annotation situations arise variety domains including speech recognition information extraction document classification active learning model consider algorithm receives set unlabeled points unknown distribution chooses subset points receives labels associated one would like minimize number labeled unlabeled examples required learn function let start trying optimize parameter isolation merely wanted minimize number unlabeled samples would label every element revert standard sampling model hence active learning algorithm achieves guarantee must take least log unlabeled samples hand merely wanted minimize number labeled samples would take infinite number unlabeled samples samples would let learn distribution let label whatever points supp want resulting problem identical query access model know active learning algorithm must label examples hope less log labels without improving result query access model show tradeoff two criteria give active learning algorithm takes log unlabeled points requests log labels achieves query access model noise independent mean zero algorithm achieves sample complexity nonlinear function families discussion focused function families linear vector spaces polynomials degree techniques general application present one example consider estimating function family continuous signals frequencies bandlimit domain frequencies may arbitrarily close family well conditioned shown achieve family query access model using poly log samples time important step bound condition number sup sup show querying points according nonuniform distribution biased towards positions analogous fact identical relevant condition number changes sup dimension linear space one could show could replace sample complexities signals instead show still better bound turn leads factor improvement sample complexity use simple net argument show exponential time recovery algorithm thing hold fast algorithms going rather long machinery final sample complexity bound obtain presumably optimal pieces proof remain inefficient might actually smaller still argument demonstrates sampling improve sample complexity nonlinear function spaces expect similar sampling scheme necessary achieve optimal sample complexity continuous signals related work many different query strategies considered active learning literature see references therein uncertainty sampling designed serial setting algorithm iteratively requests labels updates model estimate algorithm queries single batch still given lower bounds within log factor optimal variance reduction sampling techniques literature also serial one could consider procedure performing form variance reduction result similar agnostic form active learning considered although problems quite different proposed algorithm similar strategies proposed importance sampling literature importance sampling aims find sampling distribution much like minimizes variance estimating given function one key difference alternative distribution importance sampling typically defined respect class functions done enormous literature least squares regression variety settings justice see example overview many results analyzed fixed design setting considers deterministic set observation points wants minimize error points one relevant work analyzes ridge linear regression aims robust recovery given samples distribution error measured sample complexity depends properties another body closely related work involves leverage score sampling goal approximately solve finite linear regression problem essentially fixed design linear regression problem sample distribution seen continuous limit leverage score sampling distribution condition number bound show theorem similar result setting previously observed gives bound solve linear spaces bound shown actually somewhat stronger truncating output condition number replaced fkf actual function learned sample complexity depends conditioning actual function rather function family good question whether one get similar result without factor loss discussed earlier shows log samples suffice linear function spaces also show univariate polynomials sampling chebyshev measure leads log sample complexity improved via similarly explicit sampling distribution behaves analogously chebyshev measure univariate polynomials extends result arbitrary linear function spaces interesting question whether log term get general linear spaces also improved situation somewhat similar spectral sparsifiers graphs log sparsifiers follow appropriately weighted sampling using matrix chernoff sparsifiers exist via clever constructions perhaps clever choice queries also avoid log get matrix chernoff setting organization outline proofs main results section introduce notation tools section consider number samples empirical norms linear families section next study query access model section active learning section demonstrate lower bounds query complexity sample complexity section finally provide application nonlinear function spaces section proof overview consider observations form necessarily linear family arbitrary possibly random function improved conditioning better sampling start noiseless case query access model consider problem estimating high probability sample points distribution estimate empirical norm correct expectation show concentrates would like apply chernoff bounds depend maximum value summand particular concentration depends reweighted condition number sup sup define minimize quantity making inner term every namely pick sup sup shows sampling rather number estimate condition sup improves sup sup chernoff bound samples let estimate within accuracy probability fixed function able estimate every basic solution would apply union bound linear function families let improve result two ways first show dimension linear function space second replace union bound matrix log concentration bound showing samples suffice estimate within every probability particular log samples suffice estimate every within constant factors probability results appear section effect noise consider actual problem estimate nonzero noise given samples empirical risk minimizer function closest empirical norm linear family solution linear projection acts independently previous discussion showed log samples empirical norm good estimator every function indicates projection samples linear subspace equals hence error projection onto empirical norm first suppose orthogonal true norm instance independent random variable case expected value projection zero time bound variance projection single random sample drawn condition number indicates projection independent samples squared norm less gives following theorem proven section theorem consider dimension linear space functions domain let joint distribution arg min exists queries outputs satisfying efficient algorithm makes log probability noise function orthogonal expectation let denote decomposition orthogonal part theorem indicates provides desired kgkd via pythagorean theorem result appears corollary section active learning next consider active learning setting know distribution receive samples choose receive labels suppose function family linear space let condition number give algorithm uses log unlabeled samples log labeled samples achieves guarantees query access model intermediate step suppose algorithm knew distribution simulate algorithm would need simulate samples could using rejection sampling labeling sample probability proportional supf make probabilities actual probability labeling would value divided maximum chance random sample receives label would become kdf sup hence log unlabeled samples get log labeled samples need run algorithm problem know fortunately way algorithm uses estimate functions least log unlabeled samples matrix concentration inequalities show empirical estimate approximation every suffices get desired bound giving following theorem proven section theorem consider dimension linear space functions domain let joint distribution arg min let log sup unlabeled exists efficient algorithm takes samples requests log labels output satisfying probability lower bounds first prove lower bound query complexity using information theory theorem indicates gaussian noise function every point observation obtains information dimension indicates queries necessary recover function next construct distribution linear family condition number sample complexity achieving log first term comes coupon collector problem second query bound summarize results table query complexity sample complexity lower bound theorem log theorem upper bound log log table lower bounds upper bounds different models signals fourier transform consider nonlinear family functions fourier transform defined distribution discussed even nonlinear function families sampling improves condition number sup effectively replacing supx describe bound let revisit bound shown key expressed linear combination constant coefficients gives upper bound terms bounds integrating give better bound show expressed linear combination elements arithmetic sequence sides elements required much shorter version lets show log leads appears theorem section note passing two bounds analogous markov brothers inequality bernstein inequality polynomials space univariate polynomials markov brothers inequality shows bernstein inequality shows preliminaries use denote subset denote indicator function event first state chernoff bound real numbers lemma chernoff bound let independent random variables assume always let exp exp corollary let independent random variables expectation expectation satisfies exp vector let denote norm given sequence allowing prepetition corresponding weights let denote weighted norm convenience omit uniform distribution given matrix let kak denote operator norm kak denote eigenvalues given matrix let denote smallest eigenvalue largest eigenvalue state matrix chernoff inequality theorem theorem consider finite sequence independent random matrices dimension assume random matrix satisfies define state theorem information theory theorem theorem let random variable mutual information log given function domain distribution use denote expected norm weights different distributions given distribution estimate function random samples use following notation denote reweighting definitionq distribution domain function define function definition let random samples distribution always choose keep expectation condition number reweighted sampling main result section distribution estimation functions given linear family finite dimension theorem let linear family functions dimension domain distribution sup sup distribution khkd khkd sequence log random samples corresponding weights guarantees probability least generality prove sample complexity arbitrary distribution estimate norm functions bound number samples condition number sup sup lemma let linear family dimension domain distribution let distribution sup sup sequence random samples log guarantees probability least corresponding weights prove sup khkd section theorem follows lemma condition number kdf focus proof lemma rest section condition number kdf indicates log random samples corresponding weights could guarantee property union bound improve applying matrix chernoff bound get log samples suffice first restate guarantee bounding eigenvalues matrix determined corresponding weights let orthonormal basis inner products taken distribution work always use denote following matrix definition given sample points weights define matrix based let denote conjugate transpose prove property equivalent bounding eigenvalues lemma given weights let matrix defined every eigenvalues next bound eigenvalues matrix chernoff bound lemma let arbitrary distribution sup sup exists absolute constant sample independently distribution log choose matrix defined satisfies probability least finish proof lemma defer proofs lemma lemma section proof lemma define using sample points weights first apply lemma bound eigenvalues probability lemma indicates property condition number describe distribution kdf provides almost optimal sample complexity first observe condition number always family necessarily linear claim family distribution domain let distribution defined condition number kdf sup sup proof domain supf next use linearity prove kdf let orthonormal basis inner products taken distribution lemma linear family dimension sup khkd sup condition number kdf moreover khkd exists efficient algorithm sample compute weight algorithm sampledf procedure generatingdf span sample uniformly sample distribution set weight end procedure proof given orthonormal basis khkd exists domain inequality sup sup tight always exist hence khkd sup calculation let denote claim sup khkd definition kdf sup sup sup sup khkd khkd present sampling procedure algorithm proofs lemma lemma first prove lemma prove lemma proof lemma function let denote vector coefficients khkd first prove direction eigenvalues definitions prove direction exists eigenvalue exists vector discussion khks khkd indicates hold function apply matrix chernoff bound theorem bound eigenvalues proof lemma let orthonormal basis definition matrix let denote coefficients time fixed thus def sup point weight vector indicates sup sup let denote jth row matrix defined always notice eigenvalue time equals identity matrix size expectation entry apply theorem given log results query access model prove theorem corollary section theorem consider dimension linear space functions domain let joint distribution arg min exists efficient algorithm makes log queries outputs satisfying probability give corollary theorem specific kinds noise first case consider noise functions representing independently noise position gaussian noise second consider arbitrary noise functions corollary consider dimension linear space functions domain distribution let observed function denotes noise function exists efficient algorithm observes log points outputs probability random function independent random variable kgkd deterministic function generality prove lemma applies resampling according arbitrary distributions let denote conditional distribution denote distribution first generates generates lemma consider dimension linear space functions domain let joint distribution arg min let distribution define sup sup khk exists efficient algorithm takes log samples outputs satisfying probability theorem follows lemma distribution lemma condition number kdf introduce several notations algorithm let orthonormal basis inner product taken marginal distribution function let denote coefficients coefficients function arg min satisfy indicates given weights let denote matrix definition number samples log large enough assume eigenvalues satisfy happens probability lemma let denote definition hence empirical distance equals function minimizing min pseudoinverse present algorithm algorithm defer proofs lemma claim section proof corollary section algorithm learning linear subspace procedure subspacelearning let orthogonal basis log constant sample independently set sample independently return end procedure remark running time algorithm time obtain orthonormal basis compute inverse given basis procedure compute time could obtain orthonormal basis time gramschmidt orthogonalization running time computing matrix multiplication constant correctness function let denote vector denote weights def first prove following lemma bound claim let arg min proof recall function independently randomly generated first write last step uses property independence different eliminate cross terms simplify swapping summations max notice using definition hand bound use last step discussion bound next prove lemma proof lemma let log large constant lemma probability constitute orthonormal basis markov inequality probability least probability discussion claim proof corollary section finish proof corollary proof corollary let distribution linear subspace condition number kdf lemma apply procedure subspacelearning obtain first part let joint distribution expectation every arg min lemma log large constant kfe probability second part let projection orthogonal let denote coefficients fixed orthonormal basis decompose therefore distance kfe equals lemma probability thus kgk let denote kgk rewrite kgkd kgkd kgkd discussion kgkd results active learning section investigate case know distribution receive random samples finish proof theorem linear families bounds number unlabeled samples condition number number labeled samples dim find truth theorem consider dimension linear space functions domain let joint distribution arg min let log sup unlabeled exists efficient algorithm takes samples requests log labels output satisfying probability present algorithm algorithm first takes log unlabeled samples defines distribution uniform distribution samples use simulate could apply procedure subspacelearning algorithm require labels algorithm regression unknown distribution procedure regressionunknowndistribution set large constant log take unlabeled samples let uniform distribution let vdpbe orthonormal basis set sample efficient distribution apply procedure subspacelearning algorithm parameters end procedure proof still use denote denote lemma probability least khkd every separately let denote random label including unlabeled samples algorithm labeled samples stage applying procedure subspacelearning let arg min claim let constant large enough markov inequality probability uniform distribution let denote unlabeled samples let sample efficient distribution condition number denote labeled samples generated lemma definition generated empirical minimizer satisfies kfe kfe notice hence kfe let constant large enough markov inequality probability kfe discussion rescaling gives desired result corollary let family functions domain dimension distribution bounded condition number sup let observation exists efficient algorithm takes log unlabeled samples require log labels output probability kfe random function independent random variable kfe kgkd otherwise proof proof essentially identical corollary random function independent random variable let joint distribution arg min theorem algorithm outputs satisfying kfe probability second part let projection onto orthogonal part first part satisfies kfe kgkd thus kfe kgkd similar proof corollary prove pkf kgkd follows let denote rewrite kgkd kgkd kgkd lower bounds present two lower bounds number samples section first prove lower bound query complexity based dimension prove lower bound sample complexity based condition number sampling distribution theorem exist distribution linear family functions dimension gaussian noise algorithm observes outputs satisfying fekd probability needs least queries notice lower bound almost matches upper bound theorem rest section focus proof theorem let uniform distribution first construct packing set claim exists subset following properties kfi kfi kfi distinct proof construct procedure constructm notice loop time procedure constructm removes functions every time indicates thus algorithm construct procedure constructm set choose remove functions end return end procedure finish proof theorem using theorem proof theorem yao minimax principle assume deterministic algorithm given gaussian noise let denote mutual information random function output given observations output satisfies closest function indicates third property fano inequality log log log log time data processing inequality algorithm makes queries sees indicates query denote algorithm given first observations apply theorem bounds log max log log apply second property second step bound hence bound implies next consider sample complexity linear regression theorem exist distribution linear family functions dimension whose condition number sup equals noise function sup orthogonal algorithm observing needs least log samples output satisfying kfe kgkd probability proof fix integer set domain functions choose uniform distribution let denote family functions equals condition number sup sup provides lower bound time indicates upper bound first consider case log let orthogonal notice kfe kgkd indicates every hence algorithm needs sample every sampling uniform distribution lower bound coupon collector problem takes least log samples otherwise algorithm needs samples without loss generality assume truth let theorem find satisfying algorithm needs least queries hence needs random samples uniform distribution application continuous fourier transforms consider nonlinear function space containing signals fourier transform continuous setting let uniform distribution bandlimit frequencies fix family section main result section estimation signals sup fourier transform better condition number used sup sup theorem signals fourier transform sup moreover exists distribution log log guarantees first state condition number result lemma lemma log sup main technical lemma upper bound every position lemma log sup main ingredient interpolation lemma complex numbers lemma given pany exists laurent polynomial every every proof let whose coefficients let denote power largest coefficient arg max set statement satisfies three conditions proof lemma pgiven frequencies set let laurent polynomial lemma rewrite linear combination zjl zjl roots provides linear relationship coefficients zjl zjl hence indicates without loss generality assume integrate log choosing upper bound log bound sup sup sup log log log log discussion proof theorem next define sup lemma lemma sup description follows lemmas claim condition number corollary observation exists algorithm takes log random samples outputs vej satisfying probability proof corollary let first state main tool previous work lemma denote net frequencies constant algorithm recover procedure sparseft log sample independently set corresponding weights query observation constant possible frequencies find span minimizing yks update yks kfe yks end return end procedure signal exists signal satisfying whose frequencies rewrite kgkd goal recover construct khkd first pick frequencies construct linear subspace hence size net span kck consider number random samples estimate signals based condition number theorem chernoff bound corollary union bound indicates log log log random samples would guarantee signal net property net sufficient recover present algorithm algorithm bound fekd follows expectation fekd random samples fekd feks yks feks kgkd markov inequality probability fekd kgkd acknowledgements authors would like thank adam klivans david zuckerman many helpful comments work references balcan alina beygelzimer john langford agnostic active learning proceedings international conference machine learning pages acm joshua batson daniel spielman nikhil srivastava sparsifiers siam journal computing albert cohen mark davenport dany leviatan stability accuracy least squares approximations foundations computational mathematics herman chernoff measure asymptotic efficiency tests hypothesis based sum observations annals mathematical statistics xue chen daniel kane eric price zhao song interpolation without frequency gap focs david cohn neural network exploration using optimal experiment design advances neural information processing systems pages petros drineas michael mahoney muthukrishnan cur matrix decompositions siam journal matrix analysis applications robert fano transmission information statistical theory communications cambridge massachusetts press stuart geman elie bienenstock doursat neural networks dilemma neural computation michael kohler adam krzyzak harro walk theory nonparametric regression springer science business media ralph hartley transmission information bell system technical journal daniel hsu sham kakade tong zhang random design analysis ridge regression foundations computational mathematics david lewis william gale sequential algorithm training text classifiers proceedings annual international acm sigir conference research development information retrieval pages new york malik row sampling matrix algorithms via bernstein bound arxiv preprint art owen monte carlo theory methods examples burr settles active learning literature survey computer sciences technical report university claude shannon communication presence noise proc institute radio engineers george seber alan lee linear regression analysis volume john wiley sons sebastian seung manfred opper haim sompolinsky query committee proceedings fifth annual workshop computational learning theory pages acm daniel spielman nikhil srivastava graph sparsification effective resistances siam journal computing robert tarjan lecture chernoff bounds sampling chernoff union bound princeton class notes probability computing pages joel tropp tail bounds sums random matrices foundations computational mathematics david woodruff sketching tool numerical linear algebra arxiv preprint
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joint sum rate error probability optimization finite blocklength analysis mahdi haghifam mohammad robat mili behrooz makki masoumeh aug tommy svensson abstract study tradeoff sum rate error probability downlink wireless networks using recent results achievable rates codewords problem cast joint optimization network sum rate error probability moreover develop efficient algorithm based technique simultaneously maximize network sum rate minimize maximum users error probability evaluate effect codewords length system performance results show scenarios optimizing error probability plays key role achieving high throughput ntroduction fifth generation wireless communication must support novel traffic types low latency high data rate ultra reliability interest particularly many applications communications traffic video processing augmented reality codewords required short order channel uses stringent requirements latency reliability therefore interesting optimize performance wireless networks presence codewords presented accurate approximations achievable rates finite blocklength codes using performance wireless networks short packets mahdi haghifam mohammad robat mili masoumeh electrical engineering department sharif university technology tehran iran emails haghifam mahdi mnasiri behrooz makki tommy svensson chalmers university technology gothenburg sweden emails studied various papers cases cognitive radio relay networks hybrid automatic repeat request technique letter consider wireless network access point serving multiple users using short packets transmits packets downlink users different target error probability requirements particularly using recent results propose joint sum rate error probability optimization problem investigate effect codeword length system performance solve joint sum rate error probability optimization problem develop algorithm based approach also derive expression optimal error probability theorem finally find efficient power allocation algorithm terms sum rate error probability based augmented lagrange method algorithm simulation analytical results show proposed algorithm reach almost performance exhaustive approach considerably less implementation complexity figs also throughput sensitive length short packets sensitivity packet length decreases long packets fig finally optimal error probability assignment power allocation achieves higher throughput compared using optimal power allocation equal error probability assignment figs ystem odel consider downlink communication model users served assumed user allocated orthogonal channel instance could separated frequency time domain let denote instantaneous channel gain user channel gain expressed represents small scale fading average channel gain obtained considering path loss effects shadowing thus user located distance signal power gain distance meter path loss exponent moreover power allocated signal user denoted thus instantaneous ratio snr received user noise power density characterize network performance employs packets short length specifically user message encoded packets length transmitted power way maximum achievable information rate nats per channel use npcu user decoded block error probability greater given thm log log achievable rate increases unboundedly error probability tends towards one hand rate decreases significantly cases strict error probability requirements small also achievable rate increases signal length monotonically letting achievable rate converges shannon capacity formula cases asymptotically long codewords motivated tradeoff achievable rates error probability consider joint sum rate maximization error probability minimization problem assuming perfect channel state information csi study optimization problem maximize log minimize max subject pmax also goal minimize maximum error probability users maximum error probability constraint user indicates supporting quality service qos requirement user also total power constraint denoted pmax way interest emerging applications calling heterogeneous qos requirements data rate reliability instance massive communication low latency communication scenarios demand packet exchange stringent requirement reliability moderately low rate requirements aforementioned services illustrate framework utilized balance conflicting performance objectives namely sum rate maximum error probability moreover seen following discussions well applicable cases optimizing network sum throughput defined product rate successful message decoding probability however opposed throughput optimization flexible optimizing rates error probabilities individually based qos requirements depending number users may solution thus follow method convert problem single objective optimization using weighted sum method normalizing objectives also seen section proposed approach reach almost performance optimal exhaustive scheme loss generality assume also guarantee consistent comparison objectives normalize log max normalization factor found plugging power allocation expression shannon capacity formula provides upper bound use weighted sum method rewrite optimization problem maximize subject weighting parameter note ranging scenarios strict rate requirements relaxed error probability requirements scenarios low rate requirements addressed iii roposed lgorithm optimization problem belongs class problems multimodal objective function finding global optimal solution computationally infeasible reason apply primal decomposition approach optimize separately way solve use following iterative approach initialization optimal solution optimal error probability power allocation vectors iteration maximum number iterations considered network designer details proposed optimization approach follows error probability optimization given power allocation given power allocation find optimal error probabilities user iteration denoted setting max assuming given power allocation rephrased minimize subject theorem gives expression optimal error probability assignment user terms theorem optimal error probabilities users given times otherwise define intervals log proof since constraints affine functions sufficient prove objective function convex second derivative given exp therefore considering fact objective function sum convex functions affine function hence convex optimization problem optimal solution found considering kkt conditions thus write lagrangian function dual variables associated constraints respectively according kkt conditions optimal solution denoted satisfy exp equal verified used dqdx exp either otherwise must equal zero contradict assume according note must equal since also due fact inferred thus cases concluded way expressed exp also plugging upper branch found way depending region solution provided given straightforward show objective function decreasing function lower branch provides optimal solution note strict convexity optimal solution found searching branches optimal power allocation given error probability consider given relaxed maximize subject since function problem belongs class nonconvex optimization problems problem nonzero duality gap primal dual problems use augmented lagrange approach sec deal optimization reduces duality gap augmenting quadratic term added lagrangian function sec proved augmented lagrangian locally convex penalty parameter sufficiently large contrast penalty functions approach augmented lagrangian function largely preserves smoothness require asymptotically large penalty parameter method converge meaning penalization exact augmented lagrangian algorithms based successive maximization augmented lagrangian function multiplier estimates penalty parameter fixed iteration updated iterations applying augmented lagrangian method eliminates constraints adds objective function gives augmented lagrangian function log max pmax positive coefficient denoting penalty parameter lagrangian dual variable associated stage power allocation problem solve maximize approximates find power allocation iteration denoted moreover variables updated according max pmax respectively way increases violations introduced constraints penalized severely maximizer penalty function gives results closer feasible region sec shown constraints nonlinear convergence rate augmented lagrangian method linear iterative joint error probabilities power allocation algorithm summarized algorithm order analyze complexity order proposed algorithm note optimal error probabilities found complexity also complexity power allocation iteration thus complexity algorithm algorithm error probabilities assignment power allocation every given pmax initialize converges calculate via given initialize converges calculate via given update via end end umerical esults onclusion study sum rate maximum error probability set noise power number users error probability constraints often assumed communications also assumed users equidistant also consider rayleigh fading mean finally set algorithm numerical results consider cases channel uses approximation tight enough also compare method three baseline algorithms power allocation error probabilities users set minimum required error probabilities called minmax error probability assignment proposed method power allocation minmax error probability assignment equal power allocation proposed method error probabilities assignment finally results obtained averaging different channel realizations figure shows tradeoff sum rate error probability different algorithms pmax channel uses performance metric define sum throughput user codeword rate error probability given respectively fig demonstrates sum throughput versus total power constraint pmax setting channel uses finally fig evaluates effect codeword length sum throughput results lead following conclusions scheme power allocation error probability assignment based theorem achieves tradeoff region close proposed method optimizing error probability power allocation fig short codewords throughput remarkably affected length codeword however effect increasing codeword length decreases long codewords fig also optimal error probability assignment power allocation achieves higher throughput compared optimal power allocation equal error probability assignment moreover performance minmax error probability assignment close scheme proposed power allocation minmax error probability assignment figs short codeword say channel uses proposed algorithm leads considerable sum rate npcu exhaustive search proposed method power error pmax max figure sum rate maximum error probability throughput improvement comparison schemes instance performance proposed method improvement however performance difference schemes decreases cases long codewords finally observed figs gap developed algorithm exhaustive algorithm diminishes increasing pmax thus proposed algorithm effectively applied jointly optimize sum rate error probability networks applications eferences boccardi heath lozano marzetta popovski five disruptive technology directions ieee commun vol polyanskiy poor verdu channel coding rate finite blocklength regime ieee trans inf theory vol may makki svensson zorzi finite analysis spectrum sharing networks using rate adaptation ieee trans vol gursoy velipasalar throughput wireless channels queueing constraints finite blocklength codes proc ieee isit barcelona spain july makki svensson zorzi finite analysis incremental redundancy harq ieee wireless commun vol bertsekas nonlinear programming athena scientific belmont tan tomamichel term normal approximation awgn channel ieee trans inf theory vol sum throughput npcu exhaustive search proposed method proposed power error power error power error subplot total power log pmax sum throughput npcu exhustive search proposed method proposed power error power error power error pmax subplot codeword length channel uses figure sum throughput considered algorithms subplot sum throughput total power constraint subplot sum throughput codeword length musavian tradeoff analysis joint optimization energy efficiency effective capacity toward green communications ieee trans wireless vol may palomar chiang tutorial decomposition methods network utility maximization ieee sel areas vol
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using table valued functions sql server implement spatial data library jim gray microsoft research alex szalay gyorgy fekete johns hopkins university august technical report microsoft research advanced technology division microsoft corporation one microsoft way redmond using table valued functions sql server implement spatial data library jim gray microsoft contact author alex szalay johns hopkins university gyorgy fekete johns hopkins university august abstract article explains add spatial search functions point polygon microsoft sql using functions possible use library add spatial search application without writing special code library implements hierarchical triangular mesh htm algorithms johns hopkins university library connected sql server via set functions functions act spatial index resources article illustrated examples downloaded http sample package includes sample spatial database united states cities gauges sample queries article visual studio project sql code paper article provides manual page routine article explains hierarchical triangular mesh algorithms detail article explains htm algorithms used astronomy article also explains two approaches zones comparisons regions boolean algebra areas public domain implementations approaches implemented sql server used skyserver popular astronomy website sloan digital sky survey http several astronomy data servers table contents abstract introduction table valued functions key idea using functions add spatial index datasets usgs populated places cities usgs stream gauges instruments spatial index table digression cartesian coordinates typical queries find points near point find towns near place find places inside box find places inside polygon advanced topics complex regions references appendix basic htm routines htm library version fhtmversion returns versionstring generating htm keys fhtmlatlon lat lon returns htmid latlon xyz fhtmlatlontoxyz lat lon returns point xyz latlon fhtmxyztolatlon returns point lat lon viewing htm keys fhtmtostring htmid returns htmstring htm trixel centerpoint fhtmtocenterpoint htmid returns point htm trixel corner points fhtmtocornerpoints htmid returns point computing distances fdistancelatlon returns distance finding nearby objects fhtmnearbylatlon type lat lon radius returns spatialindextable finding nearest object fhtmnearestlatlon type lat lon returns spatialindextable circular region htm cover fhtmcovercirclelatlon lat lon radius returns trixeltable general region specification htm cover fhtmcoverregion region returns trixeltable general region simplification fhtmregiontonormalformstring region returns regionstring cast regionstring table fhtmregiontotable region returns regiontable find points inside region fhtmregionobjects region type returns objecttable general region diagnostic fhtmregionerror region returns message introduction spatial data searches common commercial scientific applications developed spatial search system conjunction effort build skyserver http astronomy community skyserver multiterabyte database catalogs million celestial objects many questions astronomers want ask involve spatial searches typical queries include near point objects inside area areas overlap area article added terrestrial sphere earth grid astronomer right celestial sphere sky grid two grids lot common correspondence exact traditional order corresponds order reversal forces explicit coordinate system call terrestrial coordinate system latlon coordinate system library supports three coordinate systems greenwich called latlon astronomical called cartesian called cartesian astronomers use arc minutes standard distance metric nautical mile arc minute distance translation natural many concepts quite similar demonstrate article show use spatial library build spatial index two usgs datasets cities gauges using indexes spatial functions article provides examples search cities near point find stream gauges near city find stream gauges cities within state polygonal area believe approach generic spatial data spine schema spatial data functions added almost application allow spatial queries ideas also apply indexing schemes example techniques would work searching color space metric space table valued functions key idea key concept relational algebra every relational operator consumes one relations produces output relation sql syntactic sugar idea allowing define relations data definition language manipulate relations syntax defining scalar functions lets make extensions relational database send mail messages execute command scripts compute scalars aggregate values tax median however create tables become part relational engine producer consumer relational tables idea oledb allows data source produce data stream also idea behind sql server table valued functions implementing table valued functions really easy create function returns points table float float begin insert points values insert points values return end fine function done entirely implementing oledb data sources table valued functions outside real challenge sql server common language runtime clr integration sql server makes easy create function create list array ienumerable object anything foreach cast table sqlfunction tabledefinition float float fillrowmethodname fillpair public static ienumerable cspoints int points return ienumerable points compile visual studio click deploy function installed database using functions add spatial index lot confusion indexes indexes really simple tables special properties sql server one kind associative value index btree keys first field carries selectivity conceptually index table consisting key fields base table key fields included fields want add index indexes sorted according index key code customer lookup sequential scan key fast indexes often smaller base table carrying important attributes looking index involves many fewer bytes examining whole table often index much smaller fit main memory thereby saving even disk accesses think index lookup either searching index alone vertical partition base table searching index joining qualifying index rows rows base table via primary key bookmark lookup central idea spatial index gives small subset data index tells look often carries helpful search information called objects oarse subset caref test good false included columns covering columns experts selectivity index tells big initial reduction coarse subset figure subset located careful test examines member subset discards false positives process indicated diamond figure good index false positives use figure metaphor coarse subset careful test throughout article figure key idea spatial index gives small subset data least smaller careful test discards false positives index good relatively false positives use idea coarse subset careful test throughout article functions combined follows let build spatial index produces coarse subsets create function generates keys cluster related data together example items related keys nearby key space create function given description subset interest returns list key ranges cover containing pertinent values always get every key near relatives keys sorted one dimension relatives near space higher however come close ratio correct answers measure well nwnw nwne nenw nene standard approach find space filling curve thread key space along curve using standard mercator map example assign everyone northwest northwest key range assign everyone southeast southeast key range figure shows order curve traverses quadrants assigning keys sequence everyone quadrant key prefix nwsw area like circle shown figure look key range key nwsw nwse search space eight times smaller whole table percent false positives indicated area outside circle inside two boxes great improvement conveys idea better index would use finer cell division fine enough cells converging area could false positives detailed review curves trees found books hanan samet samet nwsw nwse nesw nese figure start space filling peano hilbert curve one recursively divides cell systematic way cells labeled points cell nwse key prefix find nwse section btree right nwne right nwsw circle area interest overlaps two cells going define curve hierarchical triangular mesh htm works particularly well sphere earth round celestial sphere round spherical system convenient geographers astronomers could similar things metric space curve gives keys basis spatial index someone region interest table valued function give good set look coarse filter figure key ranges cover region spherical triangles called trixels much two boxes figure cover circle search function need look objects key ranges trixels see qualify careful test figure make concrete assume table objects create table object objid bigint primary key lat float latitude lon float longitude htmid bigint htm key distance function gives distance nautical miles arcminutes two points assume following function gives list key ranges htmid points within certain radius point define function fhtmcovercirclelatlon lat float lon float radius float returns trixeltable table htmidstart bigint htmidend bigint following query finds points within nautical miles san francisco lat lon select fhtmcovercirclelatlon trixeltable join object coarse test lat careful test must define htm key generation function distance function htm cover function next using two united states geological spatial datasets example skeptical scales billions objects http look around site web site uses code spatial lookup astronomy database article use sql table valued functions curve like htm build spatial index treat htm code black box documented elsewhere szalay focus adapt needs within sql application datasets geological survey gathers publishes data united states figure shows locations stream gauges measure river water flows levels usgs also publishes list place names populations figure graphical display latitude longitude usgs stream gauges usgs places two datasets items use motivate spatial search examples usgs populated places cities usgs published list place names attributes newer lists usgs website fragmented state difficult get nationwide list old list suffice demonstrate spatial indicies data following format create table place placename varchar state char population int households int landarea int waterarea int lat float lon float htmid bigint null city name null char state code null number residents null number homes null area sqare null water area within land area null latitude decimal degrees null longitude decimal degrees null primary key spatial index key speed name lookups add name index data clustered spatial key nearby objects clustering thus nearby disk pages create index place placename except htmid data downloaded usgs web site sql server data import wizard used import data already done sample database htmid field computed lat lon update place set htmid lat lon usgs stream gauges instruments usgs maintaining records river flows since jan accumulated thousand years measurement data six thousand active stations active four thousand online gauges described detail http noaa site shows data hundred popular stations convenient way http database stations continental united states see figure also stations guam alaska hawaii puerto rico virgin islands included database stream gauge station table create table station stationname varchar state char lat float lon float drainagearea float firstyear int yearsrecorded int isactive bit isrealtime bit stationnumber int htmid bigint null null null null null null null null null null null usgs station name state location latitude decimal longitude decimal drainage area first year operation record years active internet usgs station number htm spatial key based primary key htmid stationnumber htmid field computed lat lon fields update station set htmid lat lon stations one location primary key must include station number make unique however htm key clusters nearby stations together speed lookups add station number name index create index station stationname create index station stationnumber spatial index table ready create spatial index could added fields base tables make stored procedures work many different tables found convenient mix objects together one spatial index could choose type htmid key segregate different types objects chose htmid key key nearby objects types cities steam gagues clustered together spatial index create table spatialindex htmid bigint null htm spatial key based lat float null latitude decimal lon float null longitude decimal float null cartesian coordinates float null derived float null type char null place gauge objid bigint null object table primary key htmid objid cartesian coordinates explained later topic enough say function fhtmcenterpoint htmid returns cartesian unit vector centerpoint htm triangle limit point htm center subdivided infinitely small trixels spatialindex table populated place station tables follows insert spatialindex select lat lon type htmid objid place cross apply fhtmlatlontoxyz xyz insert spatialindex select lat lon type objid station cross apply fhtmlatlontoxyz xyz clean database execute dbcc dbcc dbcc dbcc dbcc dbcc dbcc indexdefrag indexdefrag indexdefrag indexdefrag indexdefrag indexdefrag shrinkdatabase spatial spatial spatial spatial spatial spatial spatial station station station place place spatialindex spare space digression cartesian coordinates skip like needed use library htm code heavily uses trick avoid spherical geometry moves surface sphere allows quick tests inside polygon nearby point queries every point sphere represented unit vector threedimensional space north south poles respectively represents axis rotation plane represenst prime greenwich meridian longitude longitude formal definitions cos lat cos lon lat sin lon sin lat lat lon lat lon lat figure cartesian coordinates allow quick tests point corresponding unit vector cartesian coordiates used follows given two points unit sphere dot product cosine angle two points distance metric looking points within nautical miles arc minutes point degrees away dot product points less radians nearby test becomes quick test cartesian coordinates also allow quick test polygons cos edges edges lie along plane intersecting sphere edges defined unit vector normal plane figure great small circle intersection shift along vector plane circle point inside circle dot product plane normal vector less example equator vector shift zero latitude cos circle diameter defined vector shift circle around baltimore defined vector shift place within baltimore idea lets decide point inside outside htm triangle evaluating three dot products one main reasons htm code efficient fast implemented several helper procedures convert latlon cartesian coordiantes fhtmxyz htmid returns xyz vector centerpoint htmid fhtmlatlontoxyz lat lon returns xyz vector fhtmxyztolatlon returns lat lon vector used documented api spec intellisense fekete library defaults htm keys first level divides sphere faces subsequent level divides speherical triangle table indicates trixel fairly small code modified deep representation runs bits table htm level subdivdes sphere level table shows area square degrees arc minutes arc seconds meters trixel colum shows charactic sizes default trixels arc usgs data object per trixel htm depth sphere deg arc min area arc earth trixel objects trixel sdss usgs typical queries assuming get functions defined ready queries find points near point find towns near place common query find places nearby certain place point consider query find towns within nautical miles baltimore htm triangles covering nautical mile circle arc minutes baltimore obtained select find htm cover around baltimore fhtmcovercirclelatlon returns trixel table right baltimore circle htm cover fhtmcovercirclelatlon function returns set htmidstart htmidend htm triangles cover circle case single trixel htm keys objects inside circle also inside one triangles need look triangles discard false positives careful test figure order answer set distance baltimore want closest place select top distance want exclude baltimore closest declare lat float lon float select lat lat lon lon place state select objid lat lon lat lon distance spatialindex join fhtmcovercirclelatlon lat lon htmid htmidstart htmidend coarse test type lat lon lat lon careful test order distance asc cover join returns rows coarse test within air miles careful test gives false positives within milliseconds common tasks standard functions fhtmnearbylatlon type lat lon radius fhtmnearestlatlon type lat lon query becomes select objid distance fhtmnearestlatlon find places inside box applications often want find objects inside square displaying square map window colorado almost exactly square corner points corner corner state center point one cover square circle centered point declare radius float set radius select station stationnumber select objid fhtmcovercirclelatlon radius join spatialindex htmid htmidstart htmidend lat lon type option force order example returns stream gauges milliseconds five colorado gauges right border wanders south nautical mile extra stations appear southern latitude adjusted cover circle returns triangles join spatialindex table returns gauges percent false positives next section shows improve using htm regions specify polygon cover rather cover circle force order clause embarrassment missing query runs ten times longer optimizer join spatial index outer table perhaps tables larger millions rows optimizer would pick different plan count paradoxically optimizer chose correct plan without hints queries previous section gis systems astronomical applications often want buffer zone around region htm code includes support buffer zones much used real applications look reference szalay see done find places inside polygon htm code lets specify area circle rectangle convex hull union regions particular htm library allows specify region using following linear syntax circlespec rectspec hullspec convexspec areaspec regionspec circle latlon lat lon radius circle dec radius circle cartesian radius rect latlon lat lon rect dec rect cartesian chull latlon lon lat chull dec chull cartesian convex cartesian rectspec circlespec hullspec convexspec region areaspec areaspec give examples region specifications circle point specification nautical mile arc minute radius latlon cartesian rect two corner points defining minimum maximum lat lon longitude coordinates interpreted sense degree wide range latitudes must north south pole rectangle edges constant latitude longitude lines rather edges chull convex latlon chull three point specifications define spherical convex hull edges convex hull connecting adjacent points great circles points must single hemisphere otherwise error returned order points irrelevant latlon convex number including zero constraints form cartesian vector fraction unit length vector region region union zero circle rect chull convex areas region convex circle latlon region descriptions fed fhtmcoverregion routine returns trixel table describing set trixels triangular areas covering region simpler code colorado query select select objid fhtmcoverregion latlon loop join spatialindex htmid htmidstart htmidend lat lon type join station option force order unusual query format required tell optimizer exactly order perform join make force order option work correctly difficult modify optimizer way functions statistics estimated expensive force inner loop join query returns stream gauges candidates cover percent false positives note rectangle cover better circular cover false positives polygon syntax states article table valued functions htm algorithms see htm code project also documentation project similar query cast convex hull select select objid fhtmcoverregion latlon loop join spatialindex htmid htmidstart htmidend lat lon type join station option force order query returns stream gauges candidates cover percent false positives convex hull cover even better equivalent rectangular cover case advanced topics complex regions previous examples gave syntax regions discussion searches regions get quite complex boolean combinations convex areas space explain regions detail htm library accompanying project logic boolean combinations regions simplify regions compute region corner points compute region areas many features ideas described fekete gray szalay give hint ideas consider state utah boundaries approximately defined union two rectangles declare utahregion varchar max set utahregion latlon main part latlon ogden salt lake find stream gauges utah query select select objid fhtmcoverregion utahregion loop join spatialindex htmid htmidstart htmidend lat lon lat lon type join station option force order careful test inside one two boxes cover returns trixels join returns stations careful test finds stations utah two wyoming stations right border percent false positives states require much complex regions example region string approximate california declare californiaregion varchar max set californiaregion latlon latlon select stationnumber fhtmcoverregion californiaregion loop join spatialindex htmid htmidstart htmidend careful test type join station objid option force order nortwest corner center lake tahoe arena lake tahoe start colorado river lake havasu yuma san diego san nicholas san miguel arguelo sur monterey rayes cover returns trixels cover stations inside california false positives percent careful test nontrivial query done places rather stations careful test included looks like select place htmid select distinct fhtmcoverregion californiaregion loop join spatialindex htmidstart htmidend join place cross join fhtmregiontotable californiaregion poly group min option force order uses representation california techniques described gray quickly test point inside california convex hull returns places seven arizona border california polygon approximates california runs seconds processor leave option force order clause runs slower taking seconds common requirement code tricky added procedure fhtmregionobjects region type returns object ids spatialindex procedure encapsulates tricky code two california queries become select get california river stations station stationnumber inside region select objid fhtmregionobjects californiaregion select get california cities place htmid inside region select objid fhtmregionobjects californiaregion colorado utah queries also simplified using routine summary htm spatial indexing library presented interesting useful right convenient way index data queries sphere library also good example sql server database systems extended adding class library substantial computation language like visual basic java ability implement powerful functions scalar functions integrate queries persistent data database powerful extension mechanism starts deliver promise databases first step next decade programming languages database query languages likely get even better data integration boon application developers references gray goes neighborhood relational algebra spatial data search jim gray alexander szalay gyorgy fekete wil mullane maria nietosantisteban aniruddha thakar gerd heber arnold rots april szalay indexing sphere hierarchical triangular mesh alexander szalay jim gray george fekete peter kunszt peter kukol aniruddha thakar appear included project fekete sql server htm interface release george fekete jim gray alexander szalay may included project applications spatial data structures computer graphics image processing gis hanan samet reading design analysis spatial data structures hanan samet addisonwesley reading isbn appendix basic htm routines section describes htm routines companion document szalay manual page routine routines annotated support intellisense follows lat lon decimal degrees southern western latitudes negative distances nautical miles arc minutes htm library version fhtmversion returns versionstring routine returns nvarchar max string giving htm library version example use print returns something like august generating htm keys fhtmlatlon lat lon returns htmid routine returns htm latlon point example use update place set htmid lat lon also fhtmxyz fhtmeq functions astronomers latlon xyz fhtmlatlontoxyz lat lon returns point routine returns cartesian coordinates lat lon point example use identity function select fhtmlatlontoxyz xyz cross apply fhtmxyztolatlon latlon also fhtmeqtoxyz functions astronomers xyz latlon fhtmxyztolatlon returns point lat lon routine returns cartesian coordinates lat lon point example use identity function select fhtmlatlontoxyz xyz cross apply fhtmxyztolatlon latlon also fhtmxyztoeq functions astronomers viewing htm keys fhtmtostring htmid returns htmstring given htmid routine returns nvarchar form triangle number describing htm trixel depth triangular mesh example use print server development returns also fhtmxyz fhtmeq functions astronomers htm trixel centerpoint fhtmtocenterpoint htmid returns point returns cartesian center point htm trixel specified htmid example use select fhtmtocenterpoint htm trixel corner points fhtmtocornerpoints htmid returns point returns three cartesian corner points htm trixel specified htmid example use select fhtmtocornerpoints computing distances fdistancelatlon returns distance computes distance nautical miles arc minutes two points example use declare lat float lon float select lat lat lon lon place placename state select placename lat lon lat lon distance place also fdistancexyz fdistanceeq functions astronomers following routines return table serves spatial index returned spatial index table data definition spatialindextable table htmid bigint null htm spatial key based lat float null latitude decimal lon float null longitude decimal float null cartesian coordinates float null derived float null type char null place gauge objid bigint null object table distance float null distance arc minutes object primary key htmid objid finding nearby objects fhtmnearbylatlon type lat lon radius returns spatialindextable returns list objects within radius distance given type distance given point list sorted nearest object example use select distance place fhtmnearbylatlon join place order distance also fhtmgetnearbyeq fhtmgetnearbyxyz functions astronomers finding nearest object fhtmnearestlatlon type lat lon returns spatialindextable returns list containing nearest object given type point example use select distance place fhtmnearestlatlon join place also fhtmgetnearesteq fhtmgetnearestxyz functions astronomers following routines return table describing htmidstart htmidend set trixels htm triangles covering area interest table definition trixeltable table htmidstart htmidend bigint null min htmid trixel bigint null max htmid trixel circular region htm cover fhtmcovercirclelatlon lat lon radius returns trixeltable returns trixel table covering designated circle example use declare answer nvarchar max declare lat float lon float select lat lat lon lon place state set answer using fhtmcovercirclelatlon finds select answer answer cast varchar max str lat lon arcmintes distant spatialindex join fhtmcovercirclelatlon lat lon htmid htmidstart htmidend coarse test type place lat lon lat lon careful test join place order lat lon asc print city within arcminutes baltimore highlands arcminutes away also fhtmcovercircleeq astronomers general region specification htm cover fhtmcoverregion region returns trixeltable returns trixel table covering designated region regions described earlier topic select select objid fhtmcoverregion latlon loop join spatialindex htmid htmidstart htmidend lat lon type join station option force order general region simplification fhtmregiontonormalformstring region returns regionstring returns string form region convex redundant halfspaces removed convex convex simplified described fekete print latlon following routine returns table describing htmidstart htmidend set trixels htm triangles covering area interest table definition regiontable convexid bigint null halfspaceid bigint null float float float float null null null null convex halfspace within convex cartesian coordinates halfspace plane displacement halfspace along unit vector cast regionstring table fhtmregiontotable region returns regiontable returns table describing region union convexes convex intersection halfspaces convexes simplified described fekete section article describes use function select latlon find points inside region fhtmregionobjects region type returns objecttable returns table containing objectids objects spatialindex designated type inside region select find colorado places places join htmid select objid dbo fhtmregionobjects latlon general region diagnostic fhtmregionerror region returns message returns region definition valid otherwise returns diagnostic saying wrong region definition followed syntax definition regions print latlon sql server htm interface release alex szalay gyorgy fekete jim gray august document describes sql server interfaces htm functions also explains install modify procedures tutorial hierarchical triangular mesh htm http table contents htm concepts reviewed find version installed htm code fhtmversion geometric conversion functions fhtmxyztolatlon fhtmxyztoeq fhtmlatlontoxyz lat lon fhtmeqtoxyz dec compute htmid point fhtmlatlon lat lon fhtmeq dec fhtmxyz htm triangle functions fhtmtocenterpoint htmid fhtmtocornerpoints htmid distances points fdistancelatlon fdistanceeq fdistancexyz regions region specifications examples region specifications fhtmregiontonormalformstring regionspec fhtmregiontotable regionspec fhtmregionobjects regionspec type fhtmregionerror regionspec htm covers compute htmid ranges region fhtmcoverregion regionspec fhtmcovercirclelatlon lat lon radiusarcmin fhtmcovercircleeq dec radiusarcmin fhtmcovercirclexyz radiusarcmin installing htm code compiling modifying debugging htm code table htm depths approximate areas htm concepts reviewed hierarchical triangular mesh multilevel recursive decomposition sphere top depth eight spherical triangles four northern southern hemispheres four triangles share vertex pole sides opposite pole form equator imagine orienting regular octahedron two vertices poles four equally spaced equator spherical polygons projection edges octahedron onto circumscribing sphere eight unique integers represent triangles triangles mesh scheme called trixels trixel split four smaller trixels introducing new vertices midpoints side adding great circle arc segment connect new vertices existing one trixel division repeats recursively indefinitely produce smaller smaller trixels trixel level number corresponds number times original octant triangle split points decomposition represented leading bit level trixel number successive numbers gives trixel unique bit identifier called htm htmid represents particular trixel htm hierarchy smallest valid htmid level htmid triangle htmids numbered level term depth tells many levels involved depth though division process continue indefinitely representation runs bits depth depth good enough meter surface earth arc seconds code defaults depth arc seconds note numbering scheme complete cover positive integers bit patterns form valid htmid numbers functions described return region area sphere return value functions referred trixel tables tables htmid ranges trixels overlap cover region structure tables quite simple rows two numbers bigints starting ending values htmids trixels range htmids context always trixels depth library supports three coordinate systems latlon greenwich meridian spherical coordinate system latitude longitude lat lon used geographers equatorial celestial right ascension declination dec spherical coordinate system used astronomers vector pointing center milky way defines intersection prime meridian equator cartesian unit vector representation sphere point sphere either latlon corresponding unit vector north pole prime meridian intersection equator library installed sample spatial database uses latlon coordinates many examples geared astronomy examples drawn tables http find version installed htm code fhtmversion returns string describing version htm code typical description version timestamp july returns version varchar max version string installed code example use declare version varchar max select version print version version produces installed version july geometric conversion functions fhtmxyztolatlon given cartesian coordinates returns table containing corresponding latlon coordinates parameters float null float null float null returns vertextable lat float lon float equivalent given point example use select errors empty table returned close within zero fhtmxyztoeq given cartesian coordinates returns table containing corresponding dec coordinates parameters float null float null float null returns vertextable float dec float equivalent given point example use select errors empty table returned close within zero fhtmlatlontoxyz lat lon given latlon point returns table containing corresponding cartesian coordinates parameters lat float null latitude lon float null longitude returns vertextable float float float example use select errors none extreme latitude values truncated fhtmeqtoxyz dec given equatorial point returns table containing corresponding cartesian coordinate parameters float null right ascension dec float null declination returns vertextable float float float example use select errors none extreme declination values truncated compute htmid point fhtmlatlon lat lon given latlon point returns htmid point earth parameters lat float null latitude converted range lon float null longitude returns htmid bigint null htmid dec point example use declare htmid bigint select htmid lat lon place placename baltimore state errors none fhtmeq dec given equatorial point returns htmid point celestial sphere parameters float null right ascension degrees converted range dec float null declination degrees converted range returns htmid bigint null htmid dec point example use declare htmid bigint select htmid dec stars errors none fhtmxyz given cartesian point returns htmid point sphere parameters float null vector galaxy center prime meridian intersection equator float null vector normal galactic plane normal prime meridian float null vector normal galactic plane equator xyz normalized converted dec returns htmid bigint null htmid point example use declare htmid bigint select htmid errors none htm triangle functions fhtmtocenterpoint htmid given htmid return cartesian centerpoint vertex table parameters htmid bigint null htmid triangle returns vertextable float float float example use select fhtmtocenterpoint errors none fhtmtocornerpoints htmid given htmid returns three cartesian corner points triangle vertex table htmid less shallow depth large triangle example htmid returns corner points entire octant parameters htmid bigint null htmid triangle returns vertextable float float float example use select fhtmtocornerpoints errors none distances points fdistancelatlon given two latlon points fdistancelatlon returns distance arc minutes nautical miles parameters float null latitude degrees truncated float null longitude degrees truncated float null latitude degrees truncated float null longitude degrees truncated returns float null distance arc minutes example use print errors none fdistanceeq given two equatorial points fdistanceeq returns distance arc minutes parameters float null right ascension degrees truncated float null declination degrees truncated float null right ascension degrees truncated float null declination degrees truncated returns float null distance arc minutes example use print errors none fdistancexyz given two cartesian points fhtmxyz returns distance arc minutes parameters float null vector galaxy center prime meridian intersection equator float null vector normal galactic plane prime meridian float null vector normal galactic plane north pole normalized converted returns float null distance arc minutes example use print errors none regions region area interest celestial sphere specify region polygon convex hull polygon rectangle circle inside kernel htm engine regions represented union convexes turn intersections halfspaces information see article geospatial project syntactically region list convexes furthermore convex list halfspaces halfspace region specifications syntax region cover specifications circlespec rectspec polyspec hullspec convexspec areaspec regionspec latlon cartesian latlon cartesian latlon cartesian latlon cartesian latlon cartesian circlespec rectspec polyspec hullspec convexspec areaspec areaspec dec rad lat lon rad rad dec lat lon dec lat lon dec lat lon dec lat lon examples region specifications region number convexes including zero region convex region convex region convex convex region convex latlon region convex number including zero constraints convex cartesian convex circle point specification like dec arc minutes radius angles degrees represented convex consisting single constraint circle circle cartesian rect followed two angular point specs defining minimum maximum dec latmin must smaller latmax similar case longitudes interpreted sense means wide range rect poly followed optional single coordinate specification number corresponding point specifications two three numbers spherical polygon created connecting points great circle segments restricted convex polygon order matter must consistent bowtie pattern points polygon convex error result poly chull followed optional single coordinate specification number corresponding point specifications two three numbers spherical convex hull created connecting adjacent points great circles least three points needed points within single hemisphere otherwise error returned order points irrelevant chull fhtmregiontonormalformstring regionspec given string describing region fhtmregiontonormalformstring returns normalized representation region union convex hulls redundant constraints halfspaces discarded convex parameters regionspec nvarchar max null see syntax region specifications returns nvarchar max null returns normalized region spec form region convex null error example use declare regionspec nvarchar max select regionspec errors regionspec syntax error returns empty string see fhtmregionerror fhtmregiontotable regionspec given string describing region fhtmregiontotable returns tabular representation region union convex hulls redundant constraints halfspaces discarded convex tabular representation schema described parameters regionspec nvarchar max null see syntax region specifications returns regiontable convexid bigint null halfspaceid bigint null float float float float null null null null convex halfspace within convex cartesian coordinates halfspace plane displacement halfspace along unit vector empty table error example use select fhtmtonormalform errors regionspec syntax error returns empty see fhtmregionerror fhtmregionobjects regionspec type routine particular sql server sample spatial database library place station spatialindex tables functions installed given string describing region type place station fhtmregionobjects returns tabular list spatialindex objects type inside region parameters regionspec nvarchar max null see syntax region specifications type char places place table stations station table returns objecttable objid bigint null primary key object type type empty table error example use select find colorado places place htmid select objid fhtmregionobjects latlon errors regionspec syntax error returns empty see fhtmregionerror fhtmregionerror regionspec returns valid regionspec else returns syntax error message parameters regionspec nvarchar max null see syntax region specifications returns nvarchar max null diagnostic message example use declare diagnostic nvarchar max select diagnostic errors none htm covers compute htmid ranges region suite routines given region specification returns table trixels trixels cover specified region trixels described htm pair points within trixel startstop htm pair fact closed interval resulting table definition trixeltable bigint bigint simple regions described standard geometric shapes circle rectangle giving parameters typically regions described using linear syntax described enumeration htmids tends form locally connected intervals interface unifies contiguous triangles one large trixel fhtmcoverregion regionspec given string describing region fhtmregioncover returns trixel table covering region parameters regionspec nvarchar max null see syntax region specifications returns trixeltable bigint bigint example use select fhtmcoverregion errors case error returns empty table use fhtmregionerror regionstring get diagnostic fhtmcovercirclelatlon lat lon radiusarcmin given latitude longitude point radius arc minutes fhtmcovercirclelatlon returns trixel table covering circle parameters lat float null latitude degrees converted range lon float null longitude degrees radiusarcmin float null circle radius arc minutes radius positive less degrees minutes arc returns trixeltable bigint bigint example use select fhtmcovercirclelatlon errors none fhtmcovercircleeq dec radiusarcmin given dec point radius arc minutes fhtmcovercircleeq returns trixel table covering circle parameters float null right ascension degrees converted dec float null declination degrees converted range radiusarcmin float null circle radius arcminutes radius positive less degrees minutes arc returns trixeltable bigint bigint example use select fhtmcovercircleeq errors none fhtmcovercirclexyz radiusarcmin given string describing region fhtmcovercirclexyz returns trixel table covering circle parameters float null vector galaxy center float null vector normal galactic plane float null vector normal galactic plane radiusarcmin float null circle radius arc minutes xyz normalized converted north pole radius range limited returns trixeltable bigint bigint example use select fhtmcovercirclexyz errors none installing htm code install sql server samples following instructions installing samples topic sql server books online default sample installed drive sql drive system drive compile provided solution using visual studio microsoft framwork sdk using command similar following framework sdk command prompt msbuild attach spatial database data directory using sql server management studio executing command file command prompt window done already execute script sql server management studio executing command similar following command prompt window sqlcmd local spatial sql script requires sql server default database server local system command part contents work sql server earlier versions script enables common language runtime clr drops existing htm assembly replaces current assembly defines htm functions assembly compiling modifying debugging htm code database project able use deploy debug features breakpoints work table htm depths approximate areas library defaults htm keys first level divides sphere eight faces subsequent level divides speherical triangle four subtriangles table indicates trixel fairly small code modified deep representation runs bits floating point representation transcendental functions lose precision near level table htm level subdivdies sphere level table shows area square degrees arc minutes arc seconds meters trixel colum shows characteristic sizes default trixels arc usgs data object per trixel htm area depth degees minute area sec earth meters trixel sdss usgs sphere
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congruence topology grothendieck duality thin groups sep alexander lubotzky venkataramana abstract paper answers question raised grothendieck grothendieck closure integral linear group proves conjecture first author made done detailed study congruence topology arithmetic groups obtaining along way arithmetic analogue classical result chevalley complex algebraic groups application also deduce group theoretic characterization thin subgroups arithmetic groups introduction polynomial map two complex varieties general image zariski closed subset necessarily closed classical result theorem chevalley polynomial homomorphism two complex algebraic groups closed every closed subgroup arithmetic analogue issue let group let prime ring finite topology induces congruence topology compact open subgroup called congruence subgroup defines congruence topology subgroups subgroup closed topology called congruence closed subgroup commensurable called arithmetic group two surjective homomorphism groups image arithmetic subgroup arithmetic subgroup theorem image congruence subgroup necessarily congruence subgroup well known sln congruence subgroups whose images adjoint map sln psln aut congruence subgroups see ser proposition exposition explanation lubotzky venkataramana direct analogue chevalley theorem hold still case congruence subgroup sln normal subgroup congruence closure psln quotient finite abelian group first technical result says general case similar especially important simply connected image congruence subgroup congruence subgroup see proposition stating result give following definition set notations rest paper let linear algebraic group connected component solvable radical largest connected normal solvable subgroup say essentially simply connected gss simply connected given subgroup gln throughout paper denote intersection connected component zariski closure therefore always finite index normal subgroup notion essentially simply connected play important role paper due following proposition considered arithmetic analogue chevalley result proposition surjective algebraic homomorphism two algebraic groups every congruence closed subgroup image normal congruence closure finite abelian group essentially simply connected congruence subgroup image congruence subgroup congruence closed analogue chevalley theorem result nori weis enable prove proposition gln congruence closed subgroup closed congruence topology zariski closure exists congruence subgroup essentially simply connected image actually congruence subgroup apply proposition two directions duality discrete groups congruence topology group theoretic characterization thin subgroups arithmetic groups grothendieck closure gro grothendieck interested following question question assume homomorphism two finitely generated residually finite groups inducing isomorphism profinite completions already isomorphism tackle question introduced following notion given finitely generated group commutative ring identity let cla group automorphisms forgetful functor category moda finitely generated action moda preserving tensor product grothendieck strategy following showed conditions question induces isomorphism moda moda hence also cla cla asked question natural map clz isomorphism finitely generated residually finite group affirmative answer question would imply affirmative answer question grothendieck showed arithmetic groups strict congruence subgroup property indeed satisfy clz question basically asks whether recovered category representations lub first author phrased question framework tannaka duality asks similar question compact lie groups also gave concrete description clz clz modz continuous extension aut original representation aut however also shown lub answer question negative counterexamples provided arithmetic groups weak congruence subgroup property holds strict one congruence kernel finite conjectured lub conj arithmetic group lubotzky venkataramana clz strict congruence subgroup property conjecture left open even almost years since lub written various counterexamples given question also give counterexamples question even settled whether clz finitely generated free groups answer fact prove following surprising result gives essentially complete answer question theorem let finitely generated subgroup gln satisfies duality clz congruence subgroup property consequently every faithful representation glm zariski closure essentially simply connected every finite index subgroup closed congruence topology gln case image group simply connected quotient congruence arithmetic group theorem surprising shows cases proved grothendieck motivated suggest duality holds general essentially cases duality holds let note assumption really restrictive lemma show every gln find representation glm whose zariski closure essentially simply connected theorem implies conjecture lub corollary simply connected semisimple group congruence subgroup clz satisfies strict congruence subgroup property particular corollary clz every finitely generated free group least two generators furthermore clz fact follow results clz uncountable moving last application let say words proposition helps prove result like theorem congruence topology description clz equation implies clz lim limit finitely generated abelian group representation aut aut aut inverse limit countable discrete groups one say much unless connecting homomorphisms surjective general case congruence closure aut proposition shows corresponding maps almost onto even surjective modules call simply connected representations namely cases torsion free hence isomorphic zariski closure aut gln essentially simply connected show category modz saturated modules see lemma deduce one compute clz equation considering simply connected representations use proposition get fairly good understanding clz enables prove theorem addition also deduce corollary simply connected representation induced map clz aut onto congruence closure corollary deduce last application thin groups recent years following sar lot interest distinction thin subgroups arithmetic subgroups algebraic groups let recall definition subgroup gln called thin infinite index gln zariski closure gln general group say thin group thin representation exists representation gln thin last five decades lot attention given study arithmetic groups many remarkable results especially higher rank mar references therein much less known thin groups example known exists thin group property also given subgroup arithmetic group say given set generators difficult lubotzky venkataramana decide whether thin arithmetic finite infinite index integral zariski closure therefore interest perhaps even surprising results enable give purely group theoretical characterization thin groups gln stating precise result make topology clz explicit take class simply connected representations computing group clz one show clz closed subspace product given discrete topology topology quotient space clz following theorem state theorem let finitely generated group thin group satisfies least one following conditions namely finite index subgroup infinite abelianization clz compact warning groups realized arithmetic groups well thin groups example free group arithmetic subgroup time thin subgroup every semisimple group well known result tits terminology thin group thanks math department hebrew university great hospitality major part work done would also like thank bose fellowship support period authors thank math department university marseilles conference oberwolfach work completed would especially like thank bertrand remy many interesting discussions warm hospitality indebted erc nsf bsf support preliminaries algebraic groups recall definition essentially simply connected group congruence topology definition let linear algebraic group maximal connected normal solvable subgroup radical identity component say essentially simply connected part simply connected note essentially simply connected quotient group unipotent radical product hss hss simply connected torus example connected group essentially simply connected simply connected group sln essentially simply connected however radical group gln group scalars gln sln gln essentially simply connected show later lemma iii every group finite cover essentially simply connected lemma suppose subgroup product two essentially simply connected linear algebraic groups suppose projection surjective also essentially simply connected proof assume may connected let radical projection normal since surjective moreover image group latter zariski dense compact subgroup hence therefore reductive commutator hence hence radical let since radical follows solvable normal subgroup hence connected component contained since follows precisely connected component identity inclusion projections surjective assumption simply connected moreover connected thus inclusion projections surjective simply connected let kernel map identity component surjective map connected algebraic groups finite kernel simple connectedness implies hence normal lubotzky venkataramana write simple simply connected closed normal subgroup must equal subset simply connected therefore simply connected preceding two paragraphs simply connected hence since connected component simply connected follows hence simply connected completes proof lemma arithmetic groups congruence subgroups introduction defined notion arithmetic congruence subgroup using adelic language one define notion arithmetic res congruence group concrete terms follows given linear algebraic group sln defined say subgroup arithmetic group commensurable sln intersection finite index well known notion arithmetic group depend specific linear embedding sln ser may define arithmetic completion completion group respect topology topological group obtained designating arithmetic groups fundamental systems neighbourhoods identity given sln preceding paragraph say arithmetic group congruence subgroup exists integer contains principal congruence subgroup sln sln kernel residue class map sln sln get structure topological group group designating congruence subgroups fundamental system neighbourhoods identity completion respect topology denoted notion congruence subgroup depend specific linear embedding sln since every congruence subgroup arithmetic group exists easily seen surjective map kernel compact profinite subgroup called congruence subgroup kernel one says congruence subgroup property trivial easily seen equivalent statement every arithmetic subgroup congruence subgroup congruence topology known see last one paragraph solvable groups congruence subgroup property moreover every solvable subgroup gln polycyclic group every subgroup intersection finite index subgroups every solvable subgroup arithmetic group congruence closed use facts frequently sequel another equivalent way viewing congruence completion see ser remarque follows let ring finite adeles equipped standard adelic topology let closure group also locally compact group contains group congruence completion may viewed closure lemma let linear algebraic groups defined suppose surjective let representation defined exists faithful subrepresentation surjective map defined image arithmetic subgroup map arithmetic subgroup iii connected exists connected essentially simply connected algebraic group surjective homomorphism finite kernel surjective homomorphism algebraic essentially simply connected image congruence subgroup congruence subgroup proof let faithful representation linear algebraic group defined clearly faithful contains proves part statement theorem prove iii write product radical group let hss simply connected cover hence hss acts via covering map define hss product clearly map finite kernel satisfies properties iii prove may assume connected unipotent radicals assumptions change quotient groups moreover since product similarly lubotzky venkataramana unipotent group congruence subgroup property suffices prove reductive assumption essentially simply connected hss hss tori hss hss simply connected groups thus connected reductive surjective map derived groups simply connected abelianization torus similarly hss simply connected group hence product simply connected algebraic groups factor hss group hss product smaller number renumbering indices may assume hss product map hss projection first factors hence image congruence subgroup hss congruence subgroup hss tori congruence subgroup property result chevalley already stated beginning section true solvable algebraic groups hence image congruence subgroup congruence subgroup thus need prove every subgroup reductive group form hss congruence subgroups congruence subgroup use adelic form congruence topology suppose compact open subgroup ring finite adeles image quotient map congruence subgroup torus hence hss possibly smaller open subgroup proves note part iii prove proposition arithmetic chevalley theorem section prove proposition assume surjective morphism groups prove contains commutator subgroup congruence subgroup containing starting proof let note general image congruence subgroup need congruence subgroup following proposition gives fairly general situation happens congruence topology proposition let finite covering algebraic groups defined simply connected assume dense write kernel kernel map let congruence subgroup closure image congruence subgroup proving proposition let note finite group product infinitely many finite abelian groups central implies corollary infinitely many congruence subgroups subgroups unbounded finite index congruence closures image contains commutator subgroup normal abelian quotient prove proposition proof let image rational points define subgroup subgroup inverse image congruence subgroup note subgroups exactly images congruence subgroups routine check declaring subgroups open get structure topological group topology weaker equal arithmetic topology however strictly stronger congruence topology last assertion follows fact completion quotient congruence completion whereas completion respect congruence topology let congruence subgroup let congruence closure open topology denote completion respect topology denote closures equalities hence proves proposition lubotzky venkataramana proof shows normal since central abelian quotient true corollary also proved continue proof proposition assume may replacing zariski closure characters defined suppose zariski closure let therefore surjective homomorphism defined image arithmetic group zariski dense arithmetic group however arithmetic groups finite zariski dense therefore also assume connected start proving proposition case congruence subgroup write radical may assume essentially simply connected lemma iii without affecting hypotheses conclusion proposition hence semi direct product clearly every congruence subgroup contains congruence subgroup form congruence subgroups similarly write since easily seen map onto onto enough prove proposition separately first recall solvable linear algebraic group defined congruence subgroup property holds every arithmetic subgroup congruence subgroup reference see last one paragraph consequently lemma image congruence subgroup arithmetic group hence congruence subgroup thus dispose solvable case case groups denote simply connected cover map lifts map simply connected groups surjective map sends congruence subgroup congruence subgroup lemma thus reduced situation simply connected cover assumptions connected simply connected semisimple claim factor congruence topology compact otherwise image arithmetic group finite zariski dense connected strong approximation theorem theorem gives dense proposition applied finish proof proposition case congruence subgroup need show true also general case congruence closed end let formulate following proposition independent interest proposition let gln zariski closure der congruence closed der congruence subgroup der proof toral factors proved fact case congruence closed zariski dense subgroup congruence subgroup note stated general assumption toral factor mistakenly omitted proof shows toral factor assume connected unipotent torus zariski dense congruence closed congruence subgroup direction note image solvable always congruence closed proposition follows end proof proposition congruence closed subgroups looking zariski closure apply proof der also proves proposition course proposition general form following result based nori weis fact core proposition proposition suppose zariski dense simply connected subgroup closed congruence topology congruence subgroup grothendieck closure grothendieck closure group definition let representation lattice space get continuous hob group denotes momorphism lubotzky venkataramana profinite completion extends denote subgroup profinite completion preserves lattice fact since det every hence also every clg hence subgroup denote subgroup lattices therefore runs integral representations group suppose finitely generated abelian group necessarily lattice necessarily also torsion finite subgroup finite exponent say torsion free since acts finite also acts group finite group via say follows finite group via thus suppose definition hence equality quotient group shows shows preserves finitely generated abelian groups clz mean grothendieck closure finitely generated group essentially result lub grothendieck closure clz group defined lub group considered closure respect finitely generated modules also modules whereas consider finitely generated modules modules argument preceding paragraph shows closures identify grothendieck closure clz foregoing group notation let group finitely generated abelian group corresponding denote zariski closure image connected component identity linear algebraic groups defined congruence topology since profilet denote subgroup nite topology induces congruence topology congruence closure denote intersection derived subgroup thus exact sequence extension finite group abelian group image abelianization connected component simply connected representations definition say simply connected group essentially simply connected unipotent radical quotient product semisimple simply connected torus easy consequence lemma simply connected representations closed direct sums lemma let two simply connected representations abstract group direct sum also simply connected also lemma let representation simply connected map surjective proof image contains image proposition congruence subgroup algebraic group map surjective map simply connected groups therefore part lemma image congruence subgroup proposition congruence closed hence equal congruence closure surjective general lemma every integral representation subrepresentation faithful representation simply connected lubotzky venkataramana proof let representation let der derived subgroup identity component zariski closure lemma iii exists map finite kernel connected hss simply connected group denote space lemma may considered faithful representation covering group lemma image arithmetic subgroup arithmetic group moreover virtually torsion free one may choose normal arithmetic subgroup map splits particular map splits normal subgroup finite index thus may considered representation group consider induced representation since representation follows trivn since first paragraph proof modules follows hence write representation normality implies restriction representation contained direct sum varies finite set write zariski closure image since zariski closure group hss simply connected composed conjugation simply connected representation follows lemma simply connected since simple connectedness representation subgroups finite index follows representation simply connected proved exists embedding module lattices spaces basis lattice combination basis finite generation implies exists integer inclusion embedding clearly module isomorphic isomorphism given multiplication hence lemma follows congruence topology following main technical result section main results paper derived proposition group inverse limit groups runs simply connected representations congruence closure moreover simply connected map surjective proof denote temporarily subgroup elements stabilize lattice simply connected representations let arbitrary finitely generated lattice also denote action lemma exists simply connected representation contains since stabilizes follows dense particular thus group definition set elements stabilize stable torsion free latprofinite completion tices follows particular elements stabilize lattices associated simply connected representations hence preceding paragraph implies proves first part proposition see equation enumerate simply connected integral representations since finitely generated write sequence simply connected representations write direct sum lemma simply connected moreover simply connected representation contained lemma follows inverse limit totally ordered family moreover maps onto taking inverse limits follows maps onto group every follows lemma every homomorphic image hence proves second part proposition lubotzky venkataramana definition let finitely generated group say abelianization finite every finite index subgroup corollary every simply connected representation congruence closure congruence subgroup inverse limit totally ordered set simply connected representations congruence groups groups simply connected moreover maps surjective hence maps surjective proof simply connected representation finite index subgroup image connected zariski closure assumption torus simply connected since group fab follows hence der proposition implies congruence subgroup corollary immediate proposition take proof proposition prove theorem let first prove direction claiming congruence subgroup property implies proved arithmetic groups grothendieck follow proof lub works general indeed gln faithful simply connected representation satisfies congruence subgroup property means map gln injective gln last exactly congruence closure assumption congruence closed equal summary injective opposite direction assuming description follows every finite index subgroup see lub proposition faithful simply connected representation also proposition congruence closed case means every finite index subgroup congruence closed congruence subgroup property thin groups let finitely generated group gln let zariski closure gln gln congruence topology say thin subgroup otherwise arithmetic subgroup general given say given set generators difficult question determine thin arithmetic next result gives still group theoretic characterization abstract group thin first warning abstract group sometimes appear arithmetic subgroup sometimes thin subgroup example free group two generators finite index subgroup arithmetic time well known result tits asserting sln contains copy zariski dense sln also thin precise let define definition finitely generated group called thin group faithful representation gln infinite index gln zariski closure gln called thin representation assumed gln assume also may see lemma representation simply connected preserves proposition group subgroup lattices totally ordered set respect relation sub representation faithful simply connected integral representations maps surjective congruence closure hence inverse limit varies congruence closed subgroups inverse limit images equip discrete topology consequently closed subspace tychonov product topology considered following theorem theorem let finitely generated group glm thin group satisfies following two properties group every finite index subgroup finite group compact proof assume first thin group done assume must prove compact know faithful thin representation lubotzky venkataramana gln addition simply connected induces surjective map see proposition congruence closure gln congruence subgroup corollary thin infinite index thus mapped onto discrete infinite quotient space hence compact assume thin group implies every faithful integral representation finite index integral zariski closure claim finite otherwise finitely generated mapped group zariski dense integral representation torus take integral matrix sln neither unipotent whose semisimple part infinite order unipotent semisimple part zariski closure non trivial contain subgroup finite index since commensurable factors non trivial infinite representation faithful integral representation give thin representation proves finite similar argument using induced representation works every finite index subgroup hence satisfies prove compact already know corollary congruence groups surjective homomorphisms note faithful integral representation assume representations sequence faithful lim implies lim assumption finite index inverse limit finite sets hence compact grothendieck closure let finitely generated group say integral superrigid exists algebraic group glm embedding finite index subgroup every integral representation gln exists algebraic representation gln agree finite index subgroup note integral finite index subgroup integral congruence topology example groups first irreducible arithmetic lattices high rank semisimple lie groups also arithmetic lattices rank one simple lie groups see mar cor shows groups thin groups let subgroup glm whose zariski closure essentially simply connected say satisfies congruence subgroup property csp natural extension glm glm finite kernel theorem let glm finitely generated subgroup satisfying compact arithmetic group integral finite arithmetic group satisfying congruence subgroup property remarks finiteness implies particular compactness theorem recovers well known fact see bms congruence subgroup property implies explained based ser simple connectedness necessary condition csp hold lemma embedding gln also simply connected one prove theorem proof assume first compact case theorem must arithmetic subgroup algebraic group without loss generality using lemma assume connected simply connected call representation let representation let direct sum group subgroup surjective projections since embeddings group thin representations follows corollary projection yields isomorphism arithmetic groups assume may arithmetic group every arithmetic group virtually product form unipotent semisimple parts respectively note torus quotient since hence may also described lubotzky venkataramana virtually maximal normal nilpotent subgroup similarly proves groups isomorphic arithmetic groups proves isomorphism otherwise ker normal subgroup would infinite intersection arithmetic group therefore arithmetic groups isomorphic isomorphism induced projection since simply connected assumption factor follows product group defined zariski dense isomorphism arithmetic groups shows group finite means finite therefore isomorphism map also isomorphism since surjective morphism groups dimension since simply connected proves group lub proved satisfies super rigidity simply connected group finite index correspondence ker finishes proof parts remark situation theorem arithmetic group satisfying difference parts whether also satisfies csp known arithmetic group simply connected group satisfies without satisfying csp conjecture serre congruence subgroup problem predicts arithmetic lattices rank one lie groups fail csp include lie groups like shown serre made conjecture potentially arithmetic subgroups groups compact finite experts seem believe groups satisfy csp anyway know subgroup gln compact finite references bms bass milnor serre solution congruence subgroup problem sln publ math bass lubotzky nonarithmetic superrigid groups counterexamples platonov conjecture ann math congruence topology borel introduction aux groupes arithmetiques actualite scientifiques industrielles hermann paris bridson grunewald grothendieck problems concerning profinite completions representations groups ann math chahal solution congruence subgroup problem solvable algebraic groups nagoya math cor corlette archimedean superrigidity hyperbolic geometry ann math golod finitely approximable russian izv akad nauk sssr ser mat gri grigorchuk burnside problem periodic groups russian funktsional anal prilozhen gro grothendieck representations lineaires compactification profinie des groupes discrets manuscripta math gromov schoen harmonic maps singular spaces superrigidity lattices groups rank one publ math lub lubotzky tannaka duality discrete groups american math mar margulis discrete subgroups lie groups ergebnisse der mathematik und ihrer grenzgebiete berlin nori nori subgroups gln invent math platonov rapinchuk algebraic groups number theory series pure applied mathematics academic press pps platonov tavgan grothenkieck problem profinite completions groups soviet math dokl platonov tavgan grothendieck problem profinite completions representations groups pyber groups intermediate subgroup growth problem grothendieck duke math raghunathan discrete subgroups lie groups ergebnisse der mathematik und ihrer grenzgebiete band new yorkheidelberg raghunathan congruence subgroup problem publ math sar sarnak notes thin matrix groups thin groups superstrong approximation math sci res publ cambridge university press cambridge ser serre groupes congruence bass matsumoto mennicke milnor moore bourbaki vol exp soc math france paris tits free subgroups linear groups algebra venkataramana remark extended congruence subgroups imrn lubotzky venkataramana weis weisfeiler strong approximation zariski dense subgroups ann math institute mathematics hebrew university jerusalem israel tata institute fundamental research homi bhabha road colaba mumbai india venky
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quantitative generalization theorem minimal abelian topological groups jun taras banakh memory ivan prodanov abstract topological group defined compact exponent number set compact closure number called compact exponent principal result states complete abelian topological group compact exponent equal injective continuous homomorphism topological group every exists positive number equal result many interesting implications abelian topological group compact complete weaker hausdorff group topology minimal abelian topological group precompact famous prodanovstoyanov theorem topological group complete compact exponent closed hausdorff paratopological group containing topological subgroup confirms old conjecture banakh ravsky introduction paper prove theorem considered quantitative version fundamental theorem precompactness minimal abelian topological groups topological groups considered paper hausdorff recall topological group precompact completion uniformity compact happens totally bounded sense neighborhood unit exists finite subset subset topological space denote closure shall say topological group compact exponent number set compact closure case number called compact exponent observe topological group compact compact exponent first time concept compact exponent without naming appeared banakh ravsky powertopological semigroup understand semigroup endowed hausdorff topology map axn continuous clear topological semigroup powertopological powertopological semigroup semitopological means shift axb continuous paper contains unique principal result theorem let number complete abelian topological group following conditions equivalent compact exponent equal injective continuous homomorphism topological group every point exists number equal continuous homomorphism powertopological semigroup every point exists number equal theorem many interesting implications mathematics subject classification key words phrases abelian topological group minimal topological group paratopological group topological semigroup topological group compact exponent taras banakh corollary abelian topological group compact injective continuous homomorphism topological group image closed corollary reformulated follows corollary abelian topological group compact complete weaker hausdorff group topology corollary implies famous theorem precompactness minimal abelian topological groups recall topological group minimal admit strictly weaker hausdorff group topology corollary prodanov stoyanov minimal abelian topological group precompact proof precompactness equivalent compactness completion corollary compactness follow soon check continuous injective homomorphism topological group image closed lose generality assuming topological group complete dense minimality restriction topological isomorphism uniquely extends topological isomorphism completions topological groups topological isomorphism closed another corollary theorem confirms old conjecture banakh ravsky corollary complete abelian topological group following conditions equivalent compact exponent continuous homomorphism hausdorff powertopological semigroup image closed injective continuous homomorphism topological group quotient group torsion group isomorphic topological embedding hausdorff paratopological group image closed proof equivalences follow immediately corresponding equivalences theorem proved ravsky remark noticed dikranjan megrelishvili theorem fails nilpotent groups group nilpotent lie group nilpotence degree minimal compact see also nonetheless dikranjan uspenskij proved following two extensions corollary nilpotent solvable topological groups theorem dikranjan uspenskij nilpotent topological group compact every continuous homomorphism topological group image closed theorem dikranjan uspenskij solvable topological group compact every continuous homomorphism closed normal subgroup topological group image closed proof theorem follows line proof theorem additional care non completeness compact exponent quite long technical separate proof several steps section establish preliminary results related bounded sets topological groups properties topological groups pre compact exponent section abelian topological group define weaker group topology dependent sequence points prove two key lemmas first key lemma produces sequence quantitative generalization theorem additional properties second key lemma shows properties ensure topology hausdorff shall apply key lemmas proof bounded version theorem three times unbounded version arbitrary theorem follows bounded unbounded versions proved theorem respectively preliminaries section fix standard notations used paper also recall known results prove simple lemmas standard notations denote set natural numbers positive integer numbers set integer numbers finite ordinals denote additive group integer numbers denote set prime numbers number let set prime divisors subset topological space denote closure point called accumulation point sequence topological space neighborhood contains infinitely many points group denote unit unique element two subsets group put powers defined induction also sequence subsets group denote subgroup generated union subset group natural number let set powers elements observe general subgroup abelian group subgroup topological group complete complete uniformity generated entourages runs neighborhoods unit see completion topological group uniformity carries natural structure topological group contains dense subgroup bounded sets topological groups section establish properties bounded precompact sets topological groups subset group called set finite subset subset topological group called totally bounded neighborhood unit subset complete topological group compact closure totally bounded totally bounded subsets topological groups also called precompact sets see following simple important lemma shows failure total boundedness recognized countable subgroups lemma subset group set countable subgroup set proof since every finite set hence zorn lemma exists maximal subset sense distinct points maximality guarantees exists hence consequently choice ensures set infinite choose countable subgroup infinite intersection claim set assuming opposite could find finite subset since set infinite two distinct points contradict choice set localizations topological group called prime number divisible set dense abelian topological group union taras banakh subgroups largest closed subgroup see abelian discrete topological group set coincides subgroup shall need following fact whose proof found lemma abelian topological group union compact subgroups subgroup dense lemma compact abelian topological group prime number number ntp pnp largest number pnp divides lemma let abelian topological group number pnp largest number pnp divides moreover union compact subgroups pnp dense proof every implies pnp ntp taking account group conclude pnp assume union compact subgroups see pnp dense suffices open set intersecting find element pnp since exists point choose neighborhood since union compact subgroups closure cyclic group compact lemma subgroup dense find element lemma implies ntp pnp ntp pnp pnp bohr topology theorem bohr topology semitopological group weakest topology continuous homomorphisms compact topological groups remain continuous bohr topology necessarily hausdorff topological groups hausdorff bohr topology called maximally almost periodic proof theorem shall apply following fundamental theorem whose proof found theorem let abelian semitopological group every map continuous group subset neighborhood set neighborhood unit bohr topology two key lemmas section prove two lemmas following ideas proof lemma first lemma help construct nice sequence abelian topological group sequence determines weaker group topology second lemma find conditions weaker topology hausdorff lemma let abelian group sequence decreasing sequence sets sequence subgroups satisfying following conditions every set contains unit either exists sequence points satisfying conditions numbers xnm max proof construction sequence inductive let assume points constructed consider finite set defined condition lemma let set pairs condition every set quantitative generalization theorem group abelian amenable hence admits invariant finitely additive invariant probability measure defined boolean algebra subsets see claim set zwk measure proof using zorn lemma choose maximal subset distinct points maximality guarantees every exists hence means since set set infinite claim distinct points sets disjoint assuming intersection contains point conclude contradicts choice set therefore family disjoint family zwk consider morphism observe family zwk zwk disjoint additivity invariantness measure ensures set zwk zwk measure additivity measure claim imply set measure zero consequently find point clear point satisfies condition lemma let abelian topological group sequence points every positive real number consider set let topology largest precompact group topology group bohr topology group endowed discrete topology let families open symmetric neighborhoods unit topological groups respectively easy see family neighborhood base unit group topology topology called topology sequence sequence called topology topology hausdorff sequence called case hausdorff topological group consider completion complete topological group following lemma detects sequences lemma let abelian topological group sequence decreasing sequence neighborhoods unit sequence closed subgroups satisfying following conditions every either closed subgroup compact every neighborhood unit exists sequence points satisfies condition lemma set totally bounded topological group proof separate proof lemma four claims claim every taras banakh proof derive contradiction assume set contains point write number integer numbers since thus let largest number maximality guarantees every hence contradicts property lemma claim exists since claim real number required property derive contradiction assume find point ywk write integer numbers latter inequality implies inclusion ywk implies follows max let largest number number exists since maximality guarantees follows contradicts condition lemma hence proof write integer numbers claim topology hausdorff proof given element find belong closure subgroup generated set find neighborhood unit conclude contain next consider case hence since subgroup closed exists neighborhood unit done assume case exists follows applying claim find claim opposite case contradicts choice finally consider case choose neighborhood unit condition lemma exist compactness exists finite set consider neighborhood group observe theorem set neighborhood unit bohr topology group endowed discrete topology baer theorem extensions homomorphisms divisible groups implies bohr topology group coincides subspace topology inherited bohr topology group consequently exists bohr neighborhood let claim neighborhood contain point assuming find points conclude according claim contradicts choice neighborhood claim set totally bounded topological group proof given neighborhood need find finite set assume neighborhood basic form let maximal set distinct elements maximality guarantees point exists point claim set finite derive contradiction assume infinite quantitative generalization theorem choose neighborhood total boundedness topology yields finite set pigeonhole principle exists point set contains two distinct points last inequality ensures hence hand contradicts definition set contradiction shows set finite set totally bounded topological groups compact exponent recall topological group compact exponent set compact closure lemma complete abelian topological group compact exponent contains sequence accumulation point completion topological group every power belongs set precompact proof let smallest number precompact using lemma find countable subgroup every set precompact put every prime number let largest number psp divides pcompactness exponent implies union compact subgroups lemma subgroup psp dense claim prime divisor subgroup psq precompact proof consider number every let largest number tpthat divides observe lemma subgroup dense assuming precompact would conclude subgroup ptp psp precompact closure contradicts minimality choose neighborhood unit every divisor subgroup psp choose decreasing sequence neighborhoods unit easy see sequences satisfy conditions lemma lemma applying lemmas obtain sequence satisfying conditions lemma set totally bounded hausdorff topological group hence accumulation point completion claim every power belongs subgroup precompact proof set precompact closure compact hence closed dense subset closed hence contains point consequently next assume set precompact case prove derive contradiction assume first observe number divide otherwise would precompact consequently prime number power divide since every neighborhood unit neighborhood intersects set xnk implies exists number observe neighborhood topology let since accumulation point sequence xnk exists number max xnk hence xnk yws write point last inequality implies integer numbers taras banakh follows xnk claim since divisible number divisible let largest number divisible since number greater maximality guarantees hence let consider power zls wss psp contradicts condition lemma see last every prime number let largest number ptp divides follows lemma closure set coincides psq closure set precompact contains set precompact according claim set precompact hence contained precompact set lemma implies bounded rather quantitative part theorem theorem number complete abelian topological group compact exponent following conditions equivalent subgroup totally bounded closed subsemigroup continuous homomorphism powertopological semigroup injective continuous homomorphism topological group proof prove assume subgroup totally bounded hence compact closure complete topological group fix closed subsemigroup continuous homomorphism powertopological semigroup compact set compact hence closed image powertopological semigroup continuity power map implies set closed since closed set contains closure point belongs hence implication trivial follows lemma remark theorem extended semitopological semigroups take infinite discrete topological group compact exponent consider compactification extend group operation semigroup operation letting easy check compact hausdorff semitopological semigroup containing discrete subgroup topological groups hypocompact exponent shall say topological group hypocompact exponent neighborhood unit exist finite subset clear topological group precompact exponent hypocompact exponent lemma topological group hypocompact exponent finitely generated abelian subgroup precompact quantitative generalization theorem proof given neighborhood unit need find finite subset since hypocompact exponent exist since group abelian finitely generated quotient group finite find finite subset hence finite set required property lemma complete abelian topological group hypocompact exponent subgroup compact every exists proof see closed set compact suffices check totally bounded complete topological group given open neighborhood unit use hypocompactness exponent find finite set hence means precompact hence compact claim opposite case every could choose point taking account topological group hypocompact exponent see set precompact complete topological group hence sequence accumulation hand every implies desired contradiction shall say subset topological group precompact modulo set open neighborhood unit set lemma let two subsets abelian topological group precompact modulo set precompact modulo proof given neighborhood unit find neighborhood unit since precompact modulo exists finite subset follows every points hence means precompact modulo topological group neighborhood unit exists countable subset clear separable topological group lemma assume complete abelian topological group hypocompact exponent prime number compact exponent contains sequence accumulation point proof first prove following statement claim every set precompact modulo proof derive contradiction assume set precompact modulo lemma implies every set precompact modulo hence precompact modulo claim precompact indeed every neighborhood unit hypocompactness exponent exist finite subset ntp let largest number divides since subgroup subgroup ntp dense consequently ntp since precompact modulo exists finite subset means precompact hence closed subgroup complete group compact exponent contradicts assumption lemma using claim choose decreasing sequence neighborhoods unit every following conditions satisfied taras banakh topological group contains countable subset zwk every let since subgroup every ntp largest number divides observation choice sequence lemma guarantee sequences satisfy conditions lemma lemma applying lemmas construct sequence satisfying conditions lemma applying lemma conclude sequence set totally bounded topological group choose accumulation point sequence exists since set totally bounded complete topological group claim every power belong proof derive contradiction assume let largest number divides subgroup claim guarantee ntp since exists number let since sequence accumulates neighborhood contains point max consequently write integer numbers last inequality implies follows follows hence taking account conclude divisible number divisible assume largest possible number property follows max let largest number divides taking account divided conclude xnm hence contradicts condition lemma next treat case topological groups hypocompact exponent whose compact exponent lemma assume complete abelian topological group hypocompact exponent every prime number precompact exponent fails compact exponent contains sequence accumulation point proof lemma subgroup compact every neighborhood unit exists number assumption every prime number precompact exponent consequently exists psp precompact assume number smallest possible subgroup precompact psp precompact psp claim subgroup psp precompact proof given closed neighborhood unit choose neighborhood unit hypocompactness exponent exist finite set every subgroup implies mtp since subgroup also get quantitative generalization theorem precompactness groups psp implies precompactness finite sum find finite subset finite set desired inclusion psp psp psp witnessing subgroup precompact claim set infinite proof derive contradiction assume set finite case shall prove number psp set precompact contradict assumption lemma indeed every group implies stp psp lemmas subgroup dense continuity map subgroup stp dense closed subgroup stp psp coincides claim subgroup compact witnessing precompact exponent contradicts assumption lemma compactness subgroups imply compactness subgroup shorten notations every prime number denote subgroup psp observe precompact psp psp precompact claim exists strictly increasing sequence prime numbers decreasing sequence neighborhoods unit satisfying following conditions every spn kwn proof construction sequences inductive start inductive construction let smallest prime number since group compact subgroup precompact neighborhood unit assume prime number neighborhood constructed hypocompactness exponent lemma exists number choose prime number max prime number number divided group subgroup mtp dense mtp since subgroup also get since group compact group spn precompact neighborhood unit completes inductive step since topological group exists countable subset zwk every let closure subgroup spn claim numbers inclusion holds proof nspi spi spi spi spi hence spi next assume number divide tpi implies nspi spi contains subgroup spi condition claim hand condition claim group kspj contained kwk kwi hence kwi taras banakh let show sequences satisfy conditions lemma condition coincides condition claim hence satisfied check condition fix two numbers assume claim divisible prime number spi nspi condition claim set spi hence since conditions lemma follow soon check every neighborhood unit exists lemma exists choose large every prime number divide hence spi tpi ntpi kspi therefore conditions lemma satisfies conditions lemma satisfied apply lemmas construct sequence satisfying condition lemma set totally bounded topological group hence accumulation point completion topological group claim every power proof derive contradiction assume let smallest number prime number divide since exists number let since sequence accumulates neighborhood contains point max consequently write integer numbers last inequality implies follows also hence let smallest number exists number mpsuch divisible prime number number exists since number divisibleqby minimality guarantees every number divisible number hence according claim let largest number divisible maximality guarantees divisible follows hence contradicts condition lemma claim treating topological groups hypocompact exponent following lemma proved ravsky shall give alternative proof lemma deriving key lemmas lemma abelian topological group hypocompact exponent contains sequence accumulation point proof since hypocompact exponent exists neighborhood unit every set choose sequence unit claim exists countable subgroup every subgroup quantitative generalization theorem proof using zorn lemma every choose maximal subset distinct elements maximality implies every exists hence nmn hence nmn since set set infinite choose countable subgroup every intersection infinite claim every subgroup assuming opposite could find finite subset since set infinite two distinct points contradict choice set let clear sequences satisfy conditions lemma lemma applying two lemmas obtain sequence satisfying conditions lemma set totally bounded topological group hence accumulation point completion claim every point belong proof assuming would conclude every neighborhood unit neighborhood intersects set xnk implies exists number let since sequence xnm accumulates topological group neighborhood contains point xnk max consequently xnk pmsome write integer numbers last inequality implies follows xnk xnk claim follows pkm since number equal zero let largest number follows xnk hence contradicts condition lemma joining pieces together section combine lemmas prove unbounded part theorem first observe three lemmas imply following corollary corollary complete abelian topological group compact exponent admits weaker hausdorff group topology completion topological group contains element prove promised unbounded part theorem theorem complete abelian topological group following conditions equivalent complete compact exponent continuous homomorphism powertopological semigroup every point exists number injective continuous homomorphism topological group every point exists number proof implication follows theorem trivial prove assume compact abelian topological group compact exponent every set precompact using lemma find closed separable subgroup every subgroup precompact means compact exponent taras banakh corollary admits weaker hausdorff group topology completion topological group contains element let family open neighborhoods unit topological group easy family satisfies pontryagin axioms hence neighborhood base unit hausdorff group topology subgroup remains closed subspace topology inherited coincides topology completion topological group contains completion topological group consequently element required property acknowledgements author expresses thanks alex ravsky fruitful discussions topic paper michael megrelishvili helpful remarks comments text special thanks due dikran dikranjan told author genuine story proof theorem differs bit original proof theorem suggested dedicate paper memory ivan prodanov died heart attack age precisely age author moment writing paper references arhangel skii tkachenko topological groups related structures atlantis studies mathematics atlantis press paris world scientific publishing pte hackensack banakh ravsky paratopological groups ukrainian congress mathematics kyiv dikranjan recent advances minimal topological groups topology appl dikranjan megrelishvili relative minimality subgroups topological groups topology appl dikranjan megrelishvili minimality conditions topological groups recent progress general topology iii hart van mill jan simon eds springer verlag atlantis press berlin dikranjan prodanov stoyanov topological groups characters dualities minimal group topologies edn monographs textbooks pure applied mathematics vol marcel dekker new york dikranjan shakhmatov selected topics structure theory topological groups pearl open problems topology elsevier dikranjan uspenskij categorically compact topological groups pure appl algebra engelking general topology heldermann berlin generalization theorem bogoliouboff topological abelian groups math scand guran topological groups similar groups dokl akad nauk sssr megrelishvili generalized heisenberg groups shtern question georgian math paterson amenability amer math providence prodanov stojanov every minimal abelian group precompact acad bulgar raikov completion topological groups izv akad nauk sssr russian ravsky paratopological groups visnyk lviv ser robinson course theory groups new york roelcke dierolf uniform structures topological groups quotients mcgrawhill banakh ivan franko national university lviv ukraine jan kochanowski university kielce poland address
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international journal foundations computer science technology ijfcst november new hybrid coaw method solving problems zeinab borhanifar elham shadkam department industrial engineering faculty eng khayyam university mashhad iran abstract article using cuckoo optimization algorithm simple additive weighting method hybrid coaw algorithm presented solve problems cuckoo algorithm efficient structured method solving nonlinear continuous problems created pareto frontiers coaw proposed algorithm exact good dispersion method high speed finding pareto frontiers identifies beginning end points pareto frontiers properly order validation proposed algorithm several experimental problems analyzed results indicate proper effectiveness coaw algorithm solving problems keywords cuckoo optimization algorithm coa simple additive weighting saw pareto frontier optimization problem mop introduction many methods solving nonlinear constrained programming problems newton genetic algorithm algorithm birds paper using emerging cuckoo optimization algorithm simple additive weighting method solve problems presented optimization assumed decision makers communicate one goal like profit maximization cost minimization waste minimization share minimization real world possible consider single goals usually one goal examined example control projects time factor considered objectives cost quality ignored results reliable necessary use optimization problems ehrgott gandibleux presented detailed approximation method regarding problems related combinatorial optimization klein hannan multiple objective integer linear programming problems moilp presented algorithm additional restrictions used remove known dominant solutions sylva crema offered method find set dominant vectors multiple objective integer linear programming problems arakawa used combined general data envelopment analysis genetic algorithm produce efficient frontier optimization problems deb analyzed solution problems evolutionary algorithms reyesseerra coello coello analyzed solution problems particle swarm cooper worked solution problems dea presenting international journal foundations computer science technology ijfcst november application pham ghanbarzadeh solved problems bee algorithm nebro analyzed new method based particle swarm algorithm solving multiobjective optimization problems gorjestani proposed coa multi objective algorithm using dea method optimization problems usually possible obtain optimal solution simultaneously optimizes targets question therefore try find good solutions rather optimal ones known pareto frontier given far simple additive weighting method used especially cuckoo algorithms paper presents combined method first section introduces cuckoo optimization algorithm second section simple additive weighting saw method discussed combined method solving multiobjective described finally fourth section provides proposed implemented approach numerical results comparison made methods cuckoo optimization algorithm cuckoo optimization algorithm developed yang suash deb thence cuckoo optimization algorithm presented ramin rajabioun cuckoo algorithm flowchart figure algorithm applied several researches production planning problem portfolio selection problem evaluation organization efficiency evaluation coa information algorithm refer simple additive weighting method saw one practical methods designed multiple criteria presented hong eun method also known weighted linear combination scaling decision matrix weighted coefficients criteria free scale weighted decision matrix obtained according scale score option selected important feature method simple application mathematical logic assuming multiple target model defining parameters weight objective functions defined based importance functions decision maker model converted models max max models decision maker objective functions weight defined importance international journal foundations computer science technology ijfcst november figure cuckoo optimization algorithm flowchart presentation hybrid coaw algorithm section present method coaw proposed paper steps algorithm follows also flowchart coaw algorithm figure different random generated subject summation two values equals one step present locations cuckoos determined randomly step number eggs allocated cuckoo step laying radius cucko cuckoo determined step cuckoos hatch nests hosts within laying radius step eggs detected host birds destroyed step eggs identified cuckoos nurtured step habitats new cuckoos evaluated saw method determined weights step maximum number cuckoos living location determined ones wrong areas destroyed step cuckoos clustered best cluster cuckoos determined residence step new population cuckoos moves toward target location step stop condition established ablished otherwise step value determined best solutions pareto frontier gained based international journal foundations computer science technology ijfcst november saw module determination weights problem evaluation cost function based determined weights figure flowchart coaw algorithm implementation coaw algorithm test problems problem section order validation validat coaw algorithm test problems analyzed test problems presented esented table table test problems number problem objectives constraints international journal foundations computer science technology ijfcst november given determining input parameters one effective problems algorithms parameters algorithm presented follows number initial minimum number eggs maximum number eggs cuckoo maximum iterations cuckoo number clusters want lambda variable coa accuracy answer maximum number cuckoos live control parameter egg cuckoopopvariance solution test problems section experimental problems previous section solved proposed algorithm results compared examined algorithm first problem figure pareto frontiers created coaw algorithm first problem figure pareto frontiers created ranking method dea method gdea method first problem international journal foundations computer science technology ijfcst november second problem figure pareto frontiers created coaw first problem figure pareto frontiers created ranking method dea method gdea method second problem third problem figure pareto frontiers established coaw third problem international journal foundations computer science technology ijfcst november figure pareto frontiers created ranking method dea method gdea method third problem implementation proposed approach test problems pareto frontiers obtained according figures order compare coaw method methods ranking method dea method gdea method implemented problems results show figures figures indicate created pareto frontiers coaw proposed algorithm exact good dispersion method high speed finding pareto frontiers identifies beginning end points pareto frontiers properly coaw algorithm solves problems lower initial population also presents better exact answers fewer repetitions similar methods conclusion paper hybrid coaw algorithm presented solve problems hybrid approach includes cuckoo algorithm simple additive weighting method algorithm analyzed number experimental problems compared several similar methods results indicate accuracy finding pareto frontiers also pareto frontier better similar methods result coaw proposed method reliable fast simple solve optimization problems references ehrgott gandibleux bound sets objective combinatorial optimization problems computers operations research vol issue klein hannan algorithm multiple objective integer linear programming problem european journal operational research sylva crema method finding set vectors multiple objective integer linear programs european journal operational research arakawa nakayama hagiwara yamakawa multiobjective optimization using adaptive range genetic algorithms data envelopment analysis deb optimization using evolutionary algorithms john wiley sons coello coello multiple objective particle swarm optimizers survey international journal computational intelligence research cooper seiford tone data envelopment analysis comprehensive text models applications references dea solver software springer new york international journal foundations computer science technology ijfcst november pham ghanbarzadeh optimization using bees algorithm third international virtual conference intelligent production machines systems iproms whittles dunbeath scotland nebro durillo coello coello luna alba smpso new metaheuristic optimization ieee symposium computational intelligence multi criteria mcdm ieee press new york gorjestani shadkam parvizi aminzadegan hybrid method solving problems international journal computational science applications rajabioun cuckoo optimization algorithm applied soft computing vol akbarzadeh shadkam study cuckoo optimization algorithm production planning problem international journal technologies shadkam delavari memariani poursaleh portfolio selection means cuckoo optimization algorithm international journal computational sciences applications shadkam bijari optimization bank branches efficiency means response surface method data envelopment analysis case iran journal asian finance economics business vol shadkam bijari evaluation efficiency cuckoo optimization algorithm international journal computational sciences applications
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transactions neural networks learning systems vol month deep neural networks salient object detection mar guanbin yizhou convolutional neural networks become key element recent breakthrough salient object detection however existing methods based either patchwise training inference fully convolutional networks methods former category generally timeconsuming due severe storage computational redundancies among overlapping patches overcome deficiency methods second category attempt directly map raw input image predicted dense saliency map single network forward pass though efficient arduous methods detect salient objects different scales salient regions weak semantic information paper develop hybrid deep neural networks overcome aforementioned limitations deep networks composed two complementary components including fully convolutional stream dense prediction spatial pooling stream sparse saliency inference propose attentional module learns weight maps fusing two saliency predictions two streams tailored alternate scheme designed train deep networks finetuning baseline models finally customized fully connected crf model incorporating salient contour feature embedding optionally applied step improve spatial coherence contour positioning fused result two streams extensive experiments six benchmark datasets demonstrate proposed model significantly outperform state art terms popular evaluation metrics index contrast network salient object detection conditional random fields ntroduction visual saliency detection aims locate conspicuous regions images according human visual system recently received increasing research interest image saliency detection traditionally approached form either prediction salient object detection former focuses natural mechanism visual attention aims accurately predicting human eye attended image locations however previous research pointed salient object detection concerned integrity predicted object regions conducive series computer vision tasks including semantic segmentation object localization detection contentaware image editing visual tracking person reidentification although numerous valuable models work supported part national natural science foundation china grant also sponsored open research fund sun university guangzhou china liguanbin department computer science university hong kong yizhouy preliminary version paper appeared cvpr proposed salient object detection remains challenging due variety complex factors scenarios perceptual studies shown visual contrast key factor affects visual saliency series conventional salient object detection algorithms based local global contrast modeling successfully proposed previous research efforts visual contrast modeling generally focused differences among various handcrafted features coupled heuristic saliency priors although handcrafted features tend perform well simple cases robust enough challenging scenarios example hard local contrast models accurately segment large homogeneous regions inside salient objects global contrast information may fail handle images cluttered background although exist machine learning based algorithms salient object detection basically focused integrating various handcrafted features merging multiple saliency maps computed different methods recently deep convolutional neural networks widely used salient object detection powerful feature representations achieved substantially better performance traditional methods methods based deep convolutional neural networks roughly divided two categories methods first category generally perform training inference specifically image first divided set regions patches deep cnn based regression classification models trained independently map image patch region saliency score binary class label salient however results serious storage computational redundancies making training testing example training patchoriented cnn model takes two gpu days requiring hundreds megabytes storage save deep features extracted one single image inspired latest trends developing fully convolutional neural networks pixellevel image understanding problems methods second category train models directly map input image arbitrary size saliency map size performing dense feedforward computation backpropagation entire image type methods rapidly become cornerstone field achieve favorable performance also efficient however still arduous methods detect salient objects different scales salient regions weak semantic information moreover correlation typically considered fully convolutional networks fcns usually give rise incomplete salient transactions neural networks learning systems vol month regions blurry contours work develop hybrid deep neural networks overcome aforementioned limitations two types contemporary salient object detection methods deep networks composed fully convolutional stream dense prediction spatial pooling stream sparse saliency inference devise fully convolutional network first stream receives entire image input directly learns map dense saliency prediction pixellevel accuracy learn feature representations also accurately judge saliency every pixel mining visual contrast information hidden receptive fields spatial pooling stream computes another sparse saliency map superpixels modeling contrast every superpixel spatially adjacent regions extracts regional features efficiently performing feature masking feature map intermediate layer end produce final saliency map merging saliency maps streams weight maps generated proposed attentional module deep network msfcn also generate contour map salient objects contour map used improve contour localization fused saliency map via fully connected crf summary paper following contributions propose deep neural networks localizing salient objects using contextual information incorporate fully convolutional stream dense prediction spatial pooling stream sparse inference tailored alternate scheme designed train deep networks baseline models network pretrained image classification fully convolutional stream infer dense saliency prediction directly raw input image single forward pass fully convolutional network also retrained infer salient object contour map represented feature embedding incorporated fully connected crf model improve contour localization final result also devised spatial pooling stream complementary fully convolutional stream deep network stream efficiently masks features one designated feature map accurately models visual contrast among superpixels well captures saliency discontinuities along region boundaries rest paper organized follows section reviews related work salient object detection section iii introduce proposed deep neural networks complete algorithm presented section section provides extensive performance evaluation well comparisons models finally conclude paper section elated ork traditional salient object detection categorized approaches handcrafted features approaches incorporating knowledge methods usually based center bias background priors infer saliency maps global local contrast represented combination handcrafted features color texture image gradient computational models primarily based scheme compute saliency maps using linear combination lowlevel features color intensity texture orientation edges methods general require machine learning scheme incorporate knowledge process originally limited specified objects assumptions graph based methods also widely used enhance spatial consistency refine detected saliency maps recently deep learning based methods widely used salient object detection promoted research new phase since focus paper deep learning based salient object detection highlight relevant previous work following discussion recent years successful application deep convolutional neural networks triggered revolution machine learning artificial intelligence yielded significant improvement variety visual comprehension tasks including image classification object detection semantic segmentation closing gap performance motivated several attempts also made apply deep neural network models salient object detection han first attempted develop stacked denoising autoencoders learn powerful representations salient object detection unsupervised manner weighted sparse coding framework proposed image saliency detection recently widespread application convolutional neural networks image analysis comprehension tasks surprising see surging number research papers good results achieved salient object detection via application cnns trained fully connected network deriving saliency value every superpixel contextual cnn features wang proposed two deep neural networks take account features objectness salient object detection patch level deep cnn framework incorporating global local contexts presented however methods include fully connected layers infer saliency maps isolated manner crucial spatial information input image ignored however since image patches treated independent samples network training inference shared computation among overlapping image segments results significant redundancies excessive computational cost transactions neural networks learning systems vol month training testing address issues inspired seminal work developing deep networks semantic image segmentation variant fully convolutional neural networks introduced solve problem salient object detection since publication earlier conference version proposed explore correlations saliency detection semantic image segmentation using fully convolutional neural network liu propose hierarchical recurrent cnn progressively refine details saliency maps coarse prediction result generated forward pass fully convolutional network kuen proposed recurrent attentional network racdnn consists recurrent neural network spatial transform module recurrently attend selected image saliency refinement wang introduced recurrent fully convolutional network rfcn iteratively refine saliency map incorporated prior knowledge fcn based models greatly improved accuracy efficiency saliency detection still three aspects flaws first models mostly based topmost feature map network saliency inference regional semantic feature may result pool detection performance salient region weak semantic information second methods consider feature modeling single scale may accurately detect salient objects different sizes finally value position saliency map generated models derived context fixed size receptive field contours salient objects hardly well detected generated saliency maps usually inadequate spatial consistency proposed method instead delves nature saliency prediction capturing key aspect problem contrast learning proposed method able infer saliency probability map contrast information multiscale deep cnn also contrast information addition proven fully connected crfs formulated recurrent neural networks rnns however experimental results show rnns hardly trained achieve comparable results crfs proposed method therefore exploits effectiveness crf experimental results demonstrate superiority proposed method comparison existing fcn based salient object detection techniques note initial deep contrast network reported cvpr viewed first piece work aims designing fully convolutional network visual contrast modeling certain extent inspired subsequent development models field updated deep neural network salient object detection several improvements initial version first adapt network image classification fully convolutional network use replace network original fully convolutional stream achieving better performance second fully convolutional stream run multiple scaled versions original input image segmentwise spatial pooling stream trained using segments image segmentation strategies make deep model accurately detect salient objects different scales third propose add attentional module learns soft weights fusing two saliency maps respectively generated two streams fourth discover proposed fully convolutional stream deep network detect salient region contours integrated fully connected crf model improve contour localization final saliency map finally present comprehensive experimental comparison among multiple model variants report improved results benchmarks using evaluation metrics iii eep ontrast etwork illustrated fig proposed deep neural network composed two complementary components fully convolutional stream dense saliency prediction spatial pooling stream sparse saliency inference specifically first component fully convolutional network receives entire image input trained map input dense saliency map mode exploiting visual contrast across multiple levels feature maps spatial pooling stream trained infer saliency map segment level discovering contrast among spatially adjacent regions basis features masked one designated feature map first stream perceptron end two intermediate saliency predictions two network streams merged according weight maps prescribed trained attention module merged map becomes final saliency map fully convolutional network inspired groundbreaking application fully convolutional networks image comprehension focus constructing pixelwise regression network directly map raw input image dense saliency map considering centrality contrast modeling saliency detection following considerations designing structure network first network deep enough accommodate features multiple levels since visual saliency relies modeling contrast among appearance features well semantic features second network needs able explore visual contrast across multiple feature maps detect salient objects various scales finally due lack training images labeling much desired existing network instead training scratch vgg resnet two representative widely used deep classification networks publicly available models choose networks adapt requirements transactions neural networks learning systems vol month sfm output fig overall architecture proposed deep neural network consists fully convolutional stream upper part spatial pooling stream lower part attentional module fuse intermediate saliency maps two streams sfm refers segment feature masking layer refers spatial pooling operation describe detail transformation network similarly transformed satisfy requirements network dense saliency map generation first convert two fully connected layers convolutional ones described moreover original network consists max pooling layers stride resulting network yield prediction maps input resolution make resulting saliency map higher resolution remove downsampling operation last two layers simply setting stride results downsampling factor instead time maintain size receptive fields convolutional layers follow refer apply dilation operation corresponding filter kernels dilation algorithm also called trous algorithm originally proposed improve computational efficiency undecimated wavelet transforms recently incorporated caffe framework dilated convolution efficiently control resolution feature maps within deep cnns without need learn extra parameters works inserting zeros filter weights specifically consider applying dilated version convolutional filter input feature map generating output feature map output value position calculated dilation rate corresponds stride sample input feature map equivalent applying convolution input feature map filters inserting zeros two originally adjacent filter elements along dimension dilated convolution allows explicitly control density feature responses customized fully convolutional networks implementation setting stride last two pooling layers replace subsequent convolutional layers dilated convolutional layers dilation rate three consecutive convolutional layers penultimate layer last two newly converted convolutional layers five max pooling layers performing downsampling operations start pooling layer closest input image pooling layers increasingly larger receptive field containing contextual information design deep convolutional network capable mining visual contrast information crucial saliency inference develop multiscale network fully convolutional version shown left part fig connect three extra convolution layers first four layers first extra layer uses convolution kernels channels second one uses convolution kernels also channels third extra layer one kernel single channel used produce output saliency map make output feature maps four sets extra convolutional layers size downsampling resolution stride first layer four sets set transactions neural networks learning systems vol month respectively although four resulted feature maps size computed using receptive fields different sizes hence represent contextual features different scales stack four feature maps last output feature map customized fully convolutional conversion stacked feature maps channels fed final convolution layer kernel single output channel modulated sigmoid activation function produce saliency probability map though resulting saliency map network stream downsampling factor comparison input image smooth enough allows use simple bilinear interpolation restore resolution original input negligible computational cost call resized saliency map note network hidden fully connected layers adapt dense saliency prediction simply replace linear classification layer linear convolutional layer kernel single output channel similar resolution feature maps linear convolutional layer original input image original consists one pooling layer convolutional layers stride call five layers layers described layers divided five groups feature maps computed different layers group share resolution increase resolution final saliency map replace last two layers dilated convolution layers skip subsampling setting stride correspondingly increase dilation rate subsequent convolution kernels enlarge receptive fields therefore features maps last three groups resolution original resolution network transformation develop multiscale version extension shown right fig connect extra three convolutional layers final layers first four groups additional layers structure added similar multiscale extension four output feature maps four subnetworks stacked together final output feature map transformed fed final convolutional layer kernel single output channel final saliency map inference saliency inference salient objects images usually presented variety irregular shapes corresponding saliency map often exhibits discontinuities along object boundaries multiscale fully convolutional network operates subsampled pixel level equally treats pixel input image without explicitly taking account saliency discontinuities better model visual contrast regions visual saliency along region boundaries design segmentwise spatial pooling stream network first divide input image set superpixels call superpixel segment mask computed every segment feature map generated one selected convolutional layer named feature masking layer choose convolutional layer feature masking layer based last convolutional layer fourth layer group feature masking layer based suggested since activations location feature masking layer controlled receptive field input image first project every location feature masking layer center receptive field segment input image first generate binary mask within bounding box mask pixels inside segment labeled others labeled pixel labeled binary mask first assigned closest receptive field center backprojected onto feature masking layer thus location feature masking layer collects multiple labels backprojected receptive field ratio number collected labels location number pixels input image closest receptive field center recorded yield binary mask segment feature masking layer previously computed ratio every location thresholded set locations nonzero values thresholding form segment mask event ratio locations set locations nonzero ratios thresholding form segment mask resulting segment mask applied output feature map feature masking layer simply multiplying binary mask channel feature map call resulting features features method note feature map generated feature masking layer downsampling factor instead original network original network since subsampling skipped last two downsampling layers described section therefore resolution feature map generated feature masking layer sufficient segment masking since segments irregular shapes variable sizes projected onto feature masking layer perform spatial pooling operation produce feature vector fixed length segment simplified version spatial pyramid pooling described specifically divide bounding box projected segment cells perform valid positions mask label grid cell results feature vectors size number convolutional filters feature masking layer afterwards concatenate feature vectors extracted grid cells segment obtain final feature vector dimensions segment discover visual contrast represent segment concatenation three feature vectors respectively three nested increasingly larger regions masked designated feature map three regions include bounding box considered segment bounding box immediate neighboring segments well entire feature map feature masking layer considered segment excluded indicate position transactions neural networks learning systems vol month conv conv pooling max conv conv conv conv conv conv fig architecture based fully convolutional network left based fully convolutional network right connect three extra convolutional layers first four layers convert multiscale version divide layers five groups connect extra three convolutional layers final layers first four groups form multiscale version segment feature representation segment fed two fully connected layers output second fully connected layer fed sigmoid layer employs sigmoid function perform logistic regression produces distribution binary saliency labels call saliency map generated way fact spatial pooling stream network accelerated version previous work proposed although share identical idea inferring saliency contrast among multiscale contextual regions feature extraction processing current method much efficient hundreds segmental features image instantaneously masked feature map generated single forward pass moreover spatial pooling stream also achieves better results segment features extracted multiscale fully convolutional network finetuned salient object detection instead original model image classification attentional module saliency map fusion merge predicted saliency scores two different streams three straightforward options average pooling convolution however strategies image content independent two network streams complementary strengths saliency map prediction inspired design trainable attentional module generate weight maps fusing results two streams let probabilistic saliency maps two network streams respectively final saliency map deep contrast network calculated weighted sum two maps spatially varying weights adaptively learned therefore called weight maps let fused saliency map weight map saliency map generated stream weight map saliency map generated second stream merged saliency map calculated summing product probability map resized input image resolution corresponding weight map refer call attention weights reflect much attention paid individual network streams well saliency scores different spatial locations two attention weights also considered feature maps size predicted saliency maps thus jointly trained fully convolutional network work employ differentiable attention module deep network infer attention weights illustrated fig proposed attention module receives input output feature map feature masking layer contains two convolutional layers first layer filters kernel size second layer two convolutional filters kernel size output feature map two channels fed softmax layer generates two score maps corresponding aforementioned two attention weights deep contrast network training propose alternate training scheme train network specifically initialization phase segments training images train segmentwise spatial pooling stream alone convergence obtain transactions neural networks learning systems vol month initial network parameters saliency labeling performed thresholding average labeling inside segment segment features extracted using image classification model imagenet dataset initialization alternately update weights two network streams first fix weights second stream train well attention module one epoch note weights attention module adaptively merging predicted saliency maps two streams trained simultaneously stream mode next fix weights well attention module parameters second stream one epoch using segment features extracted updated network embedded stream alternately train two streams times epochs total whole training process converges define following classbalanced loss function training fully convolutional steam attention module network log log represents class balancing weight denoted respectively indicate total number pixels salient pixels ones image represents groundtruth annotation represents collection network weights stream attention module spatial pooling stream use batch images unit update parameters minimizing summed squared errors accumulated segments batch training images omplete lgorithm superpixel segmentation spatial pooling stream network requires input image decomposed segments order better avoid artificial boundaries generated saliency map segment perceptually homogeneous region time strong contours edges still well preserved earlier version use geodesic distance based slic algorithm superpixel generation work discover graph based image segmentation produces segments better edge preservation slic algorithm using segments generated multiple levels image segmentation improve performance therefore refer employ graph based image segmentation algorithm therein generate three levels segments different parameter settings train single spatial pooling stream segments across three levels segmentation instead learning different model parameters segments different levels segmentation generating saliency map spatial pooling stream apply stream infer saliency map level segmentation simply average three resulting saliency maps salient contour detection cases proposed deep contrast network works well sometimes produces saliency maps salient region boundaries accurately localized particularly images containing small salient regions meanwhile find fully convolutional network described section using annotated salient region contours also capable detecting contours salient regions detected contours encoded feature vectors embedded crf framework enhance spatial coherence preservation salient region contours saliency maps prepare training data salient region contour detection boundary pixels salient regions groundtruth saliency maps labeled pixels labeled salient region contour maps taken groundtruth annotations trained salient region contour detection weight updated according fraction pixels salient region contours given detected salient region contour map apply normalized cut algorithm generate feature vectors used fully connected crf improve boundary localization final saliency map first construct sparse graph every pixel connected pixels neighborhood affinity matrix graph defined follows wij exp max wij denotes affinity pixels represents pixels along line segment connecting pixels indicates probability pixel salient region contour constant scaling factor set experiments idea two pixels similar saliency value salient region contour crossing line segment connecting two pixels given affinity matrix define dii wij solve generalized eigenvectors following system use eigenvectors additional features improve spatial coherence experiments use eigenvectors corresponding smallest eigenvalues spatial coherence since streams deep contrast network independently infer saliency score individual pixel segment without considering impact correlation among pixels segments saliency prediction resulting saliency maps contain less incomplete false positive salient objects mitigate issue adopt fully connected conditional random field crf transactions neural networks learning systems vol month step enhance spatial coherence energy function crf model formulated log binary label prediction pixels salient salient indicates probability pixel labeled initialization refers predicted probabilistic saliency value pixel saliency map generated deep contrast network pairwise potential defined kii kpi exp kvi kpi exp zero otherwise involves summation two gaussian kernels first kernel based observation neighboring pixels assigned similar saliency scores similar colors intervening salient region contours therefore depends pixel positions pixel intensities contour feature embedding discussed section importance color similarity spatial closeness salient region contours controlled three parameters respectively second kernel dependent pixel positions hyperparameter controlling scale gaussian function pointed helps enhance label smoothness remove small isolated regions proved energy minimization process modeled efficient approximate probabilistic inference adopting approximation original crf filtering employed speed computation adapt publicly available implementation minimize energy function optimization process takes less second image pixels crf model optimization saliency map scrf generated pixelwise posterior probabilities saliency labels visualize effectiveness crf fig seen source crf crf contour saliency contour crf contour fig examples saliency maps generated without crf including crfs without contour feature embedding original saliency maps proposed method without crf rather coarse integrity spatial coherence detected salient regions hardly maintained though saliency maps generated traditional crf without contour feature embedding enhance spatial coherence detected salient regions extent salient region contours still may well positioned may false detections smooth background third row fourth column figure demonstrates salient region contours detected proposed method seen usually possible accurately capture boundaries salient regions corresponding embedded features enhance consistency saliency prediction across salient region contours correct prediction errors quantitative analysis crf based saliency refinement provided section xperimental esults experimental setup datasets evaluate proposed method widely used saliency detection benchmarks including msrab ecssd sod includes images holds single salient object hkuis proposed previous work images images include multiple separate salient objects based validation set pascal segmentation challenge contains natural images challenging images relatively complex diversified contents sod images originally designed image segmentation challenging images contain multiple objects low contrast cluttered background train proposed deep neural networks based combination training sets images images two validation sets also combined final validation contains total images test model trained combined training set datasets verity model adaptability evaluation criteria employ curves mean absolute error mae quantitatively evaluate performance method well salient object detection methods given saliency map continuous values normalized range compute binary masks using every possible fixed integer threshold pair values computed comparing binary mask ground truth precision defined ratio detected groundtruth salient pixels predicted salient pixels binary mask recall ratio detected groundtruth salient pixels groundtruth salient pixels pairs binary maps computed curve plotted averaging pairs precision recall values saliency maps given dataset defined harmonic mean transactions neural networks learning systems vol month source drfi pisa bsca legs mdf dhsnet rfcn dcl fig visual comparison methods dcl methods source input images ground truth saliency maps dcl crf refinement consistently achieves best results variety complex scenarios data set ecssd sod metric maxf mae maxf mae maxf mae maxf mae maxf mae maxf mae drfi pisa bsca legs mdf rfcn dhsnet dcl table quantitative comparison terms maximum larger better mae smaller better three best performing algorithms marked red blue green respectively testing set dataset used part training set released model dhsnet rfcn part dataset also used training dhsnet model exclude corresponding results legs pisa bsca legs pisa mdf mdf dhsnet dhsnet ecssd rfcn recall recall pisa mdf rfcn legs rfcn pisa mdf legs bsca precision bsca precision precision bsca precision recall recall fig curves method algorithms benchmark datasets dcl crf consistently performs better methods across benchmarks average precision average recall calculated recision recall recision recall set place emphasis precision recall suggested evaluation report maximum maxf among scores computed pairs curve also use twice mean value every saliency map threshold generate corresponding binary map report average precision recall binary maps complement also calculate mean absolute error mae follows quantitatively measure average absolute difference estimated saliency map corresponding groundtruth saliency map implementation proposed model implemented top open source code deeplab based caffe platform trained gtx titan gpu cpu training resize images corresponding groundtruth saliency maps perform data augmentation horizontal flipping training transactions neural networks learning systems vol month pre rec fmea mdf legs pisa drfi bsca ecssd pre rec rec rec fmea fmea fmea mdf legs drfi pisa bsca pre rfcn dhsnet pre rfcn dhsnet mdf legs pisa drfi bsca rfcn mdf legs drfi pisa bsca fig precision recall achieved using adaptive threshold every image proposed method consistently performs best among different methods datasets methods various challenging cases salient regions touching image boundary first fifth rows low contrast salient objects background third sixth rows images multiple separate salient objects last three rows method significantly outperforms methods including fully convolutional network based deep models published earlier conference version large margin public datasets terms curve fig well average precision recall fmeasure fig moreover purpose quantitative evaluation report comparison maximum mae table complete model clearly outperforms previous method terms maximum skipping rfcn dhsnet dataset ecssd skipping dhsnet sod respectively time respectively lowers mae also observed proposed method dcl without postprocessing already outperforms evaluated methods considered datasets also compare efficiency among considered algorithms shown table dcl model needs around second generate saliency map testing phase comparable fully convolutional methods rfcn dhsnet much efficient regionbased cnn models legs mdf precision stream set learning rate newly added layers learning rate rest layers employ poly learning rate updating iter learning rate scaled max iter iteration power set weight decay momentum parameter training spatial pooling stream refer obtain segments image levels image segmentation achieved different parameter settings set grid size performing spatial pooling segment aggregated feature dimensions based dimensions based feature fed consisting two fully connected layers contains neurons determine parameters fully connected crf performing cross validation validation set finally actual value respectively set evaluation use dcl respectively represent best saliency detectors without refinement takes approximately hours train model costs around second dcl process image size nvidia titan gpu cpu note far efficient deep saliency detectors independently treat image patches superpixels saliency estimation however expensive requires additional seconds since need compute generalized eigenvectors used crf model experimental results reported following section show dcl alone without crf refinement already performs better existing methods specific comparison computational cost different methods summarized table dcl msfcn pre rec fmea dcl msfcn recall comparison state art compare models dcl algorithms including drfi pisa bsca legs mdf rfcn dhsnet last three fully convolutional neural network based methods published publication earlier conference version qualitative evaluation figure provides visual comparison saliency detection results results proposed method achieve much improvement algorithms specifically method capable highlighting salient regions missed fig validation proposed model effectiveness crf based refinement ablation studies effectiveness deep contrast network validate necessity effectiveness two components contained deep contrast network take based version representative compare saliency maps inferred first stream msfcn saliency maps second stream well fused ones based shown fig transactions neural networks learning systems vol month time drfi pisa bsca legs mdf rfcn dhsnet dcl table comparison running time gpu time msfcn attentional module input slic superpixel segmentation crf contour metric contour maxf mae table iii performance evaluation different model factors dataset source msfcn dcl fig sample visualizations demonstrating componentwise efficacy deep contrast network fused saliency map consistently performs best evaluation metrics testing set dataset fully convolutional stream contributes merged prediction far spatial pooling stream two streams deep contrast network complementary capable discovering global local contrast collaboratively multiscale feature aggregation streams validate effectiveness also generated saliency maps last scale msfcn comparison illustrated fig single scale msfcn may lead significantly inferior performance compared full version terms curve well average precision recall fig shows sample visualizations demonstrate complementary nature two streams inside dcl network shown figure although fully convolutional stream spatial pooling stream produce promising saliency maps far perfect tends generate smooth saliency maps well maintain integrity salient regions stream predicts saliency maps unit superpixels hardly capture global contrast well handle images complex background however fused dcl model exploits advantages produces accurate saliency predictions confirms complementarity two particular examples second image fig two streams different mistakenly predicted regions proposed network still preferentially integrate respectively predicted salient pixels produce accurate results demonstrates robustness network strong complementarity two network streams effectiveness contour guided crf described section incorporate fully connected crf embedded contour features improve spatial coherence contour positioning saliency maps generated deep contrast network compare performance generated saliency maps without crf postprocessing shown fig crf significantly increases accuracy saliency maps generated testing images dataset also show visual comparison figure illustrate effectiveness conventional crf crf incorporating salient region contours shown figure conventional crf improves spatial consistency predicted results certain extent incorporating salient region contours enhances confidence saliency predictions especially pixels near detected salient region boundaries improvements conference version conference version work made following five major modifications method adding attention module infer spatially varying weights saliency map fusion employing network fully convolutional stream running fully convolutional stream multiple scaled versions original input image fusing results using training testing spatial pooling stream using segments image segmentation performing salient region contour detection incorporating detected contours fully connected crf postprocessing table iii evaluate factors affects maximum mae dutomron dataset shown table five factors together contribute improvement maximum fmeasure decline mae comparison transactions neural networks learning systems vol month best reported results earlier conference version paper effectiveness attention module described section instead simply adding convolutional layer top saliency maps two network streams design attention module infer spatially varying weight maps validate effectiveness conduct performance comparison deep contrast network trained attention module another deep contrast network simple convolutional layer shown table iii adopting attention module saliency map fusion improves maximum dataset lowering mae effectiveness mechanism always integrate module network subsequent experiments effectiveness described section attempted replace network transformed network fully convolutional stream deep network demonstrate effectiveness trained new deep contrast network model comparison new model trained using setting section except transformed network replaced transformed shown table iii adopting instead significantly improves maximum dataset lowering mae also reached conclusion vgg based dcl network single scale setting generates saliency maps prediction errors performs much worse version side branches shown second third columns fig proposed dcl network generates much confident cleaner results dcl original effectiveness multiple scaled inputs inspired adopt input strategy generating saliency map fully convolutional stream specifically obtain three scaled versions original input image scaling factor respectively set independently feed scaled images fully convolutional stream three resulting saliency maps fused taking maximum response across scales source dcl resnet dcl dcl multiscale input fig effectiveness dcl model position max pooling shown table iii input brings extra improvement maximum lowering mae sample visualizations shown fourth column fig fusing saliency predictions inputs gives rise accurate saliency maps especially exists multiple salient objects different scales testing image effectiveness image segmentation described section final saliency map revised spatial pooling stream average three saliency maps computed using superpixels one levels image segmentation shown table iii image segmentation improves maximum lowers mae effectiveness salient region contours described section revise step version integrating additional feature vector computed detected salient region contours salient region contours detected using separately trained contour detection model network structure msfcn stream compare saliency maps computed without crf crf without contour saliency features contour guided crf respectively shown table iii saliency maps dense crf always yields performance improvement based deep contrast network running crf step boosts maximum lowers mae based deep contrast network already achieves much better performance adding dense crf still brings improvement maximum decrease mae worth noting contour guided crf results accurate saliency maps improvement maximum fmeasure decrease mae onclusions work proposed deep neural networks salient object detection deep networks contain two complementary capable extracting wide variety visual contrast information first based multiscale fully convolutional network intended infer saliency looking contexts receptive field multiple scales around pixel second designed capture contrast information among adjacent regions maintain consistency saliency prediction within homogeneous regions also better detect discontinuities along salient region boundaries attentional module learnable weights introduced adaptively fuse two saliency maps two finally produce accurate saliency predictions incorporate crf contour feature embedding enhance spatial coherence contour localization produced saliency map experimental results show proposed model achieves performance six public benchmark datasets various evaluation metrics transactions neural networks learning systems vol month eferences deep contrast learning salient object detection proc ieee conf cvpr june wei liang chen shen cheng feng zhao yan stc simple complex framework semantic segmentation ieee trans pattern anal mach navalpakkam itti integrated model attention optimizing detection speed proc ieee conf cvpr vol wang chen lin image recognition recurrently discovering attentional regions proc ieee conf iccv avidan shamir seam carving image resizing acm transactions graphics tog vol acm luo weighted attentional blocks probabilistic object tracking visual computer vol person using multiple experts random subspaces journal image graphics vol contribute saliency map overt visual attention european journal neuroscience vol parkhurst law niebur modeling role salience allocation overt visual attention vision research vol cheng mitra huang torr global contrast based salient region detection ieee trans pattern anal mach vol yang zhang ruan yang saliency detection via manifold ranking proc ieee conf cvpr wang yuan yan visual saliency selective contrast ieee transactions circuits systems video technology vol mahadevan vasconcelos learning optimal seeds salient object detection proc ieee conf cvpr jiang ling peng salient region detection ufo uniqueness focusness objectness proc ieee conf iccv liu yuan sun wang zheng tang shum learning detect salient object ieee trans pattern anal mach vol mai niu liu saliency aggregation approach proc ieee conf cvpr visual saliency based multiscale deep features proc ieee conf cvpr june zhao ouyang wang saliency detection multicontext deep learning proc ieee conf cvpr wang ruan yang deep networks saliency detection via local estimation global search proc ieee conf cvpr long shelhamer darrell fully convolutional networks semantic segmentation proc ieee conf cvpr chen papandreou kokkinos murphy yuille semantic image segmentation deep convolutional nets fully connected crfs arxiv preprint xie edge detection proc ieee conf iccv gao vasconcelos saliency discriminant process proc ieee conf iccv achanta hemami estrada susstrunk salient region detection proc ieee conf cvpr klein frintrop divergence feature statistics salient object detection proc ieee conf iccv ieee perazzi pritch hornung saliency filters contrast based filtering salient region detection proc ieee conf cvpr zhu liang wei sun saliency optimization robust background detection proc ieee conf cvpr wang yuan yan saliency detection multipleinstance learning ieee transactions cybernetics vol judd ehinger durand torralba learning predict humans look proc ieee conf iccv chang liu chen lai fusing generic objectness visual saliency salient object detection proc ieee conf iccv ieee goferman tal saliency detection tpami vol shen unified approach salient object detection via low rank matrix recovery proc ieee conf cvpr liu cao lin adaptive partial differential equation learning visual saliency detection proc ieee conf cvpr jia han saliency detection proc ieee conf iccv ieee hou koch rehg yuille secrets salient object segmentation proc ieee conf cvpr hou zhang saliency detection spectral residual approach proc ieee conf cvpr lei wang fang lin callet ling hou universal framework salient object detection ieee transactions multimedia vol krizhevsky sutskever hinton imagenet classification deep convolutional neural networks advances neural information processing systems girshick donahue darrell malik rich feature hierarchies accurate object detection semantic segmentation proc ieee conf cvpr visual saliency detection based multiscale deep cnn features ieee transactions image processing vol zhao wei yang zhuang ling wang deepsaliency deep neural network model salient object detection arxiv preprint xie lin salient object segmentation proc ieee conf cvpr xie lin weakly supervised salient object detection using image labels proc conf aaai feb han zhang wen guo liu learning predict human eye fixations via sdaes ieee transactions cybernetics vol sun weighted sparse coding framework saliency detection proc ieee conf cvpr liu han dhsnet deep hierarchical saliency network salient object detection proc ieee conf cvpr kuen wang wang recurrent attentional networks saliency detection arxiv preprint wang wang zhang ruan saliency detection recurrent fully convolutional networks proc conf eccv springer zhang ren sun deep residual learning image recognition arxiv preprint simonyan zisserman deep convolutional networks image recognition arxiv preprint zhao wang highly efficient forward backward propagation convolutional neural networks pixelwise classification arxiv preprint mallat wavelet tour signal processing academic press girshick fast international conference computer vision iccv zhang ren sun spatial pyramid pooling deep convolutional networks visual recognition proc conf eccv springer bahdanau cho bengio neural machine translation jointly learning align translate proc conf iclr chen yang wang yuille attention scale semantic image segmentation arxiv preprint deng dong socher imagenet hierarchical image database proc ieee conf cvpr criminisi sharp rother geodesic image video acm transactions graphics tog transactions neural networks learning systems vol month felzenszwalb huttenlocher efficient image segmentation ijcv vol shi malik normalized cuts image segmentation ieee trans pattern anal mach vol koltun efficient inference fully connected crfs gaussian edge potentials arxiv preprint shotton winn rother criminisi textonboost image understanding object recognition segmentation jointly modeling texture layout context international journal computer vision vol yan shi jia hierarchical saliency detection proc ieee conf cvpr martin fowlkes tal malik database human segmented natural images application evaluating segmentation algorithms measuring ecological statistics proc ieee conf iccv vol everingham van gool williams winn zisserman pascal visual object classes voc challenge international journal computer vision vol jia shelhamer donahue karayev long girshick guadarrama darrell caffe convolutional architecture fast feature embedding proceedings acm international conference multimedia acm liu rabinovich berg parsenet looking wider see better arxiv preprint wang lin shi pisa pixelwise image saliency aggregating complementary appearance contrast measures coherence ieee transactions image processing vol oct qin wang saliency detection via cellular automata proc ieee conf cvpr guanbin currently research associate professor school data computer science sun university received phd degree university hong kong recipient hong kong postgraduate fellowship current research interests include computer vision image processing deep learning authorized papers academic journals conferences serves area chair conference visapp serving reviewer numerous academic journals conferences tpami tip tmm yizhou received phd degree university california berkeley currently professor university hong kong faculty member university illinois recipient national science foundation career award nnsf china overseas distinguished young investigator award served editorial board iet computer vision ieee transactions visualization computer graphics visual computer international journal software informatics also served program committee many leading international conferences including siggraph siggraph asia international conference computer vision current research interests include deep learning methods computer vision computational visual media geometric computing video analytics biomedical data analysis
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distributed estimation relative measurements heterogeneous uncertain quality oct chiara ravazzi nelson chan paper studies problem estimation relative measurements graph vector indexed nodes reconstructed relative pairwise measurements differences values nodes connected edge order model heterogeneity uncertainty measurements assume measurements affected additive noise distributed according gaussian mixture original setup formulate problem computing estimates propose two novel algorithms solution main difference two algorithm one distributed former algorithm said distributed allows node compute estimate value using information directly available node immediate neighbors prove convergence algorithms present numerical simulations evaluate compare performance ntroduction whenever measurements used estimate quantity interest measurement errors must properly taken account statistical properties errors identified enable effective estimation paper look specific case broad issue within context network systems namely consider problem distributed estimation relative measurements defined follows assume real vector indexed nodes graph nodes allowed take pairwise measurements differences vector entries neighbors graph estimation problem consists reconstructing original vector additive constant prototypical problem applied variety contexts one example relative localization mobile automated vehicles vehicles locate using distance measurements another example statistical ranking set items needs sorted according quality evaluated comparatively scenarios noise affecting measurements research leading paper partly developed ravazzi department electronics telecommunications det politecnico torino italy chan frasca department applied mathematics university twente enschede netherlands chiara ravazzi national research council cnr institute electronics computers telecommunication engineering ieiit politecnico torino italy nelson chan university groningen groningen netherlands paolo frasca univ grenoble alpes cnrs inria grenoble france research associate torino italy paolo frasca drastically heterogeneous importantly distribution may known priori instance vehicle localization distances vehicles may measured less accurate sensors ranking system items upon evaluation compared less trustworthy entities thus important identify unreliable measurements weight differently estimation order model uncertainty paper assume measurement noise sampled mixture two gaussian distributions different variances representing good poor measurements respectively solution problem builds classical expectationmaximization approach likelihood maximized alternating operations expectation maximization particularly interested finding effective distributed algorithms solve problem precisely say algorithm distributed requires node use information directly available node immediate neighbors actually many distributed algorithms relative estimation available best knowledge assume quality measurements known beforehand time large literature robust estimation also covers estimation relative measurements often provides algorithms distributed see instance references therein fill gap proposing distributed algorithm way contribute growing body research distributed algorithms estimation problems heterogeneous unknown measurements distributed algorithms based consensus ranking procedures proposed approximate maximumlikelihood estimates authors used estimate gaussian mixtures parameters problems distributed inference sensor networks works network given node independently performs local observations information suitably propagated collaboratively perform also key instrument learning unreliable observations context social sensing goal designing reliable inference systems based unreliable information reported users contribution paper define problem robust estimation relative measurements measurement noise drawn gaussian mixture design two iterative algorithms solve algorithms based combining classical weighted least squares wls popular tool statistical estimation problems involving incomplete data first algorithm centralized whereas second algorithm distributed requires know approximately two variances gaussian mixture knowledge necessary centralized version algorithms proved converge performance compared synthetic data observe centralized algorithm better performance achieving smaller estimation errors centralized algorithm also requires less iterations converge iteration involves computations organization paper formally present problem relative estimation section centralised lsem algorithm described section iii distributed algorithm section section contains numerical examples section conclusions details proofs postponed appendix stimation relative measurements set nodes cardinality considered endowed unknown scalar quantity relative estimation problem consists node estimating scalar value based noisy measurements differences set available measurements conveniently represented oriented graph edge represents measurement orientation edges conventionally assumed let edge incidence matrix graph defined follows rows columns indexed elements respectively vertex edge incident otherwise according originates terminates respectively precisely aew otherwise every let vector collecting measurements mutually independent random variables distributed ber provided value associated measurement unreliable formulation random variables gaussian mixtures whose model completely described three parameters convenience consider known knowledge entail significant restriction analysis main goal obtain robust estimate state vector suitably taking account different quality measurements thus consider joint maximum likelihood estimation bml argmax log exp exp computational complexity optimization problem makes brute force approach infeasible large graphs estimation via weighted least squares problem becomes much simpler assume know distribution produced noise term measurement using noise source information zee zee realization mlestimation becomes bml argmax log zee exp zee exp noticing log log binary vector easy see equivalent solving weighted least square wls problem argmin argmin diag zee zee following lemma describes solutions lemma wls estimator let graph connected set solutions let denote weighted laplacian graph following facts hold true exists unique bwls xwls bwls denotes weighted laplacian useful properties collected following result gradient descent algorithm converges wls solution provided details found iii entralized algorithm fig network nodes considered example lemma moments wls estimator provided connected holds true xwls wls wls wls wls section tackle likelihood maximization problem full generality since closed form solution shall design iterative algorithm solution preliminarily designing algorithm convert maximum likelihood problem minimization problem following result whose proof postponed appendix theorem following optimization problems solutions min min min vector length whose entries stressed determining state vector relative measurements possible additive constant bwls span ambiguity avoided assuming centroid nodes origin cartesian coordinate system view comment results shall assume connected next provide simple example illustration example consider connected network figure nodes set edges let incidence matrix vector measurements easily constructed vector xwls computed bwls instead position estimated unweighted least squares given bls obtain xwls shown lemma wls solution explicitly known easily computed solving linear system furthermore following distributed computation also possible using gradient descent algorithm observe gradient cost function given set initial condition fix consider max max log log natural entropy function log log note respect original problem problem explicitly introduces variable represents estimated probabilities edges large variances actually instead solving problem solve suitably modified problem going define next modification marks key difference classical approaches namely shall solve min min min defined log log compared optimization problem introduces positive variable goal avoid possible singularities one gaussian components mixture collapses one point constraint set forces algorithm take account least measurements reliable become clear proofs additional variables instrumental guarantee convergence algorithm following lemma summarizes main properties minimization problems involve one variable time proposition partial minimizations argmin argmin argmin argmin denote diag use mixture parameters procedure iterated suitable stopping criterion satisfied maximum number iteration tmax fixed tmax algorithm run estimate stops changing tol tol algorithm require data parameters initialization computation weights wls solution diag holds true posterior distribution evaluation projection zeroes smallest components given vector proof statements directly verified differentiating using insights obtained proposition propose alternating method minimization resulting method combination iterative reweighted least squares irls expectation maximization algorithm detailed algorithm based following four fundamental steps iteratively repeated convergence wls solution given relative measurements current parameters new estimation variable obtained solving wls problem weights expectation posterior distribution edges evaluated based current projection smallest components posterior probabilities zeroed ensure least measurements trusted used weighted least squares problem vector best posterior probabilities maximization given projected posterior probabilities best regularization parameter dim ker log min parameters estimation end although algorithm modified version classical fact sufficient guarantee convergence proposed method fact observed generic algorithm guaranteed converge limit point produce sequence points along function decrease hence explicit convergence proof required specific cases algorithm also includes regularization sequence appears maximization step designed monotonic zero upon convergence algorithm presence regularization actually instrumental prove convergence local maximum function order state denote algorithm seen map produces sequence iterates theorem convergence whole sequence converges diag converge point fixed point algorithm local minimum locally maximizes proof theorem based observing function lyapunov function increasing along sequence iterates details postponed appendix istributed algorithm section design study distributed algorithm solve problem starting centralized one proposed previous section preliminary let examine steps algorithm order identify whether amenable distributed computation steps require information depending edge thus inherently decentralized furthermore already know least squares problem step computed distributed procedure instead steps involve global information easily distributed based discussion propose following simple effective variation algorithm detailed algorithm new algorithm based two design choices first one inspired distributed gradient dynamics instead fully solving wls problem iteration perform one step corresponding gradient iteration second one assume known thus removing need estimation advantage keeping fixed evolution algorithm avoids certain difficulties convergence analysis appendix namely removes need regularization projection steps even though knowledge restrictive assumption observed algorithm fairly robust uncertainties values quality discussed remark algorithm distributed require data parameters initialization computation weights gradient step diag posterior distribution evaluation end convergence algorithm proved similarly theorem condition parameter belongs certain range details postponed appendix theorem distributed convergence sequence generated algorithm converges diag limit point fixed point algorithm local minimum umerical results section provide simulations illustrating performance proposed algorithms mainly interested comparing terms convergence time final estimation error time would like examine performance depends parameters problem parameters gaussian mixture topology measurement graph measure performance consider normalized quadratic error nqe defined denotes vector let begin describing baseline simulation setup details first generate synthetic data define estimation problem number nodes set components state vector generated randomly according uniform distribution interval mean subtracted yielding state vector mean topology generated random graphs edge probability pedge ranging iteration fig nqe plotted iteration count randomly chosen trials obtained using algorithm color represents result trial parameter set pedge note color chosen trial matches counterpart fig iteration fig nqe plotted iteration count randomly chosen trials obtained using distributed algorithm color represents result trial parameter set pedge note color chosen trial matches counterpart fig edge created two arbitrary nodes probability pedge sample realization fig extreme case pedge graph generated complete graph nodes connected others fix range also probability getting bad measurement taken noise vector sampled normal distribution using combination parameters next simulate iterative algorithms initialize state vector zeros also vector zeros meaning measurements initially presumed good initial value specified randomly chosen set order meet constraint held fixed different trials distributed fixed values taken true values stopping criterion chosen according tolerance tol verified small enough represent numerical convergence simulate different trials whereby trial vector regenerated fig example generated random graph folowing details pedge number edges realization order illustrate evolution algorithms plot nqe iteration count algorithm fig algorithm fig chosen four trials set note trials random graphs measurements chosen algorithms observe plots algorithm converges faster algorithm two algorithms achieve similar final errors majority trials comparison algorithms algorithm explored fig distributions final nqes trials mentioned summarized via boxplot command matlab default settings showing lower edge median central mark upper edge percentiles order make comparison complete provide benchmarks also include weighted least squares per uses exact values parameters naive unweighted least squares estimator bls assume measurements good observe approaches median clearly lower median approach actually bulks error distributions similar wls benchmark except trials distributed lsem perform poorly careful inspection trials shows large errors due incorrect classification type small number edges phenomenon observed possibly thanks better diffusion information centralized algorithm also performed parameter study order quantify behavior mean nqe respect pedge fig observe mean nqe decreases increasing pedge graph becomes connected measurements available estimate state variables figure also observe starting pedge performance distributed similar fig mean nqe increases increasing due presence bad measurements similar reasoning explains increase nqe increasing ratios fig wls fig boxplot showing nqe different approaches parameter set pedge nqe considered outliers boxplot command fig boxplot nqe different initialization obtained using distributed consider known actual values set times real values parameter set pedge ave pedge fig mean nqe respect pedge parameter set number trials wls distributed fig boxplot nqe different initialization obtained using distributed consider known let actual ratio parameter set pedge ave fig mean nqe respect parameter set pedge number trials wls distributed remark parameter uncertainty distributed crucially algorithm values assumed known priori practice one would usually able information hence want explore sensitivity algorithm incorrect choices ave dependencies parameters consistent intuition three figures clear distributed lsem larger average error centralised turn larger error wls estimate however recall mean error distributed driven aforementioned sporadic errors comparison median values would show smaller gap centralized approach fig mean nqe respect parameter set pedge number trials wls distributed fig assume know actual values ratio observe choosing smaller actual value yields higher median larger values real one yield similar results fig assume know observe incorrect large value increases presence large errors even though bulk error distribution remains similar overall conclude algorithm fairly robust moderate uncertainties knowledge parameters oncluding remarks paper studied problem estimation relative measurements heterogeneous quality introduced novel formulation problem proposed two novel algorithms based method one algorithms important feature fully distributed thus amenable applications communication limited expensive algorithm also distinguishes standard approaches due presence regularization variables projection step help use coping nature problem besides designing algorithms proved convergence local maximum function approximation regularization employed also presented number simulations support good performance algorithms robustness uncertainties choice parameters despite generally good performance algorithms particularly distributed one may perform poorly instances could explained local nature optimality results several interesting problems remain open mention three first one could investigate role topology measurement graphs determining performance algorithms namely topologies could effective number measurements taken second one could look distributed algorithms need assume knowledge mixture parameters third one could explore algorithms perform hard classification measurements opposed soft classification done paper measurements assigned probability type preliminary results designs hard classification available eferences barooah hespanha estimation relative measurements electrical analogy large graphs ieee transactions signal processing vol estimation relative measurements algorithms scaling laws ieee control systems magazine vol jiang lim yao statistical ranking combinatorial hodge theory mathematical programming vol osting brune osher optimal data collection improved rankings expose graphs journal machine learning research vol dempster laird rubin maximum likelihood incomplete data via algorithm journal royal statistical society series vol moon algorithm ieee signal processing magazine vol giridhar kumar distributed clock synchronization wireless networks algorithms analysis ieee conference decision control san diego usa bolognani del favero schenato varagnolo distributed sensor calibration parameter identification wsns international journal robust nonlinear control vol rossi frasca fagnani distributed estimation relative absolute measurements ieee transactions automatic control vol online available http carron todescato carli schenato asynchronous algorithm estimation noisy relative measurements ieee transactions control network systems vol freris zouzias fast distributed smoothing relative measurements ieee conference decision control maui usa ravazzi frasca tempo ishii ergodic randomized algorithms dynamics networks ieee transactions control network systems vol frasca ishii ravazzi tempo distributed randomized algorithms opinion formation centrality computation power systems estimation european journal control vol jul carlone censi dellaert selecting good measurements via relaxation convex approach robust estimation graphs int conf intelligent robots systems chiuso fagnani schenato zampieri gossip algorithms simultaneous distributed estimation classification sensor networks ieee journal selected topics signal processing vol fagnani fosson ravazzi distributed algorithm sensor networks siam journal control optimization vol nowak distributed algorithms density estimation clustering sensor networks ieee transactions signal processing vol rabbat nowak distributed optimization sensor networks third international symposium information processing sensor networks ipsn april distributed algorithm gaussian mixtures sensor networks ieee transactions neural networks vol wang kaplan abdelzaher maximum likelihood analysis conflicting observations social sensing acm transactions sensor networks vol wang abdelzaher kaplan social sensing building reliable systems unreliable data morgan kaufmann bishop pattern recognition machine learning springer gupta chen theory use algorithm foundations trends signal processing vol rossi frasca fagnani transient limit performance distributed relative localization ieee conference decision control maui usa chan analysis design algorithms robust estimation relative measurements master thesis university twente enschede netherlands mar online available http ppendix properties likelihood theorem converts problem minimization problem proving recall expression introduce useful notation fbe exp exp fbe exp exp fbe exp exp fze therefore using definition function obtain fbe fbe log fbe log fbe log fbe jensen inequality get log log min log max log log fbe log log log max proof theorem differentiating respect write optimality condition log log log obtain exp exp exp fze using expression get fze log fze log fbe let therefore conclude inequality actually log equality therefore log min log last expression obtained using definition exp function conclude log max max min min min exp log log proof theorem section describe proof theorem log log log showing convergence algorithm lemma monotonicity function defined inequality true nonincreasing along iterates function extended continuity proof repeatedly applying proposition see every time proving result following lemma implies algorithm converges numerically lemma asymptotic regularity sequence generated algorithm proof consequently assertion verified instead neither converge zero exists constant sequence integers min holds true general log log log last inequality follows lemma log log since argmin get ckx suitable positive constant last inequality true since positive semidefinite multiplicity eigenvalue equal fact suppose contradiction exists min needs exist subsequence diverge implies step algorithm obtain deduce exists get contradiction case analogous compute ckx diag log log letting obtain lemma sequence bounded proof first prove upper bounded contradiction exists subsequence lim monotonicity function see lemma inequality log log log since conditions immediate continuity order verify need prove supp exp exp exp supp exp exp step step algorithm get exp lim lim lim exp exp exp supp deduce exists since since lim exists supp exist hand means defined following set verified asymptotic regularity must bounded contradiction since upper bounded constant conclude supp exp exp exp exp exp exp supp distinguish following two cases either zero eventually converges zero asymptotically case exists tsuch supp satisfied case exists strictly positive lim otherwise since time exists noticed together converges tsupp letting conclude lim lim exp lim lim exp exp consequently converges exp lim exp exp guarantees bounded well lemma accumulation point fixed point algorithm satisfies equalities proof accumulation point sequence exists subsequence converges show since sequence bounded see lemma exists subsequence xtj lemma get proving convergence finally lemma ensures converges fixed point since proof theorem section prove theorem showing convergence distributed algorithm let consider function defined fixing variables together surrogate function argmin combining get diag let reader verify following lemma lemma partial minimizations argmin argmin proposition function defined section nonincreasing along iterates proof lemma every time thereby proving result following lemma implies algorithm converges numerically lemma sequence generated algorithm proof define maxt since assumption take sum cated series telescopic last inequality follows fact argmin finally observe log max log last inequality holds letting obtain series convergent deduce inequality claim proved lemma sequence bounded proof since lemma kkx mint maxt notice belongs finite set matrices conclude lim lim lemma sequence bounded exists subsequence xtj lemma get since continuous function also fixed point
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computational topology graphs surfaces colin sep introduction chapter surveys computational topology results special case ambient space surface surface topology comparably simpler counterparts many computational problems undecidable general homotopy questions solved efficiently surfaces leads distinct flavor computational topology dedicated techniques revisiting topological problems surfaces computational viewpoint topological surfaces graphs drawn appear various fields mathematics computer science aspects surveyed topology manifolds also connection recent resolution conjecture combinatorial algebraic structures defined surfaces often relevant via study mapping class groups spaces topological graph theory branch structural graph theory graphs surfaces studied combinatorial point view also relation theory robertson seymour graph minors example colorability questions graphs surfaces generalizing theorem planar graphs wellstudied enumerative combinatorics natural problem count exactly asymptotically maps given properties plane surfaces help generating series moreover typical properties random maps investigated various applications involve surface meshes particular geometry processing computer graphics approximation topological simplification compression parameterization techniques general surfaces apply also subsets plane thus relevant vlsi design map simplification chapter organized follows first review basic concepts properties topological surfaces graphs embedded sections consider three categories topological problems mostly computational perspective drawing abstract input graph surface section homotopy questions variations section optimization curves graphs surfaces also author affiliation cnrs ligm france email part work done author cnrs informatique normale paris france colin homological point view section survey techniques allow solve general graph problems faster case input graph embedded fixed surface section finally collect miscellaneous results section surfaces surfaces considered topological point view two homeomorphic surfaces regarded equivalent surfaces sphere disk topologically uninteresting focus surfaces closed curves deformed point continuous motion surface glossary homeomorphism given two topological spaces map homeomorphism bijective inverse continuous surface topological definition chapter surface compact manifold possibly boundary equivalently compact topological space hausdorff two distinct points disjoint neighborhoods every point neighborhood homeomorphic plane closed set points surface neighborhood homeomorphic plane boundary surface combinatorial definition equivalently surface topological space obtained finitely many disjoint triangles identifying pairs edges triangles quotient topology boundary union edges identified edge path path continuous map two endpoints connectedness surface connected two points surface endpoints path inclusionwise maximal connected subsets surface form connected components orientability surface subset induced topology homeomorphic strip defined figure otherwise orientable properties classification surfaces every connected surface homeomorphic exactly one following surfaces orientable surface genus boundary components concisely boundaries obtained sphere removing disjoint open disks attaching handle defined figure resulting circles finally removing open disks disjoint closures chapter computational topology graphs surfaces disk orientable sphere entable orig strip nonorientable handle orientable figure examples surfaces surface top row comes polygonal schema bottom row polygon labeled directed edges surface obtained identifying pairs edges labels respecting direction genus number boundary components specified well whether surface orientable surface genus boundary components boundaries obtained sphere removing disjoint open disks attaching strip defined figure resulting circles finally removing open disks disjoint closures every surface obtained identifying pairs edges disjoint triangles concisely every surface defined polygonal schema polygon labels directions edges specifying must identified particular one define canonical polygonal schema every connected surface without boundary canonical polygonal schema orientable surface genus whose successive edges labeled edge directed clockwise edge directed counterclockwise identifying edge indicated directions gives orientable surface genus see figure similarly canonical polygonal schema surface genus whose successive edges labeled edges directed clockwise figure double torus system loops left surface cut along loops middle disk shown form canonical polygonal schema right colin surface sphere disk annulus cylinder pair pants torus handle double torus projective plane strip klein bottle orientable yes yes yes yes yes yes yes genus boundary components table common surfaces examples table lists common connected surfaces see also figures graphs surfaces glossary let surface loop loop path whose two endpoints equal single point called basepoint loop closed curve closed curve continuous map unit circle almost loop except closed curve distinguished basepoint closed curve sometimes called cycle although contrary standard terminology graph theory cycle may curve curve either path closed curve purposes parameterization unimportant example path could regarded equivalent bijective increasing simplicity path closed curve simple injective loop simple restriction injective graph chapter unless specified otherwise graphs finite undirected may loops multiple edges curve graph curve graph also called walk terminology graph theory sequence directed edges target equals source repetitions vertices edges allowed endpoints curve source target equal curve closed graph embedding topological definition graph naturally leads topological space defined follows one considers disjoint set segments chapter computational topology graphs surfaces one per edge identifies endpoints correspond vertex gives topological space obtained adding one isolated point per isolated vertex special case loop multiple edge simplicial complex associated topological space embedding continuous map homeomorphism onto image graph embedding concrete definition equivalently embedding drawing maps vertices distinct points edges paths whose endpoints images incident vertices image edge intersect image another edge vertex endpoints confusion arises identify embedding image embedding face faces embedded graph connected components complement image degree degree vertex number edges incident counted multiplicity edge loop degree face number edges incident counted multiplicity edge face sides cellular embedding morphic open disks graph embedding cellular faces triangulation graph embedding triangulation cellular faces degree three triangulation may fail simplicial complex triangle necessarily incident three distinct vertices even three distinct edges cutting given embedded graph without isolated vertex operation cutting along results possibly disconnected surface boundary denoted sometimes sqg similar connected component corresponds face identifying pieces boundaries components obvious way one recovers surface similarly one cut along set disjoint simple closed curves technically nonempty boundary additional condition needed intersection edge boundary either entire edge two endpoints one endpoints planarity graph planar embedding plane equivalently sphere dual graph dual graph cellularly embedded graph assumed without boundary graph embedded one vertex inside face edge edge crosses edge dual graph cellularly embedded combinatorial map see uniquely determined combinatorial map euler genus euler genus connected surface genus equals orientable euler characteristic euler characteristic cellularly embedded graph equals number vertices edges faces respectively colin properties euler formula consequences euler formula cellularly embedded connected surface euler genus boundary components particular depend consequently called euler characteristic number vertices faces graph cellularly embedded connected surface linear number edges particular combinatorial complexity linear number edges conversely let necessarily cellular graph embedding connected surface euler genus boundaries assume face degree one two open disk numbers edges vertices satisfy data structures problems shall consider exact embedding graph surface irrelevant actual combinatorial data associated embedding meaningful graph cellularly embedded surface without boundary need information together facial walks namely closed walks encountered walking along boundary faces information called combinatorial map allows reconstruct surface attaching disks every facial walk conditions walks needed ensure resulting space indeed surface boundaries one specify corresponding faces orientable loop edge instead facial walks one could well specify cyclic ordering edges incident vertex however complicated data structures needed perform basic operations efficiently example one able compute degree face time linear degree count number faces linear time determine whether surface orientable linear time etc last two operations together counting number vertices edges allow identify topology surface linear time using euler formula also done logarithmic space figure map data structure edge bears four flags drawn parallel three operations allow move flag nearby flag chapter computational topology graphs surfaces one data structure map gem representation uses flags equivalently incidences vertex edge face see figure three involutive operations applied flag move incident flag alternative data structures designed general situations allow surfaces boundaries take advantage special situations case triangulation orientable see survey however choice data structure irrelevant theoretical design asymptotic analysis algorithms conventions chapter henceforth assume surfaces connected several works mentioned following orientable surfaces considered cases surfaces easy handle sometimes lead additional difficulties refer original articles determine whether results hold surfaces also problems studied chapter surfaces boundaries harder handle surfaces without boundary algorithm surfaces without boundary immediately implies algorithm surfaces boundary running time replacing complexity genus number boundary components reason mostly focus computational problems surfaces without boundary finally consider cellularly embedded graphs algorithmic problems implicitly assume specified form data structure described map embedding drawing graphs surfaces able build embeddings graph surface small genus important almost algorithms graphs embeddable fixed surface require embedding input graph exceptions discuss algorithmic results related problem embedding graph surface consider general drawings crossings allowed embedding graphs surfaces let abstract graph embedded surface given unordered list edges incident every vertex assume connected let denote combinatorial complexity total number vertices edges general facts embedding orientable surface minimum possible genus cellular embeddable orientable surface genus embeddable orientable surface genus every colin general bound cellularly embedded orientable surface genus planar case algorithm deciding embeddability sphere equivalently plane also time graph embedded segments plane see also loop multiple edge see chapter results graph drawing time complexity given graph surface specified euler genus whether orientable determining whether embeds nphard done time poly polynomial linear fixed embedding computed amount time exists space complexity every fixed determining whether input graph embeds surface orientable euler genus done space logarithmic input size approximation given input graph integer one polynomial time either correctly report embeds surface euler genus compute embedding surface euler genus except planar case algorithms rather complicated implementing real challenge example seems available implementation algorithm testing embeddability torus publicly available implementation algorithm decide whether graph embeds double torus attempts implementing known embedding algorithms even simplest cases unveiled difficulties hand recent approach promising practice graphs moderate size using integer linear programming boolean satisfiability reformulations contrast determining maximum genus orientable surface without boundary graph cellularly embedded done polynomial time also results embeddability simplicial complexes surfaces less algorithmic side field topological graph theory lot known embeddability classes graphs surfaces see sect references therein glossary drawings let graph surface drawing drawings general embeddings allow finite set crossing points exactly two pieces edges intersect actually cross formally recall associated topological space topological drawing continuous map preimage every point cardinality zero one except finite set points crossings whose preimages cardinality two moreover crossing point disk neighborhood contains exactly images two pieces edges form homeomorphism two crossing straight lines chapter computational topology graphs surfaces arrangement let drawing arrangement graph embedded image obtained inserting vertex degree four crossing subdividing edges accordingly similarly one consider arrangement set curves drawn crossing number crossing number respect minimum number crossings drawing pair crossing number pair crossing number respect minimum number pairs edges cross drawings odd crossing number odd crossing number respect minimum number pairs edges cross odd number times drawings drawing graphs surfaces crossings crossing numbers computing planar crossing number graph even special cases planar graph single additional edge exists algorithm approximation guarantee better certain constant however every fixed one linear time determine whether input graph planar crossing number although problem admits polynomial kernel approximation algorithms planar crossing number known restricted cases bounded maximum degree variations crossing numbers relations various notions crossing numbers fully understood let denote planar crossing number planar pair crossing number planar odd crossing number respectively graph clear known left inequality strict widely believed best bound known far follows essentially see details wide survey various notions crossing numbers theorem weak theorem however states furthermore holds plane arbitrary surfaces graph drawn surface way every pair edges crosses even number times embedded planar case actually suffices assume every pair independent edges share endpoints crosses even number times whether generalizes arbitrary surfaces open except projective plane refer surveys details homotopy isotopy works computational topology surfaces take input given abstract graph previous section instead consider already embedded graph colin given combinatorial map glossary let surface reversal reversal path path defined concatenation concatenation two paths path defined homotopy paths given two paths homotopy continuous deformation keeps endpoints fixed formally continuous map constant maps equal respectively paths homotopic homotopic equivalence relation partitioning paths given endpoints homotopy classes fundamental group homotopy classes loops given basepoint form group concatenation loops accounts multiplication reversal accounts inverse operation denotes homotopy class path homotopy closed curves also called free homotopy given two closed curves homotopy continuous deformation namely continuous map contractibility loop closed curve contractible homotopic constant loop closed curve isotopy isotopy two simple paths loops closed curves homotopy create simple path loop closed curve isotopy graph continuous family embeddings vertices edges move continuously ambient isotopy ambient isotopy surface continuous map homeomorphism minimally crossing family closed curves minimally crossing every family closed curves homotopic number intersections larger covering space let possibly connected surface continuous map covering map every point connected neighborhood disjoint union open sets homeomorphism say covering space lift path path finally chapter computational topology graphs surfaces loop contractible universal covering space essentially unique precisely universal covering spaces homeomorphism basic properties two paths homotopic contractible loop two loops basepoint freely homotopic viewed closed curves without basepoint homotopy classes loops conjugates fundamental group fundamental group surface without boundary genus best understood looking canonical polygonal schema surface orientable group generated generators single lation corresponding boundary polygonal schema similarly group generated generators single relation fundamental group surface least one boundary component free group surface homotopy type graph let covering space every path admits lifts moreover lift unique lift two paths homotopic homotopic lifts particular two paths homotopic admit lifts endpoints universal covering space deciding homotopy isotopy homotopy one first studied problems regarding curves surfaces concerned homotopy tests contractibility problem given closed curve equivalently loop contractible free homotopy problem two given closed curves freely homotopic problems translate central problems group theory special case fundamental groups surfaces given finitely generated group presented form generators relations given word generators represent trivial element group word problem two given words generators represent conjugate elements group conjugacy problem computational geometry problems studied following context input cellularly embedded graph one two closed curves represented closed walks exist thus optimal algorithms contractibility free homotopy problems earlier article claims results reported algorithm free homotopy article subtle flaw approaches rely construction part universal covering space results small cancellation theory group theory remark dehn algorithm implemented linear time assuming surface colin fixed graph single face algorithms mentioned require isotopy deciding whether two simple closed curves isotopic also done linear time equivalence relation simple refinement homotopy simple closed curves deciding isotopy graph embeddings complicated also done efficiently since essentially reduces homotopy tests closed curves homotopies often known two curves homotopic one would like compute reasonable homotopy relevant questions include finding homotopy sweeps minimum possible area discretized sense minimum possible number steps homotopy maximum length intermediate curves minimal height homotopy homotopy maximum distance traveled point first second curve minimal width related homotopic distance etc several questions studied case plane extensions surfaces still open elementary moves uncrossing figure four reidemeister moves ambient isotopy pictures represent intersection union curves small disk particular pictures regions bounded curves homeomorphic disks parts curves intersect parts curves shown elementary moves every family closed curves general position made minimally crossing finite sequence reidemeister moves described figure closed curve reidemeister moves needed tight curve homotopic simple curve general subexponential upper bound seems known actually one deform family curves continuously make minimally crossing without increasing total number crossings step moreover minimally crossing family curve minimally pair curves minimally crossing see also characterizations curves minimally crossing position making curves simple let graph cellularly embedded surface one decide whether input curve represented closed walk homotopic simple closed curve time generally one compute minimum number curve homotopic input closed walk minimum number intersections two curves respectively homotopic two input closed walks quadratic time chapter computational topology graphs surfaces untangling curves homeomorphism given two families disjoint simple curves one try minimize number crossings changing one homeomorphism surface bounds known number crossings one achieve simultaneous graph drawing also relates problem embedding two input graphs surface way embeddings cross times also results known one also require combinatorial maps fixed number homotopy classes many simple closed curves different homotopy classes one draw pairwise cross times given integer orientable surfaces genus without boundary answer pants decomposition see together contractible closed curve problem interesting larger values recently proved fixed number curves one draw polynomial genus optimization shortest curves graphs problem computing shortest curves graphs satisfying certain topological properties surfaces widely considered leads problems flavor combinatorial optimization problems meaningful metric must provided computational geometry one could naturally consider piecewise linear surfaces euclidean space perhaps however efficient algorithms computing shortest paths surfaces need additional assumptions distances involve square roots leads deep unrelated questions complexity comparing sums square roots furthermore context graph problems specific case graphs section model would insufficient notions combinatorial surfaces defined developed avoid technical distractions suitable various settings hand oracle shortest path computations several results section extend geometric settings example piecewise linear surfaces euclidean space see sect glossary discrete metrics surfaces combinatorial surface combinatorial surface data cellular graph embedding positive weights edges allowed curves walks length curve sum weights edges traversed curve counted multiplicity algorithmically curves stored closed walks complexity combinatorial surface complexity embedding asymptotically number edges colin surface surface also data cellular graph embedding surface positive weights edges however contrast combinatorial surface model curves drawn surface general position respect length curve sum weights edges crossed curve counted multiplicity algorithmically family curves graph surface stored combinatorial map arrangement family curves graph together complexity surface complexity embedding asymptotically number edges without loss generality one could draw curves neighborhood dual graph pushing completely onto would transform curves combinatorial surface defined however surface defined retains information combinatorial surface defined latter case curves share edges automatically overlap model allows make disjoint except crossing points point still possible define notion crossing two curves combinatorial surface still insufficient algorithms described figure left closed curves surfaces splitting right pants decomposition surface types simple closed curves let simple closed curve interior surface see figure curve surface cut along denoted two connected components one homeomorphic disk separating curve separating two connected components splitting curve essential curve splitting separating essential component disk annulus topological decompositions cut graph cut graph graph embedded surface homeomorphic closed disk system loops system loops surface without boundary cut graph single vertex see figure chapter computational topology graphs surfaces canonical system loops system loops surface without boundary canonical edges polygon appear order canonical polygonal schema see section pants decomposition pants decomposition orientable surface family simple disjoint closed curves disjoint union pairs pants see figure octagonal decomposition octagonal decomposition orientable surface without boundary family closed curves self intersection point crossing exactly two closed curves face arrangement octagon disk eight sides homology context graphs surfaces homology surfaces field used described somewhat concisely general homology theories let surface assume graph embeddings piecewise linear respect fixed triangulation homological sum previous assumption closure symmetric difference images two graph embeddings image graph embedding called homological sum defined subdivision edges vertices insertion isolated vertices reverse operations graph embeddings considered operations homology cycle graph embedded homology cycle every vertex even degree set homology cycles forms vector space field empty graph trivial element addition homological sum homology boundary graph embedded homology boundary faces colored two colors say black white boundary two colors exactly one side edge incident black face set homology boundaries forms vector space every homology boundary homology cycle homology group space denoted quotient homology cycles homology boundaries graph embedding homologically trivial homology boundary homology sets loops closed curves defined similarly loops closed curves images graph embedding using advanced theory singular homology one remove restriction dealing piecewiselinear graph embeddings basic properties simple closed curve contractible simple closed curve separating homologically trivial colin homology group surface without boundary dimension euler genus generated loops appearing boundary canonical polygonal schema shortest curves deciding whether simple closed curve combinatorial surface separating done time linear size data structure used store cellular graph curve boils determining whether graph connected whether surface disk easy using euler formula consider optimization version looking shortest curves given topological type combinatorial surface curves particular interest cutting along curve simplifies topology surface use shorthand either undirected log log log log log log log log log log directed log log log log log log see gnk gnk table algorithms shortest closed curves surfaces without boundary depending whether graph weighted whether directed mean respectively size output best complexities known date bold several category due tradeoff course undirected case reduces directed case unweighted case reduces weighted case cell repeat algorithms available general scenarios structural properties combinatorial surface shortest noncontractible loop based vertex made two shortest paths single edge condition follows globally shortest closed curves repeat vertices edges also shortest nonseparating closed curves generally algorithms mentioned typical tool prove bound number crossings unknown shortest curve shortest path chapter computational topology graphs surfaces different scenarios shortest curves table summarizes running times known algorithms problems relevant look efficient algorithms case genus smaller compared complexity graph defining surface standard scenario one considered elsewhere chapter combinatorial equivalently surface undirected weighted case upper left corner table one also aim faster algorithms unweighted case unit weights finally one extend techniques case directed graphs edges combinatorial surface directed used specified direction equivalently edges surface crossed specific direction topological types shortest simple closed curves topological types investigated well following denotes complexity surface shortest splitting curves computable log time fixed genus shortest essential curves log time log fixed genus number case surfaces boundary require sophisticated techniques curves shortest unspecified homotopy class log shortest homotopic curves slightly different problem computing shortest curve homotopic given curve either path closed curve also doable small polynomial time using octagonal decompositions build part universal covering space earlier algorithms dealt simple curves iterated shortening process leads global optimum shortest paths algorithms rely shortest path computations combinatorial surfaces done log time using dijkstra algorithm classically speeded fibonacci heaps primal dual graph actually computes shortest paths single source vertices combinatorial surface algorithms available computing multiple shortest paths quickly conditions locations endpoints shortest decompositions decompositions surfaces central topology example standard proof classification theorem transforms arbitrary cut graph canonical system loops many algorithms described previous subsection rely topological decompositions properties shortest cut graph problem computing shortest cut graph crossmetric surface extensively studied computing shortest cut graph algorithm runs log time moreover every one compute time function one looking shortest cut graph specified vertex set example shortest system loops given basepoint algorithm running time log root several articles lies treecotree property cellular graph embedding exists partition colin edges spanning tree edges dual form spanning tree dual graph contracting deleting transforms system loops loop corresponding element topological decompositions canonical system loops orientable surfaces without boundary computed time octagonal decomposition pants decomposition made closed curves short possible respective homotopy classes computed log time general complexity computing shortest decompositions open hand bounds maximum length decompositions assuming combinatorial surface unweighted triangulation dually surface unweighted vertex degree three stretch let surface let associated embedded graph stretch minimum product lengths closed curves crossing exactly quantity related planar crossing number size largest toroidal grid minor computed small polynomial time homology relation cuts flows hinted homology useful simple closed curve separating algorithms computing shortest closed curves actually compute shortest closed curves turn simple homology natural concept particular interesting look family closed curves minimum total length homology classes generate homology group efficient algorithms given purpose also connection algorithm compute minimum cycle basis graph another reason importance homology relation cuts given graph cellularly embedded surface without boundary dual subgraphs fixed homology class surface obtained removing faces containing thus computing minimum cuts amounts computing shortest homologous subgraphs property exploited study general graph problems better algorithms designed specific case graphs embedded fixed surface compute minimum time best algorithm runs log time genus relies homology cover particular type covering space compute maximum faster exploiting duality flows cuts count sample minimum efficiently compute global minimum cuts efficiently without fixing chapter computational topology graphs surfaces deal problems compute edge expansion connectivity measures bound space complexity bipartite matching algorithms graphs embedded fixed surface general graph problems solved faster special case graphs embedded fixed surface examples include cut flow problems see previous section multicommodity problems domination independence problems connectivity problems steiner tree traveling salesman problem etc disjoint paths problems shortest paths problems subgraph problems sometimes problems solvable arbitrary graphs goal obtain faster algorithms graphs many cases problems considered arbitrary graphs algorithms obtained graphs embeddable fixed surface occasionally fixing parameters problem typically optimization problems considered case relevant look approximation algorithms methods involved usually combine topological aspects described techniques structural algorithmic graph theory glossary minor graph minor another graph obtained removing edges isolated vertices contracting edges family family graphs every minor graph also tree decomposition tree decomposition graph tree node labeled subset set nodes whose labels contain induces connected subtree edge connecting vertices label least one node contains width width tree decomposition maximum cardinality labels minus one treewidth treewidth graph minimum width tree decomposition survey techniques central algorithmic structural graph theory study families graphs deep result robertson seymour family colin finite set graphs graph minor refer survey structural aspects graphs embeddable fixed surface form family benefit studied using topological techniques robertson seymour provide decomposition theorem families graphs involving graphs embeddable fixed surface efficient algorithms graphs sometimes extended families graphs different family graphs impossible list results algorithms graphs focus general methods several algorithms based topological techniques described previous sections particular shortest curves shortest decompositions several cases advanced algorithmic techniques sometimes techniques led new results planar graphs methods applicable several algorithmic problems also emerged many cases extending previous ones invented planar graphs graph separators treewidth let graph vertices embedded surface genus linear time one compute balanced separator size namely set vertices whose removal leaves graph without connected component vertices also treewidth dynamic programming small treewidth implies efficient algorithms using dynamic programming arbitrary graphs graph embedded one exploit fact obtain algorithms smaller dependence treewidth problems irrelevant vertex technique several graph problems enjoy following property input graph large treewidth exists irrelevant vertex whose removal creates equivalent instance problem vertex center large grid minor property widely used structural graph theory exploited several times context algorithms graphs approximation schemes ptass baker introduced technique designing approximation schemes optimization problems local constraints planar graphs showed one delete small part input graph without changing much value solution resulting graph small treewidth technique extended graphs embeddable fixed surface graphs drawn fixed surface bounded number crossings per edge general problems contraction instead deletion must used crucial step making latter technique effective construction spanner case minimization problem subgraph input graph containing solution whose weight linear optimal solution brick decomposition technique builds spanners problems originally planar graphs also sometimes graphs surfaces bidimensionality theory applies minimization problems unweighted graphs contracting edge graph chapter computational topology graphs surfaces increase value solution value solution grid graphs generalizations large leads algorithms graphs embeddable fixed surface running time form value solution input size also provides ptass cases problems bidimensionality applies ptass sometimes also obtained weighted graphs using different framework stochastic embeddings let positively edgeweighted graphs metric embedding mapping represent shortest path distances respectively distortion maximum see chapter every graph embeddable orientable surface genus admits probability distribution metric embeddings planar graphs one log expectation distribution reduces several optimization problems graphs problem planar graphs loss log factor actually distribution computed polynomial time even embedding known models rather large number results relate concepts described chapter would impossible cover provide selection miscellaneous results consider models representing graphs surfaces computational topology plane obstacles plane minus finitely many points polygons obstacles forms surface taking cellular graph embedding makes combinatorial crossmetric surface topological algorithms apply however much natural consider arbitrary curves whose length defined euclidean metric model defined obstacles finite set disjoint simple polygons simplicity exposition curves arbitrary polygonal lines avoiding interior obstacles problems defined previous sections related problems studied model homotopy isotopy tests efficient algorithms test whether two curves freely homotopic whether two graphs isotopic shortest homotopic paths computed efficiently well see also section variant several simple disjoint paths must shortened preserving homotopy class keeping neighborhoods simple disjoint paths thick also investigated shortest disjoint paths goal compute disjoint paths minimum total length precisely paths since limit case colin solution may consist overlapping paths endpoints lie boundary bounded number obstacles problem solvable polynomial time results include approximation algorithm shortest pants decomposition case obstacles points algorithm computing homotopic distance measure similarity curves takes obstacles account topologically simple disjoint curves graphs model defined cellularly embedded graph one think curves drawn neighborhood intuitively curves drawn share vertices edges simple pairwise disjoint natural especially topological graph theory forbid overlaps set disjoint simple curves repeat vertex edge many problems mentioned previous sections make sense setup turns generally difficult handle model model following results known circuit mean closed curve graph without repeated vertex containing least one edge determining whether exists separating splitting circuit npcomplete determining contractible circuit circuit exists possible linear time even one requires circuit pass given vertex computing shortest contractible circuit possible polynomial time one requires circuit pass given vertex problem becomes nphard computing shortest separating circuit combinatorial characterization whether curves made simple disjoint graph homotopy surface case planar surface boundaries leads algorithm turn algorithmic consequences problem computing paths planar graphs see also normal curves surfaces let family disjoint simple closed curves surface general position respect triangulation natural way represent described previous sections arrangement normal curves economical representation price mild condition every triangle intersection image must set disjoint simple paths called normal arcs connecting different sides triangle one stores three integers recording number normal arcs connecting three pairs sides chapter computational topology graphs surfaces overall described integers number triangles conversely given vector integers one unambiguously reconstruct normal isotopy ambient isotopy leaves edges globally unchanged store vector normal coordinates log bits needed number crossing points contrast representing curves surface requires least store constant amount information per vertex arrangement total normal curve representation exponentially compressed compared one despite time polynomial input size one count number connected components normal curve note normal curve connected partition components according normal isotopy classes given multiplicities normal coordinates representative decide whether two normal curves isotopic compute algebraic geometric intersection number two normal curves algebraic intersection number sum crossings sign crossing crosses left right crossing point otherwise welldefined surface orientable since invariant isotopy geometric intersection number minimum number crossings curves isotopic problems initially studied using programs concise encoding words finite alphabet many algorithms words solved efficiently using program representation particular programs represent exponentially long words leads efficient algorithms normal curves problems revisited using topological techniques normal curves analog normal surfaces widely used topology resources books graphs surfaces combinatorial viewpoint treated detail see also basic surface topology recommend survey surveys optimization problems graphs providing details large fraction section course notes unpublished material provides notes computational topology strong emphasis graphs surfaces survey algorithms optimization graphs curves surfaces emphasizes graph algorithms graphs colin related chapters chapter polyhedral maps chapter shortest paths networks chapter graph drawing acknowledgments many thanks sergio cabello vincent jeff erickson francis lazarus arnaud mesmay dimitrios thilikos careful reading preliminary versions numerous comments greatly improved chapter references alliez gotsman recent advances compression meshes dodgson floater sabin editors advances multiresolution geometric modelling pages berlin archdeacon topological graph theory survey congr armstrong basic topology undergraduate texts mathematics springerverlag berlin baker approximation algorithms problems planar graphs acm borradaile chambers fox nayyeri minimum cycle homology bases surface embedded graphs proc sympos comput vol lipics article schloss dagstuhl beyer chimani hedtke practical method minimum genus graph models experiments proc sympos experimental algorithms vol lncs pages springer cham borradaile demaine tazari approximation schemes problems graphs algorithmica burton elder kalka tillmann recognition logspace comput borradaile eppstein nayyeri minimum cuts time graphs proc sympos comput vol lipics article schloss dagstuhl bespamyatnikh computing homotopic shortest paths plane algorithms bettinelli topology scaling limits positive genus random quadrangulations ann berg van kreveld schirra topologically correct subdivision simplification using bandwidth criterion cartography gis computing sums radicals polynomial time proc ieee sympos found comp pages chapter computational topology graphs surfaces bonsma surface split decompositions subgraph isomorphism graphs surfaces proc sympos theoret aspects comp vol lipics pages schloss dagstuhl cabello finding shortest contractible shortest separating cycles embedded graphs acm trans algorithms cabello hardness approximation crossing number discrete comput colin klein mathieu meierfrankenfeld approximating connectivity domination weighted graphs proc acm sympos theory pages mesmay fixed parameter tractable approximation scheme optimal cut graph surface proc european sympos algorithms vol lncs pages springer berlin cabello chambers multiple source shortest paths genus graph proc sympos discrete algorithms pages cabello chambers erickson shortest paths embedded graphs siam cabello chimani computing stretch embedded graph siam discrete chambers colin erickson lazarus whittlesey splitting complicated surfaces hard comput chambers colin erickson lazard lazarus thite homotopic distance curves walking dog woods polynomial time comput cabello colin lazarus finding cycles topological properties embedded graphs siam discete cabello colin lazarus algorithms embedded graph comput cabello colin lazarus finding shortest cycles directed graphs surfaces comput cabello devos erickson mohar finding one tight cycle acm trans algorithms cheng dey poon hierarchy surface models irreducible triangulations comput chang erickson untangling planar curves proc sympos comput vol lipics article schloss dagstuhl chambers erickson nayyeri minimum cuts shortest homologous cycles proc sympos comput pages acm press chambers erickson nayyeri homology flows cohomology cuts siam chambers fox nayyeri counting sampling minimum cuts genus graphs discrete comput chen han shortest paths polyhedron internat comput geom chuzhoy algorithm graph crossing number problem proc acm sympos theory pages colin chambers letscher height homotopy proc canad conf comput pages see also erratum cabello liu mantler snoeyink testing homotopy paths plane discrete comput cabello mohar finding shortest cycles topologically embedded graphs discrete comput cabello mohar adding one edge planar graphs makes crossing number hard siam cairns nicolayevsky bounds generalized thrackles discrete comput colin shortest cut graph surface prescribed vertex set proc european sympos algorithms part vol lncs pages springer berlin colin topological algorithms graphs surfaces habilitation thesis normale paris available http colin algorithms embedded graphs course notes http colin multicuts planar graphs bounded number terminals algorithmica colin erickson tightening nonsimple paths cycles surfaces siam colin hubard mesmay discrete systolic inequalities decompositions triangulated surfaces discrete comput colin tancer direct proof strong theorem projective plane proc sympos graph drawing network visualization pages vol lncs springer cham colin lazarus optimal system loops orientable surface discrete comput colin lazarus optimal pants decompositions shortest homotopic cycles orientable surface acm colin mesmay testing graph isotopy surfaces discrete comput chambers wang measuring similarity curves via homotopy area proc sympos comput pages acm press dehn transformation der kurven auf zweiseitigen math demaine fomin hajiaghayi thilikos subexponential parameterized algorithms graphs graphs acm dey guha transforming curves surfaces comput system chapter computational topology graphs surfaces datta gopalan kulkarni tewari improved bounds bipartite matching surfaces proc sympos theoret aspects comp vol lipics pages schloss dagstuhl demaine hajiaghayi bidimensionality new connections fpt algorithms ptass proc sympos discrete algorithms pages demaine hajiaghayi bidimensionality theory algorithmic applications computer journal demaine hajiaghayi mohar approximation algorithms via contraction decomposition combinatorica demaine hajiaghayi thilikos bidimensional theory graphs siam discrete dijkstra note two problems connexion graphs lazarus computing geometric intersection number curves proc sympos comput vol lipics article schloss dagstuhl demaine mozes sommer tazari algorithms planar graphs beyond course notes http erickson fox nayyeri global minimum cuts surface embedded graphs proc sympos discrete algorithms pages erickson optimally cutting surface disk discrete comput elberfeld kawarabayashi embedding canonizing graphs bounded genus logspace proc acm sympos theory computing pages efrat kobourov lubiw computing homotopic shortest paths efficiently comput erickson nayyeri computing replacement paths graphs proc sympos discrete algorithms pages erickson nayyeri minimum cuts shortest cycles via homology covers proc sympos discrete algorithms pages erickson nayyeri shortest walks plane proc sympos discrete algorithms pages erickson nayyeri tracing compressed curves triangulated surfaces discrete comput eppstein diameter treewidth graph families algorithmica eppstein dynamic generators topologically embedded graphs proc sympos discrete algorithms pages eppstein squarepants tree sum subtree clustering hyperbolic pants decomposition acm trans algorithms colin epstein curves isotopies acta erickson maximum flows parametric shortest paths planar graphs proc sympos discrete algorithms pages erickson shortest cycles directed surface graphs proc sympos comput pages acm press erickson combinatorial optimization cycles bases zomorodian editor advances applied computational topology vol proc sympos appl pages ams providence erickson computational topology course notes http erickson sidiropoulos approximation algorithm asymmetric tsp embedded graphs proc sympos comput pages acm press erickson whittlesey greedy optimal homotopy homology generators proc sympos discrete algorithms pages erickson worah computing shortest essential cycle discrete comput erickson whittlesey transforming curves surfaces redux proc sympos discrete algorithms pages furst gross mcgeoch finding graph imbedding acm farb margalit primer mapping class groups princeton university press fox shortest cycles directed undirected surface graphs proc sympos discrete algorithms pages fredman tarjan fibonacci heaps uses improved network optimization algorithms acm grigoriev bodlaender algorithms graphs embeddable crossings per edge algorithmica gilbert hutchinson tarjan separator theorem graphs bounded genus algorithms gao jerrum kaufmann mehlhorn storb continuous homotopic one layer routing proc sympos comput pages acm press gersten short small cancellation theory automatic groups invent graaf schrijver making curves minimally crossing reidemeister moves combin theory ser gross tucker topological graph theory wiley new york guskov wood topological noise removal proc graphics interface pages canad inf process toronto yau global conformal surface parameterization proc sympos geom processing pages chapter computational topology graphs surfaces chojnacki hanani wesentlich kurven dreidimensionalen raume fund chimani approximating crossing number graphs embeddable orientable surface proc sympos discrete algorithms pages crossing number hard kernelization proc sympos comput vol lipics article schloss dagstuhl henle combinatorial introduction topology dover publications mineola hubard mesmay tancer shortest path embeddings graphs surfaces proc sympos comput vol lipics article schloss dagstuhl nayyeri salavatipour sidiropoulos walk dog mountains magic leash discrete comput hass scott intersections curves surfaces israel hass scott shortening curves surfaces topology hershberger snoeyink computing minimum length paths given homotopy class comput hopcroft tarjan efficient planarity testing acm italiano nussbaum sankowski improved algorithms min cut max flow undirected planar graphs proc acm sympos theory computing pages kelner spectral partitioning eigenvalue bounds circle packings graphs bounded genus siam kettner using generic programming designing data structure polyhedral surfaces comput kawarabayashi klein sommer approximate distance oracles planar graphs proc int coll automata languages part vol lncs pages springer berlin klein approximation scheme planar weighted tsp proc ieee sympos found comp pages kawarabayashi mohar recent progress applications graph minor theory graphs kawarabayashi mohar reed simpler linear time algorithm embedding graphs arbitrary surface genus graphs bounded proc ieee sympos found comp pages kawarabayashi reed computing crossing number linear time proc acm sympos theory computing pages kawarabayashi sidiropoulos beyond euler characteristic approximating genus general graphs proc acm sympos theory computing pages colin thilikos contraction checking graphs surfaces proc sympos theoret aspects comp vol lipics pages schloss dagstuhl kutz computing shortest cycles orientable surfaces bounded genus almost linear time proc sympos comput pages acm press lins maps combin theory ser leiserson maley algorithms routing testing routability planar vlsi layouts proc acm sympos theory computing pages lazarus pocchiola vegter verroust computing canonical polygonal schema orientable triangulated surface proc sympos comput pages acm press lazarus rivaud homotopy test surfaces proc ieee sympos found comp pages lando zvonkin berlin string graphs separators pellegrini geometry structure randomness combinatorics pages scuola normale superiore pisa miermont tessellations random maps arbitrary genus annales scientifiques normale myrvold kocay errors graph embedding algorithms comput system mitchell mount papadimitriou discrete geodesic problem siam mohar minimal genus graph theory mohar linear time algorithm embedding graphs arbitrary surface siam discete makarychev sidiropoulos planarizing unknown surface proc workshop approximation vol lncs pages springer berlin sedgwick tancer wagner untangling two systems noncrossing curves israel mohar thomassen graphs surfaces johns hopkins university press negami crossing numbers graph embedding pairs closed surfaces graph theory nishizeki rahman planar graph drawing world scientific singapore patel determining edge expansion connectivity measures graphs bounded genus siam pilipczuk pilipczuk sankowski van leeuwen network sparsification steiner problems planar graphs proc ieee sympos found comp pages graphs surfaces applications chapter computational topology graphs surfaces przytycki arcs intersecting geom funct pelsmajer schaefer odd crossing number crossing number discrete comput pelsmajer schaefer stasi strong projective plane siam discrete pelsmajer schaefer removing even crossings surfaces european robertson seymour graph minors xvi excluding graph combin theory ser robertson seymour graph minors wagner conjecture combin theory ser richter salazar two maps large representativity one surface graph theory robertson seymour graph minors xxii irrelevant vertices linkage problems combin theory ser sau thilikos asymptotic enumeration partitions surfaces discrete sau thilikos dynamic programming graphs surfaces acm trans algorithms schnyder embedding planar graphs grid proc sympos discrete algorithms pages schrijver homotopic routing methods korte schrijver editors paths flows pages springerverlag berlin schrijver disjoint circuits prescribed homotopies graph compact surface combin theory ser schrijver combinatorial optimization polyhedra efficiency vol algorithms combinatorics berlin schaefer graph crossing number variants survey electron dynamic surveys article updated schaefer toward theory planarity planarity variants graph alg schaefer related results fejes editors discrete convex tribute fejes vol bolyai society math studies pages springer berlin sidiropoulos optimal stochastic planarization proc ieee sympos found comp pages schaefer sedgwick algorithms normal curves surfaces proc conf computing combinatorics vol lncs pages springer berlin schaefer sedgwick computing dehn twists geometric intersection numbers polynomial time proc canad conf comput pages stillwell classical topology combinatorial group theory edition new york colin thilikos graph minors parameterized algorithm design bodlaender downey fomin marx editors multivariate algorithmic revolution beyond vol lncs pages springer berlin thilikos bidimensionality parameterized algorithms proc sympos parameterized exact computation vol lipics pages schloss dagstuhl thomassen graph genus problem algorithms thomassen embeddings graphs short noncontractible cycles combin theory ser better bound number pach editor thirty essays geometric graph theory pages springer new york tutte toward theory crossing numbers combin theory wood hoppe desbrun removing excess topology isosurfaces acm trans
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behavior identification diagnosis juan biondi gerardo silvia osvaldo jan laboratorio desarrollo neurociencia cognitiva instituto investigaciones iiie departamento computadoras diec universidad nacional del sur uns conicet blanca argentina laboratorio vyglab departamento ciencias dcic universidad nacional del sur uns blanca argentina investigaciones provincia buenos aires cic argentina abstract present work develop approach differentiate behavior people neurodegenerative diseases healthy control subjects reading subjects without alzheimer disease read previously validated sentences including sentences proverbs data derive information consisting descriptors capture reading behavior subjects information train set denoising build deep neural network using trained autoencoders softmax classifier allows identifying patients alzheimers disease accuracy results encouraging show models promise helpful understand dynamics eye movement behavior relation underlying neuropsychological processes keywords alzheimer disease introduction alzheimer disease nonreversible neurodegenerative disease characterized progressive impairment cognitive memory functions develops period years prevalent cause dementia elderly subjects initially people experience memory loss confusion may mistaken kinds memory changes sometimes associated normal aging waldemar subtle changes behavior response early manifestation disease make difficult diagnose using classical neuropsychological tests state examination use advanced diagnosis tools mri pet results critical early diagnosis since nonreversible early treatment improve patient life delaying full manifestation disease last years study eye movement known reading proved help performing task reading cognitive activity received considerable attention researchers evaluate human cognitive performance requires integration several central cognitive subsystems attention oculomotor control word identification language comprehension eye movements show reproducible pattern normal reading eye movement ends fixation point allows brain process incoming information program following saccade different neuropsychiatric pathologies produce abnormalities eye movements disturbances reading particular pattern registered measured holzman iacono riby hancock kellough eye movements classified three groups corresponding author email movement maintaining image fovea area retina acuity vision compensating head object movements movements shifting eyes attention changes one object another subtypes shifting movements saccades looking new center visual attention monitoring vergence slower saccades responsible carrying image interest foveae allowing stereoscopic vision movements binocular fixation also prevent fading image movements three variations tremor drift microsaccade saccades rapid big eye movements particularly important cognitive point view since cognitive processes direct influence movements saccade direction people depending language read left right saccadic eye movements oriented accordingly normal reading movements called forward saccades reading movements going right left called regressions saccade movement alternates fixation made eyes directed particular target see rayner review shown patients early alzheimer disease show alterations execution tasks reading alterations related impairment working memory fact shown differentiation possible infer diagnosis use diagnosis key challenge since growth computational power permits creation complex models models used create biomarkers help disease identification since popularization neural networks schmidhuber deng many efforts made use field medicine technique commonly used conjunction imaging diagnoses pet mri mainly feature representation technology provides may help even data incomplete specifically advances detection pattern differentiation physical brain alterations neurodegenerative diseases produce mild cognitive impairment mci suk shen suk even advances early diagnostic liu problem brain physical alteration observable damages made brain irreversible even though disease early stage may cause deterioration quality life patient technique allows find subtler changes made brain alleviate small memory deficits patient changes noticed patient small changes way read set sentences found technique presented paper work use neural network trained reading information extracted controls patients probable order identify patterns made reading process later cluster respective groups throughout work may use patients patients probable interchangeable manner nature diagnosis hypothesis using feature identification key characteristics patient eye behavior reading sentences may lead correct classification used infer diagnosis using type technology may improve results obtained since provides smaller granularity detection disease consequently better performance additionally technology allows improve effectiveness classification collect ground truth subjects methods ethics statement investigation adhered principles declaration helsinki approved institutional bioethics committee hospital municipal agudos blanca buenos aires argentina patients caregivers control subjects signed informed consent prior inclusion study participants twenty six patients mean age years years diagnosis probable recruited hospital municipal blanca buenos aires argentina clinical criteria diagnose early stages remains debate mckhann present work diagnosis based criteria dementia outlined diagnostic statistical manual mental disorders patients underwent detailed clinical history revision examination thyroid function test presented apo genotype magnetic resonance images obtained twelve patients computerized tomography scans patients underwent biochemical analysis discard common pathologies hemoglobin full blood count erythrocyte sedimentation rate urea electrolytes blood glucose data provided precise diagnosis patients excluded suffered medical conditions could account interfere cognitive decline evidence vascular lesions computed tomography fmri evidence axis diagnosis major depression drug abuse defined eligible study patients least one caregiver providing regular care support patients taking cholinesterase inhibitors included none subjects taking hypnotics sedative drugs major tranquilizers control group consisted elderly adults mean age years years known neurological psychiatric disease according medical records evidence cognitive decline impairment daily activities anova showed significant differences ages control individuals participants diagnosed suffering ophthalmologic disease glaucoma visually significant cataract macular degeneration well visual acuity less excluded study mean scores controls patients state examination mmse folstein respectively latter suggesting early mental impairment anova evidenced significant differences mmse patients controls mean score patients adenbrook cognitive examination revised mioshi mean school education trajectories patients controls years years respectively anova showed significant differences education control individuals apparatus eye movement data single sentences presented center line lcd monitor pixels resolution font regular new courier points height participants sat distance monitor head movements minimized using chin rest correction viewing distance performed using corneal reflection system assessed changes gaze position measuring reflection infrared illuminator cornea pupil size means video camera sensitive light infrared spectrum eye movements recorded desktop mount research eye tracker sampling rate eye position resolution arc recordings calibration binocular eye movement data participants reading sentences resulted total fixations control people data cleaned blinks track losses prior removing analysis fixations shorter longer fixations first last word sentence see kliegl description analytic procedure measured patient elapsed time instant sentences first presented instant participants looked final spot mean reading time sentences controls controls low procedures participant gaze calibrated standard grid eyes validating calibration trial began appearance fixation point position first letter sentence presented soon eyes detected within radius equal fixation spot sentence presented reading participants looked dot lower right corner screen gaze detected final spot trial ended occasionally external factors minor movements slippages head gear could cause small drifts avoid performed drift correction presentation spot assess whether subjects comprehended texts presented three alternative question sentence progress sentence trials participants answered questions moving mouse choosing response mouse click overall mean accuracy control anova showed significant differences comprehension answers controls patients latter marginally less accurate control subjects probably early stage pathology indicated mmse values comprehension test ended next trial started presentation fixation spot extra calibration done sentences eye tracker detect eye initial fixation point within sentence corpus sentence corpus composed short sentences line low predictable sentences sentences proverbs maria always laughing good mood worthwhile think talking bird hand worth two bush sentences comprised number content function words similar grammatical structure word sentence lengths sentences ranged minimum words maximum words mean sentence length words low predictability sentences words high predictability sentences words proverbs words ranged letters mean word length sentences proverbs respectively word frequencies used spanish lexical lexesp corpus assigning frequency word sentence corpus word frequencies ranged per million transformed requency mean requency low predictability sentences high predictability sentences proverbs word predictability measured independent experiment researchers electrical engineering computer science department universidad nacional del sur used incremental cloze task procedure participants guess next word given prior words sentence participants guessed first word unknown sentence entered via keyboard return computer presented first word original sentence screen responding participants entered guess second word period indicated end sentence correct words stayed screen participants years old participate reading experiment academic background reading experiment group cloze task group similar word predictabilities ranged mean average predictability measured cloze task transformed using logit function pred predictabilities zero replaced among five perfectly predicted words represents number complete predictability protocols mean logit predictability low predictability sentences high predictability sentences proverbs languages find strong correlations spanish word length word frequency word predictability long words low frequency low high predictability sentences respectively frequent words highly predictable low high predictability sentences respectively highly predictable words tend short words low high predictability sentences respectively used information information used work compaction original data keep descriptors mean reading behavior subjects read sentence measured saccade amplitude fixation duration duration fixation single word subject reading sentence kept mean standard deviation addition measured total number fixations classified first pass fixations refixations unique fixations total fixations first pass fixations first fixation specific word sentence unique fixations fixations occur word skipped first pass multiple fixations multiple fixations word first pass refixations fixations take place word already first pass fixation unique fixation implying regression categorical data diagnosis used training labels replaced numerical values order unify data types improve classification process integer two possible values used diagnosis information construction control identification diagnosis information subject kept apart data detail variables used input model construction shown table since tag control associated patient sentence since use classification approach subject tag applied sentences read following approach may introduce noise classifying stage use classification approach control subject could example distracted reading specific sentence thus making read person anyway system able detect ignore artifacts many samples used training stage table used variables model construction name gaze gaze ntf ntm dfp dfp fpp nfu dfu dfu description number words sentence global sentence mean sum fixation durations word standard deviation gaze mean saccade amplitude sentence standard deviation count total number fixations sentence count number multifixations sentence mean duration first pass fixations sentence standard deviation dfp count number first pass fixations sentence count refixations sentence count unique fixations sentence mean duration unique fixations sentence standard deviation dfu data previously establishing dropout policy order use cleaner dataset outlier check policy consisted finding mean condition group checking two groups separately trials standard deviation bigger two times standard deviation group considered outliers dropped resulting samples lost sentence identication order type kept separate training information standard deviations information patients proverbs high predictability sentences appear particularly high data causing highly unbalanced dataset resulting dataset consisted trials mean trials per subject dataset splitted two groups one training network data subjects testing data subjects randomly picked finally training dataset consisted trials subjects control mean trials testing dataset consisted trials subjects control mean trials splitting data way ensures network infer condition way avoids ensures testing data totally unknown network deep learning denoising work used codification stage work regular autoencoders neural networks supervised learning targets set equal input identity case average number activations per neuron restriction applied hidden layer penalizing average number activations different desired known sparsity proportion adding penalty term cost function restriction introduced neuron specializes particular feature lower sparsity proportion specific feature resulting trained neural network thought encoder involving input hidden layer decoder involving hidden output layer case set activation restriction equal idea force hidden layer discover robust features prevent simply learning identity training autoencoder reconstruct input corrupt version altered version input generated introducing noise obtained clamping fields zero corrupt data used input clean unaltered data target using type data corruption mechanism forces network learn way reconstructing field based others combined sparsity restriction results robust features neural network built using two stages denoising stage train autoencoder corrupting encoded clean data obtained previous stage providing next input end two stages set softmax layer classifier training uncorrupted data corresponding tag used classification approach patient diagnosis extended sentences read classifier trained data target softmax layer multiclass generalization binary logistic regression output probability class quoted word probability shape depends regularization used training stage diffuse peaky results several configurations generated varying sparsity proportion number units layers shape network decreasing number units layers adopted one produced best results consisted two layers denoising hidden units hidden layer using sparsity proportion training network series tests performed data included training dataset dataset mentioned composed sentences subjects control mean trials used softmax layer classification trained using condition translation control subjects subjects means single class since output classifier real number read sentences classified network values close small probability read patient high probability read control patient ground truth values known split output groups observe number sentences misclassified network based show output network values considered classified control higher values considered classified see figure seen output network consistent expected values figure classification results histogram representing number sentences split ground truth values values considered classified control higher values considered classified figure classification results values considered classified control class higher values considered classified class round values plot confusion matrix approximate number misclassified sentences order measure performance network figure see confusion matrix output axis represents expected output values axis rounded output network seen overall performance network good giving sentences performance network using sentences read control patients correctness approximates performance observed using sentences read patients correctness figure number misclassified sentences type split ground truth label hand misclassified sentences concentrated particular type sentence seen figure see original concentration sentences types testing dataset correctness classification following mentioned method figure parallel coordinates plot two subsets one composed patients control subjects trials similar values input field codification different stages network expected similar values encoded together control subjects encoded closer subjects may due high within group variability group result added fact neither concentration misclassication particular sentence may suggest misclassications occurred due presumably stochastic processes addition trained networks evaluated using spread result test order determine softness model tests checked similar information encoded similar way subsequent stages network significant differentiation later stages encoding may show network different stages two subsets one composed patients control subjects trials similar values input field shown figure shown similar input values map similar encoded values stage autoencoders modeled function smooth furthermore data subsequent stages codification tends group results shown output information encoding different stages reliable hand show certain neurons later stages tend specialize detection specific control input features conclussions future work results showed using architecture identifying characteristic eye movements patterns neurodegenerative diseases like alzheimer disease good approach since technology focused pattern finding suitable work moreover high performance classification approach leads conclude since single patient reads many sentences assertion rate per patient higher accuracy reported work using policy network outputs higher classified control tag patient using majority voting read sentences network gets classification accuracy testing set well classified subjects total expected since test set total number missclassified sentences every patient dataset cleaning mean sentences table comparison mean value given network severity disease score given psychiatrists idpat mean score mean difference additionally asked head psychiatrists patients included training process elaborate score overall severity disease patient traditional tests using scale without knowing results given network process creating score required physicians deep knowledge psychiatric history patient recompilation results every neuropsychological tests made patient comprehension table shows scores given psychiatrists compared mean value obtained network sentences read subjects standard deviation seen patients values obtained similar scores given psychiatrists mean value obtained results show created marker reasonably close score involves much simpler process finding better way interpret output classifier left future work improvement required values near determined identified control equal probabilities policy chosen work first rough approach reflect actual power network using encoder obtain overall diagnosis patient might lead accurate result determining whether number given classier related severity disease left future improvement task particularly difficult since ground truth measurements corroborate information due current psychological testing methods anyway although adopted classification approach easy think overall diagnosis may related measurement extracted entire test single trial shown even policy used work simply using mean scores controlling label network behaved expected references deng dong deep learning methods applications foundations trends signal processing gerardo pablo mandolesi nora rotstein oscar colombo osvaldo agamennoni luis politi eye movement alterations reading patients early alzheimer disease eye movement behavior alzheimer disease patients investigative ophthalmology visual science gerardo jochen laubrock pablo mandolesi oscar colombo osvaldo agamennoni registering eye movements reading alzheimer disease difficulties predicting upcoming words journal clinical experimental neuropsychology gerardo diego shalom reinhold kliegl mariano sigman eye movements reading proverbs regular sentences incoming word predictability effect language cognition neuroscience gerardo liliana castro marcela schumacher osvaldo agamennoni diagnosis mild alzheimer disease analysis eye movements reading journal integrative neuroscience gerardo marcela schumacher liliana castro david orozco osvaldo agamennoni patients mild alzheimer disease produced shorter outgoing saccades reading sentences psychiatry research gerardo salvador guinjoan marcelo sapognikoff david orozco osvaldo agamennoni contextual predictability enhances reading performance patients schizophrenia psychiatry research gerardo facundo manes luis politi david orozco marcela schumacher liliana castro osvaldo agamennoni nora rotstein patients mild alzheimer disease fail using working memory evidence eye tracking technique journal alzheimer disease gerardo marcelo sapognikoff salvador guinjoan david orozco osvaldo agamennoni word processing reading sentences patients schizophrenia evidences eyetracking technique comprehensive psychiatry marshal folstein susan folstein paul mchugh state practical method grading cognitive state patients clinician journal psychiatric research philip holzman leonard proctor deborah levy nicholas yasillo herbert meltzer stephen hurt dysfunctions schizophrenic patients relatives archives general psychiatry william iacono margaret moreau morton beiser jonathan fleming lin smoothpursuit eye tracking psychotic patients relatives journal abnormal psychology jennifer kellough christopher beevers alissa ellis tony wells time course selective attention clinically depressed young adults eye tracking study behaviour research therapy reinhold kliegl antje nuthmann ralf engbert tracking mind reading influence past present future words fixation durations journal experimental psychology general rongjian wenlu zhang suk wang jiang dinggang shen shuiwang deep learning based imaging data completion improved brain disease diagnosis medical image computing pages springer siqi liu sidong liu weidong cai sonia pujol ron kikinis dagan feng early diagnosis alzheimer disease deep learning biomedical imaging isbi ieee international symposium pages ieee guy mckhann david drachman marshall folstein robert katzman donald price emanuel stadlan clinical diagnosis alzheimer disease report work group auspices department health human services task force alzheimer disease neurology eneida mioshi kate dawson joanna mitchell robert arnold john hodges addenbrooke cognitive examination revised brief cognitive test battery dementia screening international journal geriatric psychiatry keith rayner eye movements reading information processing years research psychological bulletin deborah riby peter hancock looking movies cartoons evidence williams syndrome autism journal intellectual disability research schmidhuber deep learning neural networks overview neural networks lexesp informatizado del edicions universitat barcelona suk dinggang shen deep feature representation classification medical image computing pages springer suk lee dinggang shen alzheimer disease neuroimaging initiative hierarchical feature representation multimodal fusion deep learning diagnosis neuroimage waldemar dubois emre georges mckeith rossor scheltens tariska winblad recommendations diagnosis management alzheimer disease disorders associated dementia efns guideline european journal neurology
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mutual information ldgm codes jul jan van den brand jaafari bstract provide matching upper lower bounds mutual information noisy reconstruction parity check codes thereby prove conjecture montanari ieee transactions information theory besides extending prior concentration result abbe montanari theory computing case odd check degrees precisely determine conjectured formula code ensembles arbitrary degree distribution thus capturing broad class capacity approaching codes ntroduction sparse random binary matrices provide natural way encoding messages without exhausting transmission rate let number larger blocklength message choosing random generator matrix field obtain codeword simple matrix multiplication given noisy observation obtained binary memoryless symmetric bms channel likely recover solving system linear equations provided sufficiently large matrix imposes adequate redundancy averaged bits properly structured sparse random matrices induce class low density generator matrix ldgm codes ldgm codes known many decades although describe lines date status research hardly advanced partly due fact simplifying encoding decoding process sparsification also severely exacerbates analysis unsurprising remarkable know least code constructions codes perform remarkably well paper aims break first ground proving precise formula mutual information ldgm codes previously conjectured montanari optimal codes graphs may expected structured sparse code ensembles readily constructed bipartite graphs also known tanner graphs factor graphs consist variable nodes representing bits signal side check nodes factor nodes representing parity check equations side general context factor graphs used model constraint satisfaction problems csps factor nodes impose constraints participating variable nodes ldgm codes variable node participates least one parity equation via neighboring check nodes check node constrained adjacent bits codewords need satisfy intersection codewords satisfying parity check equations modeled graph form linear code crux local interactions constructed simple fashion global structure code perform complex interplay ensure efficient coding decoding importantly placement number check nodes chosen sophisticated way keep encoding decoding complexity low maintaining sufficient amount redundancy nutshell standard code ensembles achieve performance imposing specific degree distribution nodes factor graph example constrain check nodes perform parity check constant number variable nodes degree distribution mutual information ensembles rigorously studied previous work pioneering paper montanari derives upper bound mutual information even conjectures bound tight subsequent work abbe montanari able prove existence limit mutual information scenario result comprehensively determines limit mutual information ensembles even odd work submitted ieee possible publication copyright may transferred without notice version may longer accessible mutual information last decade critical endeavor analyze standard code ensembles respect decodeability mutual information captures much information output channel contains input channel thus provides essential measure quantify limits decodeability artlessly entails bounds error probabilities mutual information proves key many related areas decoding noisy signals importantly analyses random linear codes setting bms channels noise bit independently correctly transmitted probability flipped probability computing exact mutual information configuration highly task despite substantial amount research one usually merely able provide bounds occasionally tailor formulas individual scenarios general case analysis ldgm codes given variable degrees montanari derives upper bound mutual information subject condition check degrees satisfy convexity assumption conjectures bound sharp matches explicit formulas auspicious calculations theory statistical mechanics paper building upon indispensable groundwork montanari establish cavity computation standard code ensembles derive matching lower bound prove conjectured bound tight furthermore introduce new technique drop assumptions variable degree distribution extend results comprehensive class standard code ensembles results following theorem proves montanaris conjecture particularly first result precisely determine mutual information random ldgm codes given variable degrees without imposing restrictions degree distribution magnitude noise predicted formula comes form stochastic equation state theorem denote set probability distributions mean zero let fix degree distribution number let independently identically distributed samples chosen write outcome random bernoulli experiment parameter let sequence independent copies let theorem let random blocklength variable degree distribution check degree let message chosen uniformly random message obtained passing memoryless binary symmetric channel error probability lim sup argument develop prove mutual information standard graph ensembles reasonably general expect extend ldpc code ensembles similar problems conditional random fields possibly approach may facilitate simpler proofs analyses spatially coupled codes important mention spatial coupling invented engineer codes technique applied beyond context coding theory section therefore actually prove general version main theorem include broader class factor graph ensembles encompassing many problems related conditional random fields theorem deals natural application problem noisy reconstruction standard ldgm code ensembles background related work following shannon work early codes based algebraic constructions analyzed realize codes saturate capacity limit far margin progress stalled subsequent decades early introduction turbo codes reignited interest turbo codes able deliver performance close shannon limit ensuing generalizations uncovered power codes based graphs low density parity check ldpc codes originally put forward robert gallagher phd thesis rediscovered neglected quite time ldpc codes reemerged broad audience series papers luby mitzenmacher shokrollahi spielman steman followed work richardson shokrollahi urbanke proved gallagher parity check codes perform rates close shannon capacity designed efficient decoding possible crucially constructing codes approach capacity one keep mind performance decoder recently parity check codes become widely used successfully implemented context satellite communication wifi transmission data protocols time significant progress made regard design optimal codes achieve capacity also allow efficient coding decoding usually achieved prescribing degree distribution variable nodes driven success spatial coupling analysis standard code ensembles gathered tremendous momentum nonetheless construction analysis spatially coupled codes particular notoriously complicated analysis linear codes sourlas work dating back early provided crucial link physics theory spin systems would betoken path numerous fruitful results many methods therefore base developments rigorous theory mean field spin glasses far promising analyses standard ldgm code ensembles utilize guerra toninelli interpolation method provide general bounds inference threshold graphical models subject condition generating function left degree distribution convex abbe montanari show entropy transmitted message conditional received one concentrates around well defined deterministic limit previous work montanari interpolation method employed lower bound entropy asymptotic expression derived obtained using heuristic statistical mechanics calculations recent paper krkazala perkins entropy derived type ldgm graph ensembles however merely fragmentarily applicable codes corresponding adjacency matrices unstructured may exhibit empty columns roof utline section outline pillars proof derive mutual information noisy observations broad class random factor graph models make slightly general assumptions needed codes result includes number problems related conditional random fields model viewed natural generalization retrieval problem ldgm codes instead consider arbitrary finite set possible bit values generalize parity check constraints finite set positive weight functions fixed basis probability space prior distribution weight functions write random choice specify unweighted factor graphs bipartition variable nodes check nodes well neighborhood structure weighted factor graph additionally carries weight functions factor node locally evaluate signals assignments variable nodes graph set variable nodes size write omit index apparent context commonly integer write identify set check node denote neighborhood vector neighbors ascending order exert supplementary methods results throughout paper require satisfy two assumptions sym pos one hand easily verified class ldgm codes number related applications hand directly imply assumptions end let denote set probability distributions let denote set probability measures whose mean corresponds uniform distribution sym let pos following true chosen chosen chosen mutually independent linear codes satisfy sym pos see section general story goes follows teacher chooses ground truth uniformly random finds unable directly convey students uses model may set random variable nodes check nodes parity checks chosen proportionally local evaluation students get see random graph ground truth stand task deciphering much information possible thus limit amount deductable information quantified mutual information theorem chosen chosen chosen chosen uniformly random mutually independent let let graph pair model lim sup mutual information model previously derived type ldgm graph ensembles include standard ensembles delicate extension techniques modified poisson approximation inspired montanaris ensembles technical proofs involved carried arbitrarily precise approximations random bridged original model way coupling following proposition establishes precision approximative model proposition let bounded function coupling add missing check nodes expected number check nodes different neighborhood words obtain random first generating approximation rewiring small number edges prove proposition section coupling exact instance chosen configuration model track incremental influence small perturbations construction process prove approximation error negligible coupling similar procedure yet introducing additional approximation parameter achieve error asymptotically independent outline proof theorem suppose jointly distributed tuple model underlying factor graph random mutual information expressed marginal entropies write easily calculated derivation requires sophisticated approach making use several methods physicist toolbox sparse apart inherent properties model prove make use called nishimori property physics property allows identify reweighed posteriori measure induced graph commonly named gibbs measure samples ground truth results imply given nishimori property perform slight perturbations without significant impact mutual information allow factorize marginals novel pinning technique crucial new ingredient facilitates proof montanaris conjecture therewith layed foundation derive upper bound conditional entropy lower bound mutual information means scheme proof carried section essence procedure boils estimating change entropy model variable nodes one variable nodes indicated figure denote change conditional entropy going representation clearly implies converges limit also limit unfortunately able compute limit directly settle using representation obtain upper bound entropy limit lim sup lim sup igure change entropy induced introducing another variable random factor graph model achieved generating graphs random graph process rigorously coupling underlying models result explicit calculation carried section upper bounded conjectured formula translates following lower bound mutual information proposition lim inf sup section using interpolation method derive matching lower bound conditional entropy interpolate original model much simpler graph model gibbs measure easily understood effortlessly verify entropy coincides conjectured formula random graph approximation assembled sequence layers depending approximation parameter layer contributes many check nodes refine interpolation argument segmentation layers portrayed figure igure interpolation scheme layer originally consisting factor nodes dashed neighborhoods one two factor nodes layer split unary nodes beginning surface layer split factor nodes one time replacing unary nodes weight functions simulating complex underlying structure thus peel apart intricate composition forest unary variable components independently splitting factor node layer probability sequentially layer ensure continuous interpolation ultimately aim show entropy latter model indeed upper bound original conditional entropy amounts controlling within interpolation interval involves yet another clever coupling argument carried full detail section taken whole derivative respect positive entire interpolation obtain desired lower bound conditional entropy following upper bound mutual information proposition lim sup sup together propositions immediately imply assertion planted csp ensemble given variable degree distributions theorem follows explicitly calculating quantities case ldgm codes verifying sym pos hold simple technical computations put section eacher tudent model ymmetry section set tools utilized proof formalize notion retrieval constructively defining model ensembles well poissonian approximation end verify number expedient properties brought model setting analyze approximative random graph model terms free energy quantity closely related mutual information main object interest many physics models moreover derive tight bounds expressions obtain free energy employ pinning lemma finally show bounds free energy imply general main result following section lower upper bound derived sections respectively preliminaries notation factorqgraph associate partition function gives rise gibbs distribution functional write referring average regard gibbs distribution additionally define empirical distribution graph average gibbs marginal probability measure write denote product measure moreover write random sample standard graph ensembles follow predefined distribution considering degree sequences chosen distributions expedient tool construction random graphs given ensemble call degree sequence finite supx explicitly stated otherwise assume number variable nodes given graph degree sequences finite maximal degree real number let max formally specify random construction process let define process samples graphs specific sequences commonly known configuration model definition let degree sequence denote random unweighted factor graph obtained following process let initiate vector setting choose random measure defined update set multiple add new check node set neighborhood resulting graph satisfies degree sequence long divides also write number variable nodes apparent context definition allows partition random unweighted factor graph obtained choosing partition uniformly random setting write random choice times beneficial generate random graph sequence batches factor nodes according specified structure rather one time facilitated means following model use lay poissonian approximation definition given degree sequence well vector msmax integer entries define random graph obtained following experiment sake clarity omit subscript unambiguous let initiate vector setting layers smax choose random neighborhoods add factor nodes respective neighborhoods graph denote number times variable node chosen neighbor round update given degree distribution say degree sequence random uniformly random variable degrees satisfy definition given degree distribution vector msmax integer entries let random thereby define approximation scheme obtain approximation introduce two additional parameters aid laying appropriate vector definition degree sequence fixed let smax let sequence independent distributed numbers obtain setting msmax letting analogously define first choose random generate thereafter note procedure stated definition fail smax however succeed tends infinity probability adding factor nodes construction tends towards zero particularly tend becomes arbitrarily close approximation sense obtain rewiring node sockets enabled following observation fact fixed smax let degree sequence definitions let bounded integer valued function variable nodes let consider distributions proof simply write total variation distance bound smax obtain light lemma consider generating approximative graph independently generate missing factor nodes fashion exact procedure definition verify distribution resulting factor graph matches distribution rewire finite number factor nodes importantly number asymptotically independent proof proposition sufficiently small layers contain one check node contain constant number check nodes sake simplicity suppose layer consists one check node argument extends constant number check nodes first observe creation step layer approximative graph matches original graph except factor nodes different entries degree sequences fact degree sequence exact model updates immediately approximative degree sequences adds incremental deviations creation one check node results total variation distance choice next neighbor couple configuration model process graphs inductively suppose process optimal coupling graphs differ exactly check nodes coupling lemma graphs coincide probability thus choose neighborhood next check nodes hence process complete geometrically distributed random variable parameter total number different check nodes minimum number independent random variables tends chosen geometric distribution infinity random variables arbitrarily well approximated random variables chosen exponential distribution parameter consequence modeled function asymptotically independent assumed layer smax consists one check node argument generalizes sequence layers independent many check nodes tails poisson distribution show decay guarantee layer coupling differs check nodes induction write function model derive mutual information transmitted message received one analyze free energy density within model end generalize scheme arbitrary degree distributions define random factor graph via following construction choose uniformly random choose random graph check node choose random distribution let denote resulting weighted random factor graph omit parameters sake readability write limit pinning symmetry another indispensable tool make use pinning derive bounds free energy confronted drawing chosen gibbs measure due confinement weight functions certain configurations tuples favored within gibbs measure hence cases variable node assignments sampled gibbs measure nodes connected common check node highly correlated means weight function thus far independent correlation ought persist nodes within finite distance graph hand choose two variable nodes random typically far apart hope gibbs measure almost product measure captured notion generally definition let probability measure let denote marginal distribution chosen say simply speak fortunately following pinning lemma guarantees degree exchange small modifications original measure lemma every every probability measure following true obtain random probability measure follows draw sample independently choose number uniformly random obtain random set including probability independently measure defined probability least applying procedure lemma gibbs measure graph perturbation equates pinning color variable node coloring adding constraint variable nodes definition given factor graph subset variable nodes assignment obtain adding unary check nodes weights variable node moreover let random subset vertices generated first choosing uniformly random including variable node probability chosen independently uniformly random variable node let following lemma establishes perturbations performed pinning procedure yield measure probability least lemma lemmata let gibbs measure sufficiently large meausre probability least moreover also note pinning procedure merely constant additive impact free energy factor graph model weight functions check nodes positive variable degrees case pinning single variable node changes free energy graph expected number pinned variable nodes following lemma lemma random factor graph lemma together nishimori property key tools utilized proof nishimori property stands testament fact prior satisfies certain symmetry condition gibbs measure teacher student model indeed resemble posteriori distribution planted assignment given graph outcome use argument way done proofs nevertheless symmetry condition imposes much weaker restriction prior direct implication sym condition merely yields mutual contiguity measures class models covers codes transmitted binary memoryless channels obtain following stronger formulation lemma suppose satisfies sym particular distribution coincides gibbs measure proof random graph model event positive probability write conditional random graph factor nodes observe choice event see suffices write weights event get left verify summing graphs factor nodes gives established let derive write distribution distribution moreover consequently write sample gibbs measure joint distribution satisfies multiplying denominator numerator applying sym final step following lemma shows applying pinning process order preserves nishimori property lemma set vertices following distributions pairs factor graphs assignments identical choose choose output choose choose output choose choose output choose pin choose output proof lemma pairs well identical see coincide suffices prove holds event arbitrary set unpinned factor graphs pinning variables denote sample gibbs measure write restriction pinning neighborhood evaluation model remains thus equals proof main result prove conjectured formula showing mutual information per bit converges solution stochastic equation state result general setting write convergence terms free energy proposition sym pos hold lim prove proposition section performing interpolation forest isolated variable nodes taking account details come along understood simply splitting factor node unary factors probability easy compute show resulting expression also upper bound verifying positive proposition sym holds lim section prove lower bound proposition via previously described scheme bound difference free energy coupling joint distribution large common subgraph compute expected change free energy given additional constraints going graphs possible due lemma constraints generated empirical gibbs marginals significantly differ writing mutual information propositions translate propositions thus revealing main theorem proof theorem theorem corollary propositions writing mutual information lemma get chosen coincides given obtain chosen uniform distribution assertion follows theorem readily implies theorem verifying sym pos case code proof theorem let uniform distribution clearly sym holds see pos write uniform choice writing independent samples independent samples every obtain setting writing logarithmic series expansion clearly sufficient show xlk ylk kxl case odd immediate even thus xlk ylk kxl lower bound section perform technical computations scheme remainder section let arbitrary fixed let random graph obtained experiment tch vertices let chosen tch vertices moreover let signify respective graphs obtained performing pinning procedure definition establishing following proposition way generalization argument pave way proving lower bound proposition let lim sup lim sup sup let write limt lim proposition immediately implies sup taking lim sup subsequently obtain proposition coupling prove proposition construct coupling two graphs play first sampling maximal common subgraph obtain graphs mirror respectively let chosen uniformly choose random set subdomain denote random sample distribution let ned min smax moreover let smax independently chosen variables choose uniformly obtain weighted factor graph performing following setting beginning empty graph consisting variable nodes add factor node neighborhood chosen weight chosen set counts number times drawn neighbor round increase abort smax pwhen smax check nodes obtain graph consists probability independently pin variable node vertex attach unary constraint node identify random graph resulting pair former experiment perform another rounds smax smax smax add new weighted factor nodes updating increment independently pin yet pinned probability identify resulting graph smax first experiment extend choosing independently uniformly random independently experiment create graph adding factor nodes follows let smax add factor node neighborhoods weights xik xik xij set denotes number times drawn neighbor round halt smax consequently set reset add factor node neighborhoods weights xik xik xij set denotes number times drawn neighbor round halt finally pin probability identify resulting graph note lemma replace first drawing neighborhoods respectively subsequently adding weights proportional evaluation lemma sufficiently large proof clearly large enough min ned choice neighborhoods independent pinning process hence might well switch order perform experiments add many factor nodes distribution neighborhoods chosen independently everything planted coloring neighborhood consequently suffices compare processes generate random neighborhoods number rounds perform model tch vertices perfectly couple occurences round moreover process performing variable gets pinned independently probability therefore procedure results graph remains prove second distributional equality event see fashion procedure yields suppose even though first consider last step choice ensure degree expectation outcome poisson random variables chosen vertices neighborhood distribution finally due variable node independently pinned probability therefore distributed lemma characterization side approachable calculating contributions added partition function going either assure mediates assertion establish following fact lemma proof fixed chosen uniformly denotes random set pinned variable chosen probability sets coupled coincide probability moreover whereas variance least hence factor nodes optimally coupled neighborhoods apart total variation distance proceed calculate want capture coupling typically claim let event given process pin additional vertices let set vertices belonging neighborhood chosen let denote number factor nodes added given sufficiently large given event occurs probability least proof validate first part observe graph variable node independently pinned probability thus equation immediate weights added process strictly positive neighborhood chosen product measure proportional slight perturbation smax neighbors chosen set size expected number new factor nodes whence moreover first part lemma sufficiently large graph probability least combining second part lemma find given event occurs probability least long sufficiently large lemma let empirical distribution gibbs marginals probability least choice proof event graph obtained simply adding check nodes neighborhoods weights chosen respectively factor nodes independently chosen distribution claim expression simplifies hence chosen independently uniformly random using distribution writing empirical distribution obtain claim let event given process yields factor nodes pin sufficiently small let independently uniformly chosen indices uniformly random choices neighborhoods subject condition let set neighbors bsb chosen without let sample mutually contiguous given event occurs probability least proof probability pinned least finite support sufficiently small thus first assertion follows given probability drawing neighbor upper bounded therefore size asymptotically almost surely moreover bounded small constant mutually contiguity sufficiently large may apply lemma obtain sequence occurs probability least lemma let empirical distribution gibbs marginals probability least choice proof proof previous lemma given occurs claim implies obtained simply adding weighted check nodes therefore hence natural extension uniformly chosen let chosen together statement claim equation becomes writing probabilities independently chosen check nodes sym get finally prove proposition going make use following fact immediate fact corollary moreover note performing continuous transformation functional utilized preserve following property fact lemma functional weakly continuous proof proposition lemmata established bridge gap carve arbitrarily close set fact together yields assertion taking limits specified order uniform distribution fact gives therefore using lemma obtain whence expectation measure converges uniform distribution total variation distance measure convex combination closes gap upper bound section carry calculations interpolation given set family graphs interpolate original graph graph free energy proving derivative positive entire interval obtain throughout section assume degree distribution arbitrary fixed interpolation interpolation utilize fact graph consists smax layers poissonian degree case smax define interpolation follows codeword interpolation model consists layers parity checks followed layer probability parity check replaced repetition codebits contains final smax layers simple blocks repetition code altogether satisfying degree distribution make precise define random factor graph model follows draw random let chosen chosen let sequence random variables mutually independent define vectors letting smax add factor node neighborhood add unary factor neighbors chosen set denotes number times drawn neighbor round increase abort smax factor node graph independently assign weight function chosen unary factor node graph independently assign unary weight function follows choose indenpendently choose uniform distribution choose iid let map established interpolation define original interpolation model reweighing signal define distribution letting signal chosen uniformly random write notice smax yields original graph model corresponds graph simple repetition code interpolation split interpolation interval smax intervals equal length finally ensure symmetry apply pinning procedure purpose fix choose uniform distribution choose random graph choose uniformly random let random connect unary check node weight function let write resulting graph note lemma depends gibbs measure probability least thus fix sufficiently large performing interpolation guarantee throughout process family graphs remains tally total factor nodes partition function interpolation accounted correction term end independent samples let following lower bound main ingredient interpolation argument tethers derivative arbitrarily closely zero proposition let smax prove proposition section let first see proposition implies proposition proof proposition fundamental theorem calculus write smax max max sequence entries independently chosen entries independently chosen independently uniform choices random sample simple calculation unfolds also definition smax plugging consequently taking lim inf lim inf lim inf weights strictly positive hence proposition allows compare free energy exact model approximative free energy thus extends exact model ultimately taking approximation limit lim sup sup lim sup lim sup sup proof proposition prove proposition derive practical expression derivative comparable expression pos proposition let chosen experiment let chosen chosen mutually independent smax let chosen definition set chosen uniformly set variable nodes let uniformly smax prove proposition begin rewriting distribution parameters satisfies similar expression note poisson hence independent add new neighbors layer put together differentiating respect yields term cancels write consider graph model slightly alter procedure follows construct graph described instead increasing moving onto add another unary check nodes neighborhoods chosen update accordingly equip new check nodes constant weight functions afterwards increase continue short differs additional unary check nodes within layer neutral weight letting write nishimori property naturally extends case interpolation model choosing neutral weights includes lemma lemma remains true replace either proof analogous lemma lemma let chosen experiment let check node chosen chosen moreover let unary check node chosen assign weight function defined index weight function chosen proof begin showing note differs additional unary check nodes neutral weight assigned layer course sockets chosen neighbors additional check nodes induce perturbation distributions however coupled neighborhood assigned additional check node remaining choices neighborhoods weight functions chosen distribution within graphs given coupling taking logarithm integrating immediately gives show couple similar fashion layer neighborhood chosen equally assigned additional unary check nodes result positive entry unary nodes independently assigned weight functions chosen couple remaining random variables trivially copying choice joint distribution write taking logarithm integrating gives initial gibbs measure necessarily factorize marginals lemma guarantees pinned measure fact probability random set contiguous respect uniformly random choice implies thus proves assertion verify proposition write simple technical computation claim assumptions proposition definition gives proof begin showing interpolation model check node chooses neighborhood obtains weight function weight functions take values writing independent samples expanding logarithm obtain lemma pairs identically distributed thus write moreover sym gives thus simplifies lemma implies calculate recall layer unary check node added chosing neighbor equipping weight function chosen write check node chosen way conditioned event sym normalization equation get writing utilize fact weight functions take values within expand logarithm write simplify expression using distributions coincide obtain employing sym final equation finally let derive definition write independent samples previous cases perform procedure expanding logarithm simplifying telescopic sum obtain proof proposition assertion immediate claim yield complete proof proposition comparing expressions value given pos end let sequence independently samples let independently chosen define let chosen lemma pairs identically distributed thus choice mean empirical marginal distribution given uniform distribution pos holds expanding shows expression proof proposition consideration assertion follows proposition verifying choice first third summand use pinning lemma sufficiently regularize underlying graph models exchangeability corresponding term possible time consider first suammnd observe independently uniform choices among variable nodes write hence triangle inequality inequality yield rewrite ratio andp vector expression maximized sparse vectors vector contains maxv entries hence bound factor moreover evaluates lemma implies way bound lemma guarantee probability implies probability acknowledgement thank amin helpful discussions eferences abbe community detection stochastic block models recent developments arxiv preprint abbe montanari conditional random fields planted constraint satisfaction entropy concentration theory computing achlioptas jia moore hiding satisfying assignments two better one journal artificial intelligence research achlioptas hassani macris urbanke bounds random constraint satisfaction problems via spatial coupling proceedings annual symposium discrete algorithms soda aizenman sims starr extended variational principle model physical review banks 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thresholds cavity method proceedings annual acm sigact symposium theory computing stoc feldman perkins vempala complexity random satisfiability problems planted solutions proceedings annual acm symposium theory computing stoc franz leone montanari dynamic phase transition decoding algorithms physical review clauset ghasemian moore peel zhang detectability thresholds optimal algorithms community structure dynamic networks physical review giurgiu macris urbanke spatial coupling proof technique three applications ieee transactions information theory guerra toninelli thermodynamic limit mean field spin glasses communications mathematical physics hassani macris urbanke threshold saturation spatially coupled constraint satisfaction problems journal statistical physics jia moore strain generating hard satisfiable formulas hiding solutions deceptively journal artifical intelligence research kabashima krzakala sakata phase transitions sample complexity matrix factorization ieee transactions information theory kudekar macris proof replica formulas high noise regime communication using ldgm codes information theory workshop kudekar macris sharp bounds optimal decoding codes ieee transactions information theory kumar young macris pfister threshold saturation spatially coupled ldpc ldgm codes bms channels ieee transactions information theory lelarge miolane fundamental limits symmetric matrix estimation proceedings conference learning theory pmlr lelarge reconstruction labeled stochastic block model ieee transactions network science engineering luby mitzenmacher shokrollahi spielman analysis low density codes improved designs using irregular graphs proceedings annual acm symposium theory computing stoc luby mitzenmacher shokrollahi spielman efficient erasure correcting codes ieee transactions information theory luby mitzenmacher shokrollahi spielman improved codes using irregular graphs ieee transactions information theory luby mitzenmacher shokrollahi spielman stemann practical codes proceedings annual acm symposium theory computing stoc manoel krzakala generalized linear estimation arxiv preprint community detection thresholds weak ramanujan property proceedings annual acm symposium theory computing stoc montanari tight bounds ldpc ldgm codes map decoding ieee transactions information theory moore computer science physics community detection landscapes phase transitions hardness arxiv preprint richardson shokrollahi urbanke design irregular codes ieee transactions information theory richardson urbanke capacity parity check codes decoding ieee transactions information theory richardson urbanke modern coding theory cambridge university press shannon mathematical theory communication bell system technical journal sourlas models codes nature sourlas statistical mechanics codes statistical mechanics neural networks lecture notes physics edited garrido springer new york sourlas statistical mechanics codes statistical physics statistical inference back edited grassberger nadal kluwer academic dordrecht jan van den rand janvdb kth royal nstitute echnology tockholm weden jaafari jaafari oethe niversity athematics nstitute rankfurt ermany
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aug ramification valuations local rings positive characteristic steven dale cutkosky introduction paper consider birational properties ramification excellent local rings suppose finite separable field extension excellent local ring quotient field excellent local ring dim dim dominates maximal ideals satisfy valuation dominates valuation ring dominates let restriction notation use paper explained detail section local monomialization definition monomial regular local rings dimension exist regular systems parameters units matrix aij natural numbers nonzero determinant equicharacteristic zero algebraically closed residue fields within extension regular parameters giving form generally ask given extension local monomialization along valuation definition local monomialization commutative diagram vertical arrows products monoidal transforms local rings blowups regular primes monomial proven theorem local monomialization always exists algebraic function fields necessarily algebraically closed field characteristic algebraic local rings respectively algebraic local ring essentially finite type also define weaker notion weak local monomialization requiring conclusions definition hold vertical arrows required birational necessarily factorizable products monoidal transforms partially supported nsf leads following question extensions defined beginning paper exist local monomialization least weak local monomialization extension excellent local rings dominated valuation commented question positive answer within algebraic function fields arbitrary field characteristic zero theorem question positive answer positive characteristic mixed characteristic course necessary form resolution singularities true certainly true equicharacteristic zero known true generally dimension positive characteristic dimension recent papers going beyond dimension three case two dimensional algebraic function fields algebraically closed field positive characteristic considered shown monomialization true defectless extension two dimensional algebraic local rings algebraically closed field characteristic theorem theorem discuss important concept defect later introduction give example showing weak monomialization hence monomialization exist general extensions algebraic local rings dimension field char prove following theorem theorem theorem counterexample local weak local monomialization let field characteristic least elements let exists finite separable extension dimensional function fields valuation restriction algebraic regular local rings respectively dominates dominates exist regular algebraic local rings dominates dominates dominates dominates monomial defect example theorem strong form local monomialization established within characteristic zero algebraic function fields call strong local monomialization theorem theorem form stable appropriate sequences monoidal transforms encodes classical invariants extension valuation rings show strong local monomialization true defectless extensions two dimensional algebraic function fields theorem theorem give example theorem showing strong local monomialization generally true defect extensions two dimensional algebraic function fields field positive characteristic paper establish local monomialization strong local monomialization hold defectless extensions two dimensional excellent local rings first state theorem proven section define discuss defect theorem suppose dimensional excellent local domain quotient field suppose finite separable extension two dimensional excellent local domain quotient field dominates let valuation dominates let restriction suppose defect exists commutative diagram vertical arrows products quadratic transforms along monomial proof theorem actually produces stable strong monomialization define defect extension valuations role concept local uniformization observed kuhlmann good introduction role defect valuation theory given brief survey well suited purposes given section suppose finite galois extension fields characteristic splitting field smallest field property extension defect defined identity corollary theorem section chapter case finite separable define defect extension galois closure defect equal zero residue field characteristic zero theorem section chapter dvr corollary theorem section chapter associated graded rings valuations semigroup respect valuation group generated valuation group well understood semigroup however extremely complicated perverse associated graded ring defined ideal elements value ideal elements value ring plays important role teissier approach resolution singularity completely realized abhyankar valuations arbitrary characteristic always proven exists strong local monomialization defectless extensions two dimensional algebraic local rings two algebraic function field algebraically closed field property induced extension associated graded rings along valuation finite type even toric extension result extended case two dimensional algebraic function fields arbitrary field characteristic zero proofs make use technique generating sequences valuation local ring developed spivakovsky two dimensional regular local rings algebraically closed residue fields extended arbitrary regular local rings dimension two unfortunately technique special dimension two extend well higher dimension local rings even normal local rings dimension two examples strange semigroups show interesting construction generating sequences within valuation ring exhibits defect extension valuations given different general construction generating sequences given general extension finite type even equicharacteristic zero algebraic regular local rings dimension two example blowing reach good stable form required obtain good form difficult show extension finite type toric abhyankar valuation equality holds abhyankar inequality theorem dimq dim special case theorem recalled theorem paper give precise statement stable strongly monomial forms obtained rational rank valuation dimq case extension algebraic local rings extension algebraic function fields arbitrary field characteristic zero case regular parameters regular parameters unit paper give simple proof theorem case extension algebraic local rings extension algebraic function fields arbitrary field characteristic zero rational rank strongly monomial extension property respective classes degree extension quotient fields particular extension associated graded rings along valuation finite toric show theorem paper stable strongly monomial forms found theorem defectless extensions two dimensional excellent local rings dominated valuation rational rank extension associated graded rings along valuation form since stable forms abhyankar valuations finite type toric extension commented conclude stable strongly monomial forms defectless extension two dimensional excellent local rings always finite type toric extension contrast nice stable form extension associated graded rings along valuation positive defect shown example theorem analyzed section paper using notation section explained following subsection invariants stable forms along valuation follows remark graded domains integral finite contrast situation defect zero theorem quotient fields equal extension theorem degree conclusions theorem invariants stable forms along valuation suppose inclusion regular two dimensional algebraic local rings within function fields respectively algebraically closed field positive characteristic finite separable dominates valuation restriction dominates dominates residue field algebraic value group rational rank isomorphic ordered group subgroup shown corollary theorem sequences quadratic transforms along vertical arrow product quadratic transforms constructed algorithm section simplified notation writing rrn ssn algebraic regular local ring algebraic regular local ring dominates quadratic transform factors regular parameters regular parameters stable forms xnap xbnn unit mod natural valuation dvr depend depend theorem example given tower two artin schreier extensions particular shows strong local monomialization fails extension however also shown example local monomialization true extension considering different sequences quadratic transforms corollary theorem shown constant integers defined associated stable forms extension valued two dimensional algebraic function fields shown constant defect extension however shown sum constant defect extension remark asked numbers computed stable forms eventually constant case give examples section equations showing case even within defect artin schreier extensions examples found considering factorization example theorem product two artin schreier extensions computing generating sequences intermediary rings notation preliminaries local algebra rings commutative identity ring essentially finite type local ring finitely generated denote maximal ideal local ring quotient field domain require local ring noetherian suppose inclusion local rings say dominates local ring domain say local ring algebraic function field field assume algebraically closed local ring essentially finite type say algebraic local ring suppose finite field extension local ring local ring say lies localization integral closure local ring denote completion maximal ideal suppose regular local ring monoidal transform local ring form regular prime ideal regular local ring prime ideal called quadratic transform valuation theory suppose valuation field denote valuation ring denote value group good treatments valuation theory chapter contain references original papers valuation ring algebraic function field field insist vanishes say valuation valuation field local ring say dominates valuation ring dominates suppose dominates monoidal transform called monoidal transform along dominates suppose finite separable extension valuation restriction ramification index reduced degree defect defined introduction paper basic properties developed section chapter section call ring dvr valuation ring value group galois theory local rings suppose finite galois extension local ring local ring lies splitting group splitting field inertia group defined basic properties developed section galois theory valuations galois theory valuation rings developed section chapter section basic results need surveyed section take valuation restriction obtain splitting group splitting field inertia group section chapter written called decomposition group written ramification group defined section chapter surveyed section denote group semigroups associated graded rings local ring respect valuation suppose valuation field dominates local ring denote semigroup values suppose define ideals define associated graded ring respect birational geometry two dimensional regular local rings recall basic theorems make frequent use theorem theorem suppose field regular local ring dimension two suppose another dimensional regular local ring dominates exists unique sequence quadratic transforms factor lemma lemma suppose two dimensional regular local ring field valuation dominates let infinite sequence quadratic transforms along also make use fact embedded resolution singularities true within regular local ring dimension theorem fact resolution singularities true two dimensional excellent local rings local monomialization two dimensional defectless extensions section prove theorem establishing local monomialization defectless extensions two dimensional excellent local domains extends result extensions two dimensional algebraic local rings two dimensional algebraic function fields algebraically closed field theorem indicate differences proof analogue theorem algebraic local rings algebraically closed field proof theorem paper essential case rational rank valuations theorem essential ingredient proof computation complexity proposition generalizes proposition steps proof individual calculations require different methods coefficient fields general situation paper completions local rings longer extensions power series rings field analogue theorem deduced consequence detailed analysis stable forms theorem makes essential use assumption residue field extension ground field algebraically closed paper give different direct argument deduce theorem proof theorem actually produces stable strong monomialization first establish strong monomialization two essential cases theorem give proof theorem end section make use list good properties excellent rings given scholie degree formulas subsection generalize formulas proposition proposition suppose two dimensional regular excellent local rings dominates finite separable extension regular system parameters regular system parameters expression unit unit exist inclusions local rings following properties two dimensional normal local ring essentially finite type lies let natural valuation dvr mod proof first suppose let gcd ideal integral closure let normal since integrally lclosed powers integrally closed theorem appendix ring integrally closed local ring normal scheme proj elements lemma exists rank valuation dominates let integral closure let ams theorem finite finite normal dimension two since excellent normal since version zariski main theorem thus lies let dvr neither unit divisable thus prime ideal height one thus dvr let regular parameter valuation ring valuation ordt since monomial ideal value group ordt generated ordt ordt thus gcd ordt ordt since ordt ordt ordt since ramification index integrally closed ideal generated monomials since gcd ordx monomial provided thus unless since residue corollary section chapter local fields theorem section chapter fact regular parameter inclusion finite extension complete dvrs prime ideal lying prime ideals since regular local rings normal since normal excellent also finite extension since lies thus finite field extension integral closure obtain unique dvr dominates thus theorem page since analytically unramified excellent thus suppose taking rur sxs simpler variant proof shows proposition suppose two dimensional excellent regular local rings dominates finite separable extension regular system parameters regular system parameters expression units exist inclusions local rings following properties two dimensional normal local ring essentially finite type lies proof conclusions proposition follow proposition one zero may assume positive possibly interchanging assumed proof proposition generalization proof proposition give outline proof indicating essential differences let gcd gcd let respectively integral closure ideal respectively bii normal local ring products integrally closed ideals regular local ring integrally closed theorem appendix let let dvr let dvr let let height one prime thus dvr dominated thus proof proposition conclude residue natural valuation calculate proof proposition rational rank throughout subsection following assumptions suppose dimensional excellent local domain quotient field suppose finite separable extension dimensional local domain quotient field dominates suppose valuation dominates residue field algebraic value group rational rank isomorphic ordered group subgroup let restriction theorem suppose defect exists commutative diagram vertical arrows products quadratic transforms along two dimensional regular local rings regular system parameters regular system parameters exists unit class generator group join follow proof exists diagram conditions theorem stable appropriate sequences quadratic transforms proposition exists local ring essentially finite type dominated dominates commutative diagram regular local ring essentially finite type dominates regular local ring essentially finite type dominates regular system parameters regular system parameters expression unit unit mod natural valuation dvr proof first prove proposition assumption galois galois group let number extensions writing gcd gcd corollary theorem page since argument top page exists diagram normal local rings respectively essentially finite type dominates lies applying proposition extension satisfying dominates commutative diagram normal local rings lies let unramified theorem since thus proposition page since unique local ring lying combining obtain formula proof valid general situation since excellent establish proposition general case assumed finite separable let galois closure let extension galois case exists normal local ring giving property proposition within extension normal local ring giving property proposition within extension choose lies suppose given diagram proposition local rings lies let integral closure let thus since multiplicative formula follows proposition exists local ring essentially finite type dominated dominates commutative diagram regular local ring essentially finite type regular local ring essentially finite type dominates regular system parameters regular system parameters expression unit unit exists unit generator mod natural valuation dvr proof let local ring conclusions proposition let classes basis let regular local ring dominates essentially finite type integral closure suppose diagram conclusion holds since generator generator finally holds proposition since dominates corollary let assumptions proposition assume let local ring conclusions proposition suppose commutative diagram proof generator proposition divides thus since proposition contains basis proposition since thus equations giving conclusions corollary give proof theorem let ring conclusions proposition may assume replacing appropriate sequences quadratic transforms regular dominates regular parameters regular parameters generator unit corollary let smallest sequence quadratic transforms along prime ideal possible since rational rnak rationally dependent let smallest sequence quadratic transforms along prime ideal show dominates regular parameters unit since conclusions hold sequences quadratic transforms established conclusions theorem remark stability following statement theorem polynomial ring let monic generator exists residue regular parameters substitute obtain let gcd sequence substitutions obtain expression det thus set without loss generality may assume expression polynomial ring let monic generator exists residue regular parameters expression inclusion induces natural homomorphism let residue residue residue residue nonzero thus proposition completes proof theorem rational rank following theorem rational rank valuations dominating two dimensional excellent local domain whose simple proof page statement extension defectless follows argument proposition establishes replacing references proposition proposition obtain formula instead since obtain related results proven valuations maximal rational rank algebraic function fields theorem theorem suppose dimensional excellent local domain quotient field suppose finite separable extension local domain quotient field dominates suppose valuation dominates residue field algebraic value group rational rank let restriction defect exists commutative diagram vertical arrows products quadratic transforms along two dimensional regular local rings regular system parameters regular system parameters exist units det classes generators group join exists diagram conditions theorem stable appropriate sequences quadratic transforms proof theorem give proof theorem dimq abhyankar inequality theorem dvrs algebraic local rings respectively monomial mapping obtained respectively sequence quadratic transforms along exists sequence quadratic transforms along algebraic thus zariski main theorem type valuation must theorem section chapter suppose dimq algebraic abhyankar inequality case exists monomialization theorem also always type valuation theorem remaining case algebraic dimq existence monomialization case follows theorem extensions associated graded rings valuations section extend results calculating extension associated graded rings valuation defectless extensions dimensional algebraic function fields algebraically closed field dimensional algebraic function fields necessarily closed characteristic zero field refer introduction paper discussion problem recall following theorem strong monomialization rational rank valuations extension characteristic zero function fields theorem rational rank case theorem theorem theorem let algebraic function field field characteristic zero finite algebraic extension rational rank valuation suppose algebraic local ring quotient field dominated algebraic local ring quotient field dominated let restriction exists commutative diagram vertical arrow product monoidal transforms along regular local rings dimension equal trdegk regular system parameters regular system parameters exists unit class generator group conclusions theorem continue hold whenever exists commutative diagram vertical arrow product monoidal transforms along holds show assumptions conclusions theorem simple description extension associated graded rings valuations theorem let assumptions theorem let conclusions theorem natural isomorphism graded rings respective classes degree extension quotient fields proof let residues basis establish following formula suppose min without loss generality may suppose minimum classes impossible since basis assumptions theorem thus established formula suppose let positive integer expression mts gij gij next establish following formula min gij assumption gij set gij since class order formula follows homomorphism defined map proof thus graded subalgebra suppose let exists unique satisfies thus class classes relations divisible relation theorem let assumptions theorem let conclusions theorem natural isomorphism graded rings respective classes degree extension quotient fields proof proof exactly proof theorem also obtain theorem following result showing even positive mixed characteristic associated graded rings abhyankar valuations dominating stable extension two dimensional excellent regular local rings nice form theorem let assumptions theorem let conclusions theorem natural isomorphism graded rings respective classes degree extension quotient fields non constancy corollary theorem shown constant integers defined associated stable forms extension valued two dimensional algebraic function fields shown constant defect extension however shown sum constant defect extension remark asked numbers computed stable forms eventually constant case give examples showing case even within defect artin schreier extensions example theorem tower two artin schreier extensions algebraically closed field characteristic divides defined valuation defined generating sequence valuation defined generating sequence theorem shown stable forms let define local ring show section stable forms theorem theorem conclusions theorem even odd even odd theorem even odd even odd prove statements make use notation introduced section construction example theorem proof makes essential use theory generating sequences valuation dominating two dimensional regular local ring developed extended define prescribing generating sequence starting ujp ujp odd even establish determines unique valuation define odd even induction let odd even odd even let group generated remark theorem proof generating sequence unique valuation order precisely odd even order since holds generating sequence determines valuation following propositions little stronger proposition proposition let sequence quadratic transforms notation definition let generating sequence determining proposition qpj exist generating sequences regular parameters local equation exceptional locus spec spec regular parameters local equation exceptional locus spec spec generating sequence defined generating sequence defined follows proposition regular parameters defined proposition let sequence quadratic transforms notation definition let generating sequence determining corollary pip exist generating sequences determining regular parameters local equation exceptional locus spec spec regular parameters local equation exceptional locus spec spec generating sequence defined generating sequence defined follows proposition regular parameters defined proposition sequence generating sequence unique let sequence notation definition exist generating sequences determining regular parameters local equation exceptional locus spec spec odd even regular parameters local equation exceptional locus spec spec generating sequence defined odd even generating sequence defined odd even odd even odd odd even even exist units mod mak odd odd even even odd even proof fact sequence generating sequence unique shown proposition remainder proposition proved induction theorem exists generating sequence defined odd even odd even odd even prove conclusions proposition hold use induction assumption appropriate equations stated first verify units mod verifying generating sequence appropriate equations hold finally appropriate equations hold state prove equations exists unit mod equation follows fact theorem regular parameters defined exists unit mod odd even simultaneously verify equations first verify regular parameters defined odd even odd even follow theorem thus power times unit equivalent mod thus exists unit mod ukp ukp odd even thus odd ukp even ukp lemma odd divides degy even divides degy proof xcp verifying xcp verifying prove equations induction first assume odd ujp formula follows since assume even ujp formula follows since proposition unique extension proof since suffices show fact need show equation obtain odd even thus lemma thus proposition odd even odd even fact unique extension follows since proposition unique extension theorem theorem finite extension odd expressions units even expressions ysp units proof units times power units times power unit odd unit even odd even odd equations proposition expression unit even equations proposition expression unit unit odd unit even extension finite since quasi finite unique extension theorem finite extension odd expressions units even expressions units proof finite since finite normal expressions follow theorem theorem references abhyankar valuations centered local domain amer math abhyankar local uniformization algebraic surfaces ground fields characteristic annals math abhyankar ramification theoretic methods algebraic geometry princeton univ press abhyankar resolution singularities embedded algebraic surfaces second edition springer verlag new york berlin heidelberg benito villamayor techniques study singularities application resolution schemes math ann bravo villamayor singularities positive characteristic stratification simplification singular locus advances math cossart jannsen saito canonical embedded resolution singularities excellent schemes cossart piltant resolution singularities threefolds positive characteristic reduction local uniformization purely inseparable coverings algebra cossart piltant resolution singularities threefolds positive characteristic algebra cutkosky local factorization monomialization morphisms cutkosky resolution singularities positive characteristic amer math cutkosky counterexamples local monomialization positive characteristic math ann cutkosky ghezzi completions valuation rings contemp math cutkosky piltant ramification valuations advances math cutkosky pham vinh valuation semigroups two dimensional local rings proceedings london mathematical society cutkosky pham vinh ramification local rings along valuations appear journal pure applied algebra cutkosky teissier semigroups valuations local rings mich math jong smoothness alterations inst hautes etudes sci publ math ghezzi huy kashcheyeva toroidalization generating sequences dimension two function fields algebra ghezzi kashcheyeva toroidalization generating sequences dimension two function fields positive characteristic pure appl algebra grothendieck vol publ math ihes hauser problem resolution singularities positive characteristic proof waiting bull amer math soc hironaka three key theorems infinitely near singularities francojaponaises congr soc math france paris knaf kuhlmann every place admits local uniformization finite extension function field adv math knaf kuhlmann abhyankar places admit local uniformization characteristic ann sci norm sup kuhlmann valuation theoretic model theoretic aspects local uniformization resolution singularities research textbook tribute oscar zariski hauser lipman oort quiros progress math kuhlmann value groups residue fields bad places algebraic function fields trans amer math soc kuhlmann classification artin schreier defect extensions characterization defectless fields illinois math lipman desingularization schemes annals math maclane construction absolute values polynomial rings trans amer math soc maclane schilling branches rank algebraic varieties annals math moghaddam construction class valuations field large value group algebra serre corps locaux hermann spivakovsky valuations function fields surfaces amer math teissier valuations deformations toric geometry valuation theory applications kuhlmann kuhlmann marshall editors fields institute communications amer math providence teissier overweight deformations affine toric varieties local uniformization preprint temkin inseparable local uniformization algebra famille admissible valuations une extension algebra zariski samuel commutative algebra volume van nostrand zariski samuel commutative algebra volume van nostrand steven dale cutkosky department mathematics university missouri columbia usa address cutkoskys
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channel hardening favorable propagation massive mimo stochastic geometry oct zheng chen emil massive mimo alternative topology massive mimo networks large number access points aps distributed coverage area cells users jointly served aps using network mimo methods prior work claimed massive mimo inherits basic properties cellular massive mimo namely channel hardening favorable propagation paper evaluate one rely properties realistic stochastic deployment results show level channel hardening depends strongly propagation environment generally little hardening except pathloss exponent small show using antennas per instead one greatly improve hardening level favorable propagation affected propagation environment distance users spatially well separated users exhibit favorable propagation conclusion rely channel hardening favorable propagation analyzing designing massive mimo networks need use achievable rate expressions resource allocation schemes work well absence properties options described paper index massive mimo channel hardening favorable propagation achievable rates stochastic geometry ntroduction throughput conventional cellular networks limited uncoordinated interference mitigate interference shamai zaidel introduced concept commonly known network mimo key idea let access points aps network jointly serve users downlink well uplink thereby turning interference useful signals despite great theoretical potential lte standardization technology failed provide remarkable gains two practical issues network mimo achieve scalable channel acquisition sharing data aps former solved utilizing local channel state information csi refers knowledge channels users channels estimated exploiting uplink pilot transmission channel reciprocity duplex tdd systems chen department electrical engineering isy university sweden email part work appear globecom workshops work supported part elliit ceniit swedish foundation strategic research thus making tdd key enabler network mimo usercentric clustering aps reasonably close user transmit signals key reduce data sharing overhead concepts easily incorporated lte relies channel acquisition clustering network mimo concept recently reappeared name massive mimo refers network massive number geographically distributed aps jointly serve smaller number users particular presented better option providing coverage using uncoordinated small cells concept fundamentally network mimo paper aps perform joint transmission access data every user local csi main novelty introduced massive mimo capacity analysis takes practical pilot allocation imperfect csi account using similar methodology cellular massive mimo literature conventional cellular massive mimo systems consist aps equipped massive number colocated antennas systems deliver high spectral efficiency utilizing channel hardening favorable propagation phenomena channel hardening means beamforming turns fading channel nearly deterministic scalar channel favorable propagation means users channel vectors almost orthogonal consequences law large numbers massive mimo essentially massive mimo system antennas distributed wide geographical area makes joint channel aps user strongly spatially aps closer user others capacity massive mimo spatial correlation imperfect csi analyzed among others channel models provide channel hardening favorable propagation claimed outstanding aspect massive mimo also utilize phenomena fully demonstrated far hence clear known massive mimo capacity lower bounds useful context underestimate achievable performance example derived new downlink capacity lower bound case precoded channels estimated using downlink pilots bound provides larger values common bound estimates precoded channels relying channel hardening unclear whether indicates need downlink pilots lack channel hardening massive mimo setup paper aims answering following open questions observe channel hardening favorable propagation massive mimo aps beneficial deploy antennas aps aps antennas order achieve reasonable degree channel hardening favorable propagation important factors affect conditions two properties capacity bounds conventional cellular massive mimo appropriate use massive mimo order answer questions model distribution homogeneous poisson point process ppp equipped antennas unlike conventional regular grid model base station deployment stochastic point process model considered work capture irregular deployment real networks first conditioning specific network realization aps located fixed locations reference user point origin define channel hardening favorable propagation criteria functions distances examine spatially averaged randomly located users satisfy criteria separation randomness caused fading spatial locations aps allows study statistical performance largescale massive mimo network separation similar concept meta distribution proposed difference mainly lies definition studied performance metrics analysis carried considering different number antennas per different pathloss models model different pathloss exponents model compared conference paper focuses channel hardening aspect massive mimo paper provide thorough investigation channel hardening favorable propagation based give insights selection achievable rate expressions massive mimo remainder paper organized follows section describe massive mimo network model including distribution channel models next section iii analyzes channel hardening section analyzes favorable propagation massive mimo section considers different capacity lower bounds cellular massive mimo demonstrates ones useful systems section concludes paper ystem odel consider massive mimo system finitesized network region aps distributed twodimensional euclidean plane according homogeneous ppp intensity measured per equipped antennas generalization massive mimo considered prior works aps connected central processing unit cpu backhaul cpu codes decodes data cpu fig massive system equipped either single multiple antennas signals see fig illustration different smallcell network aps coordinated serve users simultaneously using resources number users locations generated another independent point process denote number aps specific realization ppp poisson random variable mean value denotes area network region let denote total number antennas existing given user distribution assume users specific network realization density much larger user density boundary effect caused network region weak users located network boundary still likely nearby dominant aps makes received signal distribution similar users consider typical user origin spatially averaged network statistics seen typical user represent average network performance seen randomly located users network denote channel vector antennas typical user labeled user element modeled represents fading represents pathloss function distance antenna user since every antennas assume independent rayleigh fading antenna typical user means independently identically distributed rvs first part paper consider pathloss model min distance pathloss pathloss model also studied section note include shadow fading analysis commonly used shadowing model random shadowing coefficients randomly located aps fundamental impact channel gain distribution therefore inclusion shadowing coefficients would change general trends observed paper main advantage massive mimo kgk guaranteed finite mean even infinite network size kgk increases unboundedly network size grows particularly interesting study case antenna density fixed observe mean channel gain irrespective whether high density aps smaller density aps variance however proportional thus grows similar distributed antenna systems main advantage massive mimo reduced distance user nearest aps demonstrated analyzing distribution squared norm channel vector refer channel gain depends realization ppp equipped antennas therefore sum fading coefficients antennas following gamma distribution mean variance define distance vector element denotes distance typical user origin thus squared norm written kgk cdf kgk gamma note two sources randomness studying channel distribution randomly located user natural consider distribution kgk respect sources randomness prior studies application stochastic geometry wireless networks well known sum received power randomly distributed nodes described shot noise process mean variance kgk averaging spatial distribution antennas known unbounded bounded pathloss models considered pathloss model min finite network region radius centered around typical user kgk var kgk proof see appendix note unbounded pathloss model appropriate analyzing massive mimo stochastic geometry antennas arbitrarily close user might result unrealistically high power gain using unbounded pathloss model fig cdf squared norm channel vector user respect fading ppp realizations number antennas per antenna density fixed fig shows cumulative distribution function cdf kgk respective random spatial locations fading realizations consider different numbers antennas per note horizontal axis shown decibel thus users close large values kgk majority substantially smaller values exponential distribution fading strong impact cdf also density makes difference larger longer tail distribution point achieves higher value reason behind increasing tail var kgk proportional described practical interpretation higher density reduces average distance aps macrodiversity reduces risk randomly located user large distances closest aps observation evinces key motivation massive mimo high density terms providing uniform coverage users random locations conventional cellular massive mimo deployment low density known gamma distribution provides good approximation interference distribution poisson random field pathloss expression kgk coincides definition interference power thus gamma distribution used approximate distribution kgk details omitted since outside scope work iii easure hannel ardening conditional channel distribution fixed location previous analysis characterized channel gain distribution user observe moving around large network aps deployed user fixed location located room fading varies time fading aps user remains conditioning specific network realization assuming aps network distances aps typical user basically fixed conditional distribution channel statistics respect fading distribution essential performance evaluation computing ergodic capacity massive mimo networks fixed topology users fixed random locations aps total number antennas equal number aps result exponentially distributed fading coefficient conditional distribution channel gain kgk follows hypoexponential distribution denoted hypo usually distribution coefficients distinct antennas per channel gain kgk given gamma result conditional distribution channel gain gamma due sum independent gamma rvs different scale parameters mean variance kgk conditioning distance vector kgk var kgk exact conditional probability density function pdf kgk computed using approach exact expression available looking unclear channel gain behaves example mean value channel variations grow faster cellular massive mimo antennas conditioning specific location user denote kgk pathloss antennas user squared norm channel gain kgk follows gamma mean value standard deviation increases distribution approaches normal distribution relatively speaking concentrated around mean since grows faster standard deviation different channel gain distributions cellular massive mimo highlight fundamental difference channel statistics two types networks remainder paper proceed investigate massive mimo network could observe classical massive mimo phenomena namely channel hardening favorable propagation cellular massive mimo number antennas grows channel user behaves almost deterministic property called channel hardening conditioning specific network realization distance vector channel hardening appears massive mimo following condition holds kgk kgk kgk channel gain aps typical user one way prove channel hardening mean square sense show channel gain variation var kgk kgk var kgk kgk large wireless network studying channel statistics specific location limited interest results generalized users arbitrary locations quantify channel gain variation users arbitrary locations define following channel hardening measure var kgk kgk var given certain threshold kgk probability obtained different network realizations generate different distance vector mentioned section spatially averaged probability provides percentage randomly located users experience var smaller equal notice implies kgk users channel gain variations smaller ideal case variance zero users threshold small enough larger higher possibility observe channel hardening users arbitrary locations cdf necessary conditions channel hardening antennas per channel hardening measure written since appears denominator given always increases implies regardless density antennas per always helps channel harden following fix number antennas per study impact density channel hardening criterion note convergence mean square implies convergence probability given network realization aps defining xch write channel hardening measure network radius implies xch exact distribution xch difficult analyze even joint pdf one objective work provide intuitive insights relation channel hardening density specifically approach density increases need xch since follows highly correlated though distributions trivial obtain mean variance obtained campbell theorem section pathloss model min network region radius var using var obtain using campbell theorem var results make following observations scales proportionally higher order element scales proportionally pathloss bounded finite mean increase given observations one intuitive conclusion increases implies approaching since density density satisfy measured aps per condition channel hardening small order words converge zero increases adding aps help channel harden continue investigate necessary condition channel hardening network region grows infinity large depending pathloss exponent following cases satisfied rather unrealistic condition practice case behaves propagation environment implies operator means difference expressions vanishes asymptotically observe channel hardening achieved network radius increases one example case indoor near field propagation implies equations see decreases rapidly network region eventually approach grows infinitely large suggests smaller pathloss exponents propagation indoor near field propagation likely observe channel hardening massive mimo pathloss model convergence channel hardening happens impractically high antenna density order validate analytical predictions present fig simulated cdf xch different densities obtained pathloss exponents note figure consider chosen large values order see behavior fig shows given threshold channel hardening measure change much intensity unless reach however probably practically unreasonable convergence channel hardening measure one becomes obvious density grows indicates probability observe channel hardening random locations fairly large array antennas smaller pathloss exponent required number antennas per achieve reasonably strong channel hardening also smaller fig cdf xch pathloss exponent network radius density equivalent property increasing number antennas per massive mimo always helps channel harden one antenna per increasing density lead channel hardening using typical pathloss exponents densities propagation environment small pathloss exponent channel hardening criterion higher chance satisfied density increases antennas aps aps antennas antenna density fixed whether choose larger smaller density vice versa achieve high level channel hardening inferred rewrite since denominator contains plus constant term fixed clearly obtain channel hardening antennas fewer aps limited number antennas get observations note stronger hardening comes price less macro diversity fig present cdf xch different keeping overall antenna density fixed figure confirms prediction multiple antennas per substantially help channel harden level channel hardening clearly increases curve interpreted cellular massive mimo system due massive number antennas per largest gains occur going thus achieve reasonable strong channel hardening within scope massive mimo equipped result obtained uncorrelated fading user antennas instead spatially correlated fading due insufficient scattering around slightly reduce hardening antennas still beneficial threshold threshold fig cdf xch pathloss exponent network radius antenna density distance number antennas higher density average number antennas certain area larger thus distances nearest aps user generally macro diversity effect effect larger density channel hardening criteria related reduced distance increased antenna number understand whether distance number antennas plays dominant role examine impact density assuming typical user served nearest antennas usually much smaller total antenna number network region focus case single antenna per thus nearest antennas represent also nearest aps existing results distance distribution poisson networks joint pdf distances nearest antennas typical user avoid dependence distribution distance variables consider approximately equivalent case fixed number antennas uniformly independently distributed disk centered typical user origin average radius determined equivalent antenna density result uniform distribution joint pdf points within xch approximation comes two parts consider nearest aps contribute majority channel gain usually much smaller assumption antennas uniformly dismc tributed approximation real joint distance distribution nearest antennas aps affect network statistics observed user property density grows shortened distances nearby aps user little impact channel hardening criterion increasing total number antennas environment small pathloss exponent stronger influence channel hardening system larger pathloss exponent approximation section extend analysis scenario pathloss model models fact pathloss exponent generally increases propagation distance similar consider pathloss model simulation mean value xch pathloss model simulation approximation simulation simulation density fig mean value xch approximation results obtained simulation results obtained case nearest aps transmitting user case antennas inside entire network area radius transmitting user measured fig present approximation simulation results xch different antenna densities compare approximation obtained mean value xch obtained simulations assuming nearest antennas transmitting user comparison present also simulation results xch antennas inside entire network area radius transmitting user origin figure make following observations analytical simulation results show mean value xch barely affected density depends pathloss exponent smaller smaller mean xch likely observe channel hardening evinces previous observations approximation provides accurate measure xch nearest aps affect channel gain variation pathloss exponent large smaller pathloss exponent mean value xch typical user served antennas much lower nearest antennas implies number aps affect channel gain variation much higher example based monte carlo simulations approximately nearest aps affect distribution xch pathloss exponent determines speed signal attenuation therefore smaller fixed distances slope starts change constant depends carrier frequency antenna height since constant factor affect channel hardening measure simplicity consider remainder section previously paper consider necessary condition channel hardening rdr var rdr var since var approximation holds based using larger make approach higher speed increases intuitive given previous observation smaller pathloss exponent improves hardening increase number aps small pathloss exponents increases adding antennas aps also improve channel hardening fixed total antenna density fixed fig shows cdf xch obtained pathloss model figure compare obtained different values threshold fig cdf xch pathloss model total antenna density density antennas per first antenna density increases increase substantial influential results fig fig model second comparing results obtained see number antennas per plays important role increasing density helping channel harden terms achieving small xch practically reasonable density values property pathloss model due small pathloss exponent propagation environment nearby user channel gain variance declines rather fast density compared mean value furthermore large number antennas per guarantee small channel variation makes channel hardening easier achieve favorable ropagation section define analyze favorable propagation conditions massive mimo networks similar previous section consider conventional networks aps generalization multiple antennas per recall channel vector antennas user element favorable propagation channel vectors user terminals orthogonal means gkh kgk condition satisfied user get communication performance alone network practice condition fully satisfied approximately achieved number antennas grows infinity case channels said provide asymptotically favorable propagation specific massive mimo asymptotically favorable propagation condition defined follows gkh kgk conditioned specific network realization distance vectors element represents distance antenna user recall every antennas different cellular massive mimo antennas fading coefficients antenna massive mimo user different viewed type spatial channel correlation thus gkh kgk since rvs follows gkh kgk mean value convergence holds mean square sense probability variance side goes asymptotically zero using gkh var kgk density grows increases approach positive value depends network size pathloss model thus positive value decreases increases variance channel orthogonality approach combined prove asymptotically favorable propagation condition defined holds massive mimo finite use define channel orthogonality metric xfp consider probability two users random locations xfp larger threshold xfp threshold impact antenna density channel orthogonality impact antenna density analyzed two cases fixed different fixed different mentioned xfp inversely proportional fixed increasing always helps channels become orthogonal case increases denominator xfp grows almost numerator increases almost linearly knowing poisson mean value proportional consequently xfp scale roughly inversely proportional evinces given grow density combining two cases see larger larger density help channel offer favorable propagation fig presents cdf channel orthogonality metric xfp different comparing results obtained marked circle left triangle plus sign validate increasing increasing improve channel orthogonality fig cdf xfp pathloss exponent network radius first four curves marked upward triangle cross circle left triangle antenna density distance user user impact distance channel orthogonality obvious distance two users affects variance term two users far apart channel vectors likely orthogonal smaller variance result comes fact become much smaller distance user user large addition vary much distance fixed therefore xfp becomes smaller distance increases clearly order asymptotically favorable propagation antenna density grows approach one practical purposes desirable large values threshold close zero following analyze antenna density interuser distance pathloss exponent affects channel orthogonality threshold antennas aps aps antennas see value xfp always upperbounded antenna density fixed increasing means smaller result average number aps within close distance user less thus hard predict whether beneficial antennas aps aps antennas fig present cdf xfp fixing total antenna density see increasing necessarily lead higher lower given value also observe choosing sufficiently large channel orthogonality metric xfp becomes small words sufficiently large help channels different users asymptotically orthogonal fig cdf xfp different different distances pathloss exponent fig present different densities distances first shown larger variance orthogonality metric xfp smaller second distance two users larger likely nearly orthogonal channels observation showcases importance serving spatially separated users order ensure near channel orthogonality impact pathloss exponent channel orthogonality number antennas locations fixed consider two extreme cases user user close extremely far since xch coincides xfp special case infer section iii smaller pathloss exponent also lead smaller xfp extreme case two users far apart ldenominator xfp almost independent terms increase much faster numerator especially small combining two extreme cases expect smaller pathloss exponent would help channels become asymptotically orthogonal sufficiently large consider massive mimo system singleantenna aps users assigned mutually orthogonal pilot sequences transmission divided coherence intervals samples whereof used uplink pilot signaling channels modeled previous sections diag user transmits orthogonal pilot sequence user uses transmit powers pilot data respectively since elements independent optimal estimate separately receiving antenna mmse estimate additive noise denote mean square mmse estimate follows threshold fig cdf xfp different pathloss exponent distance fig shows obtained pathloss exponents distance figure shows smaller channels become orthogonal line prediction would instead use pathloss model aps close user distance smaller pathloss exponent users larger distances pathloss exponent therefore compared pathloss model min channels two users average orthogonal summarizing analysis observations following conclusions property increasing antenna density increasing either density number antennas per help user channels offer favorable propagation smaller pathloss also helps channels become asymptotically orthogonal larger distance two users likely channels nearly orthogonal apacity ounds ell ree assive mimo key conclusion previous sections massive mimo systems exhibit little channel hardening compared cellular massive mimo hence although massive mimo equivalent massive mimo system strong spatial channel correlation must careful reusing results massive mimo literature particular capacity bounds derived relying channel hardening potentially loose applied systems section explain capacity lower bounds suitable systems compare different achievable rate expressions uplink downlink using maximum ratio processing commonly assumed massive mimo since implemented distributively expressions lower bounds capacity thus use one gives largest value accurately predict achievable performance uplink achievable rate uplink data transmission users simultaneously transmit aps using achievable rate lower bound capacity user uatf used massive mimo bound derived based use forget uatf principle channel estimates used forgotten channel hardening utilized obtain uatf special simple expression note case general expression correlated singlecell massive mimo systems apply processing alternatively achievable rate expression spatially correlated channels used constant factor given log log log log cdf perfect csi general rate uatf rate rate fig cdf uplink achievable rates obtained uatf bound general bound rate perfect csi cdf perfect csi general rate uatf rate rate fig fig carrier frequency user antenna height respectively simulation parameters summarized table fig fig show cdfs user rates random distances aps users general rate expression provides almost identical rates perfect csi case indicates estimation errors closest aps negligible contrast uatf rate much looser bound capacity particularly users support highest data rates users get almost twice rate using general expression due lack channel hardening users close aps hence general guideline use achievable rates massive mimo aps important observation fig fig multiple antennas per differences uatf bound perfect csi case becomes smaller particularly users support smallest rates comes fact multiple antennas per gives better channel hardening seen section iii however average rate reduces going many aps fewer aps obvious better choice practice table imulation etup parameters values carrier frequency antenna height uplink pilot power data power downlink power per ghz diag combining vector general bound rely channel hardening fig fig compare uplink achievable rates obtained also rate perfect csi obtained letting simulation performed network area users randomly uniformly distributed network region total number antennas results fig obtained singleantenna aps fig obtained aps antennas per aps independently uniformly distributed network region length coherence block fading coefficients antennas users generated different ppp realizations pathloss model used downlink achievable rate downlink massive mimo popular use rate expressions csi available user channel statistics channel hardening one achieve rates similar uplink precoding used one use rate expression spatially correlated fading obtain achievable rate uatf user average transmit power allocated user type expression used massive mimo using slightly different notation call uatf rate since use received signals detection forget use blind estimation instantaneous channel realizations suppose precoding vector including power allocation assigned user avoid relying channel hardening user estimate instantaneous channel precoding akh collection received downlink signals current coherence block following similar approach lemma obtain achievable rate log var first term rate perfect csi second term penalty term imperfect channel estimation user note latter term vanishes thus general rate expression good lower bound channels change slowly precoding rate obtained setting cdf perfect csi general rate uatf rate onclusion work provided thorough investigation channel hardening favorable propagation phenomena massive mimo systems stochastic geometry perspective studying channel distribution stochastically distributed aps either single multiple antennas per characterized channel gain distribution based examined conditions channel hardening favorable propagation results showed whether channel hardens increasing number antennas strongly depends propagation environment pathloss model one generally expect much hardening however one improve hardening deploying multiple antennas per several factors help channel provide favorable propagation smaller pathloss exponent higher antenna density spatially separated users larger distances well separated users generally exhibit favorable propagation since mainly communicating different subsets aps main implication work one rely channel hardening favorable propagation computing achievable rates massive mimo could lead great underestimation achievable performance good uplink rate expression spatially correlated massive mimo systems used development downlink rate expressions needed fully understand achievable downlink performance massive mimo rate ppendix fig cdf downlink achievable rates obtained uatf bound general bound rate perfect csi fig compare downlink achievable rates obtained also rate perfect csi user obtained letting simulation parameters summarized table section uplink downlink power given downlink power allocated user chosen simulations kak users fig shows cdfs user rates random distances uplink substantial gap rates achieved perfect csi curve uatf rate curve general rate middle implies users estimate downlink channels rely channel hardening massive mimo unclear whether gap perfect csi curve due limited downlink estimation quality artifact capacity bounding technique lead case need study achievable downlink rates massive mimo proof recall campbell theorem follows measurable function ppp density since homogeneous ppp finite network region radius kgk rdr note min result gamma distribution thus kgk rdr depending value kgk expression variance var since gamma var thus var kgk rdr rdr note different form unlikely happen real propagation environment thus case discussed eferences chen rely channel hardening massive mimo accepted ieee globecom workshops shamai zaidel enhancing cellular downlink capacity via transmitting end proc ieee vol venkatesan lozano valenzuela network mimo overcoming intercell interference indoor wireless systems proc ieee acssc gesbert hanly huang shamai simeone mimo cooperative networks new look interference ieee sel areas vol osseiran monserrat marsch mobile wireless communications technology cambridge university press coordinated transmission zakhour gesbert ottersten cooperative multicell precoding rate region characterization distributed strategies instantaneous statistical csi ieee trans signal vol bengtsson ottersten optimality properties distributed strategies evaluation coordinated multicell ofdma transmission ieee trans signal vol ngo ashikhmin yang larsson marzetta massive mimo uniformly great service everyone proc ieee spawc nayebi ashikhmin marzetta yang massive mimo systems proc asilomar ngo ashikhmin yang larsson marzetta massive mimo versus small cells ieee trans wireless vol nayebi ashikhmin marzetta yang rao precoding power optimization massive mimo systems ieee trans wireless vol marzetta larsson yang ngo fundamentals massive mimo cambridge university press ngo larsson downlink pilots needed tdd massive mimo ieee transactions wireless communications vol may ngo larsson marzetta aspects favorable propagation massive mimo european signal processing conference eusipco september huh caire papadopoulos ramprashad achieving massive mimo spectral efficiency number antennas ieee trans wireless vol hoydis ten brink debbah massive mimo cellular networks many antennas need ieee sel areas vol yin gesbert filippou liu coordinated approach channel estimation systems ieee sel areas vol interdonato ngo larsson frenger much downlink pilots improve massive mimo proc ieee globecom renzo stochastic geometry modeling cellular networks analysis simulation experimental validation acm international conference modeling analysis simulation wireless mobile systems november andrews baccelli ganti tractable approach coverage rate cellular networks ieee transactions communications vol november haenggi meta distribution sir poisson bipolar cellular networks ieee transactions wireless communications vol april ganti haenggi interference hoc networks general node distributions ieee international symposium information theory july haenggi ganti interference large wireless networks foundations trends networking vol haenggi stochastic geometry wireless networks cambridge university press kountouris pappas approximating interference distribution large wireless networks international symposium wireless communications systems iswcs august smaili kadri kadry hypoexponential distribution different parameters applied mathematics vol amari misra expressions distribution sum exponential random variables ieee trans vol hammarwall ottersten exploiting quantized channel norm feedback conditional statistics arbitrarily correlated mimo systems ieee trans signal vol goldsmith wireless communications cambridge university press moltchanov distance distributions random networks hoc networks vol rusek persson lau larsson marzetta edfors tufvesson scaling mimo opportunities challenges large arrays ieee signal process vol sanguinetti debbah massive mimo imperfect channel covariance information proc asilomar caire ergodic rate lower bounds applications massive mimo corr vol
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case multiple parallel rrams synaptic model training snns aditya shukla dept electrical engineering indian institute technology bombay mumbai india adityashukla sidharth prasad dept electrical engineering indian institute technology bombay mumbai india enable dense integration model synapses spiking neural networks hardware various devices considered devices besides exhibiting dependent plasticity stdp need highly scalable large endurance require low energy transitioning states work first introduce empirically determine two new specifications synapse snns number conductance levels per synapse maximum best knowledge rrams meet latter specification solution propose use multiple rrams parallel within synapse synaptic reading simultaneously read synaptic event mechanism conductance stdp initiated one rram randomly picked set second validate solution experimentally demonstrate stdp conductance show due large single fails model synapse training snn anticipated network training improved added synapse fourth discuss implementing scheme conclude requirements within bounds thus work presents specifications synaptic devices trainable snns indicates shortcomings synaptic contenders provides solution extrinsically meet specifications discusses peripheral circuitry implements solution iris dataset memristor prcamno pcmo resistive memory rram dependent plasticity spiking neural network synapse introduction decoding functioning mimicking biology using models brain significant approach taken community also perform cognitive computing tasks maintaining energy efficiency human brain grand challenge era executing tasks using spiking third generation neural networks hardware functionally structurally similar brain greatly help achieving goals led adoption approach distributes ieee sandip lashkare dept electrical engineering indian institute technology bombay mumbai india sandipl udayan ganguly dept electrical engineering indian institute technology bombay mumbai india udayan integrates processing memory electronic neurons computing units electronic synapses memory units connected required network though various vlsiamenable circuits proposed mimic synapse integrate many synapses within volume human brain requires model synapse dimensions order thickness synaptic cleft led appreciable research development plethora novel devices faithfully mimic synapse novel material based devices modeling synapses hardware resistive gained remarkable research interest resistive whose altered applying sufficiently strong strong candidates weights electronic trainable neural networks dependent plasticity stdp rule type hebbian considered essential property snn based synapse models must exhibit changing strength applied function stdp rule demonstrated various rrams extensive utilization devices synaptic array needs highly scalable excellent endurance switchability compatible cmos however important relatively unexplored requisites synaptic rram discussed work include analog range conductance ample number memory low value maximum stdp based mathematically low event requisite based fact nature stdp rules observed biology analog datasets images iris chemical assays like wine additional costs either synthetically transformed binary domain thus often necessitating analog synapse requisite comes fact stdp point maximum point time ratecorrelation value kept small stable network training paper first empirically show softwareequivalent training maximum event must less resolution least levels bits per synapse needed second show stdp demonstrations rrams todate depict thus devices meet specifications tackle problem propose use multiple parallel rrams synapse within synapse reading requires rrams dictated stdp rule brought one randomly picked rram set rrams synapse way lowered enabling learning second validate proposal using stdp measurements standalone accompanying training snn multiple parallel rrams synapse next learning performance evaluated show produces excellent peak learning performance significant fluctuations epoch epoch necessary learning comparison binary synapses needed equivalent programming improvement fourth architectural consideration circuit implementation discussed simple circuit implement random programming scheme presented thus work presents specification synaptic devices analog datasets demonstrates challenge synaptic candidates literature presents architectural solution enable learning provides circuit implementation work organized follows section overview stdp rule followed basis claim necessary condition acceptable snn training performance section report procedure stdp demonstration results continuation using show device fails synapse section validate proposal using multiple parallel rrams synapse lastly section discuss hardware requirements consideration adopting proposed approach ideal snns dependent plasticity stdp common used spiking neural networks snns gives relation time gap spikes weight change synapses illustrated figure figure biological stdp rule figure stdp rule several possible model biological ensures indefinitely positive negative strengthen causality weaken spikes example plotted figure function various function various ideal training since stdp equation matter set maximum determine training artificial neural network needs carefully chosen since work exists studies spiking neural networks trained using stdp empirically determine using snn given single layered snn trained via exponential stdp rule classify iris wine training following stdp rule used illustration use exponential rule comprising dependence term saturation factor scaling factors given figure classification accuracy versus parameters per cent figure distribution classification errors versus number conductance levels per synapse iris dataset breastcancer dataset trained linear sensors input large small classification accuracy settles maximum means training partial large small network unable learn small learning best classification accuracy settles several snns trained using stdp literature rely less though work focuses affects overall training likely performances may suffer similar degradation increased figure classification accuracies four values represents parameters conductance potentiation represents parameters conductance depression set maximum conductance change set stdp maximum conductance minimum quantitatively study effect training simulate training network classify iris dataset various values pair observe evolution classification accuracy training proceeds figure plots last five epochs training observed lower better training terms maximum accuracy stability low synaptic necessary condition high posttraining accuracy synaptic potentiation rate affects training performance lower extent depression rate figure shows four classification accuracy plots possible pairs percentages represent fraction maximum weight synapse following observations made large training never gets completed classification accuracy settle number bits per digital synapses empirically determine number bits needed digital synapse training simulated training network used classify iris datasets total conductance range divided uniformly levels event originally continuous change conductance discretized nearest conductance level figure plots mean classification error training experiments observed least discrete levels needed ensure lowest iii synapse snns stdp rrams using neural rram approximately modeled resistor whose resistance changed pulsestrength exceeds writing threshold figure change increases strength threshold increased given approximation realize stdp rram carefully shaped applied two ends synaptic rram two spiking neurons context figure pulses shaped two pulses corresponding neurons attached synapse relatively displaced time subtracted exists portion net always reaches goes threshold height net threshold compactly overdrive function relative displacement time neuron spikes pulses applied immediately response terminal synaptic rram way rram sees equal postsynaptic neuron presynaptic neuron positive negative amplitudes set less respectively decay constants sets pulse sets exponential stdp figure rram modeled function rectangular pulse height neuron connected synapse right spiking instants produce apply fresh terminal possible synaptic exponential stdp subtraction two relatively displace voltage across rram circled portion subtraction crossing threshold changes conductance rram define spike lengths time axis demonstration exponential stdp pcmo experimentally demonstrated endurable fast highly scalable analog memristive contender synaptic applications shown figure area originally reported reported used demonstrating stdp device initialized state applying constant negative compliance set writing pulses used values given table procedure followed demonstrate stdp device conductance read using small rectangular voltage pulse yielding initial conductance value figure measured structure final conductance value using voltage pulse table parameters used stdp writepulse parameter value randomly chosen subtraction relatively displaced writepulses corresponding directly applied terminal terminal grounded parameter value recorded three steps repeated times new state served new iteration pulses applied ones leading increase conductance plotted figure leading decrease conductance values normalized dividing value maximum conductance observed better visual uniformly spaced points plotted figure subtraction displaced pulses applied neurons figure implementing exponential stdp rule may given following shape figure ltp ltd lots figure observed stdp conductance various distinct colors interpolation model measurements legend represents initial conductance figure training behaviour single rram based synapse regardless initial condition classification accuracies settle low values right first training iteration epoch evolution initial condition evolution figure offered various analog rrams literature none meet specifications figure observed stdp conductance various distinct colors interpolation model measurements difference iso figure use data simulations used interpolation model isostdp curves obtained using model plotted figure figure though stdp demonstrated experimentally altered scaling portion training single synapse replaced mathematical synaptic model network mentioned sec interpolation model modified read equation mathematical conductance replaced conductance since latter normalized additional factor added leading following replacement stdp rule equation stdp rule specified replaced following equation determined interpolation model mentioned next training simulated figure plots evolution classification accuracy training proceeds clearly performance worse comparison obtained mathematical synaptic model hypothesis based observation made figure figure marked reduction performance observed large percentage change conductance equivalent large learningrates mathematical model stdp demonstrated several analog rrams observed necessary maximum conductance change less synaptic get evolution conductance however analog plasticity demonstrations maximum conductance change devices figure thus currently existing analog rrams fail produce softwareequivalent classification accuracy network second hypothesis based observation made figure training performance improved without changing network reducing maximum change conductance change lower validity hypothesis tested using like used rram lower maximum conductance since knowledge rrams literature meet constraint possible moment test hypothesis step towards better memristor based synaptic models propose using set parallel way function aggregately synapse pulses applied one next validate proposal parallel multiple pcmos synapse test ability multiple collectively act synapse continued network trained classify iris mapping mathematical synapse based synapse similar one described sec slight difference described follows replaced mathematical weight figure classification accuracies synapse initial conductance configuration stdp rule equation every spiking instant stdprule based replaced following two equations note conductance increase corresponds increase conductance synapse part let take values set figure plots training proceeds observed learning stable synapses rrams figure plots quantiles training progresses number figure number determined empirically approximately epochs needed stabilize learning since roughly inversely proportional number rram number training iterations needed proportionally large lower limit set evolution conductance synapse various plotted figure observed starting reaches softwaremaximum least training figure despite trained adequate epochs network synapses low number rrams unable learn stably figure figure increased distribution figure follows trend similar one exhibited figure showing increase ltp ltd rates decreased simultaneously higher mean lower variation conductance evolve smoothly increases discussion scheme discussed escalate circuit requirements reading rrams synapse need simultaneously read done applying reading synapse current branch associated synapse summed get current proportional synaptic weight figure quantiles number synapse varied number epochs training figure synaptic weight evolution smoothens added within synapse performing writing operation one rram randomly chosen rrams within synapse uniform probability done applying presynaptic random row among random column among synapse rram rows rram columns way rram selection uniformly random among rrams though schemes uniformly selecting rrams opinion require complicated large area peripheral circuits reading writing phases may multiplexed time using global control signal reading indicated writing writing phase allow random selection set global onehot laid along periphery array active line set changed periodically whenever neuron spikes assumed random time content selection line set copied onto rram selection register output vector figure way synapse remains selected next spike active line register reading pulse applied terminals postsynaptic terminals grounded select figure read process consists applying read pulses green rows synapse summing current virtually grounded output weight update process consists application generated blue one randomly chosen row column synapse choice determined global select lines read spiking instant clock global vector selection latch use multiple rrams within synapse clearly requires bigger array however larger implies larger attenuation voltage applied across rrams far away either input output sides array thus constant array size fundamentally number rrams synapses made arbitrarily large number also arrangement needs carefully designed maintain certain minimum level fidelity simplicity consider snn one nrram synapse rrams within conductance state due assumed linear function index within synapse read within array expressed synapse rrams maximum error current corner furthest inputs outputs one indices fixed number rrams synapse say maximum error read current figure every rising edge neural spike copies content global vector neuron unique selection latch figure writing phase switches select rram written onto reading phase rrams accessed turning switches remains reading phase selection register using control input may applied single row column via mosfet based switches shown figure gate connected selection vector thus one mosfet conducting allow applied corresponding row column however reading phase mosfets turned setting thus rrams within synapse happens synapse arrangement configuration closest square thus within synapse rrams arranged squarelike configuration conclusion work introduce empirically determine two new specifications snn based synapse number conductance levels per synapse maximum best knowledge rrams meet latter specification solution proposed use multiple rrams parallel within synapse synaptic reading rrams simultaneously read synaptic event writing pulses stdp applied one rram randomly picked set validate solution experimentally demonstrated stdp conductance showed due large single device fails model synapse training snn anticipated network training improved rrams added synapse lastly discuss implementing scheme conclude requirements within bounds references indiveri corradi qiao neuromorphic architectures spiking deep neural networks indiveri chicca douglas vlsi array spiking neurons bistable synapses dependent plasticity ieee trans neural networks vol wikipedia chemical https online available kuzum wong design considerations synaptic device neuromorphic computing proc ieee int symp circuits vol eryilmaz kuzum wong device system level design considerations based neuromorphic architectures ieee international electron devices meeting iedm paulista filho prof breve esta iris data set online available https jackson nanoscale electronic synapses using phase change devices emerg technol comput vol suri phase change memory synapse neuromorphic systems application complex visual pattern extraction int electron devices park synapse neuromorphic system pattern recognition function tech dig int electron devices meet iedm garbin oxram devices synapses convolutional neural networks ieee trans electron devices vol prezioso merrikh bayat hoskins likharev strukov plasticity memristors sci vol wang memristors diffusive dynamics synaptic emulators neuromorphic computing nat vol wang lin wang lin hou characterization modeling nonfilamentary analog synaptic device sci vol wang ambrogio balatti ielmini artificial synapse capable communication stochastic learning neuromorphic systems front vol jan chang ebong bhadviya mazumder nanoscale memristor device synapse neuromorphic systems nano vol panwar kumar upadhyay arya ganguly rajendran memristive synaptic plasticity rram programming device res conf conf dig drc june jun synaptic plasticity chalcogenide electronic synapse neuromorphic systems sci vol may ultrafast synaptic events chalcogenide memristor sci vol burr neuromorphic computing using memory adv phys vol rajendran specifications nanoscale devices circuits neuromorphic computational systems ieee trans electron devices vol poo synaptic modifications cultured hippocampal neurons dependence spike timing synaptic strength postsynaptic cell abbott nelson synaptic plasticity taming beast nat vol suppl supp breast cancer wisconsin forina wine data set https online available https wilson martinez need small learning rates large problems ijcnn international joint conference neural networks proceedings cat vol biswas prasad ganguly lashkare ganguly simple efficient snn performance robustness evaluation method enable hardware implementation diehl cook unsupervised learning digit recognition using plasticity front comput vol august song abbott cortical development remapping spike plasticity neuron vol song miller abbott competitive hebbian learning synapticplasticity nat vol tang tan spiking neural network model temporal encoding learning neurocomputing vol kheradpisheh ganjtabesh thorpe masquelier spiking deep convolutional neural networks object recognition pershin ventra spice model memristive devices threshold vol apr masquelier prodromakis indiveri stdp stdp variations memristors spiking neuromorphic learning systems front vol feb jackson modha rajendran producing dependent plasticity synapse array seong characteristics nonvolatile memory applications ieee electron device vol jung reram excellent switching speed retention characteristics international electron devices meeting kumbhare chakraborty singh chouhan panwar ganguly selectorless rram record memory window nonlinearity based trap filled limit mechanism mem technol symp nvmts vol
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classification dynamical systems approach support vector machines mar giorgio battistelli pietro tesi consider problem classifying trajectories generated dynamical systems investigate approach common approach control engineering approach based support vector machines popular method area machine learning analysis points connections two approaches relative merits ntroduction assume given two dynamical systems whose underlying dynamics might unknown interested designing classifier machine given observed trajectory generated either systems correctly identifies candidate systems generated trajectory refer problem problem classification dynamical systems problems type ubiquitous within community systems control instance fault detection one system may represent behavior nominal conditions another prescribed faulty behavior similarly networked control one system may represent behavior data successfully transmitted another system may represent behavior presence packet dropouts problems type naturally arise also dealing systems naturally exhibit multiple operating modes example switched circuits hand computer science classification core machine learning pattern recognition found several applications different fields engineering including applications intensively studied control engineering like fault detection diagnosis yet regard problem interaction two communities low within systems control community classification dynamical systems studied connection analysis switched systems often term defined problem reconstructing active mode switched system output trajectories common approach approach assuming correct model system operating mode one check whether feasible via conditions fact classifiers obtained terms rank tests functions dynamical systems theory also generalizes noisy observations types nonlinear dynamics little known approach authors dinfo university florence florence italy classification one departs hypothesis dynamics system known perfect accuracy even simplest case dynamics associated parametric uncertainty building classifier task difficulty similar one encountered adaptive control based multiple models main issue indeed guarantee modelling inaccuracies destroy learning capability control scheme computer science main paradigm classification instead paradigm classifiers designed choosing function adjustable parameters selected using number training data called examples resulting function classifier evaluated according capability generalize training dataset correctly map new examples popular methods neural networks support vector machines svm whose capability generalize training dataset quantified via suitable loss functions risk function methods intrinsic potential overcome issues related model uncertainty already proven effectiveness challenging applications prediction epileptic seizures recording eeg signals however obvious tailor analysis design methods specific context data come dynamical systems worth pointing classification data generated dynamical systems new computer science fact regarded classification time series assume existence underlying system yet approaches take standpoint still try incorporate models learning task either extract data informative features construct suitable kernel functions incorporating models natural step take leads previous question handle model uncertainty help understand performance achievable schemes similar issues related approaches pointed also context clustering paper consider autonomous linear systems approach classification problem perspectives pointing relative merits establishing connections two first consider approach derive classifier assuming knowledge system dynamics approach two fundamental merits highlight necessary conditions existence correct classifier problem feasibility guide design analysis solution connection approach shows problem feasibility one design correct classifier interpreted terms polynomial kernels building result consider approach based svm using properties stemming solution provide bounds margin classifier quantify generalization performance function systems one wishes classify rest paper follows sections iii formalize problem interest recall basic concepts regarding svm sections present main results section discusses results open problems numerical simulations reported section vii section viii provides concluding remarks ramework consider two linear dynamical systems limitations classification problem denotes time rni state output state output transition matrices assume observable sense entails loss generality observable subsequent developments apply observable subsystem obtained via kalman observability decomposition assume given sequence col measurements generated one two systems col direct information two systems generated interested determining two systems generated referring problem problem classification make problem definition precise let observability matrix order pair let rni problem interest construct correct classifiers investigate two approaches classification classifier depends knowledge matrices classification classifier depend knowledge matrices determined basis given number sample trajectories points sets case reflects situation dynamics systems known information exploited design classifier contrary case reflects situation dynamics systems unknown information directly exploited design classifier set possible nonzero trajectories samples generated system definition classifiers correctness classifier function space input data classifier dynamical systems said correct satisfies clearly knowledge system dynamics provides extra degree information used properly design classifier yet certain limitations overcome even ideal situation one perfect knowledge dynamics particular following result holds true theorem limitations classification problem correct classifier dynamical systems exists rank proof theorem definition exists correct classifier whenever implies existence trajectories compatible systems equivalent fact dynamical system resulting parallel interconnection observable observability matrix order pair column rank gives result condition theorem satisfied two systems share common pairs condition also requires means large enough observation window must chosen render classification feasible take condition standing assumption assumption rank iii upport ector achines section briefly recall concepts svm focusing case separable data material section adapted assume observations one consisting vector plus label specifying class belongs connection problem introduced section one think observation collected one two candidate systems measurement specifies two systems generated consider problem classifying vectors using hyperplanes vector adjustable weights exists vector satisfying vectors called linearly separable function defines linear classifier correct respect data linearly separable case svm searches separating hyperplane largest margin searches value maximizes min kwk cast convex program subject min reason search hyperplane largest margin related fact obtained finite set observations training set hand one would like able correctly classify also data present training set property usually called generalization performance svm guarantee good generalization performance discuss point detail section problem involves constraints unknowns convenient resort dual formulation problem called wolfe dual max subject col vector lagrange multipliers zkj symmetric matrix zkj vector ones problem involves constraints unknowns solution form vector associated positive multiplier support vector means optimal linear classifier resulting parameter given linear combination support vectors interpreted representative points training dataset kernel functions finding separating surface linear respect space input data always possible one important results svm related possibility finding separating surfaces straightforward manner looking optimization problem one sees data appears products one think mapping input space space function search function space called feature space called feature vector function satisfying called kernel function kernel functions define separating surfaces linear respect remarkable feature kernel functions need use know function order compute use fact order compute solution respect one simply use zkj moreover thus one use instead also classification task kernel functions also advantageous point view computations since operates input space usually lower dimension common kernel functions polynomial gaussian hyperbolic tangent functions sequel show classifying dynamical systems polynomial kernels good candidates odel based lassification consider approach classification following example shows correct classifier exists linear input space example consider two systems assumption clearly holds true however depicted figure exists linear classifier two candidate systems independent particular choice example indicates dynamical systems correct classifier linear respect however figure suggests correct classifier example exists given rewritten stands kronecker product see defines polynomial kernel show choice general sense applies linear dynamical system form classifier kernel function support vectors interpretation fig left pictorial representation possible observation points example trajectories generated first system correspond points red circles always falls first third quadrant cartesian plane trajectories generated second system correspond points blue circles always falls second fourth quadrant cartesian plane right pictorial representation let observability gramian corresponding system note nonsingular assumption also let following result holds true theorem classifier let let vec vec vectorization operator assumption correct classifier dynamical systems proof theorem idea show computing equivalent determining sets vector belongs consider distance minn koi notice view assumption likewise hence function theorem could stated directly terms yet form turns useful provides guidelines formulation approach fact guarantees existence solution svm formulation use input training algorithm moreover immediate verify defines homogeneous polynomial kernel means svm formulation one work directly space emplyoing kernel option possible also solution write terms support vectors interpretation simple worth mentioning reduced singular value decomposition matrix yields orthonormal columns rni diagonal matrix positive entries due assumption rni unitary thus let column let corresponding feature vector hence vec vec vec first equality follows idempotent thus concludes proof remark invariance coordinate transformations notice independent particular realization adopted dynamical systems since projection matrices second equality follows vectorization rule vec abc vec matrices appropriate dimension thus vec support vectors left singular vectors observability matrices data driven classification based upport ector achines vec defines correct classifier dynamical systems notice hence get approach suggests svm formulation problem classifying data generated dynamical systems first describe svm formulation make considerations generalization performance solution interesting result generalization performance classifier quantified function dynamics systems generate training dataset classification based svm ltr let finite nonempty subset consisting nonzero trajectories recorded thus ltr ltr training vector let ltr finally let feature vector associated following section iii theorem formulate approach problem finding hyperplane contains origin separates training datasets maximum margin subject min following result holds true theorem classifier let assumption satisfied consider arbitrary training dataset ltr solution optimization problem exists unique hence correct classifier respect proof theorem proof follows theorem fact solution guarantees thus min guarantees feasibility set constraints uniqueness follows optimization problem convex program constraint solution must contain origin simply mimic solution constraint actually needed standard formulation would still guarantee existence uniqueness solution constrain solution contain origin wolfe dual becomes max subject involve constraint notice case dimension feature space nonetheless considering implementation one remain space sequences theorem indicates one find surface separating training dataset without information underlying systems except linearity suggests feature space choice remaining part section discuss capability svm classifier generalize observations outside training set expected risk ideally one would like establish correctness classifier sense definition problem best knowledge yet solved sequel consider another way characterize generalization performance svm classifier based notion expected risk notion provide deterministic bounds merit capture situation training dataset randomly chosen hence merit describe cases one perform dedicated experiments systems consider training dataset random observations drawn according probability distribution given classifier expected risk defined sgn sgn sign function expected risk quantifies capability classifier generalize training dataset several studies devoted provide upper bounds expected risk given family classifiers interesting bound linear classifiers serves discussion reported hereafter theorem let belong sphere radius consider class functions defined kwk constant probability distributions probability least randomly vectors classifier margin least examples expected error larger log log constant related dimension linear classifiers explicit expression found theorem shows one quantify generalization performance linear classifier function margin obtained training dataset show margin svm classifier related margin solution permits quantify generalization performance svm classifier terms dynamics systems one wishes classify analysis follows holds normalized data briefly comment later general case consider normalized training data kyk feature vector holds consider optimal solution computed respect whose existence uniqueness ensured solution assume without loss generality kwd let represent margin corresponding solutions respectively min kwi holds iscussion irrespective training dataset solution margin maximizer next result shows using normalized data bounded positive quantity depends solely dynamics systems one wishes classify refer reader definition principal angles theorem bound classifier margin let assumption satisfied consider arbitrary training dataset vectors let let unique solution optimization problem computed respect holds orders dynamical systems squared sine smallest principal angle subspaces spanned columns observability matrices proof theorem since sufficient bound term kwm satisfies kwm vec vec denotes frobenius norm inequality follows kqi projection matrices consider term assume without loss generality minimum attained ltr holds min third equality comes fact ltr implies view assumption shown theorem hence proof follows theorem permits bound risk classifier based dynamics systems one wishes classify formalizes intuition risk bound becomes smaller dynamics systems classify distant one another fact higher larger coefficient inclination subspaces spanned columns maximal two spaces orthogonal data normalization ensures bounded away zero property hold general since trajectories dynamical systems arbitrarily close origin nonetheless one obtain similar bound adding extra term accounts training data margin theorem intuitive incorporating models beneficial classification task fact obvious also analysis shown paper since assumption classifier outperform classifier models exact data however mentioned approach introduces issue quantify effect modelling inaccuracies bypasses intermediate step identification thus potential applicable also accurate models difficult obtain hereafter briefly elaborate point also connection number open problems linear systems noisy observations observations corrupted noise identification may difficult require even linear systems many careful provisions contrast svm formulation address problem rather straightforward manner consider svm subject min parameter col vector slack variables account fact noise may render data one hand exist many studies aimed quantifying generalization performance svm also formulations hand even noisy data one still give separation measure linear systems margin function dynamics ratio theorem means even noisy data one quantify generalization performance svm classifier along lines section point reinforces idea linear systems polynomial kernels good candidates remains unclear better performance obtained different kernel functions nonlinear systems classification nonlinear systems another situation svm formulation bypass difficulties related system identification related capability svm find separating surfaces straightforward manner kernel trick interestingly even nonlinear case one define separation measure dynamics theorem however contrast linear case measure involves principal angles observability subspaces nonlinear systems measure involves often difficult relate underlying dynamics like linear systems deeper understanding point would beneficial figure types kernel functions suitable given class nonlinear systems fact theoretical studies classification nonlinear systems recent also within computer science approaches appear largely diversified example see interesting recent account yet also context question kernel functions suitable given class dynamics unresolved classifiers thanks simple form classifiers potential used applications thus control purposes fact noted authors introduce term classifier describe framework classifier modify online control action looking process data notion generalization considered characterizes capability classifier work small perturbations system vector fields analysis results promising remains unexplored handle general forms uncertainty ideally one provide bounds risk function classifier hold possible system trajectories relate bounds stability properties difficulty much theory generalization properties classifiers developed probabilistic setting robust stability desirable guarantee deterministic bounds interestingly architecture considered regarded supervisory control system supervisory control supervisor selects based process data candidate control law hypothesis appropriate given time done assigning candidate law score function cost function quantifies performance level achievable control law given process data supervisory control one often uses term cost detectability measure capability supervisor learn data appropriate control law even process match models used design control laws fact supervisor classifier cost detectability measure generalization performance idea adaptive control indeed new firm theoretical link realm machine learning yet established vii umerical xample consider system transfer function laplace variable improve performance system controlled proportional controller negative feedback goal design classifier detect loss control effectiveness denote system system hence transfer functions respectively finally denote corresponding systems sampling time systems using previous notation let length observation sequences number training data let number data used validation order assumption satisfied one needs condition assumption holds generic choice focus svm classifier classifier always correct assumption note classifying two systems one observe figure instance ideal case one theorem cepstral distance equal report table simulations results various choices validation data svm classifier computed theorem training validation test trajectories generated random initial conditions zero mean variance error validation test defined sgn rtest variation parameter rtest variation parameter rtest variation parameter rtest table numerical results svm classifier one sees classifier performs well reasonable choices parameters particular dependence goes zero performance clearly random guess one hand remarkably classification becomes accurate exactly soon one approaches theoretical bound performance saturates decrease error need increase one achieve error dependence performance variations less evident case suggest less critical intuition random initial conditions generically ensure excitation system dynamics even examples may suffice iii dependence sampling time play major role long avoid case tend identity matrix case tend zero matrix rag replacements psfrag replacements sampled trajectories sampled trajectories fig output trajectories two systems left right figures report normalized trajectories system generated random initial conditions zero mean variance viii oncluding emarks considered problem classifying trajectories generated dynamical systems looking approach common approach control engineering well approach based 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control vol safonov tsao unfalsified control concept learning ieee transactions automatic control vol
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sentence object notation multilingual sentence notation based wordnet abdelkrime aries djamel eddine zegour walid khaled hidouci ecole nationale informatique esi ini algiers algeria emails aries zegour hidouci jan abstract representation sentences important task used way exchange data interapplications one main characteristic notation must minimal size representative form reduce transfer time hopefully processing time well usually sentence representation associated processed language grammar language affects represent sentence avoid notations come new representation use words meanings done using lexicon like wordnet instead words use synsets syntactic relations universal much possible new notation called ston sentences object notation somehow similarities json meant minimal representative syntactic representation also want readable easy created simplifies developing simple automatic generators creating test banks manually benefit used medium different parts applications like text summarization language translation etc notation based languages arabic english franch japanese cases languages agree one representation also given diversity grammatical structure different world languages annotation may fail languages allows future improvements keywords sentence annotation sentence structure multilingual languages data exchange languages knowldge representation natural language processing introduction tagging sentences important task natural language processing one famously known methods syntactic tagging main idea detect different parts structure sentence nominal phrases verbal phrases nouns verbs etc structure expressed using languages like xml json etc bertran recasens problem syntactic tagging dependency processed language indeed good way system destined specific language comes multilingual systems better come way represent sentence structure independently languages semantic representation sentences one solution problem meaning sentence something beyond languages related different concepts words different relations words way describe concept given language example words arabic tree english arbre french japanese refer concept concept wordnet miller defined tall perennial woody plant main trunk branches forming wordnet large lexical database english nouns verbs adjectives adverbs grouped sets cognitive synonyms synsets expressing distinct concept url https distinct elevated crown includes gymnosperms angiosperms idea semantic representation represent words concepts language link words concepts semantic relations concepts extracted sentence uchida banarescu great gather concepts different languages make links one also detect semantic relations concepts sentence automatically need large amount knowledge processing power idea propose simple language help transfer information sentences applications example use send sentences summarization system translation system allows create known summarization language simple unambiguous developers easily create tools encode specific natural language one must minimal size minimize transfer time also processing one one main characteristics insist readability helps create ston representations manually end want represent syntactic relations sentences taking mind multilingualism aspect must insist term multilingual means using several languages represent world languages study interested four languages arabic english french japanese non latin scripts namely arabic japanese provide rest paper organized follows section presents related words subject text annotation section describes main proposition different parts ston section addresses cases adpositional phrases relative clauses comparison section solutions proposed concerning coordination references proper names verbs two objects complementizer passive voice section reserved discuss work grounds contribution comparison works benefits limits finally section reserved conclusion future improvements related works sentence structure represented using languages xml json lets take xml example represent sentence specify roles dtd file would contain different structures sentence subject object verb tense verb etc example recasens uses xml represent sentences order annotate corpora spanish catalan used ancorapipe bertran create corpus figure represents example xml annotation sentence europea means european commission announced use xml tags express nouns verbs nominal phrases etc tag properties suj organization agt spec article organization proper aunciar indicative main past figure example xml sentence annotation recasens interlingual machine translation uses representation sentences medium source language destination one kant mitamura example system uses interlingua represent sentences translation kant interlingua structural representation scheme american library association library congress set standards romanization represent texts writing systems using latin script url https action rep remain form finite tense past mood declarative punctuation period impersonal passive expletive subject predicate argument structure semrole generic interlingua position final clue translation object object semrole object rep time unit number singular reference definite distance near person third theme object semrole object rep default rate person third unit number singular reference definite predicate adjective phrase semrole property rep closer degree positive object unit number singular reference person third figure representing sentence default rate remained close zero kant interlingua czuba using nested frames interlingua frame contains head concept series pairs semantic slots containing nested interlingua frames concepts symbols begins asterisk followed concept prefix defining category action drive kant interlingua distinguishes many categories action object manner proper name etc uses certain features input text modality aspect discourse markers etc order generate grammatically accurate output texts mitamura semantic roles relations frames agent theme etc figure represents kant interlingua representation sentence default rate remained close zero contains concepts etc features form tense etc semantic roles theme etc despite interlingua kant shows limitations representing sentences czuba close english semantics multilingual intended english languages translation also designed technical domains thus vocabulary limited subset meanings universal networking language unl knowledge representation language represent meaning texts without ambiguity developed intermediate multilingual language used internet uchida uchida zhu major commitments unl following must represent information represent meant said said must language computers like html sgml xml etc must unl representation must depend implicit knowledge explicitly codify information unl agt affect icl present human icl animal obj affect icl present environment icl abstract thing figure representing sentence human affect environment unl must primary objective serve infrastructure handling knowledge used different tasks translation text mining multilingual document generation summarization etc must independent particular natural language unl defines tags structure text document paragraph sentence concepts represented called universal words uws natural language etc word dictionary unl expressions based binary relations binary relation two uws parameters also unl specifies attributes represent information conveyed natural language grammatical categories tense mood aspect number etc figure represents unl formulation sentence human affect environment sentence starts tag unl followed two binary relations relation agt defines thing initiates action example human plural noun one present tense action affecting relation obj defines thing focus directly affected event state example environment direct object action affecting indeed unl minimal size multilingual background covers large number languages however shows limitations comes relations must choose boguslavsky claims selection relations differs team team sometimes ambiguous one choose example phrase freedom described purpose relation pur beneficiary relation ben another team martins raises issues concerning unl one issues proper nouns treated permanent uws temporary also concept represented simple compound example physiological need food represented using hunger represented compound ness technically speaking say hard design encoder natural language unl parsing sentence find relations part sentence sometimes ambiguous select two relations manually let alone selecting automatically recent representation language amr abstract meaning representation proposed banarescu semantic representation language designed represent meaning english sentences text represented graph leaves labeled concepts boy means instance called concept boy see figure concepts english words boy propbank framesets kingsbury palmer special keywords special entity types worldregion etc quantities etc logical conjunctions etc relations concepts frame arguments following propbank conventions general semantic relations relations quantities relations relations lists amr light weight annotation heavily based english limitations comes inflectional morphology tense number deeply capture many relations also relies propbank framesets subject propbank constraints instance instance instance boy boy logic format amr format graph format figure amr representation sentence boy wants banarescu wornet wornet ston wornet wornet applications ston figure example ston purpose ston representation ston intended represent sentences syntactic structures multilingual context figure represents vision ston may used use lexical parsers several languages get different parts speech use create ston representation using ston generator wordnet lexicon language representation used multiple multilingual applications like example mentioned earlier ston parser extend handle language apart using realizer language generator reproduce readable text destination language representation look different parts sentences either actions roles actions dynamic part sentence action contains verb morphological specifications like tense negation etc roles generally nominal phrases purpose action agents themes places times etc roles nominal phrases play roles action agents themes places times etc case agents action either active passive voice themes undergo experience action example table fat man delicious food roles played action eating happened present first phrase plays role agent second plays role theme representation ston list roles appear sentences role contains following attributes see figure name role order reference table roles actions arabic english french japanese language sentence arabic english french japanese kulu fat man eats delicious meat gros homme mange viande dansei oishii niku syn synset qnt def adj adj syn synset adv synset adj adj adj adj figure roles representation syn synset number noun lexicon wordnet case set adjectives blocks modify noun adjective block contains synset set adverbs synsets qnt quantity describes amount noun example apples quantity default equals number term plural languages like arabic dual numbers simply represented number language generator handle ordinal numbers add number example means second def defined many languages arabic english french ability identify noun many using definite articles actions verb sentence clause represents action languages like arabic nominal sentences include verb nevertheless add verb express action like examples table first sentence composed subject noun predicate adjective second one subject noun predicate prepositional phrase sentences use copula action subject considered agent predicate theme case third sentence active participle origin verb movement verb like consider present continuous haak represent actions define set actions blocks block contains attributes attributes generally related verb since action verb contains also links agents themes etc main attributes contained action syn compulsory attributes see figure name action order reference syn synset number verb lexicon wordnet table example nominal sentences arabic arabic english noun adjective noun prep noun noun active participle man fat man market man going market dhahibun act act syn synset tns prg neg mod agt thm adv adv syn synset adv synset adv adv act act act act figure actions representation agt set role ids action thm set role ids receive action set adverbs blocks modify verb adverb block contains synset set adverbs synsets verb specifications tense tns progression prg perfect aspect prf negation neg modality mod tense past present future absence tense means action must try languages future tense arabic japanese case detect tense using adverbs tomorrow temporal prepositional phrases next year languages define past future example arabic tense called future expressed using auxiliaries near future use attached verb present tense soon far future use detached verb present tense later since detected using adverbs soon later ignore two aspects mostly used occidental languages progressive perfect perfect aspect refers actions prior time consideration viewed already completed ston representation tense imperfect unless add prf progressive aspect situation table example consecutive actions language sentence arabic english french japanese thumma thumma karim went market came back home watched karim est est revenu maison ensuite itchiba itte modotte terebi typ act figure sentences representation ston verb motion interval time likely action progressive unless add prg modality case express possibility may admissibility obligation must modal verb used express future tense case advice see doctor prohibition smoke certainty must rich since lives permission leave lack necessity anything represented using three modal verbs sentences representation sufficient since sentences contains consecutive actions table observe three actions going market coming back watching represent sentence three sentences instead loose information actions consecutive mention specify actions main ones actions relatives roles actions sentence part lists sentences blocks block contains type sentence affirmation aff exclamation exc question qst imperative imp attribute act list references actions sentence annotation sentences ston illustrated figure relations many relations clauses phrases sentence attributes agents themes action represent simple sentences complicated sentence contain adpositional phrases relative clauses etc figure shows view relations roles actions relation expressed adpositional relations example man car adpositional relations also used express relation car another relation relation found relative clauses man driving like adjectives relative clause modify describe nominal phrase noun described relative clause sentence man strong late example different strong man first sentence adds information past quality strength relation found adverbial clauses see comparison two roles another issue equal one less share verb work adjective stronger structure added order allow types relations man car see adverbial adpositional car adpositional role action man driving relative figure relations roles actions language arabic english french japanese table example relative clauses sentence akala yashrabu man ate meat drinking water homme qui viande boit eau tabeta dansei mizu nonde relative clauses relative clauses little challenging represented using notation notation starts roles actions since relative clauses actions describe role represented role table man drinking water one ate meat earlier one proposition use ulterior referencing roles reference action relative relationship role relative clause must specified example text table represented figure looking relative pronouns see table consider main types relative phrases subject possessive direct object indirect object categorization considered person thing difference taken consideration text generation task subject type sbj indicates main clause subject relative one example phrase man ate meat main clause man subject relative one ate meat possessive type pos indicates noun relative clause possessed main one example phrase man whose car expensive describes man possess expensive car object type obj indicates main clause direct object relative one example phrase man saw yesterday describes man object seeing deduce following information saw man yesterday indirect object clauses starts prepositions various meaning clauses follows meaning prepositions case form types using keyword followed adpositional relation next subsection example town came describes town indirect object coming deduce following information table relative pronouns person thing place subject object possessive whose whose time reason man syn rel typ sbj ref ate rel meat syn water syn act act ate syn tns obj meat act act drink syn tns prg agt man thm water act act figure example ston representation case relative clauses came town relative adverbs replace preposition example town met equivalent town met one type reason rsn expressed relative adverb adpositional phrases adpositional phrase includes prepositional phrases postpositional phrases circumpositional phrases usually used describe time place action arabic english french use prepositional phrases japanese uses postpositional phrases related action works role mother boy objective define relations possible represent meaning adpositions matter define relations extended future ago amount time back past bought years ago frm origin change state came algeria existence particular time place situation wake born snc period past till particular time worked since destination location person etc going give waited till noon amount time objective sleep hours foundation support war bef time earlier another place front something always sit front aft time another place back something behind always sleep towel behind door later specified time place besides something agent consider prepositions next besides near finish walks diffraction crystals ins something inside something else present inside something outside something else outside home blw place something treat swims abv place something treat walked btw place two things museum two thr one side another surrounded make difference across walked subject something connected something else borrowed book mathematics wth together involved possession belonging leafs tree relation used represent compound nouns printer cartridge cartridge printer role works und situation exam lot english used express exact wide locations times respectively japanese difference exact wide range locations times another aspect used distinguish prepositions first one used express existence something someone place dog second one used express place action takes place dog barks unfortunately representation take aspects consideration nevertheless believe solved text generation task case differentiation test noun generate right preposition likewise japanese postpositions use verb adverbial clauses adverbial clause dependent clause functions adverb representation represents relation action another relations used express adverbial clauses whn specific time listens whl period time listens whr place started condition come purpose tries bcs reason angry likes thg concession come although like lik manner job ftr time start job wake bfr time came door table example sentences comparative language arabic english french japanese sentence atwalu min idu aqalla min karim taller brother karim helps less brother karim est plus grand que son karim aide moins que son yori takai yori tetsudau sukunai act act cmp cmp typ adj synset ref cmp cmp act act figure comparison block comparison comparison little bit tricky mostly includes adjective shared two roles also action instance sentence karim taller brother adjective tall property karim bother represent adjectives two roles relation superlative better represented action sure relation comparison represented one roles karim reference bother cases comparison include adjectives verb instead example sentence karim helps less brother contains comparison action adjective end comparison must represented action rather role table contains two examples quoted previously must point arabic example sentence karim helps less brother fluent order fluent sentence must aqallu adatan min literally translated karim less helpful brother japanese example direct translation karim less help brother also use yorimo sukunai nagara means literally karim helps despite less brother form polite matters comparison means less represent cmp block comes text generation task language handles generated fluently idea use block cmp action block see figure three types comparison comparative superlative equality add less comparison five types less comparative least superlative equal first parameter comparison would agent second reference comparison block ston representations table examples illustrated figure act act syn tns cmp cmp typ adj ref brother cmp cmp agt karim act act act act help syn tns cmp cmp typ ref brother cmp cmp agt karim act act karim taller brother karim helps less brother figure example ston representation comparison issues solutions notation represent lot sentences ambiguities want represent others handle show problems solutions context ston coordination references one problem represent coordination references principally two main coordinations disjunction conjunction limit ambiguity representing sentence want use either disjunctions conjunctions inverse example mother son ate food father son ate food two sentences aggregated two ways mother son father son ate food disjunctions conjunctions mother father son ate food conjunctions disjunctions first one sounds appropriate second moreover second sounds like either mother alone father son ate food notation agents themes must changed taking late example agents sentence represented agt mother child father child proper names proper name phrase identifies one unique entity class entities example london distinguished common noun city specific proper names people abdelkrime aries locations algeria organizations esi etc problem proper names represent inside role ston heavily based wordnet synsets lexicon word must one synset represented proper names already synsets wordnet cities city jijel example algerian city synset number wordnet many proper names persons names exist wordnet one solution add attribute nam role section contains named entity ston role must always synset pronoun afford much information role proper name specify hypernym synset instance sentence karim lives jijel see figure two proper names karim jijel afford synset person city respectively karim syn nam karim jijel syn nam jijel act act lives syn tns agt karim rel rel typ ref jijel rel rel act act typ aff act lives figure ston representation sentence karim lives jijel table example sentences two objects language sentence arabic english french japanese man gave boy gift homme cadeau enfant okurimono verbs two objects languages like arabic english verb two objects without using preposition table represents example sentence man gave boy gift bold phrase language represents indirect object use prepositions indirect object english sentence man gave gift similarly arabic sentence hadiyyatan use adpositional phrases represent relation indirect object pronouns first time thought use references original role instead personal pronouns generation phase generate role referred many times applied normal situations information including anaphoric relations situations want represent sentences generated extractive summarization example may lot trouble recovering types relations also boost analysis natural language ston generation ston natural language tasks pronouns need pronouns classified using many features table represents eight features used pronouns classification according seah bond based features present pronouns using two attributes typ type pronoun encoded characters demonstrative pronoun etc subjective personal pronoun etc objective personal pronoun etc possessive pronoun first person second person third person singular dual plural defined number head table pronouns features accoding seah bond demonstratives entity time manner person place reason thing personal quantifier number gender case type formality politeness proximity dual plural singular feminine masculine neuter objective possessive subjective assertive elective negative reciprocal universal interrogative reflexive formal informal polite distal medial proximal syn novel typ ptsmfn qnt first figure ston representation role first novel female male neuter even source language attribute sex objects arabic french consider neuter example arabic chair masculine feminine french reserved formality politeness rude casual formal polite languages formality level pronouns choose use formal default proximity distal medial proximal defined ref reference role related pronoun pronouns attribute accompanied synset afford compact format instance figure represents case pronoun noun packed together pronoun represented ptsmfn means possessive third person singular masculine formal without proximity representation better transforming clause first novel represented passive voice passive voice specific representation ston sentence like karim ate apple see table representation apple eaten karim choice using active passive voice decided text generation task sentences agent apple eaten simply define agent representation case generate sentence use passive voice table example sentences passive voice language sentence arabic english french japanese ukilat min apple eaten karim pomme par karim act act represent syn tns agt ston thm sentence act act hope syn tns agt karim thm represent act act figure actions sentence karim hopes ston represents sentences complementizer phrase complementizer phrase subject object representation rather consider agent theme figure represents example ston annotation case complementizer phrases phrase ston represents sentences theme karim agent action hoping structure verb infinitive solved complimentizer sentence like want seen theme wanting ston tools corpora ston notation intended used nlp applications concern sentence syntax multilingual context used language generation intermediate language machines implies use mean text translation text summarization etc parsing ston language would fast since grammar well defined blocks references challenging task create tools parse text ston generate text ston corpora testing must started annotate materials already annotated unl annotation process annotating materials helpful future especially intend use ston applications since grammatical structures able represented transformation like apposition replaced steps follow order good annotation ston begin represent roles nominal phrases dependent role must last example sentence statue liberty represent role liberty role statue relation first one adjectives represented alone predicative adjectives create role noun subject adjective example man friendly would represented man friendly man proper names exist wordnet put synset cairo put value attribute nam spaces must transformed underscores afford synset type city person animal dog cat etc specific type better transform enumerations role relation example sentence many jobs web designer engineer teacher many jobs web designer engineer teacher find prepositions talk try find similar relations example preposition well considered simple omit chronological order indicators first finally represent consecutive actions try transform expressions order fit ston representation example much experimental much experimental comparison exist roles exists actions solve problem add expression amount comparison must adjective example author articles author amount articles table statistics annotated texts naguib mahfouz bio louis broglie bio original texts statistics sentences words unl statistics uws redundancy relations attributes synsets redundancy roles actions relations ston statistics started annotate already annotated unl check annotated texts nolporas aims create corpora mostly ston table represents statistics two biographies annotated biographies contains long sentences average words synsets ston equivalent uws unl number less use references roles roles sometimes repeated sentences need repeat descriptions relations unl corresponds relations adpositions relatives adverbials references themes agents concerning manual annotation sentence took half one hour since choose different senses right relations annotation process must automatic interesting started working another project called natural language parser aims generate sentence representations including ston texts uses open source text parsers stanford parser chen manning syntactic parser many languages english arabic etc licensed gpl license currently working transforming english text ston languages added finishing mean time handle sentences form subject verb object preposition noun ston parsing created parser ston project called sentence representation aim implement parsers different sentence representation languages taking ston primary focus parser abstract class extended implement cases parser finds role adjective action etc test speed parsing used processor ghz cores openjdk parser nothing finds different parts calls functions treat biography naguibmahfouz louisdebroglie executed parser times calculated time parsing milliseconds naguibmahfouz biography parsing took average almost per sentence louisdebroglie biography parsing took average almost per sentence generation generate sentences ston representation currently working another project called natural language generator nalengen aims generate text different sentence representations uses sentrep uses open source text realizers simplenlg gatt reiter http nolporas project https source https sentrep source https nalengen source https nalenpar source table example sentences generation english french born cairo naguib mahfouz began writing seventeen first novel published ten written egyptian revolution july stopped writing several years english generated text naguib mahfouz given birth cairo began writing years first novel published novels written revolution egyptians july discontinued writing several years french generated text naguib mahfouz que caire quand lui son premier nouveau nouveaux plus ont avant tour des july lequel lui pour des realization engine english licensed mpl license adapted many languages german bollmann french vaudry lapalme brazilian portuguese oliveira sripada map wordnet synsets languages use open multilingual wordnet bond foster mapping complete many languages result generation may fail synset found tried generate english french text ston annotation examine table example two first sentences naguib mahfouz biography resulted text fair english little bad french work done address issues found sometimes mapping languages done automatically leads errors translating concepts example found concept cairo mapped oss caire french see example many concepts wrong synsets contain many words words adequate others propose method order choose words based frequency use example generate directly ston changes must done fluent texts example predicative adjectives expressed role noun sentence child happy represented child happy child limitations challenges although ston represent wide range sentences shows limitations sometimes variation languages prevent representing sentences properly languages always find syntactic representation meaning instance considering table adjective hungry exact translation japanese fact japanese sentence literally means stomach emptied adjective translated noun french sentence literally means hunger arabic means thing sentence composed use sentence pronoun verb hungry conjugated past also since ston based wordnet synsets limited concepts extracted english language want represent sentence arabic example find wornet concepts close meaning sentence words ston deal text structures like paragraphs lists tables etc many languages complex predicates represent meaning singleton verb instance gave baby bath bathed baby meaning unfortunately ston fully semantic representation means handles two different forms sentence like purpose teach mathematics develop physics represented without problem case sentence purpose teach develop physics use redundancy represent purpose teach physics develop physics discussion ston representing different parts sentence independently structure natural languages meant transport sentences information different applications programs represents syntactic relations different parts sentences way syntax fails keep multilingual aspect uses semantic relations instead understand ston better know language table example sentences pos different languages language sentence romanization arabic english french japanese criteria jaw hungry faim table comparaison ston annotation formats kant unl amr ston objective interlingua machine translation english technical manuals represent meaning texts without ambiguity used language web write meanings english sentences represent sentences multilingual way without basing much semantics used interchange format applications aspects semantics morphological aspects tense etc morphology semantics pragmatics semantics morphological aspects tense etc syntax morphological aspects tense etc semantics syntax fails multilingual concepts uws unl ontology propbank frames wordnet synsets dependency domain dependent language independent english language dependent language independent relations propbank relations readability readable difficult follow less readable less readable difficult big text represent relations parts speech semantically even relations relative clauses example allow represent relations like unl beneficiary relation purpose relation format storing texts open document format odf microsoft office format based xml similarities differences ston representation languages concise comparison kant unl amr ston given table check meaning unl representation mostly represents meaning amr hand used represent meaning lacks morphological aspects verb tense kant ston annotations less depending semantics two previous ones objective represent sentences minimum cost time processing effort represent relations like even mean many things london etc semantic representation powerful tool allows represent could understood sentence eliminate redundancy sentences example banarescu soldier afraid battle soldier feared battle soldier fear battle act syn tns agt soldier thm rel rel typ ref battle rel rel act act feared syn tns agt soldier thm battle act act syn tns agt soldier thm fear rel rel typ ref battle rel rel act figure ston action representation sentences meaning sentences meaning therefore representation must contrast unl amr ston deep semantic relations comprehend sentence whole figure shows ston representation sentences clear representation makes difference verbs nouns adjectives four languages based different ontologies lexicons represent concepts kant annotation based concepts defined especially kant system generally limited technical reports domain likewise unl defines concepts base called unl ontology concept referred universal word amr ston use propbank wordnet respectively limited two bases amr kant based english unl ston seeks multilingual comes multilingual aspect uws unl powerful mean time use wordnet represent different concepts unfortunately till nowadays mapping languages complete reason problems generating french text ston annotation synset found readability important want create sentence representation manually check automatic generation kant annotation readable three representations ston developed machine language unl even want allow space readability helpful case want test system uses ston easier create test banks conclusion work proposed language ston aimed represent sentences structures multilingual context ston based somehow json representation adjustments speed parsing based assumption anything sentence either role action relations representation uses syntactic structure noun definition verb tense subjects objects etc semantic relations time place relations support multilingualism use concepts instead words case use wordnet synsets intention use ston mean communication different applications specifically language intended used automatic text summarization ston far beyond complete perfect still improvements made future problem words syntactic alignment based wordnet concepts limited english lot concepts exist english exist languages exploiting larger semantic network knowledge base like navigli ponzetto may improve representation http acknowledgement special thanks hisham omar valuable feedback concerning japanese examples references manuel bertran oriol borrega marta recasens soriano ancorapipe tool multilevel annotation issn marta recasens coreferentially annotated corpora spanish catalan language resources evaluation issn doi url http petr homola natalia klyueva annotation sentence structure language resources evaluation doi url http george miller wordnet lexical database english commun acm november issn doi url http hiroshi uchida meiying zhu tarcisio della senta unl gift millennium url http laura banarescu claire bonial shu cai madalina georgescu kira griffitt ulf hermjakob kevin knight philipp koehn martha palmer nathan schneider abstract meaning representation sembanking proceedings linguistic annotation workshop interoperability discourse pages sofia bulgaria august association computational linguistics url http teruko mitamura eric nyberg jaime carbonell efficient interlingua translation system multilingual document production proceedings third machine translation summit krzysztof czuba teruko mitamura eric nyberg practical interlinguas used difficult analysis problems proceedings workshop interlinguas hiroshi uchida meiying zhu language infrastructure toward knowledge infrastructure special speech pacific association computational linguistics pcling igor boguslavsky lexical issues unl universal networking language advances theory applications pages martins lexical issues unl universal networking language panel ebsco ebook academic collection cambridge scholars publishing isbn url https books paul kingsbury martha palmer treebank propbank language resources evaluation haak verb literary colloquial arabic functional grammar series mouton gruyter isbn url https jie seah francis bond annotation pronouns multilingual corpus mandarin chinese english japanese joint acl iso workshop interoperable semantic annotation pages reykjavik iceland danqi chen christopher manning fast accurate dependency parser using neural networks proceedings conference empirical methods natural language processing emnlp pages doha qatar october association computational linguistics url http albert gatt ehud reiter simplenlg realisation engine practical applications proceedings european workshop natural language generation enlg pages athens greece march association computational linguistics url http marcel bollmann adapting simplenlg german proceedings european workshop natural language generation pages nancy france september association computational linguistics url http vaudry guy lapalme adapting simplenlg bilingual realisation proceedings european workshop natural language generation pages sofia bulgaria august association computational linguistics url http rodrigo oliveira somayajulu sripada adapting simplenlg brazilian portuguese realisation proceedings international natural language generation conference inlg pages philadelphia pennsylvania june association computational linguistics url http francis bond ryan foster linking extending open multilingual wordnet proceedings annual meeting association computational linguistics volume long papers pages sofia bulgaria august association computational linguistics url http roberto navigli simone paolo ponzetto babelnet automatic construction evaluation application multilingual semantic network artificial intelligence
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apr stably polynomial automorphisms commutative rings shigeru abstract say polynomial automorphism variables stably tame subgroup variables contained subgroup generated affine automorphisms variables paper give conditions stably polynomial automorphisms introduction let commutative ring characteristic polynomial ring variables gan autr automorphism group identify gan elements composition defined regard gan subgroup identifying gan unique extension defined say gan affine gln set aff gan affine define gan set call haff tame subgroup elements said tame partly supported jsps kakenhi grant number subsets elements group denote subgroup generated contains haff holds derksen thm remark derksen theorem requires generated space assumption fact unnecessary field bodnarchuk proved similar result general situation different prime showed haff conjectured finite field finite subset satisfies haff edo found class gan haff contains said field element thanks jung van der kulk easy find elements gan aff field prime case field first example automorphism found recall gan said stably tame belongs known automorphisms stably tame bew following analogue stably tame automorphisms definition say gan stably haff contains equivalently haff contains clearly automorphisms stably field exist elements stably cases purpose paper study elements gan stably contains infinite field necessary sufficient condition stably cotameness corollary paper organized follows main results stated section three key results proved sections section study stably example also discuss technique useful contain infinite field main results since aff always assume unless otherwise stated take gan define generated aff definition aff following theorem holds commutative ring characteristic theorem gan stably following four cases contains unit contains contains contains exists satisfying next assume zero prime call xtnn good monomial following five cases exist mod iii exists mod exist mod exists mod let denote set coefficients good monomials appearing define ideal generated theorem assume zero prime gan satisfies stably throughout paper let field commutative kalgebra say satisfies degree condition degsxi degsxi denotes separable degree standard degree degxi polynomial separable degree polynomial one variable defined degree say gan satisfies degree condition satisfies degree condition ideal generated union define rxi satisfying degree condition since good monomial linear contained satisfies degree condition equal hence following theorem consequence theorem let field commutative gan stably satisfies degree condition stably theorems obtain following corollary corollary let infinite field commutative gan stably particular infinite field gan stably good monomial appears following three sections prove theorems proof theorem let commutative ring subgroup containing aff first study properties define identify permutation aff write following gan belongs belongs satisfy particular implies since belongs aff hence derksen showed generated aff haff holds generated space thm remark since first statement imply haff whenever lemma contains contains contains proof show contains since holds may assume monomial former case contains follows implies since contains assertion follows induction deg latter case contains hence implies similarly since contains follows contains following two implications hold conditions listed theorem lemma implies implies proof implies since unit aff hence contains similarly implies hence contains since unit implies contains prove theorem thanks lemmas suffices show haff contains write rxi aff product since contains contains hence contains since affine follows contains completes proof theorem proof theorem assume zero prime define ngn set gan good monomial appears ngn aff case following theorem obvious theorem ngn subgroup gan element ngn stably fact haff ngn holds prove theorem need lemma need zero prime consider standard grading said graded generated monomials graded recall identified substitution map defined forms monoid composition defined note closed operation holds lemma let graded closed composition gan subgroup gan proof since gan contains closed composition show belongs gan exists aff satisfies since closed composition suffices verify belongs suppose thep contrary belong write holds take minimal set let homogeneous components degree respectively belongs since belongs graded assumption minimality since belongs follows belongs implies belongs since holds thus belong contradiction therefore belongs remark hold closed composition example assume prime characteristic define xpn set let satisfy define graded xpi let prove closed composition using remark xpi first note implies since satisfy xpi xpi xpi contains hence xpi belongs clearly holds therefore let prove theorem prime clearly implies define satisfy mod mod hence good monomial similarly nonzero monomial good belong thus ngn equal vnn gan otherwise therefore ngn subgroup gan lemma note haff ngn contained otherwise since contains contains get last part theorem completes proof proper ideal induces element since surjective aff implies aff see haff implies haff gan hence theorem implies following corollary corollary let proper ideal char zero prime gan satisfies ngn haff holds char characteristic prove theorem assumption proper ideal prime contains char hence belongs ngn definition thus stably corollary affine implies stably finally remark contained hence thus monomial odd appears domain holds since field lemma assume odd coefficient belongs nilradical proof let prime ideal since domain characteristic two appear mentioned hence belongs proof theorem assume commutative prove theorem verify one theorem holds good monomial appears polynomial satisfying degree condition implies always exists satisfying take write xip regard contains distinct elements xip written combination linear algebra remark used prove following lemma lemma satisfies degree condition monomial appearing written combination proof prove lemma induction case clear ase sume write xip take monomial appearing exists appears since may find distinct written combination remarked choice monomial appearing note degsxj degsxj hence induction assumption written combination thus written combination therefore next define aff lemma xtnn good monomial appears xtnn monomial uxi cases iii case case proof case easy see monomial appears cases verified similarly let prove theorem assumption may psince find rxi follows good monomial appears coefficient satisfy degree condition since satisfies degree condition hence clearly belongs hence belongs lemma belongs monomial appearing belongs similarly lemma appears monomial xil xjl cases iii xil case case since belongs case proof lemma shows case well case must type hence contains therefore theorem holds type pror case follows nilpotent lemma since unit sum isptaken type hence contains therefore theorem holds type iii contains xil xjl contains contains hence contains thus contains therefore theorem holds completes proof theorem remarks showed neither affine mention expression slightly different original one due difference definitions composition since affine stably remark corollary belongs hence stably theorem following theorem theorem stably proof observe written define grading degw denote highest part set claim degw greater degw fact case checked directly case follows induction since since assumption claim holds respectively first three imply degxi hence satisfies degree condition since similarly monomial appears since see good mial type iii thus get therefore stably theorem finite field theorem might useful due degree condition case following technique may effective take aff set holds gan since belongs get following lemma lemma gan satisfy following condition belongs exist aff belongs assume prime take define aff put since degxi xdi rxji note ker xdi ker also ker contains generated xpi claim ker fact ker satisfies degxi hence degxi hqil holds thus get hqil induction fix set rxi xji xlnn since commute see contained rxi hence holds every gan using give sufficient conditions stably example assume belongs rxi hence satisfies belongs implies belongs lemma therefore stably theorem similarly belongs stably motoki kuroda showed stably cotame found hard decide whether stably note stably theorem tame generators problem asks gan context stably automorphisms ngn natural generalization aff following problem interest generalization tame generators problem prime characteristic see also generalizations problem prime hold gan hngn references bew berson van den essen wright stable tameness polynomial automorphisms regular ring adv math bodnarchuk generators tame invertible polynomial maps group internat algebra comput edo coordinates constructions classifications comm algebra edo kuroda generalisations tame automorphisms domain positive characteristic transform groups edo lewis affine automorphism group maximal subgroup tame automorphism group michigan math van den essen polynomial automorphisms jacobian conjecture progress mathematics vol basel boston berlin jung ganze birationale transformationen der ebene reine angew math van der kulk polynomial rings two variables nieuw arch wisk kuroda stably derksen polynomial automorphisms finite fields japanese master thesis tokyo metropolitan university january maubach willems polynomial automorphisms finite fields mimicking tame maps derksen group serdica math nagata automorphism group lectures mathematics department mathematics kyoto university vol kinokuniya tokyo shestakov umirbaev tame wild automorphisms polynomial rings three variables amer math soc smith stably tame automorphisms pure appl algebra department mathematics information sciences tokyo metropolitan university hachioji tokyo japan kuroda
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continuous monitoring norms data streams jaroslaw jian jelani nov november abstract streaming one sees sequence indices stream defines def sequence frequency vectors frequency vector seeing first items stream much work streaming literature focuses estimating function many applications though require obtaining estimates time every naively guarantee obtained devising algorithm failure probability performing union bound stream updates guarantee estimates simultaneously accurate good probability norm recent works shown union bound wasteful better space complexity possible continuous monitoring problem strongest known results work improve state art obtain via novel analysis indyk sketch introduction estimating statistics frequency vectors implicitly defined update streams defined abstract first studied flajolet martin studied distinct elements problem support size model support size equivalent number distinct appearing stream one goal streaming algorithms particular distinct elements problem well many others function estimation problems studied subsequent works minimize space consumption algorithm ideally using words memory note always trivial space algorithm storing explicitly memory two decades work estimating statistics frequency vectors streams remained dormant work estimating kxkp xpi streams integer since several works studied several problems perspective upper lower bounds including estimating kxkp necessarily integral kxkp empirical entropy quantities cascaded norms several others also general theorems classifying statistics frequency vectors admit streaming estimation algorithms taking dynamic data structural viewpoint streaming algorithms simply synonym dynamic data structures implied focus minimizing memory consumption typically striving algorithm using sublinear memory elements stream viewed updates frequency harvard university jblasiok supported onr grant chicago jianding partially supported nsf grant alfred sloan research fellowship harvard university minilek supported nsf grant career award onr young investigator award google faculty research award university vector seeing stream seen update causing change request estimate statistic query data structural language works cited previous paragraph provide guarantees following form queries starting fixed frequency vector executing fixed sequence updates probability output subsequent query fails say query fails say output good approximation particular made formal later many applications however one simply want answer one query end large number updates rather one wants continuously monitor data stream sequence data structural operations intermingling updates queries example one may threshold mind ever increases beyond data analyst alerted goal could achieved approximately querying every update determine whether updated frequency vector satisfies property indeed importance supporting continuous queries databases analogous model streaming recognized years ago several later works focused continuous stream monitoring application areas mind trend detection anomaly detection financial data analysis bio sensor data analysis one assumes query issued every update stream updates failure probability set union bound queries succeed streaming algorithms achieve space achieve failure probability point one achieve failure probability running instantiations algorithm parallel returning median estimate see example method increases space many problems estimation known least strict turnstile model update allowed positive negative promised times form space necessary nevertheless although improved space lower bounds given desiring answer single query fails probability shown necessary continuous monitoring problem one wants failure probability provide simultaneously correct answers queries intermingled updates fact contrary certain scenarios estimating distinct elements streams improved upper bounds given definition say randomized streaming algorithm provides strong tracking stream length failure probability time outputs estimate say provides weak tracking sup note monotonically increasing streams simply first tracking result aware outperformed median trick streaming roughestimator algorithm given estimating number distinct elements stream roughestimator provided strong tracking guarantee support distinct elements problem constant using space required answer single query strong tracking algorithm used subroutine main algorithm work approximating number distinct elements data stream without tracking known words memory achievable return value kxkp failure probability upper bound thus implies strong tracking algorithm space complexity tracking failure probability setting performing union bound work considered strong tracking variant streams restricted interval showed algorithms unchanged provide constant shows space bits achievable streams strong tracking space improvement standard median trick union bound stream length long small also showed update model allows deletions items turnstile streaming algorithm maintains linear sketch must use words memory constant showing median trick optimal restricted class algorithms different algorithm given strong tracking using space recently shown ams sketch though independent hash functions instead original independence proposed provides strong tracking space weak tracking space ams sketch provides weak tracking without asymptotic increase space complexity requirement correctly answer single query despite progress upper bounds tracking improvement tracking upper bound although bound provides improvement long streams provide improvement standard median trick case commonly studied case literature polynomially related contribution show indyk sketch derandomized using bounded independence provides weak tracking using words space also provides strong tracking using words space bounds thus improve space complexity achieved well range supported note known algorithm requires polynomial space even obtain single query variant problem notation use integer denote measure space words unless stated otherwise single word least bits let denote symmetric distribution scaled distribution thep property supported reals fixed vector equal distribution kxkp see reading distributions two vectors write denote coordinatewise comparison iff finite set write denote cardinality set preliminaries following lemma standard proof explicit constants found theorem lemma explicit constant depending also state results need lemma random variable finite variance corollary fixed vector vector independent random signs space written including space required store hash functions space bound assumes storage hash functions free proof follows inequality theorem theorem let sequence vectors let vector independent random signs sup universal constant theorem independent random variables every vector every pair kxkp theorem lemma let sequence independent random variables let exp min exp overview approach indyk sketch picks random matrix entry drawn according distribution maintains sketch current frequency vector sketch easily updated frequency vector changes observing index update sketch kxkp query answered returning median since storing memory explicitly prohibitively expensive generate entries row independent done seeds used generate rows independent also work discretized random variables take bounded memory together bounded independence discretization also performed allow store using low memory show instantiating indyk algorithm provides weak tracking guarantee failure probability analysis correctness algorithm follows let denote ith row first show result resembling doob martingale inequality namely section show fixed look evolution increases largest attained value good probability much larger median distribution typical magnitude counter end stream fact resembles similar facts shown independent rademachers entries though situation complicated fact random variables much heavier tails discussed section show previous paragraph implies weak tracking algorithm split sequence updates poly intervals frequency vector updates intervals order union bound poly intervals argue algorithm estimate good interval endpoints source extra factor space bound obtain failure probability union bound intervals hand within intervals counters change rapidly argument developed section finally section show given algorithm satisfying weak tracking guarantee one use get algorithm slightly larger space complexity argument already present one first identifies points input stream norm roughly doubles compared previously marked point intervals enough ensure algorithm satisfies weak tracking prefixes simultaneously order deduce algorithm fact satisfies strong tracking done union bound bad events opposed standard union bound bad events introduces extra factor space complexity compared weak tracking analysis first show two lemmas play crucial role weak tracking analysis lemma let fixed vector random vector independent entries drawn according universal constant proof let event note depends depend signs write independent random signs conditioning therefore holds corollary kxkp thus indicator random variable event integrating hand distribution moreover kxkp kxkp inequalities obtained via theorem lemma combining yields lemma let satisfy let independent entries marginally distributed according depending sup proof observe sup sup lemma directly implies hand write independent rademacher random variables independent let define vector coordinates particular sup sup condition sequence vectors satisfies assumptions theorem conclude sup moreover equivalently sup implies sup together yields sup take weak tracking kxkp section upper bound number rows needed indyk sketch boundedly independent entries achieve weak tracking lemma let sequence satisfying take random matrix entries drawn according rows independent entries within row independent every define median probability least proof consider sequence indices constructed inductively following way take smallest index given take smallest index exists one otherwise observe indeed kpp kpp kpp kpp kpp inequality kpp kpp kpp holds vectors every entries consider separately coordinate use fact nonnegative numbers api equivalently kakpp rearranging yields similarly numbers api true fixed sum maximized equal therefore kpp kpp kpp kpp kpp implies let define let vector random variables drawn according know hence universal constant similarly entries independent large constant depending thus theorem analogously hence let event ltj utj note fixed varying indicator random variables events independent thus theorem exp exp taking get hence union bound hold simultaneously except probability since number events let event construction sequence norm invoke lemma deduce eij pick sufficiently large small enough eij therefore fixed finally theorem exp exp hence sufficiently small exp hand sufficiently large exp invoke union bound deduce probability least following event holds know probability least simultaneously events hold show events hold universal constant indeed consider let assume event satisfied know event satisfied know indices triangle inequality yielding kvtj similar reasoning deduce implies median range words finally also construction sequence claim follows rescaling constant factor lemma algorithm implemented using bits memory store fixed precision approximations counters bits store proof consider sketch matrix lemma random entries rows independent entries within row independent moreover let pick consider discretization namely entry equal rounded nearest integer multiple analysis identical one shows discretization significant effect accuracy algorithm moreover one sample nearby distribution using uniformly random bits therefore store matrix succinctly using bits memory storing seed random independent hash function interpreting seed independent hash function describing row corollary hence total space complexity storing sketch matrix succinct manner bits additionally store sketch current frequency vector need store every counter need bits counters thus following main theorem section theorem streaming algorithm provides weak bits memory tracking guarantees kxkp probability using strong tracking kxkp section discuss achieving strong tracking guarantee argument appeared reduction fact general shows monotone function strong tracking problem reduces weak tracking version problem smaller failure probability lemma let monotone functon mini standard basis vectors let streaming algorithm satisfying weak tracking sequence updates probability accuracy sequence frequency vectors algorithm satisfies strong tracking probability accuracy proof define smallest index larger index exists define note algorithm fail probability satisfy conclusion theorem particular sequence vectors every probability estimate output algorithm time union bound deduce except probability construction sequence know every take smallest claim follows theorem streaming algorithm provides strong tracking guarantees estimating frequency vector probability multiplicative error space usage bounded bits proof follows lemma lemma observing sequence insertions norm frequency vector bounded references alexandr andoni robert krauthgamer krzysztof onak streaming algorithms via precision sampling proceedings ieee annual symposium foundations computer science focs pages noga alon yossi matias mario szegedy space complexity approximating frequency moments comput syst vladimir braverman stephen chestnut universal sketches frequency negative moments decreasing streaming sums proceedings international workshop approximation randomization combinatorial optimization algorithms techniques approx pages vladimir braverman stephen chestnut nikita ivkin jelani nelson zhengyu wang david woodruff bptree heavy hitters algorithm using constant memory proceedings symposium principles database systems pods vladimir braverman stephen chestnut nikita ivkin david woodruff 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may effective separability finitely generated nilpotent groups effective twisted conjugacy separability nilpotent groups jonas mark may abstract paper initiates study effective twisted conjugacy separability finitely generated groups measures complexity separating distinct twisted conjugacy classes via finite quotients focus nilpotent groups main result shows polynomial upper bound twisted conjugacy separability allows study regular conjugacy separability case virtually nilpotent groups compute polynomial upper bound well another application improve work second author giving precise calculation conjugacy separability finitely generated nilpotent groups nilpotency class introduction let finitely generated group fixed automorphism say twisted conjugate exists equivalence relation equivalence classes called conjugacy class denoted say conjugacy separable conjugate pair exists surjective group morphism finite group conjugacy separable aut equivalently say twisted conjugacy separable interest twisted conjugacy classes arises many different areas mathematics reidemeister fixed point theory selberg theory algebraic geometry twisted conjugacy separability originally introduced formally studied conjugacy classes equal usual notion conjugacy classes similarly conjugacy separable equivalent usual notion conjugacy separable leuven kulak kortrijk belgium email fellowship research foundation flanders fwo purdue university west lafayette mpengito supported postdoctoral effective separability finitely generated nilpotent groups thus notion conjugacy separability natural generalization conjugacy separability allow conjugacy classes twisted particular finitely generated group contains characteristic finite index twisted conjugacy separable subgroup twisted conjugacy separable see thm contrast conjugacy separability closed respect finite extensions finite index subgroups thus twisted conjugacy separability gives important tool studying classical notion conjugacy separability since twisted conjugacy separability implies conjugacy separability conjugacy separability along residually finiteness subgroup separability residual properties extensively studied used resolving important conjectures geometry agol work virtual haken conjecture previous work literature understand groups satisfy properties instance polycyclic groups free groups residually free groups limit groups bianchi groups surface groups fundamental groups compact orientable virtually compact special hyperbolic groups shown conjugacy separable subsequently residually finite recently considerable activity establishing effective versions separability properties see main purpose article improve effective conjugacy separability results nilpotent groups establish effective twisted conjugacy separability class virtual nilpotent groups finitely generated group finite generating subset automorphism introduce function conjg natural numbers quantifies conjugacy separability specific value natural number maximum order minimal finite quotient needed distinguish pairs elements one varies nball also quantify general property twisted conjugacy separability via function tconjg start establishing norm automorphisms define tconjg natural number maximum value conjg one varies automorphisms satisfying automorphism identity map function conjg equal function quantifies conjugacy separability introduced case denoted conjg natural consequence conjg always bounded tconjg additionally see characteristic twisted conjugacy separable subgroup index exist automorphisms conjg bounded conjh finite generating subsets respectively see theorem hence given class conjugacy separable groups closed finite index subgroups may study conjugacy separability finite extension groups studying effective twisted conjugacy separability original group far previous papers studied asymptotic behavior conjg instance finitely generated nilpotent group second author demonstrated conjg addition finitely generated nilpotent group virtually abelian exists conjg demonstrated conjg finite rank free group surface group effective separability finitely generated nilpotent groups article first study effective twisted conjugacy separability class groups effective proofs conjugacy separability classes groups polycyclic groups fundamental groups compact orientable bianchi groups etc even context surface groups free groups good asymptotic lower bound conjg one provided effective residual finiteness state results require notation two functions write exists write nilpotent group write nilpotent class recall section first result precise computation conjn torsion free finitely generated nilpotent group finite generating subset theorem let torsion free finite generated nilpotent group finite generating subset exists conjn explicit expression given section lower bound provided thm upper bound improve techniques context two step nilpotent groups obtain precise asymptotic upper bounds purpose result improve results context two step nilpotent groups demonstrate techniques used proof main theorem next result main theorem article following theorem give first effective upper bound conjugacy separability nilpotent groups first effective upper bound twisted conjugacy separability finitely generated group theorem let torsion free finitely generated nilpotent group finite generating subset let aut exist natural numbers conjn kxks particular conjn tconjn proof theorem generalizes techniques context classes introducing notion twisted centralizer corresponding automorphism inn twisted centralizer equal group elements centralize modulo term lower central series using twisted centralizers may proceed induction step length generally similar result theorem virtually nilpotent groups theorem suppose virtually nilpotent group suppose finite generating subset aut exist natural numbers conjg kxks particular conjg tconjg effective separability finitely generated nilpotent groups result write conjugacy class element finite union right translates twisted conjugacy classes finite index characteristic finitely generated nilpotent subgroup apply theorem last result article extends work second author context finite extensions nilpotent groups theorem let virtually nilpotent group finite generating subset let exist conjg kxks particular conjg virtually abelian conjg infinite finitely generated nilpotent subgroup finite index order prove upper bound theorem apply theorem lower bound follow thm using fact every conjugacy class union conjugacy classes finish working asymptotic upper bounds aut integral heisenberg group also work examples fixed automorphism acknowledgements would like thank karel dekimpe suggestion study twisted conjugacy classes context finitely generated nilpotent groups second author would like thank advisor ben mcreynolds continued support background section introduce necessary definitions paper start fixing notation let group finite generating subset order finite group denoted write kxks word length respect denote identity element denote order element group ordg define commutator subset let subgroup generated set normal subgroup set natural projection sometimes write normal subgroup clear context let inn aut associated inner automorphism inn integer prime define largest natural number divides define subgroup generated powers elements characteristic subgroup define associated projection also define abelianization effective separability finitely generated nilpotent groups gab associated projection define center centralizer norms subgroups automorphisms associate norm finitely generated subgroups following definition definition let finitely generated group finite generating subset finite subset define max kxks finitely generated subgroup define khks min finite generating subset let two generating subsets group thath equivalently finitely generated subgroups indeed take generators since elements generate statement follows particular get following relation norms subgroups different generating subsets lemma let finitely generated group finite generating subsets let finitely generated subgroup similarly define norm morphisms finitely generated groups subsequently define norm automorphisms finitely generated group definition let finitely generated groups finite generating subsets let group morphism define max automorphism assume write equivalently smallest natural number kxks let two generating subsets show case note thus conclusion follows similarly one show generating subsets particular following lemma lemma let group two finite generating subsets exists constant every automorphism aut effective separability finitely generated nilpotent groups finitely generated groups separability let finitely generated group finite generating subset let arbitrary subset following define relative depth function min understanding exists definition say subset separable say finite quotient separates recall twisted conjugacy class aut section definition let finitely generated group let automorphism say conjugacy separable twisted conjugacy class separable say twisted conjugacy separable conjugacy separable aut let finitely generated group finite generating subset let automorphism quantify conjugacy separability relative fixed define following function conjg given conjg max kyks allowing vary able quantify conjugacy separability given group automorphism define conjg given conjg max conjg kxks finally obtain method quantify twisted conjugacy separability taking automorphisms norm define tconj given tconjg max conjg aut lemma let two finite generating subsets aut conjg conjg similarly tconjg tconjg proof similar lem see also lem lem observe conjg equal conjugacy separability conjg introduced lawton louder mcreynolds subsequently conjg tconjg aut conjugacy separable similarly tconjg twisted conjugacy separable subsequently separable aut particular conjugacy separable case effective separability finitely generated nilpotent groups nilpotent groups groups work paper nilpotent groups recall basic properties see thorough account theory nilpotent groups central series group sequence subgroups two special central series play important role studying def nilpotent groups term lower central series defined inductively def def term upper central series defined def denote associated projections inductively understood context definition say nilpotent step size minimal natural number equivalently write nilpotency class define hirsch length rankz define finite characteristic subgroup finite order elements torsion free finitely generated nilpotent group say nilpotent groups subgroup always finite index lemma let let subgroup index prime proof normal subgroups result trivial using fact nilpotent groups every subgroup subnormal see meaning exists sequence normal result follows every subgroup following subgroup useful assigning numerical invariants subgroups group morphisms definition let let subgroup define isolator denoted set exists subgroup additionally follows define notion determinant subgroup finite generated nilpotent group definition let group let subgroup define determinant det det morphism nilpotent groups write det det effective separability finitely generated nilpotent groups injective map abelian groups rank det equal usual determinant matrix representative respect fixed choice basis thus generalization usual notion determinant general class groups group morphisms map abelian groups rank equal det need one invariant nilpotent groups definition let let primitive central element one dimensional quotient associated quotient central primitive central element existence associated one dimensional central quotient guaranteed prop however uniqueness guaranteed motivating following definition definition let define smallest integer every primitive exists one dimensional central quotient associated value value found statement thm originally defined defn twisted centralizers twisted determinants section introduce twisted centralizers study projections subgroups finite quotients key concept understanding finite projections twisted determinant introduce definition nilpotent group nilpotency class fixed automorphism definition let nilpotent group let automorphism every define subgroups corresponding subsets call twisted centralizer corresponding automorphism note definition inn centralizer element hence name subsets play important role studying twisted conjugacy classes determine two elements twisted conjugate differ element effective separability finitely generated nilpotent groups lemma let nilpotent group automorphism every holds proof note exists last statement equivalently twisted centralizers used define maps definition let nilpotent group let aut define map lemma map mod group morphism proof let computation mod mod mod mod order understand relates following lemma lemma notations ker proof take ker mod equivalent induces injective map abelian groups lemma define subgroup importance effective upper bound twisted conjugacy separability definition let nilpotent group let aut set subgroup since image define central subgroup effective separability finitely generated nilpotent groups one see elements take form definition let let aut define det additionally define twisted determinant twisted determinant main ingredient following proposition proposition let prime exists natural number every automorphism natural number exists number chosen exists constant primes observe independent choice automorphism proposition twisted generalization lem effective version thm proof proposition relies following two lemmas lemma let prime let group morphism abelian groups det exists since know ord proof write det element group divides det since det follows order det hence det let element desired element reproduce proof lem order estimate associated value constructed lemma let nilpotency class let prime exists exists additionally integer chosen primes proof natural number let largest integer show lemma holds value proceed induction nilpotency class length since statement evident abelian groups may assume write xip nilpotency class thm implies may write may write binomial coefficient effective separability finitely generated nilpotent groups gcd thus value divisible therefore write since nilpotent class letting write subsequently letting may write xip zip zip lem implies nilpotency class strictly less since inductive hypothesis gives construction implies proof proposition let natural number lemma proceed induction equivalently take assume sufficiently big letting consider injective morphism abelian groups induced implies lemma gives det lemma shows exists hence since follows thus induction implies theorem follows constant given finish noting lemma implies thus denote automorphism induced quotient effective separability finitely generated nilpotent groups corollary let nilpotency class let automorphism let prime natural number take proposition define natural projection define proof first assume equivalently note since group morphism hence hence inclusion let thus exists applying proposition map induced exists hence since get conclude bounding twisted determinant previous section identified twisted determinant crucial factor studying twisted centralizers finite quotients goal section bound twisted determinant automorphism terms norm automorphism start provide propositions examples relate determinates subgroups norms subgroups example consider group standard generating subset norm subgroup satisfies knzks det generalize previous example finitely generated abelian groups proposition let finitely generated abelian group finitely generating subset exists constant subgroup det khks rankz effective separability finitely generated nilpotent groups proof since always finite normal subgroup may assume moreover may assume standard generating subset since statement invariant changing generating subset let rank exist rank khi khks thus may restrict attention subgroups form since det det khks let projection first coordinates relabeling coordinates necessary may assume injective since injective follows follows injective thus since det det thus without loss generality may assume full rank let since full rank follows determinant observing det det coefficient bounded khks formula determinant gives desired bound applying proposition group morphisms following corollary let finitely generated abelian group finite generating subset exists constant every morphism abelian groups follows det rank rank proof result follows proposition since value depends image following example shows one bound norm determinant example consider group standard generating subset subgroup computation shows khks det following lemma useful estimate norm kernel group morphisms lemma let generating subset exists constant every injective group morphism every element holds kxks max det proof statement invariant changing generating subset assume standard generating subset writing cramers rule implies det det effective separability finitely generated nilpotent groups morphism given replacing column matrix representative moreover entry matrix representative bounded max thus explicit formula determinant gives result lemma let finitely generated abelian group rank finite generating subset fixed natural number exists constant every group morphism generating subset holds ker moreover exists depending bound achieved generating subset elements proof without loss generality may assume statement invariant changing generating subset may assume standard generating subset additionally assume rank taking quotient necessary first prove lemma standard generating subset assume linearly independent let det note exists constant kks construct vector ker apply lemma gives letting natural inclusion vector ker construction vectors generate finite index subgroup ker since assumed standard generating subset easy computation shows ker max therefore ker bound also follows construction assume arbitrary generating subset consider morphism maps generator composition ker maps generators kernel generators kernel moreover kxk every thus lemma follows example fix take distinct primes consider matrix given pnj corresponding linear map let standard generating sets respectively write kernel generated effective separability finitely generated nilpotent groups element computation shows ker pnj pnjn hand max example implies bound proposition optimal proposition every exists constant every nilpotent group generating subset holds proof generators consider group morphism lemma follows exists ker forpevery generator ker fix element group generated proposition let finitely generated abelian group rank finite generating subset every exists constant finite generating subset group morphism ker rankz moreover exists bound achieved generating subset elements proof assume first note subgroup ker whose norm bounded described proposition suffices bound norm kernel thus lemma gives proposition statement number generators also follows lemma theorem let finite generating subset let exists natural number constant kni aut proof proceed induction every satisfies condition theorem additional assumption number elements generating subset uniformly bounded constant thus theorem evident case assume result holds group constant integer number generators assumption exists finite generating subset effective separability finitely generated nilpotent groups quotient fix generating subset given projections commutators length denote rank fix constant proposition nilpotency class generating subset independent group morphism computation definition follows take constant ksks easy give explicit form constant hence follows consequently computation definition using proposition get follows exists bound number generators proposition important application theorem bound twisted determinant automorphism terms norm corollary let generating subset exists constant every automorphism twisted determinant satisfies proof suffices give bound every determinant bound use theorem group find generating subset whose word length particular norm next apply corollary group morphism find bound determinant final application bounds following estimate essential theorem corollary let finite generating subset let prime let constant corollary exists constant integer every automorphism proof follows immediately bounds proposition corollary precise conjugacy separability two step nilpotent groups give proof main theorem apply techniques developed far specific setting namely conjugacy separability nilpotent groups nilpotency class effective separability finitely generated nilpotent groups main goal section give precise computation asymptotic behavior conjn see theorem start preliminary results observations let nilpotency class definition inn since nilpotent group nilpotency class also observe map given note every holds inn calculation also observed fact case inner automorphisms stronger version theorem proposition let finite generating subset exists kxks every proof write finite generating subset follows exists constant kxks subgroup generated elements form follows immediately kxks hence kxks another crucial ingredient main results separability central subgroups start easy example use proof next proposition example take group standard generating subset consider nontrivial subgroup hhi seperate element finite quotient order finite quotient khks note direct sum subgroups order prime power particular separate every element quotient form khks general generating subset find exists every element separated finite quotient following result generalizes example nilpotent groups proposition let nilpotency class finite generating subset exists constant central subgroups every separate finite quotient max khks log kxks effective separability finitely generated nilpotent groups proof assume central subgroup construct separates two different cases according whether first assume taking quotient necessary assume rank quotient satisfies let generator example follows exists prime power separated generating subset exists constant khkcs take lemma consider quotient note lemma thus seperated finite quotient order quotient bounded ckhkc assume case consider quotient group nilpotent group thm follows exists constant separated quotient order log kxks since bound cases proof finished necessary tools give precise calculation conjugacy separability nilpotency class theorem let finite generating subset conjn proof lower bound given theorem case thus need demonstrate conjn end suppose kxks kyks equivalently thm implies exists surjective group morphism log since central elements conjugate thus log may assume kzks consider group morphism proposition implies proposition implies exists surjective group morphism claim suppose contradiction exists implies thus contradiction hence subsequently conjn effective twisted conjugacy separability section prove main results paper start abelian case effective separability finitely generated nilpotent groups proposition let finitely generated abelian group finite generating subset every holds conja log kxks aut subsequently conja log tconja log proof let aut take suppose satisfies kyks may write identity map thus holds observe use notations section observe generating subset thus may write ksks subsequently proposition implies exists surjective group morphism max log xks max log kxks construction holds statements theorem follow immediately still need following technical result generalization lemma lemma let finite generating subset automorphism exists constant integer every holds kxk max kyks proof proceed induction thus take assume write consider induced map use theorem find constants generating subset ksks apply lemma find exists max kyks write particular get kxks max kyks max kyks construction yks kyks kxks kyks max kyks max kyks use inductive hypothesis finish proof effective separability finitely generated nilpotent groups rest section fix automorphism work automorphisms given additionally denote induced automorphism subgroup denoted simplify notation similarly denote via theorem let finite generating subset let aut exist natural numbers conjn kxks aut particular conjn tconjn proof proceed induction nilpotency class observe base case given proposition thus may assume let aut fix suppose satisfies kyks goal construct finite quotient bound terms kxks denote automorphism induced inductive hypothesis implies exist integers satisfying following exists surjective group morphism finite group thus assume writing lemma implies lemma exists norm bounded described lemma solving get exist constants kzks kxks see thus theorem implies knx constant may write kxks subsequently kxks thus knx kxks fix constant proposition lemma implies therefore exists surjective group morphism max knx log kzks knx kzks kxks moreover may assume prime lemma implies ker subsequently effective separability finitely generated nilpotent groups using natural number notation corollary follows indeed otherwise claim thus contradiction bound order combine previous inequalities corollary implies exists constant integer therefore thus first statement holds bounds last two statements theorem follow immediately virtually nilpotent groups section broken two parts first part technical detour whereas second subsection contains main results section separability extensions twisted conjugacy separable groupss start subsection following definition definition let subset necessarily finite let defined max take max recall function introduced page function measures complexity separate elements set finite quotients function depend choice generating subset lemma let finitely generated group let subset let two finite generating subsets proof similar lem see also lem lem first give lemmas function coming main result let finite index normal subgroup let separable subset following lemma relates complexity separating complexity separating lemma let finitely generated group let finite index normal subgroup suppose finite generating subsets respectively suppose separable subset effective separability finitely generated nilpotent groups proof since finite index subgroup undistorted subgroup subsequently exists let suppose element since may pass quotient finite group assumption thus may assume note hence exists surjective group morphism since finite groups linear finite group induced morphism restricted equal original group morphism moreover thus subsequently separable subsets next lemma relate complexity separating complexity separating individually lemma let finitely generated group finite generating subset let finite collection proper separable subsets proof let follows thus exists surjective group morphism let ker selection hence conclude last lemma relates complexity separating element complexity separating lemma let finitely generated subgroup finite generating subset suppose separable subset let proof need show suppose kxks implies therefore exists surjective group morphism kyks kyks follows since right translation bijection therefore suppose finite extension following proposition shows conjugacy class element written finite union right translates twisted conjugacy classes elements following proof follows thm proposition suppose finite generated group contains finite index teristic subgroup let set representatives right cosets fix effective separability finitely generated nilpotent groups automorphism aut let automorphism induced conjugation automorphism induced proof let automorphism induced may write suppose twisted conjugacy separable group finite extension additionally assume aut following theorem relates quantification conjugacy class quantification conjugacy separability finite fixed collection automorphisms depending theorem suppose finite generated group contains finite index characteristic subgroup let set representatives right cosets fix automorphism aut let automorphism induced conjugation finite generating subsets respectively conjg proof simplicity following arguments let lemma implies since lemma implies given since fxi lemma implies conjg taking everything together conjg effective separability finitely generated nilpotent groups effective twisted conjugacy separability virtually nilpotent groups apply theorem context virtually nilpotent groups get first two main results section theorem suppose virtually nilpotent group suppose finite generating subset aut exists natural numbers conjg kxks particular conjg tconjg proof finite theorem clear thus may assume infinite assume characteristic let finite generating subset set right coset representatives consider automorphism induced conjugation theorem implies conjg theorem implies exist natural numbers thus finish give bound since subgroup undistorted suffices find bound note conjugation thus ksi kxks ckxks since finitely many implies kxks therefore kxks taking everything together may write conjg kxks last two inequalities follow immediately section last result need following proposition similar cor proposition let integral heisenberg group presentation given centrali let prime suppose surjective group morphism prime distinct natural number effective separability finitely generated nilpotent groups proof write conjugacy class let order element since gcd exists integers see reproduce proof prop proposition let conjugacy separable finitely generated groups suppose subgroup two elements proof take surjective morphism restriction subgroup separates conjugacy classes moreover following theorem gives asymptotic behavior conjugacy separability quantification function class virtually nilpotent groups theorem let virtually nilpotent group finite generating subset let exist conjg kxks particular conjg virtually abelian conjg infinite finitely generated nilpotent subgroup finite index proof note finite upper bound clearly holds therefore may assume contains characteristic finite index subgroup thus theorem implies exist integers conjid kxks since conjg conjg conjid conjg first two statements evident lower bound follow proof thm assuming virtually abelian exists finite index characteristic subgroup construct infinite sequence elements also finite conjugate kat kbt generating subset proposition implies since finite index subgroup follows undistorted particular kat kbt point last statement theorem take elements primitive using techniques assume every one dimensional central quotient associated satisfies via defn prop exists every prime every surjective morphism finite holds take replacing effective separability finitely generated nilpotent groups find subgroup isomorphic integral heisenberg group enumeration primes greater max consider eleletting ments since pairwise central elements pairwise elements claim exist elements letting set right coset representatives may write conjugacy class element union conjugacy classes elements since elements lie different conjugacy classes claim follows take claim lem follows kat kbt suffices show surjective group morphisms thus may assume thm may assume finite prime assumptions know hence also assume since isomorphic integral heisenberg group proposition implies exists therefore explained argument ends proof examples last section work explicit examples heisenberg group section work twisted conjugacy seperability function discrete heisenberg group group law given fix generating subset automorphism written let effective separability finitely generated nilpotent groups note every case uniformly bounded compute function general automorphism let two elements proposition automorphism fixed implies seperate twisted conjugacy classes finite quotient norm log always assume way also follows log note group rank corresponding dimension eigenspace corresponding eigenvalue case rank case thus implies get bounded since finite number twisted conjugacy classes log since corresponds separating central elements group using elements increasing primes easy show indeed log case rank note second eigenvalue must thus norm uniformly bounded hence fixed quotient seperate twisted conjugacy classes conclude case log case rank case case copy proof thm find conjid note three cases example let nilpotent group given central generating set let automorphism given effective separability finitely generated nilpotent groups every element uniquely expressed note subgroup every follows immediately hence facts allow prove following proposition proposition exists constant every kxks holds proof subgroup generated elements statement follows similarly proposition compute asymptotic upper bound conjn proposition group holds conjn proof let kxks kyks first assume proposition automorphism fixed implies exists constant log particular exists surjective group morphism since follows log thus may assume kzks proposition implies exists surjective group morphism max log kzks claim contradiction suppose otherwise thus exists implies may write particular contradiction calculation one see conclude max log thus conjn effective separability finitely generated nilpotent groups future questions list open problems remain main results paper theorem computed precise conjugacy separability function nilpotency class conjecture general nilpotent group function following form conjecture let nilpotency class conjn lower bound conjecture already given proposition computed upper bound effective seperability function case central subgroups yet description subgroup separability function general subgroups leads following question question compute effective subgroup seperability function finitely generated nilpotent groups section gave several explicit examples function conjn nilpotent groups examples upper bound given conjn case identity map conjecture always case conjecture let nilpotent group generating subset every automorphism aut holds conjn conjn known examples conjugacy function depends rational even real mal cev completion open question whether true general question let two abstractly commensurable finitely generated virtually nilpotent groups true references james arthur laurent clozel simple algebras base change advanced theory trace formula volume annals mathematics studies princeton university press princeton norman blackburn conjugacy nilpotent groups proc amer math mcreynolds bertrand postulate subgroup growth algebra mcreynolds asymptotic growth least common multiples groups bull lond math effective separability finitely generated nilpotent groups khalid quantifying residual finiteness algebra khalid approximating group solvable quotients new york khalid mark hagen priyam patel residual finiteness growths virtually special groups math khalid tasho kaletha quantifying residual finiteness arithmetic groups compos khalid mcreynolds extremal behavior divisibility functions geom dedicata khalid brandon seward arbitrarily large residual finiteness growth reine angew buskin efficient separability free groups sibirsk mat chagas zalesskii bianchi groups conjugacy separable pure appl algebra sheila chagas pavel zalesskii finite index subgroups conjugacy separable groups forum sheila chagas pavel zalesskii limit groups subgroup conjugacy separable algebra jonas periodic eventually periodic points affine endomorphisms discrete contin dyn fel shtyn troitsky twisted conjugacy separable groups preprint alexander fel shtyn dynamical zeta functions nielsen theory reidemeister torsion mem amer math alexander fel shtyn new directions theory topology alexander fel shtyn evgenij troitsky twisted theory discrete groups reine angew edward formanek conjugate separability polycyclic groups algebra goryaga example finite extension sibirsk mat gromov asymptotic invariants infinite groups geometric group theory vol sussex volume london math soc lecture note pages cambridge univ press cambridge effective separability finitely generated nilpotent groups grothendieck formules lefschetz volume sga lecture notes pages berlin philip hall edmonton notes nilpotent groups queen mary college mathematics notes mathematics department queen mary college london emily hamilton henry wilton pavel zalesskii separability double cosets conjugacy classes groups lond math soc jiang lectures nielsen fixed point theory volume contemporary mathematics american mathematical society providence martin kassabov francesco matucci bounding residual finiteness free groups proc amer math kharlampovich myasnikov sapir algorithmically complex residually finite groups preprint gady kozma andreas thom divisibility laws finite simple groups math sean lawton larsen louder mcreynolds decision problems complexity traces representations groups geom armando martino ashot minasyan conjugacy normal subgroups hyperbolic groups forum ashot minasyan pavel zalesskii virtually compact special hyperbolic groups conjugacy separable comment math osin subgroup distortions nilpotent groups comm algebra priyam patel theorem peter scott proc amer math priyam patel residual finiteness growths particular hyperbolic manifold groups geom dedicata pengitore effective separability finitely generated nilpotent groups preprint igor rivin geodesics one stories adv daniel segal polycyclic groups cambridge university press salahoddin shokranian trace formula volume lecture notes mathematics berlin based lectures james arthur effective separability finitely generated nilpotent groups peter stebe conjugacy separability groups integer matrices proc amer math andreas thom length laws finite groups israel
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predicting lane keeping behavior visually distracted drivers using inverse suboptimal control apr felix dietrich michael rainer driver distraction strongly contributes crashrisk therefore assistance systems warn driver distraction poses hazard road safety promise great safety benefit current approaches either seek detect critical situations using environmental sensors estimate driver attention state solely behavior however neglects driving situation driver deficiencies compensation strategies altogether determine risk accident work proposes use inverse suboptimal control predict aspects visually distracted lane keeping contrast approaches allows assessment risk posed distraction real traffic data seven drivers used evaluation predictive power approach comparison baseline built using established behavior models evaluation method achieves consistently lower prediction error speed variations additionally approach generalizes better driving speeds unseen training phase introduction motivation driver distraction psychological concept defined diversification attention away activities critical safe driving towards competing activity common examples looking display instead road ahead vehicle visual distraction dialing telephone number requires removing one hand manual distraction conversation cognitive distraction according national highway traffic safety administration crashes caused inattention form distracted fatigued drowsy drivers analysis naturalistic driving study even concludes approximately crashes near crashes involve inattentive drivers contributing factor safe driving requires predominately acquisition processing visual information lateral longitudinal vehicle control accordingly several studies found distinct negative effects visual distraction long glances driving performance slower impaired response lead vehicle braking decreased performance although conventional driver work part public project ban federal ministry economic affairs energy basis decision german bundestag schmitt bieg manstetten herman robert bosch gmbh corporate research stuttgart germany stiefelhagen institute anthropomatics robotics karlsruhe institute technology germany accepted ieee intelligent vehicles symposium assistance systems forward collision lane departure warning already partially mitigate decrements optimal assistance consider driver distraction state ultimately driver needs support abilities suffice resolve hazard hence knowledge whether whether driver currently perceptual insufficiencies spare false alerts interventions assistance system demonstrated related work motivated importance several algorithms detection assessment inattention proposed reviewed due progress estimation head orientation eye tracking current approaches address visual distraction often process driver gazedirection approximation prediction work similar authors reviewed build binary variable indicates whether driver gaze intersects region interest roi region defined plane driver look gather necessary visual information lane keeping headway control forward road scenery fig roi fig intersection driver gaze potential roi open question driver distraction research algorithmically decide driver distracted based behavior specific situation visual distraction assessment difficulties arise fact safe driving sometimes even requires look checking speedometer however assistance system would warn even intervene algorithms decision consequently driver falsely detected critically distracted frequently system would become nuisance cause driver turn nevertheless dangerous situation classified correctly several authors proposed algorithms decision critical distraction fact detect periods driver engaging secondary task primary task safe driving cases additional task subjective utility requiring visual attention ieee personal use material permitted permission ieee must obtained uses current future media including material advertising promotional purposes creating new collective works resale redistribution servers lists reuse copyrighted component work works doi doi reason long glances however approach issue particular visually distracting activities selecting radio station commonly socially accepted familiar tasks example performed driving time one naturalistic driving study many situations pose little risk drivers developed compensatory mechanisms engagements indeed utility driver rather receive interventions system alternative detection secondary task engagement subjectively perceived distraction predicted achieve distraction rating based video driver outside scenery used method captures subjective risk distraction reliably relate objective risk probability crash rapidly change seconds additionally human ratings often ambiguous prediction objective relative crash risk different algorithms investigated using data naturalistic driving study similar approaches dependence safe gaze behavior specific situation neglected furthermore hardly tractable conduct study prototype vehicles containing similar amount evaluation follow different idea instead estimating driver distraction state predict future driver trajectory vehicle driven potentially distracted driver allows assess current state wrt lane keeping estimating probability critical situations caused driver impairment methodological point view similar inverse optimal control concepts applied problems assisted autonomous driving however none suitable stochastic partially observable control problem considered work contributions main contribution work proposal inverse suboptimal control approach predict behavior distracted lane keeping consider step towards situation specific visual distraction assessment therefore introduce class partially observable markov decision processes show first result distracted lane keeping modeled strategy therein next explain driver objective inferred inverse optimal control used predict behavior new situations maximum causal entropy inverse optimal control presented approach account suboptimal driver behavior following report setting driving study conducted evaluation iii second contribution finally present numerical comparison terms predictive power approach baseline built salvucci wellknown steering model johnson model thereby investigate overall prediction error performance applying models speeds unseen training phase problem formulation partially observable markov decision processes optimal control problems belong class markov decision processes mdps mdp consists state control stochastic process model objective mdp find stochastic maximizes expected reward time horizon conditioned policy dynamics distribution problem addressed work belong class partially observable mdps pomdp extend mdps states directly observed first distribution inferred observations distributed according observation model distribution belief known pomdp transformed equivalent mdp see details motivate application pomdps idea distracted driving optimal policy combination driving performance utility arbitrary secondary task tasks conflict primary task longer solved optimally critical situations occur case especially driver behavior suboptimal considered model distracted lane keeping distracted lane keeping modeled optimal policy pomdp observations parametric reward consisting model dynamics vehicle dynamics xst ust disturbance specifically used discretization linear kinematic model given ordinary differential equations xst ust variables explained table fig reward xst ust encodes driving objectives lane keeping driver seeks keep vehicle center road smooth minimal steering squares lane position steering angle steering velocity penalized also considered lateral velocity relates impact severity case crash object change driver gaze position driver decision gaze position roi roi driver hypothesis vehicle states xot observation model xot xst belief update xot policy ust uot uot dynamics xot uot mod ust dynamics xst ust xst driver steering based expected vehicle states fig elements driver model indicated red vehicle model denoted blue tangent dynamics curve radius xot uot fig change vehicle denoted position steering wheel angle table variables inematic ehicle odel definition lateral position wrt lane center angle tangent lane vehicle longitudinal axis vehicle absolute velocity curvature lane transmission ratio mod reward xot uot account secondary task utility costs switching gaze assume constant utility every engagement progress task completion constant penalty leads reward function vehicle response steering input unit rad rad neighboring lane altogether results reward model xst ust model driver impairment visual distraction thereby xot nothing indicator purpose work attentive driver assumed perfectly perceive vehicle states driver engaged secondary task assumed perceive sense one hand steering wheel see fig mathematically implemented xot observation model xot xot identity matrix driver ability switch gaze secondary task road incorporated uot xot uot ixot iuot indicator true expression otherwise linearity pomdp easily transformed belief xst fully specified expectation aposterior covariance resulting corresponding fig visualizes final inverse optimal control behavior prediction predict future system states presented pomdp process model combined model driver policy work indirectly infer data estimating parameters driver reward assumption optimal behavior approach referred inverse optimal control ioc reward known driver behavior predicted arbitrary computation policy based optimality criterion compared direct estimation ioc advantage parameters less dependent external model variables model driver preferences potentially applying globally empirically investigate hypothesis sec maximum causal entropy inverse optimal control case linear combination reward features matching empirical feature expectation optimal policy reward results iii collected data driving study experiment hence reward observed behavior optimal although many approaches ioc mdps proposed suitable modeling driving behavior reason humans often deviate optimal behavior needs taken account modeling distracted driving therefore apply maximum causal entropy mce method stochastic policy maximum causal entropy fullfills condition computed following optimization problem max log rationale behind maximization casual entropy resulting policy effectively deviates least true behavior assumed details hence estimation robust suboptimal policies mce although reward parameters explicitly estimated mce considered inverse optimal control approach due following property likelihood estimated act like deterministic policy monotonic function evaluate modeling approach conducted study public highway decided study driving simulator possible influences participants behavior absence real risk addition evaluation prediction robustness realistic noisy signals important recruited seven drivers male female drivers took part driving safety training prior experiment experiment consisted four fixed driving speed conditions vehicle speed controlled vehicle adaptive cruise control acc prevent drivers adjusting speed compensatory action engaged secondary task conservative time gap employed ensure distance preceding vehicles least possible influence drivers behavior vehicle traveled required speed measurement periods started period either reference driver keep lane fully attentive involved visually distracting secondary task speed three secondary tasks three reference periods per participant triggered investigator task used experiment consisted incrementally reading series random numbers display typing number pad see fig thereby participants instructed perform lagrangian multiplier constraint means higher expected value likelier behavior according consequently interpreted suboptimal yet policy respect reward model parameter learning optimization problem solved optimal gradient descent problem calculate current using recursion log exp exp exp simulate mdp starting compute update using suitable fig artificial secondary task used experiment secondary task quickly correctly possible endangering driving safety artificial task chosen resembles principle variety real tasks performed driving possesses several advantages first task state fully measurable modeled easily contrast vehicle interaction system second participants needed little practice reach maximum execution performance resulting significant learning effects experiment recorded data used series system tracking lane boundaries recorded position lane angle tangent lane boundary vehicle longitudinal axis curvature via vehicles controller area network commercial infra red eye tracking system active illumination used estimation driver gaze direction roibased algorithm detected whether driver gaze road steering wheel position velocity well absolute velocity also recorded via hence beside systems relied solely signals already accessible today seriesproduction vehicles preprocessing order ensure quality dataset numerical evaluation following preprocessing filtering steps performed collected raw data selection valid trials according protocol automatically excluded lane changes preparation phases different driving maneuver lane keeping requires different driving gaze policy also situations acc controller reduced vehicle speed left due possible influence drivers behavior final dataset consisted valid segments comprising reference secondary task periods average duration filtering signals first subsampled variables vehicle model thereafter employed filtering signals using kinematic vehicle model system parameters well sensor estimated expectationmaximization approach proposed vehicle models evaluation demonstrate effectiveness approach conducted two numerical evaluations compared baseline comprising established behavior models baseline presented model visual dualtasking probability glance task logistic function uncertainty states case uncertainty present vehicle states random number either known driver seen display unknown else therefore applied following variant original approach iuo exp iuot exp iuo exp iuo uot exp uot parameters sum matrix diagonal elements modeling human predictive steering adopted model model driver steers means visual visual lane width defined angle straight vehicle center near point lane center vehicle longitudinal axis defined angle minimal magnitude tangents vehicle center boundaries fig illustration variables steering model vehicle longitudinal axis see fig computed using approximations arctan arccos arccos fed angles angle stochastic policy parameters used however less stable combination approximation due jumps models linked replacing expectations xot modeled driver looks road similar evaluation protocol numerical evaluation subdivided valid segments snippets overlap account realistic prediction horizon system approach relaxed feature matching condition equivalent regularization prevent parameters baseline inferred using maximum likelihood estimation generalized linear models implemented matlab lassoglm function every interval trajectories sampled model starting first state segment thereby incrementally simulated driver behavior responses pomdp model using preestimated parameters obtained data following metrics computed expected squared error ytd divergence empirical gaze duration prediction model log log overall prediction performance first evaluated overall prediction performance splitting dataset training set test set equal size randomly independently driver velocity afterwards roles datasets swapped better estimation error statistics procedure repeated times transfer performance investigate generalization quality unseen velocities conducted second evaluation trained random selection half data one single speed condition thereafter tested remaining half data velocities evaluation performed repetitions train test trans base train base test base base trans fig box indicates interval median horizontal line data whose distance median exceeds distance median corresponding quantile considered outliers train test denote conditions denotes test error speed transfer test error unseen speeds results results summarized table due error distributions see report total median instead mean training test methods metrics table overall rediction performance metric baseline train test train test table iii shows errors present total median errors evaluations thereby results approach bold letters table iii ransfer performance train test fig depicts error distributions evaluations discussion shows significantly lower prediction error approach based compared baseline wilcoxon ptest training test error methods metrics differed slightly wilcoxon ranksum ptest adequate regularization chosen methods possible explanation better approach explicit incorporation vehicle velocity future track curvature allows precise trajectory prediction although johnson barrier model already incorporates growing uncertainty time might worse mceioc approach lacks direct link lane keeping controller gives insight performance differences performed significantly better metrics ptest conditions except train test ptest although test differences revealed significant variations ptest pronounced baseline ptest ptest could validated significantly lower variance ptest hypothesis driver preferences estimated better transferable directly estimated policy could confirmed experiment difference error unseen speeds median test error seen speeds significantly higher baseline approach metrics wilcoxon ranksum ptest ptest however general hypothesis investigated significantly lower speeds future work onclusion presented inverse optimal control based method predict behavior potentially distracted driver lane keeping key elements model goaloriented driver behavior wrt reward explicit model driver impairment dynamics model vehicle responses reward inferred observed behavior transferred new situations estimation likely driver behavior empirically validated unseen driving speeds evaluation also showed lower prediction error method compared baseline comprising models approach used assessment visual driver distraction wrt lane keeping prediction probability critical incidents however deployment final distraction mitigation system completed prediction lateral driving behavior hence future work address inverse optimal control approaches modeling speed adjustment headway control distracted drivers additionally also want incorporate estimation observation models inference procedure necessary behavior prediction secondary task reading speedometer driver extent perceive road peripheral vision knowledge suboptimal behavior driver also exploited eferences lee young regan defining driver distraction driver distraction theory effects mitigation ranney garrott goodman nhtsa driver distraction research past present future proceedings international technical conference enhanced safety vehicles vol national highway traffic safety administration klauer dingus neale sudweeks ramsey impact driver inattention risk analysis using naturalistic driving study data national highway traffic safety administration washington tech rep dot sivak information drivers use indeed visual perception vol dingus hulse antin wierwille attentional demand requirements automobile navigation system transportation research part general vol peng boyle hallmark driver lane keeping ability eyes road insights naturalistic study accident analysis prevention vol blaschke breyer freyer limbacher driver distraction based assistance transportation research part traffic psychology behaviour vol dong uchimura murayama driver inattention monitoring system intelligent vehicles review ieee transactions intelligent transportation vol tawari martin trivedi continuous head movement estimator driver assistance issues algorithms evaluations ieee transactions intelligent transportation systems vol hansen eye beholder survey models eyes gaze ieee transactions pattern analysis machine vol liang lee yekhshatyan dangerous looking away road algorithms predict crash risk glance patterns naturalistic driving human factors journal human factors ergonomics society vol wollmer blaschke schindl schuller farber mayer trefflich online driver distraction detection using long memory ieee transactions intelligent transportation systems vol lee moeckli brown roberts schwarz yekhshatyan nadler liang victor marshall distraction detection mitigation driver feedback national highway traffic safety administration washington tech rep dot metz schoch kuhn drivers interact navigation systems real life conditions results navigation systems transportation research part traffic psychology behaviour vol metz anticipatory control processes interaction secondary tasks driving transportation research part traffic psychology behaviour vol busso predicting perceived visual cognitive distractions drivers multimodal features ieee transactions intelligent transportation systems vol shimosaka kaneko nishi modeling risk anticipation defensive driving residential roads inverse reinforcement learning proceedings ieee international conference intelligent transport systems itsc kuderer gulati burgard learning driving styles autonomous vehicles demonstration proceedings ieee international conference robotics automation icra vol ziebart bagnell dey principle maximum causal entropy estimating interacting processes ieee transactions information vol salvucci gray visual control model steering vol johnson sullivan hayhoe ballard predicting human visuomotor behaviour driving task philosophical transactions royal society biological sciences vol smallwood sondik optimal control partially observable markov processes finite horizon operations research vol abbeel apprenticeship learning via inverse reinforcement learning proceedings international conference machine learning icml acm zhifei joo review inverse reinforcement learning theory recent advances proceedings ieee congress evolutionary computation cec simon theories economics behavioral science american economic review ziebart bagnell dey modeling interaction via principle maximum causal entropy proceedings international conference machine learning icml rauch striebel tung maximum likelihood estimates linear dynamic systems aiaa journal vol stellet straub schumacher branz estimating process noise variance vehicle motion models proceedings ieee international conference intelligent transportation systems itsc johnson sullivan hayhoe ballard soft barrier model predicting human visuomotor behavior driving task proceedings annual conference cognitive science society citeseer matlab statistics toolboxtm user guide mathworks apple hill drive natick golub chase byron learning internal dynamics model control demonstration proceedings international conference machine learning icml
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type classes lightweight substructural types edward gan jesse tov greg morrisett facebook menlo park northeastern university boston harvard university cambridge tov greg linear substructural types powerful tools adding standard functional programming languages often means introducing extra annotations typing machinery propose lightweight substructural type system design recasts structural rules weakening contraction type classes demonstrate design prototype language clamp clamp supports polymorphic substructural types well expressive system mutable references time adds little additional overhead standard type system enriched type classes established type safety core model implemented type checker type inference haskell introduction type classes provide way constrain types operations support type class predicate dup indicates assumptions type subject contraction duplication drop indicates whether subject weakening dropping linear relevant affine unlimited typing disciplines enforced subset classes linear types types satisfy neither dup drop idea suggested one author dissertation forms basis prototype substructural programming language clamp clamp programs written external language weakening contraction implicit easier programmers work specify type system semantics external language elaborated internal language linear variables used exactly internal language provides explicit dup drop operations impose corresponding type class constraints arguments thus internal language one might think dup drop functions qualified types dup dup drop drop internal language nonlinear usage mediated dup drop operations example internal language term ill formed uses variable twice term let dup well typed elaboration internal language ensures resulting program linear checked using type reconstruction type classes improper duplication dropping indicated unsatisfiable type class constraints alves cervesato eds third international workshop linearity linearity eptcs gan tov morrisett fst fst drop constu dup drop drop constu constl drop constl figure prelude functions inferred signatures contributions believe clamp offers substructural types less fuss many prior approaches substructural type systems throughout design leverage standard type class machinery deal constraints imposed substructural types implementing type inference clamp straightforward also easy extend system custom resource aware structures specific contributions paper include type system design polymorphic substructural types type safety theorem flexible system managing weak strong references type checker type inference derived type checker haskell algorithm sense optimal clamp basics section introduce clamp external language dup drop operations implicit concrete syntax borrowed haskell one prominent difference clamp function type term must annotated one four substructural qualifiers unlimited relevant affine linear three examples clamp functions translated haskell standard prelude appear figure types need written explicitly inferred clamp type checker consider fst function projects first component pair would like able use library functions number times annotate arrow lambda expression qualifier annotation determines function type structural case fst satisfies dup drop note property function treats argument fst use second component pair induces drop constraint type variable particular elaboration internal language inserts drop operation make term linear drop presence drop disposes first argument returns second causes drop type class constraint inferred function constu imposes similar constraint second argument also requires type first argument unlimited constu returns unlimited closure containing first argument environment argument effectively duplicated discarded along closure inherits structural restrictions alternatively lift restriction constl returns linear closure thus allows first argument linear type classes lightweight substructural types letp inl inr case inl inr newrq releaserq swaprq dup drop inl inr unit ref strong weak unlimited relevant affine linear dup drop terms values types constraints reference qualifiers arrow qualifiers predicate constructors figure syntax formalizing validate soundness approach developed core model clamp internal language based system modifications variable bindings treated linearly arrows annotated qualifiers type class constraints added universal quantifiers example one define custom datatypes clamp also includes variety operations working mutable references type system shares many similarities tov core alms calculus unlike external clamp language prototype provides polymorphism support type inference syntax syntax appears figure language standard notably arrow types terms clamp annotated arrow qualifier annotations determine structural operations function supports well corresponding constraints imposed types closure environment unlike presentations linear logic constrains usage function usage function argument thus one call dup arrow arrow type abstractions specify type class constraints abstract bodies restricted values unlike terms type abstractions need arrow qualifier newrq releaserq forms introduce eliminate mutable references comes two flavors depending reference qualifier records whether reference supports strong merely weak updates weak conventional updates must preserve reference cell type strong updates modify value type cell form swaprq provides linear access reference exchanging contents different value release operator deallocates cell returns contents aliased store locations appear run time written programmer incorporate type classes universal types may include constraints type variables constraint denotes set atomic predicate constraints predicate constructor applied type sake current analysis either dup drop gan tov morrisett inl inr evaluation contexts case inl inr letp new releaserq swaprq swaprq dup drop stores figure runtime structures newrq fresh swaprq releasew inl releasew inr release dup incr locs drop decr locs figure relation semantics execution terms defined semantics evaluation contexts global store structures needed define small step relation given figure reference counts used track reference cells safely deallocated presence aliasing exchange properties store implicitly assumed selection small step relation rules given figure focusing rules reference cells substructural operations complexity comes reference counts swap operator exchanges contents cell heap different value release operator deallocates cell returns contents aliased however cell aliased decrements reference count returns unit dup drop operators manipulate reference counts expected metafunctions given figure necessary maintain reference counts similar functions incr decr allow increment decrement reference counts heap incrementing location straightforward decrement must defined recursively since deallocating last pointer reference cell involves decrementing reference counts cells deallocated contents originally pointed locs convenient way extracting multiset locations value uses note operator multiset operator additively combines occurrences used section locs function also designed operate terms looks one branch case expression assuming branch must share location typing context type classes lightweight substructural types incr decr locs decr locs locs locs incr incr incr decr decr decr locs locs locs case locs locs locs inr figure reference count management var dom constrainaq ase case inl inr rop dup dup drop drop figure selected term typing rules term typing variable contexts associate variables types variable appears location contexts store typings associate locations reference types ref distinguish strong weak locations weak locations carry reference count track aliasing linearity enforced via standard linear environments need operations join join operation defined pairs compatible environments written two variable contexts compatible long disjoint two location contexts compatible strong locations disjoint weak locations intersection agree types joining variable contexts appends two sets bindings together joining location contexts also involves adding reference counts shared weak locations contexts identified permutation term typing judgment assigns term type constraint variable location contexts selected typing rules core language appear figure gan tov morrisett ref ref newrq ref elease elease releasew unit releases ref ref wap ref swap ref wap ref swap ref figure term reference cell typing rules typing rules reference cells given figure consistency conditions assumed whenever contexts combined core language typing rules split share linear contexts needed otherwise natural extension system support type class constraints impose syntactic restriction similar haskell context reduction restrictions form constraints type schemes introduced rule type abstractions may constrain type variables bind compound unrelated types simplifies induction typing derivations since means constraints external type variables introduced type abstraction additionally rule variable location contexts constrained function arrow qualifier ensure values captured closure support structural operations might applied closure constraint must entailed constraint context constrainaq shorthand appropriate set dup drop constraints applied every type mapped instance constrainl imposes constraints constrainr imposes dup constraints dup drop forms constrain types parameters expected way requiring types members dup drop type classes respectively entailed constraint context since supports strong weak references different substructural properties variety typing rules governing usage swap operation needs return updated reference old contents packages pair weak strong forms reference cell operations provided safe apply weak operations strong weak references releaserq forms used deallocate reference cell possibly retrieve contents notably case weak reference since contents could linear preserve linearity allowing aliasing returning contents reference last alias cell released unit otherwise type classes lightweight substructural types dup dup dup drop drop drop dup dup dup drop drop drop dup drop dup unit drop unit dup drop dup ref drop drop ref figure dup drop instances type class instances throughout type system type class constraints propagated via entailment specifies one set type class predicates implied another context fixed background instance environment example entailment allows type system derive unit unit duplicable unit rules entailment given jones adapt naturally setting substructural essence type class system clamp set base dup drop instances appears figure since pairs sums contain values might copied ignored along pair sum value instance rules require instances components functions impose constraints closure environments assigned qualifier term typing instance rules arrows depend arrow qualifier dealing correctly references subtle seen refural clamp references support strong updates change value type mutable reference however unsafe alias reference cell whose type might change refural restrictions reference types given sizable table dup drop instances make easy express restrictions clamp clamp classifies references kind updates support strong weak specified qualifier ref type qualitatively constraints impose strong references may duplicated references droppable contents may dropped strong references support direct deallocation weak references deallocated return contents unaliased four rules capture restrictions refural references also increase expressiveness system explicitly distinguishing weak strong references allowing deallocation weak references expressed two type class instances typing judgments elease elease example kinds structures build using rules consider type ref fhandle linear file handle fhandle weak reference aliased provide shared access file handle dropped based type class instances since fhandle dropped anyone uses reference must release reference close file necessary gan tov morrisett type safety sketch part type safety proof details may found gan thesis bulk work goes proving preservation key lemma proving preservation relates constraints bindings intuitively lemma says structural constraints value type respect structural constraints everything value contains points via variable location contexts syntactic forms like dup used denote set dup constraints types similarly dup applies dup ref types mapped note lemma hold arbitrary expressions lemma constraints capture bindings suppose drop drop drop dup dup dup proof induction typing derivation lemma essential proving substitution lemma lemma lemma substitution proof induction typing derivation making use lemma case proving preservation also useful separate replacement lemma specifies exactly substitution interacts evaluation contexts lemma replacement furthermore proof induction another key lemma proving preservation relates locs function used maintain dynamic reference counts store context value requires state lemma overload locs function also return multiset occurrences multiple reference counted weak location stores locations domain store context lemma store contexts map free locations locs locs finally prove preservation type soundness need introduce store configuration typings given figure remainder type soundness proof mostly standard lemma preservation theorem type soundness either diverges reduces value configuration type classes lightweight substructural types ons ons onf figure store configuration typing implementing clamp type checker implemented type checker infers style type schemes clamp terms type checker extension jones typing haskell haskell type checker source code may found https process modifying haskell type checker support clamp straightforward illustrates one strengths clamp design requires small orthogonal additions language like haskell besides adding qualifiers arrow types made three additions haskell type checker elaboration pass inserts dups drops dup drop type classes instances substructural qualifiers constraints arrow types inferring dups drops elaboration pass bridge concise language leveraging conventional nonlinear type checking techniques pass takes input term arbitrary variable usages inserts appropriate dup drop operations renames duplicated copies resulting term variable usage strictly linear structural properties enforced constraints imposed dup drop since different elaborations lead different static dynamic semantics proven algorithm generates optimal elaboration two senses minimizes program live variables imposes minimal type class constraints follows define core linear language formalize estabilsh two points abstract linear language focus essential problems work abstraction linear calculus modeling usage binding allows focus inserting dup drop operations independently particular types term forms language syntax given figure extending results cover term forms straightforward product expression abstracts multiplicative forms pairs function pairs expressions evaluated sum expression abstracts additive forms linear logic additive conjuction relationship branches pairs expressions exactly one evaluated gan tov morrisett dup drop unannotated terms annotated terms figure syntax hoice rop dup drop figure annotated expression expressions unannotated explicitly satisfy linear usage constraints annotated expression use dup drop operations explicitly specify nonlinear usage variables dup drop operations work contexts multisets variables contexts manage scope binding restricting contraction weakening explicit dup drop annotations track types variables order avoid messy straightforward process generating names renaming variables inserting dup think contexts multisets variables equivalently functions variables natural numbers thus used denote number times appears use multiset contexts define notion figure describes annotated term properly accounts nonlinear usage variables explicit dup drop operations inference algorithm inference algorithm annotating terms given figure strategy recursively transform term based free variables recursively transformed note function always returns set denote standard set intersection difference operators dup operations inserted free variables two multiplicative form application branching discovered intersect drops added binders bound variable free scope variable used one branch say free outline key steps optimality argument additional details may found lemma states algorithm sound lemma soundness infer annotation lemma shows algorithm fact generates annotation requires minimal context lemma minimal contexts technical lemmas prove algorithm introduces unnecessary dups drops variables order compare different potential annotations term type classes lightweight substructural types infer infer infer infer otherwise infer dup infer infer infer drop infer drop infer figure inference algorithm fine function erase removes dup drop annotations annotated term straightforward way yielding unannotated term evident erase infer lemma forced drop erase contains subterm drop lemma forced dup exists subderivation aes contains subterm dup lemma unnecessary drops let infer contains subterm drop aes erase contains subterm drop aes lemma unnecessary dups let infer contains subterm dup aes erase contains subterm dup aes constraint processing dup drop insertion pass type inference proceed without needing count variable usages split contexts since insertion pass made every dup drop operation explicit exception extra constraints imposed closure environments inferring types clamp internal language like inferring types haskell constraint solver type classes dup drop instances different type class type checking system thus separated two steps first usage analysis performed elaboration second checking substructural constraints manner type class system similar division used vries integrate uniqueness typing system table present sizes components clamp implementation compares favorably implementation languages alms whose type inference engine lines haskell insertion sits top stack main addition make haskell type checker design besides included base set type class instances altered arrow kinds throughout related work existing work linear type systems focus develop general purpose polymorphic linear types research mathematical expressiveness linearity tailored gan tov morrisett component lines code insertion type class instances syntax types unification engine table type checker code breakdown use cases instance often aim broad usability polymorphism first linear type systems derive directly intuitionistic linear logic use exponential indicate types support structural operations later type systems order support parametric polymorphism linearity replace types composed qualifier pretype types languages form like similarly clean programming language makes use qualifier variables inequalities capture range substructural polymorphism though clean uses uniqueness rather linear types many design decisions applied linear settings well recent languages alms eliminate notational overhead annotating every type substructural qualifiers using distinct kinds separate substructural types thus rather working types like file file type alms defined kind like clean alms highly polymorphic makes use compound qualifier expressions function types well dependent kinds compared type systems like clean alms believe clamp offers advantages simplicity extensibility like alms clamp avoids burden clean annotating every type qualifier type classes general powerful feature language going type classes anyway clamp approach allows adding full spectrum ural types little additional complexity programmers type checkers programmers already familiar type classes well prepared understand substructural types type classes provide clean formalism constraining datatypes system weak strong mutable references found clamp finally anticipate userdefined dup drop instances yet supported clamp allow defining custom destructors copy constructors enable variety resource management strategies however compared alms clean clamp provide much polymorphism arrow assigned concrete qualifier consider instance curry function clamp unlike alms clean clamp requires different versions different desired structural properties instance two possible type schemes curry function dup drop believe extending clamp qualifier variables type class implications could increase expressiveness point curry principal typing type classes lightweight substructural types future work custom dup drop clamp current design semantics dup drop fixed allowing programmers instead define implementations dup drop types would enable scenarios similar possible via copy constructors destructors programmers could define data types automatically manage resources ways meet particular needs unlike believe could done without unsafe operations programmers could define types instances manage memory whatever way appropriate instance choosing dup operations perhaps hybrid approach eager lazy drop operations system practical important algorithm easy understand since insertion dups drops affect dynamic semantics language polymorphic arrows cases clamp allows programmers define functions inherently polymorphic substructural properties arguments cases function uses one arguments linearly accept type whether satisfies dup drop argument however case function types annotated fixed qualifiers determined function definition point alms able accommodate polymorphic arrow types introducing subtyping relation qualified arrows refines types arrows introducing usage qualifiers depend substructural properties type variables allows one write function whose qualifier inferred closure environment inherits polymorphism present environment many options increasing polymorphic expressiveness clamp without resorting complexities subtyping one could make qualifiers first class types allow quantification qualifiers arrow types implemented way current type checker idea expanding language qualifiers could also profitable clamp often means annotating arrow types types closure environments sense assigning type dup drop instances arrows could use closure environment types determine arrow type substructural properties implementation current implementation clamp type checker reflects could benefit addition standard language features found full haskell particular addition algebraic datatypes instance rules module system would allow programmers define libraries expose custom types varying substructural properties would also interesting implement compiler clamp takes advantage substructural properties memory management eagerly reusing storage occupied linear affine values may offer particular performance advantages substructural analyses enable variety optimizations well gan tov morrisett conclusion clamp introduces techniques make easier desirable add substructural types functional programming languages external internal language distinction gives programmer friendly syntax direct path type inference dup drop classes also support polymorphism ural lattice represent state aware types strong weak references type classes expressive language feature clamp shows serve base substructural types references samson abramsky computational interpretations linear logic theor comput sci amal ahmed matthew fluet greg morrisett model substructural state proc acm sigplan international conference functional programming icfp sandra alves maribel florido ian mackie linearity recursion typed proceedings international acm sigplan symposium principles practices declarative programming ppdp acm new york usa henry baker linear logic quicksort sigplan erik barendsen sjaak smetsers uniqueness typing functional languages graph rewriting semantics math struct comp sci gavin bierman intuitionistic linear logic thesis university cambridge jawahar chirimar carl gunter jon riecke reference counting computational interpretation linear logic journal functional programming luis damas robin milner principal functional programs proc annual acm symposium principles programming languages popl edward gan clamp type classes substructural types senior thesis harvard university available http simon gay vasco vasconcelos linear type theory asynchronous session types journal functional programming girard fonctionnelle elimination des coupures ordre paris mark jones qualified types theory practice cambridge university press new york mark jones typing haskell haskell proc haskell workshop karl mazurak jianzhou zhao steve zdancewic lightweight linear types system proc acm sigplan workshop types language design implementation simon peyton jones john hughes haskell purely functional language jesse tov practical programming substructural types thesis northeastern university jesse tov riccardo pucella practical affine types proc annual acm sigplansigact symposium principles programming languages popl type classes lightweight substructural types edsko vries rinus plasmeijer david abrahamson implementation application functional languages chapter uniqueness typing simplified berlin heidelberg philip wadler stephen blott make polymorphism less hoc proc acm symposium principles programming languages popl david walker advanced topics types programming languages chapter mit press cambridge keith wansbrough simon peyton jones upon polymorphic type proc acm symposium principles programming languages popl acm
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symbolic models retarded systems jan pushpak jagtap majid zamani abstract paper provide first time automated controller synthesis scheme class infinite dimensional stochastic systems namely retarded systems first construct finite dimensional abstractions approximately bisimilar original retarded systems stability property namely incremental stability second construct finite abstractions approximately bisimilar constructed finite dimensional abstractions types abstractions derived without discretization using transitivity property approximate bisimulation relations establish constructed finite abstractions also approximately bisimilar original retarded systems precision chosen given finite abstractions one synthesize controllers original systems satisfying logic requirements systematic way moreover provide sufficient conditions proposed notion incremental stability terms existence incremental lyapunov functions reduce linear matrix inequalities lmi linear systems finally effectiveness results illustrated synthesizing controller regulating temperatures building modeled delayed system introduction finite symbolic abstraction techniques gained significant attention last years since provide tools automated controller synthesis several classes control systems particular abstractions provide approximate models related concrete systems aggregating concrete states inputs symbolic ones finite abstractions one make use existing techniques synthesize hybrid controllers enforcing rich complex specifications usually expressed linear temporal logic formulae automata infinite strings original systems past years several results providing bisimilar finite abstractions various well stochastic systems results include construction approximately bisimilar abstractions incrementally stable control systems switched systems stochastic control systems randomly switched stochastic systems however abstractions obtained results based quantization suffer severely curse dimensionality computational complexity increases exponentially respect dimension concrete system alleviate issue authors proposed alternative approach constructing approximately bisimilar abstractions incrementally stable switched systems without discretizing concept extended provide finite abstractions incrementally stable stochastic switched systems stochastic control systems infinite dimensional control systems comparison discretization based free approaches refer interested readers discussion section hand retarded stochastic systems widely used model various processes finance ecology medical engineering see however construction symbolic models classes systems still unaddressed due underlying challenges functional dependency state history authors provide construction abstractions incrementally stable systems approximation functional spaces however proposed results complex implementation point view also suffer curse dimensionality respect dimension concrete system jagtap zamani motivates work paper provide scheme construction approximately bisimilar finite abstractions class infinite dimensional stochastic systems namely retarded systems without discretizing main contribution paper twofold first introduce notion incremental stability retarded systems provide sufficient conditions terms existence notion incremental lyapunov functions linear case show sufficient conditions reduce linear matrix inequality lmi second assumption incremental stability property provide finite dimensional abstractions approximately bisimilar original infinite dimensional stochastic systems provide approximately bisimilar finite abstractions constructed finite dimensional abstractions help transitivity property approximate bisimulation relations show obtained finite abstractions approximately bisimilar concrete infinite dimensional systems precision defined demonstrate effectiveness proposed results synthesizing controller keeping temperatures comfort zone building modeled linear delayed system retarded systems notations let triplet denote probability space sample space filtration probability measure filtration satisfies usual conditions right continuity completeness symbols denote set natural nonnegative integer integer real positive nonnegative real numbers respectively use denote vector space real matrices rows columns symbol denotes vector whose elements zero except ith one one let family continuous functions denoted norm denotes euclidean norm denote constant function value lft denotes family random processes notation denotes family bounded random variables matrix kak represents euclidean norm use denote minimum maximum eigenvalue symmetric matrix respectively diagonal set defined closed ball centered radius defined set called box span box defined span min represents absolute value set countable covering defining box span define note span extend notions span approximation finite unions boxes follows let box define span min span span define continuous function belongs class strictly increasing belongs class continuous function belongs class fixed map belongs class respect fixed map decreasing respect given measurable function essential supremum denoted recall ess sup identify relation map defined retarded systems let motion process assume poisson process brownian motion independent poisson process models kinds events whose occurrences assumed independent symbolic models retarded systems definition retarded system rjds tuple euclidean space subset bounded input set finite unions boxes subset set measurable locally essentially bounded functions time satisfies following lipschitz assumption exist constants kxt satisfies following lipschitz assumption exists constant kxt satisfies following lipschitz assumption exists constant kxt process said solution process exists satisfying surely drift diffusion reset terms respectively emphasize postulated assumptions ensure existence uniqueness solution process theorem throughout paper use notation denote value solution process starting initial condition input signal time also use notation denote solution process starting initial condition input signal note random variable taking values random variable taking values assume poisson processes psi rates introduce delayed system djds special case retarded system given drift diffusion reset terms respectively constants state delay drift diffusion reset terms respectively incremental stability rjds djds introduce notion incremental stability rjds resp djds definition rjds resp djds incrementally stable moment denoted exist function function two initial conditions following condition satisfied one readily verify absence delay definition reduces definition although left hand side condition based euclidean norm one also derive similar property using functional norm shown following technical lemma lemma consider retarded system resp djds two initial conditions exist function function following inequality holds max functions appearing jagtap zamani proof inequality obtain following inequalities moreover also inequalities along yield max one represent inequality max function later use provide infinitesimal generators denoted operator rjds djds using differentiation equation let function twice differentiable infinitesimal generator associated rjds operator denoted given infinitesimal generator associated djds operator denoted given symbols represent first partial derivatives respect argument argument respectively note dropped arguments sake simplicity describe terms existence lyapunov functions rjds djds using condition defined next definition consider rjds continuous function twice differentiable function called lyapunov function exist functions resp convex resp concave function iii lkft satisfying symbolic models retarded systems nonnegative function exists function satisfying function ksk kskk function satisfying ksk kskk definition consider djds continuous function twice differentiable function called lyapunov function exist constants nonnegative function functions function conditions definition hold kskk kskk provide description rjds terms existence lyapunov functions following theorem theorem rjds admits lyapunov function definition proof proof inspired proof theorem denote using lemma exist constant function whenever without loss generality assume let minimal nonnegative integer jaq let max order prove theorem need show min first show suppose inf since continuous time exist pair constants however generalized formula condition definition fot contradicts thus inequality must true show let inf using condition definition inequality implies consequently generalized formula contradicts property hence must let inf jagtap zamani continuous exists constants using similar reasoning generalized formula condition definition assumption results contradiction thus define inf similar type reasoning get particularly following jensen inequality one obtains choose function implies one readily verify inequality implies next corollary proposes similar results previous theorem djds corollary djds admits lyapunov function definition proof let considering definition lkft satisfying condition definition function moreover functions satisfy properties required condition iii definition therefore satisfies conditions definition thus following theorem obtain following lemma provide similar result corollary tailored linear delayed jumpdiffusion systems sufficient conditions boil lmi lemma consider djds given symbolic models retarded systems system exist constants satisfying gti symmetric positive definite matrix rit gti rit proof consider function given symmetric positive definite matrix one readily verify function satisfies properties definition functions considering infinitesimal generator associated considered linear delayed system lipschitz assumptions young inequality consistency norms one obtain following chains inequalities gti gti gti gti rit rit rit jagtap zamani gti thus following proof corollary lkft satisfying one obtains using generalized ito formula condition definition gronwall inequality using condition definition one obtains hence therefore introducing functions inequality satisfied noisy noiseless trajectory subsection provide important technical lemma used later construct finite dimensional finite abstractions retarded systems lemma provides upper bound distance solution taking values randomly corresponding retarded systems denoted obtained removing diffusion reset terms onwards use notation denote value solution process denote solution time started initial condition trivial input signal raise supplementary assumption lyapunov functions used show later results symbolic models retarded systems assumption lyapunov function defined definition one concave function constant condition assumption restrictive long one interested working compact subset compact subset applying mean value theorem function one obtains max provide main result subsection lemma consider rjds suppose exists lyapunov function assumption hessian positive semidefinite matrix positive semidefinite matrix trivial nonnegative valued function tends zero lipschitz constants introduced definition proof proof use notation denote hessian matrix sake simplicity drop arguments using jensen inequality properties definition jagtap zamani help assumption gronwall inequality one obtains function computed using lipschitz assumptions diffusion reset term one obtain min since lyapunov function using lemma sup kuk thus sup kuk sup kuk using inequalities one min sup kuk sup kuk defining sup kuk min sup kuk obtain definition one obtain sup sup completes proof note satisfies conditions proposed lemma symbolic models retarded systems systems approximate equivalence relations recall notion system introduced later serves unified modeling framework retarded systems finite dimensional abstractions symbolic models definition systems tuple set states possibly infinite set initial states set inputs possibly infinite transition relation set outputs output map denote alternative representation transition state called simply successor state input moreover system said metric output set equipped metric finite symbolic finite deterministic exists nonblocking exists system finite generated initial state finite sequence transitions associated finite given finite runs directly extended infinite runs well provide notion approximate simulation relation two systems introduced later used analyzing synthesizing controllers retarded systems definition let two metric systems output sets metric relation said bisimulation relation satisfies following conditions satisfying implies satisfying implies iii remove condition iii said simulation relation system bisimilar denoted exists bisimulation relation order present main results paper need employ notion system abstract representation retarded system first define metric system associated retarded system denoted set random variables defined probability space subset measurable respectively exists solution lft satisfying assume output equipped metric also note system deterministic nonblocking restrict jagtap zamani attention system control signals intervals length metric systems associated retarded systems defined measurable fih respectively exists solution lkft satisfying words finite run represented captures solutions rjds sampling times started resulting control input obtained concatenation input signals moreover corresponding finite finite dimensional abstractions rjds section introduce finite dimensional abstraction consider metric systems associated retarded systems consider triple parameters sampling time temporal horizon source state let define two metric systems abuse notation identifying input curve use similar notations rest paper well notice system resp deterministic finite dimensional necessarily symbolic unless finite set note output maps state random variable variable respectively next theorem provides one main results section construction finite dimensional abstractions approximately bisimilar rjds theorem consider rjds admitting lyapunov function form explained lemma given let sampling time temporal horizon source state sup relation bisimulation relation proof consider thus condition definition holds show condition definition symbolic models retarded systems holds consider consider let denote input sequence help jensen inequality lemma triangle inequality one obtains following chains inequalities hence thus condition definition holds similar way one show condition iii definition holds completes proof note theorem given one select temporal horizon sufficiently large enforce terms sufficiently small results enforces lower bound sampling time establish results existence finite dimensional abstraction given result theorem theorem consider results theorem select proof every always exists hence similar way show every exists completes proof next theorems provide results similar theorems using finite dimensional abstraction rather note stochastic finite dimensional abstraction theorem consider rjds given let sampling time temporal horizon source state sup relation bisimulation relation proof proof similar one theorem jagtap zamani theorem consider results theorem select proof proof similar one theorem provide results construction finite abstractions finite abstractions rjds section provide finite symbolic abstraction quantizing input set let consider tuple quantization parameter quantized input set denoted notation subsection define corresponding finite systems order provide approximate bisimulation relation sampled retarded systems symbolic models need following technical lemmas lemma consider rjds quantization parameter span relation given bisimulation relation proof let kui implies using definition one obtains first condition definition holds consider let consider choose consider obvious hence condition definition holds similarly condition iii definition holds shows approximate bisimulation relation always exists hence note existence guaranteed finite union boxes inequality span moreover choosing one readily gets hence next lemma provides results similar lemma using symbolic model symbolic models retarded systems lemma consider rjds quantization parameter span relation given lemma bisimulation relation proof proof similar one lemma provide main results section establishing approximate bisimulation relation immediate consequence transitivity property approximate bisimulation relations proposition recalled next proposition consider metric systems provide first main result section theorem consider rjds given quantization parameter span consider results theorem lemma relation given bisimulation relation note relations theorem defined theorem lemma respectively theorem assumptions theorem let choose theorem proofs theorems simple consequence results theorem lemma proposition note referring discussion theorem allowing lower bound one achieve abstraction precision theorem choosing sufficiently large temporal horizon sufficiently small input set quantization similar way help transitivity property bisimulation relations proposition provide approximate bisimulation relation sampled retarded system symbolic model probabilistic output values following theorem consider rjds given quantization parameter span consider results theorem lemma relation given bisimulation relation theorem assumptions theorem let choose theorem proofs theorems simple consequence results theorem lemma proposition jagtap zamani room room room room room room heater heater room room room figure schematic building example show effectiveness proposed results consider simple thermal model building shown schematically figure model used similar used addition modified arrangement rooms increase dimensions considered dynamic affected delays states jumps modeling door window opening dynamic considered delayed system given following delayed stochastic differential equations terms wti pti denote standard brownian motion poisson process rate respectively denote temperature room degree celsius external temperature temperatures two heaters time units time units state delays drift diffusion terms respectively control inputs amounted corresponding heaters corresponding heaters assume one heater time instance results finite input set system parameters chosen using result lemma one readily compute function considering moment constant function precision fixing sampling time time units one symbolic models retarded systems figure realizations solution process initial condition time figure evolution input signals obtain temporal horizon satisfying inequality theorem therefore resulting cardinality set states note using results theorem one obtains results bigger abstraction remark since input set finite finite dimensional abstraction also symbolic jagtap zamani time figure square root average values realizations squared distance solution process set consider objective design controller enforcing trajectories stay within comfort zone corresponds ltl specification computation symbolic model controller synthesis performed using tool quest jzst computer cpu intel core figure shows realizations solution process starting form initial condition synthesized control signals shown figure figure show square root average value realizations squared distance time solution process set namely point set distance defined inf one readily observe empirical average much lower precision table report sizes finite abstractions given number transitions sizes controllers lower bounds precision computation time required constructing finite abstractions controllers different values table performance comparison different values temporal horizon number transitions number transitions controller abstraction computation time sec controller computation time sec precision references bandyopadhyay saha pal deterministic stochastic analysis delayed allelopathic phytoplankton model within fluctuating environment nonlinear analysis hybrid systems corronc girard goessler mode sequences symbolic states abstractions incrementally stable switched systems ieee conference decision control pages dec girard approximately bisimilar abstractions incrementally stable finite infinite dimensional systems ieee conference decision control pages dec girard pappas approximation metrics discrete continuous systems ieee transactions automatic control may symbolic models retarded systems girard pola tabuada approximately bisimilar symbolic models incrementally stable switched systems ieee transactions automatic control january huang mao stability stochastic retarded systems markovian switching ieee transactions automatic control august julius pappas approximations stochastic hybrid systems ieee transaction automatic control june jzst jagtap zamani quest tool synthesis symbolic controllers international conference quantitative evaluation systems pages springer international publishing https kolmanovsky maizenberg stochastic optimal control jump diffusion excited energy harvesters american control conference pages june maler pnueli sifakis synthesis discrete controllers timed systems annual symposium theoretical aspects computer science pages springer sulem applied stochastic control jump diffusions universitext berlin pola girard tabuada approximately bisimilar symbolic models nonlinear control systems automatica october pola pepe benedetto symbolic models timevarying timedelay systems via alternating approximate bisimulation international journal robust nonlinear control pola pepe benedetto tabuada symbolic models nonlinear systems using approximate bisimulations systems control letters june shaikhet lyapunov functionals stability stochastic functional differential equations springer science business media skorokhod asymptotic methods theory stochastic differential equations american mathematical tabuada verification control hybrid systems symbolic approach springer science business media zamani abate approximately bisimilar symbolic models randomly switched stochastic systems systems control letters july zamani abate girard symbolic models stochastic switched systems discretization approach automatica may zamani esfahani majumdar abate lygeros symbolic control stochastic systems via approximately bisimilar finite abstractions ieee transactions automatic control december zamani tkachev abate towards scalable synthesis stochastic control systems discrete event dynamic systems june department electrical computer engineering technical university munich munich germany address zamani url http
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privacy amplification lossy compression duality channel coding joseph renes aug institute theoretical physics eth switzerland examine task privacy amplification points view former give characterization optimal rate privacy amplification classical adversaries terms optimal error asymmetric hypothesis testing converse significantly improves previous bounds based smooth watanabe hayashi isit turns equivalent recent formulation terms divergence yang schaefer poor latter show protocols privacy amplification based linear codes easily repurposed channel simulation combined known relations channel simulation lossy source coding implies privacy amplification understood basic primitive channel simulation lossy compression applied symmetric channels lossy compression settings construction leads protocols optimal rate asymptotic limit finally appealing notion channel duality recently detailed show linear codes symmetric channels quantum output transformed linear lossy source coding schemes classical variables arising dual channel explains curious duality problems erasure channel observed martinian yedidia allerton arxiv partly anticipates recent results optimal lossy compression polar generator matrix codes introduction packing covering core simple information processing primitives noisy channel coding instance inputs lead probability distributions output symbols one would like pack many distributions space possible output distributions two overlap significantly gives code associated inputs reliably inferred channel output covering sense dual packing goal find set distributions whose empirical average approximates covers target distribution channel simulation instance would like approximate channel output given input minimal possible amount additional randomness paper examine simple covering task privacy amplification also known randomness extraction well point view goal privacy amplification originally introduced deterministically transform given random variable may correlated largest possible new random variable independent regarding information held adversary eavesdropper eve variable held alice bob privacy amplification understood means extracting random secret key information partially correlated adversary natural question much randomness extracted answer depends setting cryptography one interested making assumptions correlations eavesdropper possible usually considers constraints formulated terms adversary related maximal probability guessing one also consider adversaries holding quantum information opposed classical information focus instead consider setting complete distribution known classical give upper lower bounds optimal rate privacy amplification setting formulated terms asymmetric hypothesis testing particular minimal error discriminating actual distribution uncorrelated distribution plays important role see theorem uniform distribution arbitrary previous work based collision entropy though see also recent converse bound reminiscient metaconverse channel coding lemma appearance also elegant proof leads tight bounds finite blocklengths converse makes substantial improvement bound based theorem turns equivalent bound recently formulated yang schaefer poor lemma turning coding theory show privacy amplification used primitive construct protocols channel simulation lossy compression first show privacy amplification based linear functions used simulating action given channel given input simulation error latter precisely equal security parameter former idea behind construction stated detail proposition consider privacy amplification channel output relative input extend function reversible say needs transmitted encoder decoder order reconstruct applying common randomness considering linear functions immediately infer size symmetric channels sufficient achieve optimal rate communication required simulation task provided amount common randomness available encoder decoder large enough shown simulating optimal channel function gives means turning channel simulation lossy compression hence privacy amplification also used perform lossy compression precise details stated corollary sources distortion functions symmetric certain sense canonical example compressing input considering hamming distortion construction achieves bound finally shows lossy compression accomplished repurposing good code appropriate dual channel recently investigated particular suppose random variable compressed reconstructed according given distortion measure optimal channel associated function show code good dual channel exists similarly good lossy compression scheme reconstructed based codewords see corollary precise details happens encoder channel code decompressor related lossy encoder unrelated channel decoder hence guarantees made efficiency lossy compressor even efficient channel decoding known possible dual channel usually quantum output one important exception erasure channel case channel erasure probability associated lossy compression problem precisely binary erasure quantization considered martinian yedidia established case channel codes converted lossy source codes vice versa thus understand forward implication resulting deeper structure duality codes channels establishing general relation precisely one main goals paper mathematical setup shall consider random variables finite alphabet treat associated probability distributions probability mass functions vectors random variable alphabet denote probability mass function consider element joint distributions labelled relevant random variables denotes uniform distribution product distributions denoted corresponds tensor product level vector representation events observables also treated elements particular set tests important purposes simply vectors whose entries lie interval test probability probability test denoted denotes euclidean inner product shall also occasion consider quantum states tests notation also employed quantum setting probability distributions replaced density operators positive operators unit trace tests positive operators space whose eigenvalues exceed unity inner product inner product variational distance two distributions defined denotes vector ones definition immediate satisfies triangle data processing inequalities easy show given two distributions set pairs probabilities achievable possible tests forms testing region unit square shall make use lower boundary region given function min interpreted minimal error asymmetric hypothesis test error constrained smaller definition immediate satisfies data processing inequality stochastic map channel since optimal test induces feasible test optimization linear program see whose dual formulation max complementary slackness conditions primal dual programs lead neymanpearson lemma optimal test satisfies values chosen error optimal value dual cutoff inverse likelihood ratio deciding clearly optimal max thus region convex hull points obtained tests form zero otherwise quantity equivalent divergence theorem see note implies therefore righthand side hand follows rerunning argument leaving optimization dual implies context also mention bound holds equation derive note feasible assumption therefore mean variance ratio log also important quantities asymptotics testing distributions let log average ratio relative entropy log variance stein lemma theorem gives behavior large following lemma following refinement lemma suppose two distributions finite log unit normal gaussian cumulative distribution function log log proof start lemma based refinement central limit theorem establishes bounds log log log log provided argument set make replacement taylor theorem pan pan two bounds equal log difficult see variational distance length longest vertical line segment one place inside diagonal inside testing region following proposition gives bounds terms lemma distributions proof supposing optimal test follows immediately upper bound let optimal test set definition next set determined later follows two cases consider smaller larger former choice ensures valid test implies desired bound since latter choosing leads valid test implies desired bound since chernoff attributes result stein note appearing proof precisely value vertical distance diagonal lower boundary variational distance lower bound corresponds upper bound proposition upper bound due dupuis mention passing lower bound claimed proposition error particular deterministic completely uniform distributions random variable respectively violates ostensible bound example also shows claimed inequality log violated bounds extractable randomness given joint distribution task randomness extraction privacy amplification relative apply function resulting distribution essentially setup depicted figure sometimes refer extractor extractor function though note cryptography community extractor refers set functions useful generating randomness source characterized terms measure closeness variational distance say protocol privacy amplification log figure schematic representation privacy amplification randomness extraction relative function produce random variable uniformly random independent measured variational distance letting largest exists log privacy amplification protocol show following result theorem joint distribution min max max show second inequality achievability statement employing hashing thus linear functions capable achieving stated bounds argument combination argument given channel resolvability hayashi channels classical lemma quantum lemma output leftover hashing lemma adapted yield bound involving first inequality converse adaptation converse involving reported based underlying achievability converse arguments also apply adversary holds quantum information quantum decidedly case hypothesis testing versions shall remark steps fail quantum side information proof converse let begin showing converse relies following lemma version statement conditioned increase application function lemma function proof using conditional distributions construct stochastic map back clearly stochastic data processing inequality observe hence completing proof suppose extractor function log privacy amplification protocol lemma equivalent whence lemma minimization gives lemma hold quantum one find counterexamples might case anticipated noticing proof make use distribution conditioned value quantum analog state specific counterexample consider qubit density operators orthonormal basis along suppose take state given marginal state averaged function mapping distribution corresponding conditional states lower bound choose upper bound take test projection onto zero eigenspace zero particular meaning smaller example shows lemma hold arbitrary open question holds proof achievability move proof direct part let arbitrary function set arbitrary test zpon define rescaled probability distributions using triangle inequality due form latter two terms combine give bound fact set first term expression use follows writing applying inequality choosing normalized distribution strictly positive probabilities determined later let deal summation first term omitting random variable subscripts taking expectation chosen uniformly random family hash functions obtain using gives finally let optimal test optimal dual formulation properties optimal test follows therefore obtain jensen inequality returning maxq ensures exists since choosing continuous may maximize strictly positive probability completes proof quantum rather classical test would take form set positive operators would immediately confronted possibility may commute conditional quantum states given quantum versions proof assume objects commute one method dealing issue done pinch quantum states remove elements restore commutation appeal bounds pinched unpinched states unclear done combination steps taken end bound terms analysis bounds asymptotics log follow immediately lemma firstorder behavior optimal rate follows theorem setting behavior originally shown theorem starting bound terms smooth see also corollary log proof upper bound set log lemma applies large enough factor disappears taking limit likewise lower bound choose lemma applies large enough publication initial version manuscript wei yang pointed bounds contained herein related recently derived achievability bound derived lemma states exists privacy amplification scheme distributions exp log log log get back observe expectation term necessarily smaller set use get converse equivalent lemma states every privacy amplification protocol must satisfy obtain expression use variational form write consider tests yield vertices specifically larger using rearranging inequality function interpolates linearly vertices therefore relation holds arbitrary setting recovers show equivalent use choose suitable optimization specifically suppose optimizer let define clearly since must positive hand must positive since second term positive indeed must rearranging expression gives therefore since previous argument implies choice optimal log extraction rate upper bound upper bound normal approximation lower bound lower bound blocklength figure comparison finite blocklength bounds randomness extraction relative case obtained uniform bsc crossover probability target security parameter asymptotic rate example shaded region indicates improvement compare bounds fixed example consider obtained uniform binary symmetric channel crossover probability target security parameter follow theorem computing assuming auxiliary variables uniform turns lower bound equation essentially identical always advantage though miniscule increase range shown figure theorem considerably better converse side equations contain small error term inside square root wei yang private communication substantially improves equation satisfying see normal approximation obtained disregarding log term lies midway upper lower bounds note example log term provably absent asymptotic expansion shown theorem repurposing randomness extraction channel simulation given channel task channel simulation reproduce joint input output statistics applied making use ideal noiseless channel common randomness encoder decoder randomness course necessary simulate stochastic nature main question characterize amounts communication randomness required given channel input stronger variant universal channel simulation simulation works input encoder constructed using knowledge particular input paper interested case let first specify setup concretely given input distribution channel leads joint distribution protocol simulation applied consists encoding map decoding map log log random variable figure simulation channel acting input random variable means ideal channel common randomness following shows privacy amplification repurposed channel simulation see also proposition let arbitrary channel arbitrary input distribution suppose linear function protocol privacy amplification relative log protocol simulating action constructed proof linear function always extended reversible linear function given joint distribution induces conditional distribution interpreted channel defines encoder requires log bits random input produces message log log bits meanwhile decoder calling output instead design meaning combined action gives back protocol combined action instead applied producing data processing inequality therefore premise latter quantity bounded proof complete since considering linear makes sense associate function linear code suppose consider output syndromes code encoding bits parity check matrix reversible output must correspond message encoded action therefore produce codeword determined offset coset determined see concretely extend invertible matrix defines say reconstruction operation apply let set matrix terms action given since meaning generator matrix action claimed channels whose input induces output construction directly leads protocols achieve optimal communication rate channel simulation asymptotic scenario assuming unlimited common randomness amount communication required case uniform log log theorem achieve mutual information rate general case one option concatenate protocol data compression though pursue similar approach applied noisy channel communication see figure lossy compression random variable test indicates zero otherwise lossy source coding task lossy source coding compress random variable smaller alphabet reconstructed close measured distortion function two main approaches specifying distortion constraint requiring either fixed average value fixed probability exceeding specified target value latter case one defines protocol lossy compression consist encoder decoder log excess distortion probability expressed expectation test function whenever zero otherwise setup depicted figure former case protocol consists encoder decoder latter stronger criterion since scheme target value converted average scheme average distortion less max see lemma channel simulation one way construct lossy compression protocol shown later quantum setting case excess distortion probability formalize statement follows proposition given distortion function target distortion suppose channel target distortion exceeded probability larger protocol channel simulation applied used construct lossy compression protocol proof let output channel simulation protocol properties variational distance therefore probability exceed target distortion larger opposed channel simulation lossy compression need randomized encoding decoding operations simulation protocols adapted lossy compression principle derandomized since output distribution mixture values distortion probability linear one may well pick value leading smallest excess distortion probability still leaves randomness mapping derandomized manner setting perhaps natural consider case average distortion directly since interested symbolwise distortion functions form dsym function dsym well constructing protocols simulating single symbol channels rather case proposition given symbolwise distortion function dsym suppose channel dsym simulation protocol acting used construct lossy compression scheme proof let output simulation protocol since protocol definition variational distance max dsym dsym thus dsym averaging gives desired result setting optimal rate given function infw optimal channel expression gives employ optimal rate channel simulation protocol construct optimal rate lossy compression procedure hence case ultimately rely privacy amplification relative linear functions perform lossy compression optimal rate fixed blocklength following corollary corollary setting proposition suppose joint distribution dsym linear privacy amplification protocol relative used construct lossy compression scheme standard example source hamming distortion dsym uniform since optimal channel binary symmetric channel crossover probability binary erasure quantization example martinian yedidia input alphabet probabilities respectively symbol distortion function dsym otherwise reported kostina equation rate distortion function case adapting notation present setting optimal channel concatenation map randomly assigns inputs leaves input values untouched binary symmetric channel crossover probability therefore uniform output value lossy compression channel coding making use duality relations channels codes show linear codes used build lossy source codes suppose random variable lossily compressed optimal joint distribution rate distortion function induces channel given marginal note channel defined opposite sense corollary establishes lossy compression constructed privacy amplification input relative output following task dual channel coding lossless compression dual channel channel whose output see precise definition restrict attention symmetric channels specifically corollary ensures code average error probability optimal decoder leads linear privacy amplification protocol relative extractor function given generator matrix acting right parity check matrix fact using theorem improve security parameter difference stems use corollary involves optimization marginal rather using actual marginal directly combining corollary obtain corollary optimal conditional distribution appearing function let dual channel according code used construct lossy compression scheme reconstruction operation outputs codewords shifted coset determined common randomness meanwhile quantizer compressor stochastic based conditional distribution distribution proposition dual channels examples mentioned explicitly given case hamming distortion optimal channel also bsc crossover probability means dual channel takes classical input pure state binary erasure quantization optimal channel concatenation bsc crossover probability erasure channel erasure probability dual channel computed example output consists two independent parts one classical one classical part output erasure channel erasure probability quantum part exactly output dual bsc thus probability input channel shows unchanged classical part output even classical part useless quantum part contains information input note latter case two quantum states identical quantum part dual output useless dual effectively erasure channel erasure probability relation code use dual code privacy amplification partly explains curious duality symbols source coding symbols channel coding observed martinian yedidia direct analysis lossy compression erasure channel show equivalence two tasks made use general theory duality shown one direction equivalence namely implies lossy compression also makes sense recent results optimality polar codes generator matrix codes lossy compression shown respectively polar codes duals sense dual polar code given using frozen bits instead information bits code constructed properties synthesized channels thus fact polar codes achieve capacity channels described follows general result implies polar codes achieve bound associated sources one simply base scheme synthesized inputs high entropy exactly done similarly unreasonable suspect parity check ldpc codes achieve capacity channels optimal decoding would imply duals generator matrix ldgm codes achieve bound associated sources precisely shown duality also helps explain ldpc codes useful lossy compression observed otherwise duals ldgm codes would good channel coding known case note implications based duality say nothing complexity encoding decoding operations either channel coding lossy compression indeed establishing low complexity bounds better part results discussion given new bounds optimal rate privacy amplification setting seen converse bound improves substantially results achievability bound tightest known literature formulation terms hypothesis testing advantages one clear connection information theory statistics particular immediately gives asymptotic expansion optimal rate second order conceptually using common quantity optimal rate bounds allow see concrete relationship covering packing problems plainly namely approximation parameter rate bounds covering problems involve packing problems involve supports notion packing dual covering open question whether approach extended covering problems involving quantum information also shown privacy amplification primitive constructing channel simulation lossy compression protocols enables extend known duality codes packing lossless compression covering privacy amplification covering problem lossy compression specifically coding duality implies duals good channel codes lead good lossy source codes least symmetric channels lossy compression setups immediate open question context whether one direction lossy source coding back channel coding observed case binary erasure channel binary erasure quantization perhaps straightforward approach demonstrating would show lossy source codes used privacy amplification since duality ensures good privacy amplification protocol implies existence good channel code dual corollary theorem one could also investigate whether duality leads improved finite blocklength bounds privacy amplification however since dual setup involves quantum output seems doubtful bounds available case tight classical output see much bigger tantalizing open question whether duality also provides link algorithms channel decoding source quantization using belief propagation link shown theorem binary erasure coding quantization problems progress general case could help finding new bounds performance first step direction would extend notion decoding duals classical channels presumably possible extending construction dealt dual bsc since symmetric classical channel regarded mixture bscs though duality used construction therein retrospect role evident particular unitaries combining quantum information check variable nodes could determined appealing convolution rules bsc theorem states dual check convolution variable convolution duals similarly dual variable hints concretely possibility duality shed light details remain seen acknowledgments thank renato renner marco tomamichel helpful discussions particular thanks dupuis providing upper bound lemma henry pfister pointing wei yang pointing relationship theorem results work despite best efforts error check node convolution initial draft persists final published version though recent arxiv version check node convolution read thanks narayanan rengaswamy pointing twice supported swiss national science foundation snsf via national centre competence research qsit well air force office scientific research afosr via grant references bennett brassard robert privacy amplification public discussion siam journal computing impagliazzo levin luby generation functions proceedings annual acm symposium theory computing nisan zuckerman randomness linear space journal computer system sciences bennett brassard maurer generalized privacy amplification ieee transactions information theory renner wolf simple tight bounds information reconciliation privacy amplification advances cryptology asiacrypt lecture notes computer science hayashi tight exponential analysis universally composable privacy amplification applications ieee transactions information theory watanabe hayashi analysis privacy amplification via entropy entropy proceedings ieee international symposium information theory yassaee aref gohari output statistics random binning applications yang schaefer poor wiretap channels nonasymptotic fundamental limits nagaoka strong converse theorems quantum information theory proceedings erato conference quantum information science eqis vol polyanskiy poor channel coding rate finite blocklength regime ieee transactions information theory hayashi quantum information theory graduate texts physics springer berlin heidelberg steinberg simulation random processes theory ieee transactions information theory winter compression sources probability distributions density operators arxiv renes duality channels codes martinian yedidia iterative quantization using codes graphs proceedings annual allerton conference communication control computing arxiv vanderbei linear programming foundations extensions springer new york neyman pearson problem efficient tests statistical hypotheses philosophical transactions royal society mathematical physical engineering sciences polyanskiy channel coding fundamental limits phd princeton university chernoff theory parametric case annals mathematical statistics tomamichel hayashi hierarchy information quantities finite block length analysis quantum tasks ieee transactions information theory dupuis faist renes renner generalized entropies xviith international congress mathematical physics edited jensen world scientific carter wegman universal classes hash functions journal computer system sciences hayashi general nonasymptotic asymptotic formulas channel resolvability identification capacity application wiretap channel ieee transactions information theory tomamichel schaffner smith renner leftover hashing quantum side information ieee transactions information theory tomamichel framework quantum information theory phd thesis eth ahlswede common randomness information theory cryptography secret sharing ieee transactions information theory renes theory information beyond information theory bennett shor smolin thapliyal capacity quantum channel reverse shannon theorem information theory ieee transactions arxiv renes renner noisy channel coding via privacy amplification information reconciliation ieee transactions information theory datta renes renner wilde lossy quantum data compression ieee transactions information theory luo devetak channel simulation quantum side information ieee transactions information theory arxiv datta hsieh wilde quantum rate distortion reverse shannon theorems separation ieee transactions information theory cover thomas elements information theory kostina verdu lossy compression finite blocklength regime ieee transactions information theory renes duality privacy amplification quantum adversaries data compression quantum side information proceedings royal society renes sutter hassani alignment polarized sets ieee journal selected areas communications korada urbanke polar codes optimal lossy source coding ieee transactions information theory aref macris vuffray approaching limit spatial coupling belief propagation decimation ieee transactions information theory wilde guha polar codes channels ieee transactions information theory mackay good codes based sparse matrices ieee transactions information theory renes belief propagation decoding quantum channels passing quantum messages new journal physics
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jan unrecognizability prime graph almost simple group pgl ali mahmoudifar department mathematics tehran north branch islamic azad university tehran iran alimahmoudifar abstract prime graph finite group denoted also called recognizable prime graph finite group isomorphic paper classify finite groups prime graph pgl particular present solvable groups prime graph pgl ams subject classification keywords projective general linear group prime graph recognition introduction let natural number denote set prime divisors also let finite group set denoted set element orders denoted denote maximal numbers divisibility relation prime graph graph whose vertex set two distinct primes joined edge write whenever contains element order prime graph denoted finite group called recognizable prime graph every finite group recognizable prime graph whenever exists fin finite group isomorphic almost simple group pgl lot results recognition prime graph proved prime number mersenne fermat prime pgl unique nonabelian composition factor isomorphic psl unique nonabelian composition factor isomorphic psl psl know pgl psl characterization simple groups refer proved odd prime odd natural number pgl uniquely determined prime graph description get characterization prime graph pgl odd prime number even open problem paper main result consider recognition prime graph almost simple groups pgl moreover construct solvable group prime graph pgl preliminary results lemma let finite group frobenius group kernel cyclic complement contained prime divisor lemma let frobenius group kernel complement following assertions hold nilpotent group mod every subgroup order necessarily distinct primes cyclic particular every sylow subgroup odd order cyclic sylow either cyclic generalized quaternion group group subgroup index isomorphic cyclic sylow using theorem following result lemma let finite group one following holds frobenius group exists nonabelian simple group aut nilpotent normal main results lemma exists frobenius group abelian pgl proof let finite field elements also let additive group multiplicative group know acts right product finite group since hand acts fixed point freely frobenius group kernel complement since multiplicative group cyclic cyclic therefore element order prime graph consists one edge edge implies pgl desired lemma exists frobenius group cyclic pgl proof let two fields elements respectively let direct product additive groups also let direct product multiplicative groups respectively know acts fixed point freely define action follows define clear definition well defined may construct finite group hand acts fixed point freely frobenius group kernel complement finally since nilpotent get contains edge pgl lemma exists group normal series pgl proof let field elements additive group know generator multiplicative group hence order also aut argument implies may construct frobenius group involution define action follows therefore frobenius group desired properties theorem let finite group pgl isomorphic one following groups frobenius group abelian frobenius group cyclic group normal series almost simple group pgl proof throughout proof assume finite group prime graph almost simple group pgl first note lemma pgl hence pgl one edge edge isolated vertex implies two connected components thus lemma get frobenius group group exists nonabelian simple group aut consider possibility let frobenius group kernel complement know nilpotent connected component prime graph also description adjacent shows either consider cases separately case let hence order complement even abelain subgroup also description since connected component edge follows pgl implies groups satisfying case let hence since nilpotent get adjacent means pgl get case let group normal series since connected components get implies case let exist nonabelian simple group aut since finite simple groups property classified table get isomorphic one simple groups psu psl subcase let know aut isomorphic alternating group symmetric group since prime graph edge hence get isomorphic thus prime graph nonadjacent least one prime numbers belongs fit let fit let sylow fit since characteristic subgroup fit fit normal subgroup hand alternating group subgroup frobenius subgroup isomorphic recall previous argument subgroup isomorphic lemma get adjacent contradiction subcase let psu edge contradiction subcase let psl isomorphic psl psl diagonal field automorphism psl particular involution field automorphism psl semidirect product psl contains element order hence neither field automorphism automorphism therefore diagonal automorphism pgl discussion pgl enough prove fit contrary let fit also let sylow fit since fit nilpotent write fit fit put fit get pgl fit since elementary abelian group without loose generality may assume fit elementary abelian pgl conclude adjacent contradiction let also let sylow pgl know cyclic hand frobenius group since isolated vertex follows cyclic impossible therefore pgl completes proof references akhlaghi khatami khosravi characterization prime graph pgl odd international journal algebra computation shariati beynekalae iranmanesh foroudi ghasemabadi quasirecognition prime graph simple group southeast asian bull wang quasirecognition prime graph southeast asian bull buturlakin spectra finite linear unitary groups algebra logic conway curtis norton parker wilson atlas finite groups oxford university press oxford gruenberg roggenkamp decomposition augmentation ideal relation modules finite group proc london math soc hagie prime graph sporadic simple group comm algebra khatami khosravi akhlaghi prime graph prime rocky mountain appear khosravi prime graph simple group algebra appl khosravi khosravi khosravi prime graph siberian math khosravi khosravi khosravi prime graph prime number acta math hungarica moghaddamfar groups order degree pattern sci china kondrat prime graph components finite simple groups math mazurov characterizations groups arithmetic properties proceedings international conference algebra algebra colloq mazurov characterizations finite groups sets element orders algebra logic moghaddamfar shi number finite groups whose element orders beitrage algebra williams prime graph components finite groups algebra zavarnitsin recognition finite groups prime graph algebra logic zsigmondy zur theorie der potenzreste monatsh math phys
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journal latex class files vol may deep convolutional neural networks predominant instrument recognition polyphonic music dec yoonchang han jaehun kim kyogu lee senior member ieee musical instruments polyphonic music recordings challenging important problem field music information retrieval enables music search instrument helps recognize musical genres make music transcription easier accurate paper present convolutional neural network framework predominant instrument recognition polyphonic music train network music excerpts predominant instrument estimate arbitrary number predominant instruments audio signal variable length obtain result aggregate multiple outputs sliding windows test audio investigated two different aggregation methods one takes average instrument takes sum followed normalization addition conducted extensive experiments several important factors affect performance including analysis window size identification threshold activation functions neural networks find optimal set parameters using dataset audio excerpts instruments evaluation found convolutional neural networks robust conventional methods exploit spectral features source separation support vector machines experimental results showed proposed convolutional network architecture obtained measure micro macro respectively achieving performance improvement compared algorithms index recognition convolutional neural networks deep learning neural network music information retrieval ntroduction usic said built interplay various instruments human easily identify instruments used music still difficult task computer automatically recognize mainly music real world mostly polyphonic makes extraction information audio highly challenging furthermore instrument sounds real world vary many ways timbre quality playing style makes identification musical instrument even harder music information retrieval mir field highly desirable know instruments used audio sample first instrument information per han kim lee music audio research group graduate school convergence science technology seoul national university seoul republic korea yoonchanghan eldrin kglee lee also advanced institutes convergence technology suwon republic korea manuscript received december revised december important useful information users included audio tags huge demand music search owing increasing number music files digital format unlike text search difficult search music input queries usually text format instrument information included tags allows people search music specific instrument want addition obtained instrument information used various applications instance tailored audio equalization applied music moreover music recommendation system reflect preference users musical instruments furthermore also used enhance performance mir tasks example knowing number type instrument would significantly improve performance source separation automatic music transcription would also helpful identifying genre music instrument recognition performed various forms hence term instrument recognition instrument identification might indicate several different research topics instance many related works focus studiorecorded isolated notes name eronen used cepstral coefficients temporal features classify orchestral instruments several articulation styles achieved classification accuracy instrument family level individual instruments diment used modified group delay feature incorporates phase information together cepstral coefficients mfccs achieved classification accuracy instruments applied sparse coding cepstrum temporal achieved fmeasure classifying instruments also reported classification result database previous works krishna sreenivas experimented classification solo phrases rather isolated notes proposed line spectral frequencies lsf gaussian mixture model gmm achieved accuracy instrument family individual instruments moreover essid reported classification system mfccs gmm along principal components analysis pca achieved overall recognition accuracy solo phrases five instruments recent works deal polyphonic sound closer music monophonic sound case polyphonic sound number research journal latex class files vol may ies used synthesized polyphonic audio single tones heittola used matrix factorization nmf model mfccs gmm synthesized polyphonic sound achieved recognition rate six polyphonic notes randomly generated instruments kitahara used various spectral temporal modulation features pca linear discriminant analysis lda classification reported using feature weighting musical context recognition rates duo trio quartet duan proposed uniform discrete cepstrum udc udc mudc spectral representation radial basis function rbf kernel support vector machine svm classify types western instruments classification accuracy randomly mixed chords two six polyphonic notes generated using isolated note samples rwc musical instrument sound database around two polyphony notes six polyphony notes shown previous works focused identification instrument sounds clean solo tones phrases recent research studies polyphonic sounds closer situation artificially produced polyphonic music still far professionally produced music music many factors affect recognition performance instance might highly different timbre depending genre style performance addition audio file might differ quality great extent depending recording production environments paper investigate method predominant instrument recognition professionally produced western music recordings utilize convolutional neural networks convnets learn spectral characteristics music recordings musical instruments perform instrument identification polyphonic music excerpts major contributions work presented paper follows present convnet architecture predominant musical instrument identification training data single labeled target data unknown number classes existing data introduce new method aggregate outputs convnets sliding windows find predominant instruments music excerpt variable length conventional method majority vote often fails conduct extensive experiment activation function neurons used convnets cause huge impact identification result remainder paper organized follows section introduce emerging deep neural network techniques mir field next system architecture section includes audio preprocessing proposed network architecture detailed training configuration explanation various activation functions used experiment section evaluation section contains information dataset testing configuration including aggregation strategy evaluation scheme illustrate performance proposed convnet section results section analysis effects activation function analysis window size aggregation strategy identification threshold analysis moreover present qualitative analysis based visualization convnet intermediate outputs understand network captured pattern input data finally conclude paper section roliferation eep eural etworks usic nformation etrieval ability traditional machine learning approaches limited terms processing input data raw form hence usually input learning system typically classifier feature representation requires extensive domain knowledge careful engineering process however getting common design system automatically discover higherlevel representation raw data stacking several layers nonlinear modules called deep learning recently deep learning techniques widely used across number domains owing superior performance basic architecture deep learning called deep neural network dnn feedforward network multiple hidden layers artificial neurons approaches outperformed previous methods speech applications phone recognition largevocabulary speech recognition speech recognition speech recognition many variants modified architectures deep learning depending target task especially recurrent neural networks rnns convnets recently shown remarkable results various multimedia information retrieval tasks rnns highly powerful approaches sequential inputs recurrent architecture enables hidden units implicitly maintain information past elements sequence since languages natively contain sequential information widely applied handle text characters spoken language reported rnns shown successful result language modeling spoken language understanding hand convnet useful data local groups values highly correlated forming distinctive local characteristics might appear different parts array hence one popular approaches recently image processing area handwritten digit recognition mnist dataset image tagging dataset addition reported outperformed approaches several computer vision benchmark tasks object detection semantic segmentation object recognition also tasks journal latex class files vol may fig schematic proposed convnet containing times repeated double convolution layers followed last layer performs global fed fully connected layer followed sigmoid outputs representation music signal composed harmonics various musical instruments human voice musical instrument produces unique timbre different playing styles type spectral characteristics music signal might appear different location time frequency image convnets usually composed many convolutional layers inserting pooling layer convolutional layers allows network work different time scales introduces translation invariance robustness local distortions hierarchical network structures convnets highly suitable representing music audio music tends present hierarchical structure time different features music might salient different time scales hence although convnets commonly used technique image processing increasing number attempts apply convnets music signal reported convnet outperformed previous approaches various mir tasks onset detection automatic chord recognition music analysis attempt apply convnets musical instrument identification found recent report park although still ongoing work predominant instrument recognition method hence instruments target instrument sounds exist research differs deal polyphonic music work based studio recording single tones addition research also differs use data training estimate data used multilabel data training phase moreover focused approach promising using raw audio signals makes system rely less domain knowledge preprocessing usually shows slightly lower performance using spectral input melspectrogram recent papers iii ystem architecture audio preprocessing convolutional neural network one representation learning methods allow machine fed raw data automatically discover representations needed classification detection however appropriate preprocessing input data still important issue improve performance system first preprocessing step stereo input audio converted mono taking mean left right channels downsampled original sampling frequency allows use frequencies nyquist frequency sufficient cover harmonics generated musical instruments removing noises possibly included frequencies range moreover audios normalized dividing signal maximum value downsampled waveform converted representation using fourier transform stft samples window size approx samples hop size approx next linear frequency spectrogram converted use number melfrequency bins following representation learning papers music annotation nam hamel reasonable setting sufficiently preserves harmonic characteristics music greatly reducing dimensionality input data finally magnitude obtained spectrogram compressed natural logarithm network architecture convnets seen combination feature extractor classifier convnet architecture generally follows popular alexnet vggnet structure contains deep architecture using repeated several convolution layers followed shown figure method using smaller receptive window size smaller stride convnet becoming highly common especially computer vision field study zeiler fergus sermanet shown superior performance although general architecture style similar successful convnets image processing area proposed convnet designed according input data journal latex class files vol may table roposed onv tructure nput ize emonstrated table nalysis indow ize econd umber filters time frequency activation unction ollowed ach onvolutional ayer ully onnected ayer nput ach onvolution ayer ero hown revity input size description convolution filters convolution filters dropout convolution filters convolution filters dropout convolution filters convolution filters dropout convolution filters convolution filters global flattened fully connected dropout sigmoid use filters small receptive field fixed stride size spatial abstraction done size stride size table illustrate detailed convnet architecture input size layer parameter values except process input convolution layer preserve spatial resolution regardless input window size increase number channels convolution layer factor every two convolution layers starting last layer eight convolutional layers perform global followed one fully connected layer recently reported use global average pooling without fully connected layer classifier layer less prone overfitting shows better performance image processing datasets mnist however empirical experiment found global average pooling slightly decreases performance global followed fully connected layer works better task finally last classifier layer sigmoid layer common use softmax layer one target label system must able handle multiple instruments present time thus sigmoid output used training configuration training done optimizing categorical crossentropy predictions targets used adam optimizer learning rate minibatch size set accelerate learning process parallelization used gtx gpu memory training regularized using dropout rate layer dropout technique prevents overfitting units training data randomly dropping units neural network training phase furthermore added dropout fully connected layer well rate since fully connected layer easily suffers overfitting addition conducted experiment various time resolutions find optimal analysis size training data fixed audio performed training dividing training audio used label divided chunk audio divided without overlap training affects validation loss used early stopping fifteen percent training data randomly selected used validation set training stopped validation loss decrease two epochs initialization network weights another important issue lead unstable learning process especially deep network used uniform distribution zero biases convolutional fully connected layers following glorot bengio activation function activation function followed convolutional layer fully connected layer section introduce several activation functions used experiment comparison traditional way model activation neuron using hyperbolic tangent tanh sigmoid function however nonlinearities rectified linear unit relu allow much faster learning saturating nonlinearities particularly models trained large datasets moreover number works shown performance relu better sigmoid tanh activation thus modern studies convnets use relu model output neurons relu first introduced nair hinton work restricted boltzmann machines relu activation function defined max input ith channel relu simply suppresses whole negative part zero retaining positive part recently several modified versions relu introduced improve performance first lrelu introduced mass compresses negative part rather make zero might cause initially inactive units remain inactive defined journal latex class files vol may parameter give small gradient negative part second parametric relu prelu introduced basically similar lrelu compresses negative part however prelu automatically learns parameter negative gradient unlike lrelu defined learned parameters ith channel choice activation function considerably influences identification performance difficult say specific activation function always performs best highly depends parameter setting input data instance empirical evaluation convnet activation functions reported performance lrelu better relu prelu sometimes worse basic relu depending dataset value moreover works regarding activation function image classification task audio processing domain hence empirically evaluated several activation functions explained tanh relu lrelu prelu find suitable activation function task lrelu leaky relu normal leaky relu used reported performance lrelu considerably differs depending value leaky relu works better used separate test audio data irmas dataset used training first sliding window used analyze input test audio size analysis window training phase hop size sliding window set half window size aggregated sigmoid outputs sliding windows summing outputs obtain total amount activation instrument summed sigmoid activations normalized range dividing maximum activation valuation irmas dataset irmas dataset includes musical audio excerpts annotations predominant instruments present intended used automatic identification predominant instruments music dataset used paper predominant instrument classification bosch includes music various decades past century hence differing audio quality great extent addition dataset covers wide variability musical instrument types articulations recording production styles performers dataset divided training testing data audio files stereo wave sampling rate training data consisted audio files excerpts distinct recordings two subjects paid obtain data pitched instruments shown table selected music tracks table ist usical nstruments sed xperiment heir bbreviations umber abels raining esting audio instruments abbreviations training testing cello clarinet flute acoustic guitar electric guitar organ piano saxophone trumpet violin voice cel cla flu acg elg org pia sax tru vio voi objective extracting music excerpts contain continuous presence single predominant instrument hand testing data consisted audio files lengths tracks training data included unlike training data testing data contained one predominant target instruments hence total number training labels identical number audio files number testing labels number testing audio files latter training testing dataset musical instruments percussion bass included annotation even exist music excerpts testing configuration training phase used fixed length window input data convnet specific fixed shape however testing audios variable lengths much longer training audio developing system handle variable length input data valuable music real life varies length performed analysis using overlapping windows obtain local instrument information audio excerpts since annotation exists per audio clip observed multiple sigmoid outputs aggregated make decision tried two different strategies aggregation average normalized sum referred throughout paper respectively simply took average sigmoid outputs whole audio clip thresholded without normalization method intended capture existence instrument mean probability might return result without detected instrument first summed sigmoid outputs whole audio excerpt normalized values dividing maximum value among classes values scaled placed zero one followed thresholding method based assumption humans perceive predominant journal latex class files vol may table iii xperiment variables activation unction ize nalysis indow aggregation trategy dentification hreshold efault ettings ndicated old variables activation func analysis win size agg strategy tanh relu prelu lrelu lrelu mean sum normalized fig schematic obtaining output test audio signal input audio analyzed sliding window multiple sigmoid outputs aggregated using two different strategies estimate predominant instrument testing audio excerpt instrument relatively scaled sense strongest instrument always detected existence instruments judged relative strength compared activate instrument majority vote one common choices number classification tasks used system majority vote first predicts classes analysis frame one vote wins however using method task would result disregarding accompaniment instruments piano example music signal composed various musical instruments usually sounds overlapped time domain presence accompaniments usually much weaker voice lead instruments target identify arbitrary number predominant instruments testing data instruments aggregated value threshold considered predominant instruments using higher value identification threshold lead better precision obviously decrease recall hand lower threshold increase recall lower precision hence tried range values threshold find optimal value measure explained next performance evaluation section values used values used threshold threshold values empirically chosen set wide enough range find best performance highest measure schematic aggregation process illustrated figure performance evaluation following evaluation method widely used instrument recognition task computed precision recall defined true positive false positive false negative addition used measure calculate overall performance system harmonic mean precision recall since number annotations class musical instruments equal computed precision recall measure micro macro averages micro averages calculated metrics globally regardless classes thus giving weight instrument higher number appearances hand calculated metrics label found unweighted average macro averages hence related number instances represents overall performance classes finally repeated experiment three times calculated mean standard deviation output esults used lrelu activation function analysis window aggregation strategy identification threshold default settings experiment possible showed best performance experiment variables listed table iii first compared performance proposed convnet existing algorithm irmas dataset effect activation function analysis window aggregation strategy identification threshold recognition performance analyzed separately following subsections comparison existing algorithms result network achieved micro measure macro measure existing algorithm fuhrmann herrera used typical timbral audio features framewise mean variance statistics train svms bosch improved algorithm source separation called fasst flexible audio source separation framework preprocessing step journal latex class files vol may fuhrmann bosch proposed micro prec rec micro prec rec macro fig performance comparison predominant instrument recognition algorithm fuhrmann herrra bosch proposed convnet terms precision fuhrmann herrera algorithm showed best performance micro macro measure however recall low around resulted low measure proposed convnet architecture outperformed existing algorithms irmas dataset micro macro measure shown figure result observed learned feature input data classified convnet works better conventional handcrafted features svms effect activation function case using rectified units activation function possible observe significant performance improvement compared tanh baseline expected shown table unlike result presented imagenet classification work prelu show performance improvement showed matching performance relu task hand using lrelu showed better performance using normal relu prelu using lrelu small gradient showed similar performance relu expected lrelu leaky alpha setting showed best identification performance matched result empirical evaluation work convnet activation function result shows suppressing negative part activation rather making zero certainly improves performance compared normal relu making whole negative part zero might cause initially inactive units never active mentioned moreover result shows using leaky relu proved work well image classification task also benefits musical instrument identification identification threshold macro fig micro macro measure analysis window size according identification threshold table nstrument ecognition erformance roposed onv various activation unctions activation func tanh relu prelu lrelu lrelu micro macro effect analysis window size mentioned conducted experiment diverse analysis window sizes find optimal analysis resolution figure shows micro macro measure various analysis frame sizes according identification threshold observed use longest window clearly performed poorer use shorter window sizes regardless identification threshold however shortening analysis frame decreased overall performance result seen optimal analysis window size task using shorter analysis frame helped increase temporal resolution found short window size identifying instrument effect identification threshold using higher value identification threshold leads better precision decreases recall contrary lower threshold results better recall lower precision journal latex class files vol may optimal setting agg strategy analysis win identification thr cel cla flu acg elg org pia sax tru vio voi fig performance instrument identification analyze effect parameter instrument compared optimal setting default setting different aggregation strategy analysis window size lrelu identification threshold table nstrument ecognition erformance roposed onv sing ifferent aggregation trategies hreshold eturned ighest easure ach trategy omparison agg strategy micro macro hence used measure harmonic mean precision recall evaluate overall performance terms measure found appropriate threshold showed best performance macro measure shown figure current system uses certain identification threshold instruments however think might room improvement using different thresholds instrument various types instruments included experiment example amplitude piano sound relatively small number music excerpts usually used accompanying instrument hand flute sound music mostly louder others usually used lead instrument effect aggregation strategy conducted experiment two different strategies aggregation convnet outputs explained testing configuration section performance demonstrated table threshold returned highest measure strategy result showed better identification performance overall slight performance gap micro measure difference notable macro measure result shows performing sum followed normalization better aggregation method predominant instrument identification taking mean values likely due training testing audios differing quality great extent depending recording production time normalization helped minimize effect quality differences audio excerpts would result generalized output analysis identification performance results demonstrated focused overall identification performance section analyze discuss result observe system performance detail shown figure identification performance varies great extent depending instruments regardless parameter setting observed system recognizes voice music well showing measure hand cello clarinet showed relatively poor performance compared instruments showing measure around results highly likely affected insufficient number training audio samples deep learning number training examples critical performance compared case using features aims learn feature input data illustrated table number training audio samples voice largest number training audio contrary audio excerpts used cello clarinet respectively least third least number training audio believe increasing number training data cello clarinet would helpful increase identification performance instruments addition number test audio samples cello clarinet much less instruments dataset test audio samples cello clarinet respectively first second least number test audio audio samples human voice evaluating system small number test data would make result less reliable less stable identification results apart issue related number audio high identification performance voice class highly likely owing spectral characteristic distinct musical instruments instruments used experiment usually produce relatively clear harmonic patterns journal latex class files vol may cel cla flu gac gel org pia sax tru vio voi fig visualization clustering result represents clustering result intermediate state proposed model left right first four plots clustering result activations end convolutional block last two plots clustering result activations hidden dense layer final sigmoid output respectably upper plots drawn sample used training lower plots validation data samples however human voice produces highly unique spectral characteristics contain much inharmonic spectrum natural vibrato regarding aggregation strategy using instead decreased identification performance organ piano result indicates showed slight advantage instruments usually used accompaniment instrument using aggregation better cases hand using analysis window instead default window considerably decreased performance especially flute acoustic guitar electric guitar violin result shows using longer analysis window disadvantage cases finally using low identification threshold caused considerable performance loss especially flute saxophone trumpet violin showed slight improvement electric guitar organ piano result understood mean using lower threshold identification performance helps detect instruments usually background using higher threshold suitable instruments frequently used lead instrument wind instruments usually show relatively strong presence music mentioned results section result indicates potential performance improvement using different identification threshold instrument qualitative analysis visualization methods understand internal mechanism proposed model conducted visual analysis various visualization methods first tried clustering layer intermediate hidden states given input data sample verify encoding behavior layer contributes clustering input samples selected stochastic neighbor embedding algorithm technique dimensionality reduction data second exploited deconvolution method identify functionality unit proposed convnet model visual analysis system basically repeats two convolutional layers followed one pooling layer grouped three components call convolutional block throughout section simplicity algorithm based stochastic neighbor embedding sne algorithm converts similarities given data points joint probability embeds data points space minimizing divergence joint probability embedding data points method highly effective especially dataset dimension high advantage algorithm accorded well condition target observations necessarily high dimension since reshaped layer filter activations single vector respectively visualization exploiting could observe layer contributed classification dataset reflecting gradually changing data points stage proposed model four intermediate activations extracted end convolutional block one hidden fully connected layer another one final output layer compression dimensionality computational efficiency pooled maximum values activation matrices unit process dimensionality layer output could diminished layer unit size visualized randomly selected training validation data samples entire dataset verify model exactly works generalizes classification capability figure clearly journal latex class files vol may shown data samples class instrument well grouped group separated farther level encoding higher particularly training set clustering clearer former case tendency clustering validation set also found similar training set condition another visualization method deconvolution recently introduced useful analysis tool qualitatively evaluate node convnet main principle method inverse every stage operations reaching target unit generate visually inspectable image consequence filtered trained subfunctionality target unit method possible reveal intuitively internal works within entire deep convolutional network tends thought black box process functionality proposed model explored generated deconvoluted images like figure arbitrary input melspectrogram unit entire model visual analysis resulting images could see several aspects proposed model units first layer tend extract vertical horizontal diagonal edges input spectrogram like lower layers convnets usual image object recognition task second layer fourth layer deconvoluted image indicates unit functionality searches particular combinations edges extracted first layer found difficult strongly declare proposed model detects specific musical articulation expression however inductive manner could see units indicate understood musical expression detector conducted visual analysis deconvoluted image two independent music signals kind sound sources differently cases activated units first layer strongly suggested primary functionality detect harmonic component input finding horizontal edges shown top figures figure however second layer higher layers highly activated units behavior appeared quite different respective input signal instance activated unit signal second layer showed functionality similar onset detection detecting combination vertical horizontal edges compared unit activated units third layer showed different functionality seems activate unstable components vibrato articulation slur singing voice part detecting particular combination diagonal horizontal edges hand model behavior signal different clearly shown second third layers output figure highly activated trying detect dense field stable horizontal edges signals composed voice acoustic guitar instrument predominant instrument signal labeled voice labeled acoustic guitar often found harmonic instruments like guitar field detected units corresponded region strumming acoustic guitar sound onclusion paper described apply convnet identify predominant instrument music trained network using data identify arbitrary number predominant instrument music clip variable length results showed deep convnet capable achieving good performance learning appropriate feature automatically input data proposed convnet architecture outperformed previous approaches predominant instrument identification task irmas dataset used input convnet use source separation preprocessing unlike existing works conducted several experiments various activation functions convnet tanh relu used baseline recently introduced lrelu prelu also evaluated results confirmed relu worked reasonably well facto standard recent convnet studies furthermore obtained better results lrelu normal relu especially leaky setting performance tanh worse rectifier functions expected prelu showed matching performance relu task paper also investigated different aggregation methods convnet outputs applied music excerpts various lengths experimented two different aggregation methods mean probability sum followed normalization experimental results showed better aggregation method effectively deals quality difference audios normalization process addition conducted extensive experiment various analysis window sizes identification thresholds analysis window size using shorter window improved performance increasing temporal resolution however short obtain accurate identification performance found optimal window size precision recall depending identification threshold hence used measure harmonic mean precision recall result threshold value showed best performance visualization intermediate outputs using showed feature representation became clearer time input data passed convolutional blocks moreover visualization using deconvolution showed lower layer tended capture horizontal vertical edges higher layer tended seek combination edges describe spectral characteristics instruments study shows many recent advances neural network image processing area transferable journal latex class files vol may layer layer layer layer layer layer layer layer fig two input signals respective deconvoluted results left two columns right two columns image denoted respectively calculated two independent music signals signals polyphonic music segment randomly cropped original music moreover signals consist mainly voice acoustic guitar sound however dominant instrument labeled voice labeled acoustic guitar row images represents deconvoluted signal overlaid original signal extracted results four intermediate stages proposed model deconvolution outputs extracted end convolutional block target signals two highest activated units point chosen deconvoluted visualized left right images arranged order decreasing absolute unit activation red region green region deconvoluted image indicate positive value negative value result respectably remaining area magnitude activation relatively lower regions range activation result normalized purpose clear visualization journal latex class files vol may audio processing domain however audio signal processing especially music signal processing many different aspects compared image processing area convnets extensively used example spectral characteristics usually overlapped time frequency unlike objects image makes detection difficult moreover music signals much repetitive continuous compared natural images present various lengths believe applying musical knowledge aggregation part adaptive thresholding instrument improve performance warrants deeper investigation acknowledgment research supported partly msip ministry science ict future planning korea itrc information technology research center support program supervised iitp institute information communications technology promotion partly national research foundation korea nrf grant funded msip eferences eronen klapuri musical instrument recognition using cepstral coefficients temporal features acoustics speech signal processing icassp proceedings ieee international conference vol ieee diment rajan heittola virtanen modified group delay feature musical instrument recognition international symposium computer music multidisciplinary research cmmr marseille france yang sparse cepstral codes power scale instrument identification acoustics speech signal processing icassp ieee international conference ieee krishna sreenivas music instrument recognition isolated notes solo phrases acoustics speech signal processing icassp ieee international conference vol ieee essid richard david musical instrument recognition solo performances signal processing conference european ieee heittola klapuri virtanen musical instrument recognition polyphonic audio using model sound ismir kitahara goto komatani ogata okuno instrument identification polyphonic music feature weighting minimize influence sound overlaps eurasip journal applied signal processing vol duan pardo daudet novel cepstral representation timbre modeling sound sources polyphonic mixtures acoustics speech signal processing icassp ieee international conference ieee goto hashiguchi nishimura oka rwc music database music genre database musical instrument sound ismir vol lecun bengio hinton deep learning nature vol deng deep learning methods applications foundations trends signal processing vol mikolov burget khudanpur recurrent neural network based language interspeech vol mesnil deng bengio investigation architectures learning methods spoken language interspeech yao zweig hwang shi recurrent neural networks language interspeech lecun jackel bottou cortes denker drucker guyon muller sackinger simard learning algorithms classification comparison handwritten digit recognition neural networks statistical mechanics perspective vol roa victorino handwritten digit recognition using convolutional neural networks gabor filters proc int congr comput intell niu suen novel hybrid classifier recognizing handwritten digits pattern recognition vol krizhevsky sutskever hinton imagenet classification deep convolutional neural networks advances neural information processing systems ngiam chen chia koh tiled convolutional neural networks advances neural information processing systems hinton deng dahl mohamed jaitly senior vanhoucke nguyen sainath deep neural networks acoustic modeling speech recognition shared views four research groups signal processing magazine ieee vol hamel lemieux bengio eck temporal pooling multiscale learning automatic annotation ranking music ismir schluter bock improved musical onset detection convolutional neural networks acoustics speech signal processing icassp ieee international conference ieee humphrey bello rethinking automatic chord recognition convolutional neural networks machine learning applications icmla international conference vol ieee bengio vincent audio chord recognition recurrent neural ismir ullrich grill boundary detection music structure analysis using convolutional neural ismir grill music boundary detection using neural networks combined features annotations proceedings international society music information retrieval conference ismir malaga spain park lee musical instrument sound classification deep convolutional neural network using feature fusion approach arxiv preprint qian wang automatic instrument recognition polyphonic music using convolutional neural networks arxiv preprint hoshen weiss wilson speech acoustic modeling raw multichannel waveforms acoustics speech signal processing icassp ieee international conference ieee palaz collobert doss estimating phoneme class conditional probabilities raw speech signal using convolutional neural networks arxiv preprint nam herrera slaney smith learning sparse feature representations music annotation ismir simonyan zisserman deep convolutional networks image recognition arxiv preprint zeiler fergus visualizing understanding convolutional networks computer springer sermanet eigen zhang mathieu fergus lecun overfeat integrated recognition localization detection using convolutional networks arxiv preprint lin chen yan network network arxiv preprint kingma adam method stochastic optimization arxiv preprint srivastava hinton krizhevsky sutskever salakhutdinov dropout simple way prevent neural networks journal latex class files vol may fitting journal machine learning research vol glorot bengio understanding difficulty training deep feedforward neural networks international conference artificial intelligence statistics wang kuen shahroudy shuai liu wang wang recent advances convolutional neural networks arxiv preprint nair hinton rectified linear units improve restricted boltzmann machines proceedings international conference machine learning maas hannun rectifier nonlinearities improve neural network acoustic models proc icml vol zhang ren sun delving deep rectifiers surpassing performance imagenet classification proceedings ieee international conference computer vision wang chen empirical evaluation rectified activations convolutional network arxiv preprint bosch janer fuhrmann herrera comparison sound segregation techniques predominant instrument recognition musical audio ismir fuhrmann herrera polyphonic instrument recognition exploring semantic similarities music proc int conference digital audio effects ono miyamoto kameoka roux uchiyama tsunoo nishimoto sagayama harmonic percussive sound separation application tasks advances music information retrieval springer van der maaten hinton visualizing data using journal machine learning research vol yosinski clune nguyen fuchs lipson understanding neural networks deep visualization arxiv preprint yoonchang han born seoul republic korea studied electronic engineering systems king college london moved queen mary university london received meng hons degree digital audio music system engineering first class honours currently phd candidate digital contents information studies music audio research group marg seoul national university republic korea main research interest lies within developing deep learning techniques automatic musical instrument recognition jaehun kim born seoul republic korea currently researcher music audio research group research interests include signal processing machine learning techniques applied music audio received english literature linguistics seoul national university received degree digital contents information studies music audio research group marg seoul national university republic korea kyogu lee associate professor seoul national university leads music audio research group research focuses signal processing machine learning techniques applied music audio lee received phd computerbased music theory acoustics stanford university
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spin group superspace hennie schepper frank sommen clifford research group department mathematical analysis faculty engineering architecture ghent university krijgslaan gent belgium apr abstract two ways describing elements rotation group first according theorem every rotation matrix written even number reflections second also expressed exponential matrix paper study similar descriptions corresponding extension superspace setting natural describe behavior bosonic fermionic particles group also extension symplectic group still connected thus longer compact consequence fully described one action exponential map lie algebra instead obtain iwasawatype decomposition group terms three exponentials acting three direct summands corresponding lie algebra supermatrices time strictly contains group generated reflections therefore lie algebra isomorphic certain extension algebra superbivectors means spin group superspace seen group generated exponentials extended superbivectors order cover also study actions spin group supervectors provide proper subset double cover finally show every fractional fourier transform bosonic dimensions seen element spin group superspace keywords spin groups symplectic groups clifford analysis superspace mathematics subject classification introduction mathematical analysis superspace super analysis important tool study several branches contemporary theoretical physics supergravity superstring theories introduced berezin see possesses huge mathematical impact since breaks several traditional patterns classical analysis commutation among coordinate variables positive dimensions indeed elements superspace defined means bosonic commuting variables fermionic variables natural way bosonic variables describe positive dimensions fermionic ones describe negative dimensions generalizations objects classical analysis appear superspace wider nature example notion supervector variable introduced variable takes values grassmann envelope corresponding definitions bosonic fermionic partial derivatives supervector derivative supergradient super dirac operator introduced well analogies lead introduction clifford analysis superspace done papers proper extension clifford analysis setting needs set rules axioms guarantee preservation clifford nature new objects although different behavior compared classical ones set rules provided notion radial algebra abstract setting algebraically describes main properties clifford environment satisfy see approach depend priori defined vector space signature particular main exigence anticommutator every pair vector variables commuting element leading definition abstract inner product formula however abstract description leaves issues unclarified particular possible describe whole set linear transformations leaving bilinear form invariant consequence proper definition spin group general abstract setting course define vector reflections wxw abstract vector variables yet enable prove linear transformations leaving invariant written composition vector reflections order find set transformations necessary work concrete representation radial algebra underlying vector space vector variables defined endowed fixed bilinear form signature example clifford polynomial representation radial algebra see euclidean space important group leaving inner product invariant set rotations spin group appears setting double cover given spin denotes unit sphere relation spin easily seen lie group representation spin sxs spin describes action every element terms clifford multiplication description spin group follows theorem states every orthogonal transformation symmetric bilinear space written composition reflections situation may however another representation radial algebra theorem longer valid paper deal one cases representation superspace main goal properly define super spin group set elements describing every superrotation clifford multiplication superspace end consider vector variables taking values grassmann envelope makes possible study group supermatrices leaving invariant define proper way spin group superspace worth noticing superstructures absorbed grassmann algebras leading classical lie groups lie algebras instead supergroups superalgebras paper organized follows start preliminaries grassmann algebras grassmann envelopes supermatrices section particular carefully recall notion exponential map grassmann numbers supermatrices elements finite dimensional associative algebras section briefly describe clifford setting superspace leading introduction lie algebra superbivectors extension algebra plays important description super spin group use exponential map extension takes radial algebra necessitates introduction corresponding tensor algebra section devoted study invariance inner product setting study several groups supermatrices particular group superrotations lie algebra combine orthogonal symplectic structures prove every superrotation uniquely decomposed product three exponentials acting special subspaces finally section study problem defining spin group setting differences classical case show compositions even numbers vector reflections enough fully describe since show orthogonal structure include symplectic part next propose alternative defining spin group exponential extended superbivectors show indeed cover whole set superrotations particular explicitly describe subset double covering contains particular every fractional fourier transform preliminaries grassmann algebras grassmann envelopes let grassmann algebra order field canonical generators subject multiplication rules implying particular every element written form fjk put define space homogeneous elements degree spank particular easily follows projection part denoted particular denote well known shows natural spaces homogeneous even odd fact defining elements respectively obtain superalgebra structure recall graded commutative sense element sum number nilpotent element fact every easily seen projection algebra satisfies homomorphism particular following property holds lemma let sometimes useful consider embedded grassmann algebra higher order particular consider algk new canonical grassmann generator embedding preserves grading structures since algebra space dimension every finite dimensional space becomes banach space introduction arbitrary norm norms equivalent particular following result obtained straightforward computation lemma norm defined satisfies every exponential denoted exp defined power series proposition series converges every continuous function proof follows kaj kakj whence absolutely converges banach space consider holds kaj converges whence weierstrass uniformly cona verges thus continuous continuous consider graded vector space standard homogeneous basis basis basis elements called even homogeneous elements elements called odd homogeneous elements grassmann envelope defined set formal linear combinations form space dimension inheriting denotes subspace vectors form denotes subspace vectors form subspaces called grassmann envelopes respectively pthere exists subspace naturally isomorphic consists pvectors form map defined useful standard basis represented columns place left place left basis elements take form supermatrices yields space end endomorphisms space isomorphic space mat block matrices form first term even part second term odd one grassmann envelope mat denoted mat consists matrices form entries namely even entries odd entries elements mat called supermatrices mat inherited mat together usual matrix multiplication provides superalgebra structure grassmann envelope precisely let mat space homogeneous supermatrices degree mat con sists diagonal block matrices entries mat consists block matrices entries subspaces define grading mat mat mat mat mat mat clear every supermatrix written sum numeric matrix mat nilpotent supermatrix mat mat accordance general ideas valid grassmann algebras grassmann envelopes define algebra homomorphism mat mat projection given set use notation refer set matrices order entries numeric projections respectively recall mat equal even subalgebra mat given set supermatrices define every supermatrix defines linear operator acts vector however supermatrix mat defining endomorphism unique fact zero endomorphism defined every supermatrix ideal mat given mat mat odd even respectively hence supermatrices define endomorphism situation changes consider endomorphisms defined supermatrices entries lemma let endomorphism admits supermatrix representation mat representation unique mat result easily proved using introduction new canonical grassmann generator show element satisfies afn study group structures mat start lie group invertible elements mat following characterization group see theorem let mat following statements equivalent invertible iii invertible addition every inverse usual definitions transpose trace determinant matrix appropriate graded case example although transpose supermatrix well defined element mat usual property hold general problem fixed introducing supertranspose transpose supertranspose operations satisfy following relations see proposition let mat lst iii every situation trace similar usual trace element mat well defined general mat notion supertrace provides solution problem defined map str mat given str following properties easily follow definition see proposition let mat str str iii str str superdeterminant berezinian function defined sdet det det det det basic properties given following proposition see proposition let sdet sdet sdet sdet sdet since mat finite dimensional vector space every two norms space equivalent whence without loss generality define mat norm kmj satisfying kklk every pair mat similarly proposition following result proven proposition every mat exponential defined absolutely converges consequence continuous function mat also supertranspose supertrace superdeterminant continuous functions proposition let mat iii denotes identity matrix eam ebm every pair ecm cem every vii etm smooth curve mat viii sdet estr etm etm remark proofs vii straightforward computations detailed proof viii found similar properties iii vii obtained exponential map also define notion logarithm supermatrix mat wherever converges proposition series converges yields continuous function near mat let neighbourhood defined let neighbourhood exp eln proof observe whence since radius convergence last series absolutely converges defines continuous function ball statement immediately follows absolutely convergence series exp formal identities eln indeterminate worth mentioning procedure repeated definitions exponential logarithmic maps possible obtain classical results known lie groups lie algebras real complex matrices exponential nilpotent matrix mat clearly given finite sum yields bijective mapping exp mat mat inverse mat mat since expansions finite number terms whence problems convergence arise recall supermatrix belongs numeric projection inverse exp unique mat superspace framework grassmann coefficients clifford setting order set clifford analysis framework superspace take canonical homogeneous basis endowed orthogonal symplectic structure multiplication rules symplectic form defined following relations elements generate infinite dimensional algebra denoted similar approach grassmann envelope leads definition algebra algr assumed elements commute elements every element written finite sum terms form ejk algebra consider corresponding generalization projection goes ejk defined ejk developing proper clifford analysis framework requires suitable realization radial algebra detailed treatment radial algebra setting refer reader realization seen set vector variables taking values grassmann envelope supervectors form satisfy fundamental axiom radial algebra every pair commuting element indeed every pair algebra generated elements called radial algebra embedded denoted algebra generated set elements even odd turns finite dimensional real space details kind realization radial algebra refer reader superbivectors superbivectors play important work following radial algebra approach space bivectors defined space combinations wedge products supervectors hence space superbivectors consists elements form observe coefficients commuting nilpotent since generated elements form belong constitutes important limitation space superbivectors means allow structure orthogonal one fact real projection every superbivector classical clifford bivector thence necessary introduce extension containing elements form extension allows consider two different structures element orthogonal symplectic one fact case remark observe finite dimensional real vector spaces dim dim extension superbivector space clearly lies radial algebra generates infinite dimensional algebra using multiplication rules defined ele ments called extended superbivectors superbivectors extended superbivectors preserve several properties classical bivectors proposition space lie algebra addition lie subalgebra proof need check lie bracket defined commutator associative algebra internal binary operation direct computation shows get aej ber aej aej well known radial algebra framework commutator bivector vector always linear combination vectors coefficients scalar subalgebra indeed vector variables obtain property easily generalized straightforward computation particular following result holds proposition let let basis let basis computations also valid supervectors extension tensor algebra exponential map since infinite dimensional definition exponential map means power series straightforward algebras mat correct definition exponential map requires introduction tensor algebra details general theory tensor algebras found several basic references see let tensor algebra vector space spanr tensor product seen subalgebra twosided ideal generated elements indeed isomorphic extension also contains infinite sums arbitrary terms form ejk exponential map exp known well defined tensor algebra see whence also well defined following mapping properties exp exp first statement directly follows definition second one obtained following standard procedure established mat since finite dimensional ortho symplectic structure invariance inner product function given supervectors leads definition symmetric bilinear form given use generalized inner product easily written terms supermatrices since diag inner product coincides euclidean inner product assume order find supermatrices mat corresponding linear operators leave inner product invariant observe thus identify supermatrices mat lemma supermatrix mat satisfies twosided ideal mat defined follows generated set supermatrices mat mat generated set supermatrices mat proof easily check straightforward computations two subspaces defined ideals mat every mat let choosing possible coordinate supervectors obtain satisfies entries taking first equation obtain every remaining coefficients need distinguish two cases case first prove spanr clear every easily follows every pair contrary would least one generators missing one canonical terms taking hence spanr holds next prove spanr fact clear since sum homogeneous terms degree least contrary would least two different grassmann generators missing one canonical terms taking hence spanr case first prove always least one grassmann generator missing one terms since even number taking hence holds prove spanr indeed clear contrary would least two different grassmann generators missing one canonical terms taking hence spanr direct consequence lemma following result corollary set supermatrices mat leaving inner product invariant characterized mat remark form suggests need whole ideal describe fact supermatrices previous form satisfy lst qst qst whence subspace lst considered definition however use important purposes paper whence continue working study algebraic structure theorem following statements hold group usual matrix multiplication iii closed subgroup proof prove every supermatrix invertible first note real projection supermatrix zero every rewritten terms real blocks implying account theorem thus invertible need prove matrix inversion matrix multiplication internal operations inversion condition implies since ideal hence implies multiplication let mjst qmj since ideal hence iii let sequence converges supermatrix mat since algebraic operations continuous mat lim mjst qmj mjst qmj every since finite dimensional subspace mat closed whence limit belongs remark theorem states lie group group partitioned natural way classes every set non empty particular every proposition following statements hold let subgroup iii binary relation equivalence relation proof already proven proof theorem subgroup prove subgroup consider directly implies iii let get thence since subgroup easily follows equivalence relation remark subgroup closed subgroup whence lie group plays crucial study since given representative element describe whole set means relation reason focus attention proposition following statements hold supermatrix mat belongs sdet every iii proof relation written terms account proposition implies sdet sdet sdet whence sdet statement follows lemma iii see proof theorem group superrotations classical way introduce set superrotations sdet easily seen lie subgroup real projection equal fact conditions sdet imply sdet means det det implies det det following proposition states classical case connected consequence identity component proposition connected lie group proof since real projection connected group suffices prove every exist continuous path inside connecting real projection end let write projection mat observe let take path continuous path addition every hence finally observe sdet every since sdet sdet investigate corresponding lie algebras theorem lie algebra given mat lie algebra coincides lie algebra given space super supermatrices mat iii supermatrix mat belongs proof mat lie algebra etx every etx qetx differentiating obtain since closed subspace hand mat satisfies computing exponential obtain etx convergent infinite sum products contain factor least one time since ideal every whence etx qetx qetx prove lie algebra suffices repeat reasoning proposition easily follows lie algebra mat str implies str fact condition implies str str str yielding str str str str hence lie algebra iii observe relation written terms follows real projection every element zero let satisfies using iii obtain implies remark remark supermatrices form satisfy lst qxs subspace replace definition connectedness allows write elements finite product exponentials supermatrices see classical case single exponential suffices description since compact consequence exp surjective property however hold group superrotations since exponential map lie group surjective whence every element written single exponential supermatrix nevertheless possible find decomposition elements terms fixed number exponentials elements every supermatrix unique decomposition real projection mat nilpotent projection separately study decompositions first consider already mentioned exp surjective exp however proved exp exp invoking following polar decomposition real algebraic lie groups see proposition proposition let algebraic lie group lie algebra every uniquely written rex sym sym subspace symmetric matrices taking proposition get every symplectic matrix uniquely written sym group isomorphic connected compact exponential map lie algebra surjective means written sym hence supermatrix decomposed sym consider element shown end section function exp mat mat bijection logarithmic function defined inverse supermatrix satisfies nilpotent properties suffice proving denote set proposition let mat proof suffices prove etz every expression etz qetz written following polynomial real variable tzst zst zst identically zero take largest subindex lim contradicting identically vanishes yielding etz every way proven following result theorem every supermatrix written sym moreover elements unique relation superbivectors theorem allows compute dimension real vector space corollary dimension real lie algebra proof since direct sum corresponding subspaces block components respectively suffices compute dimension one according theorem iii dim dim dim comparing result one remark obtain dim dim means vector spaces isomorphic isomorphism also holds lie algebra level following classical clifford approach commutator key lie algebra isomorphism proposition shows every commutator defines endomorphism represented supermatrix mat explained section supermatrix unique issue solved natural extension linear operator defined lemma map mat defined takes values particular consider canonical basis respectively obtain following basis denotes matrix entries equal except one row column equal order deduced context proof equalities directly obtained proposition whence check supermatrices obtained form basis matrices satisfy relations whence whence whence whence whence whence computations show supermatrices obtained belong direct verification shows form set linear independent elements basis theorem map defined lie algebra isomorphism proof lemma follows vector space isomorphism addition due jacoby identity associative algebra every implying lemma lie algebra isomorphism spin group superspace far seen lie algebra lie group superrotations realization lie algebra extended superbivectors section discuss proper way defining corresponding realization analogue spin group clifford superspace framework supervector reflections group generated supervector reflections briefly introduced using notion unit defined reflection associated supervector defined wxw known radial algebra setting maps vectors vectors indeed wxw wiw every supervector reflection identified unique matrix account results section option extending operator natural way wxw lemma let endomorphism represented unique supermatrix finally proof observe wxw uniqueness supermatrix mat guaranteed lemma easily seen belongs since wxw wyw algebraic operations matrices easy since define bosonic pin group superspace pinb extend map lie group homomorphism pinb proposition let sdet proof prove suffices prove thatpa pnc satisfy easily done using identity fact also hence way whence satisfy consequence prove sdet first observe since wwxww hence due theorem obtain yielding det sdet det det det compute det using formula det exp fact nilpotent matrix observe follows consequence hence det similar computations yield det shows sdet proposition states lie group homomorphism takes values restriction bosonic spin group defined spinb takes values subgroup classical case pin spin groups double covering groups respectively natural question setting whether pinb spinb cover groups answer question negative main reason real projection every vector unitary sphere real projection pinb means bosonic versions pin spin describe symplectic parts phenomenon due natural structure supervectors real projections belong space orthogonal structure symplectic structure plays nilpotent vector classical clifford vectors whence impossible generate approach real symplectic geometry also present structure chosen name bosonic pin bosonic spin groups also explains extend space superbivectors section ordinary superbivectors generated wedge product supervectors describe consequence cover classical setting see possible obtain following result shows another point view pinb completely describe proposition lie algebra pinb included proof let path pinb every tangent show summand belongs implies preserves inner product proceed similarly every proper definition group spin approach shows radial algebra setting contain suitable realization clifford superspace framework observe clifford realization given lies outside radial algebra suggests something similar happen clifford realization lie group case proper definition spin group would generated exponentials general contained elements spin ebk action group given group homomorphism spin defined restriction fact every extended superbivector maps supervectors supervectors admits supermatrix representation mat belonging summarized proposition let every proof holds every associative algebra identity remark proposition means lie algebra isomorphism derivative origin lie group homomorphism spin etb account connectedness shown group spin realization representation theorem every exist spin proof since connected lie group proposition every supermatrix exist exk see corollary taking obtain exk ebk ebk equality valid every since ebk belong mat lemma guarantees ebk spin satisfies decomposition given theorem provides exact number exponentials extended superbivectors considered spin order cover whole group consider subspaces given dim sym dim dim dim get decomposition leading subset exp exp exp spin suffices describing indeed theorem follows restriction surjective investigate explicit form superbivectors one subspaces proposition following statements hold basis basis iii consists elements form proof first recall basis lie algebra given elements matrices defined lemma holds atj hence every matrix previous equalities get basis rest proof directly follows lemma case whence basis sym rest proof directly follows lemma iii trivially follows lemma spin covering group natural question setting whether spin group still double covering group rotations classical clifford analysis order answer question investigate many times spin covers precisely cardinal set given certain fixed element proposition representation element form following decomposition given theorem get cardinal depends number extended superbivectors satisfy reduces analysis finding kernel restriction exp lie group homomorphism exp kernel given ker recall proposition may written classical real bivector components commute consequence ebo ebs consider projections algebra classical bivectors respectively equivalently hence first condition know classical clifford analysis spin double covering consequence implies let compute possible values ebs end need following linear algebra result proposition every matrix written form proof map lie group isomorphism addition infinitesimal representation lie algebra level consequence lie isomorphism inverse given every let consider matrix know every matrix unitarily eigenvalues purely imaginary see hence written diag form since previous proposition hence matrix diag cos sin hence seen equivalent connected compact exists recall leaves multivector structure invariant particular every using derivative origin get extended superbivector ebr ebr ebr implying ebr order compute ebs ebr compute following correspondences given lemma get first consequence exp exp let compute exp consider usual imaginary unit clear commuting element exp exp exp order compute exp first prove following results lemma every following relations hold xyk proof proceed induction get obviously true assume true get yxyk get xyk lemma every holds stirling number second kind corresponding remark stirling number second kind number ways partitioning set elements non empty subsets among properties stirling numbers recall following ones proof lemma proceed induction statement clearly true assume true using lemma obtain conclude exp exp remark within algebra algr elements may identified rators respectively real variables indeed identifications immediately lead weyl algebra defining relations hence may identified harmonic oscillator consequence element exp corresponds exp recall classical fourier transform one variable written operator exponential exp exp hence exp denotes identity operator whence ebs going back exp ebo ebs ker theorem set exp exp exp double covering remark shown every extended superbivector form belongs though identifications made remark see operators exp exp exp elements spin group superspace faj denotes fractional fourier transform order variable conclusions future work paper shown vector reflections superspace enough describe set linear transformations leaving inner product invariant constitutes important difference classical case algebra bivectors isomorphic special orthogonal algebra property longer fulfilled setting real projection algebra superbivectors include symplectic algebra structure present lie algebra supermatrices corresponding group super rotations fact major impact definition spin group setting set elements defined multiplication even number unit vectors superspace suffice describing spin suitable alternative case define super spin elements products exponentials extended superbivectors extension lie algebra superbivectors contains corresponding identifications harmonic oscillators way obtain spin group cover set superrotations usual representation addition every fractional fourier transform identified spin element forthcoming work prove invariance super dirac operator corresponding actions super spin group also study invariance hermitian system action corresponding spin subgroup superspace acknowledgements supported grant ghent university grant number references berezin introduction super analysis reidel publishing new york usa bie sommen clifford analysis approach superspace annals physics bie sommen correct rules clifford calculus superspace advances applied clifford algebras schepper guzman adan sommen hermitian clifford analysis superspace submitted publication hennie schepper frank sommen radial algebra abstract framework orthogonal hermitian clifford analysis complex analysis operator theory pages dienes exponential function linear algebras quarterly journal mathematics thomas friedrich dirac operators riemannian geometry volume graduate studies mathematics american mathematical society providence translated german original andreas nestke brian hall lie groups lie algebras representations elementary introduction volume graduate texts mathematics springer cham second edition joachim hilgert neeb structure geometry lie groups springer monographs mathematics springer new york roger horn charles johnson matrix analysis cambridge university press cambridge second edition anthony knapp lie groups beyond introduction volume progress mathematics boston boston second edition sommen algebra abstract vector variables portugaliae mathematica pages sommen extension clifford analysis towards clifford algebras applications mathematical physics pages springer sommen clifford analysis advances applied clifford algebras sommen analysis using abstract vector variables pages boston boston sommen clifford analysis progress analysis takeo yokonuma tensor spaces exterior algebra volume translations mathematical monographs american mathematical society providence translated japanese edition author hennie schepper clifford research group department mathematical analysis ghent university krijgslaan gent belgium clifford research group department mathematical analysis ghent university krijgslaan gent belgium frank sommen clifford research group department mathematical analysis ghent university krijgslaan gent belgium
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uniform asymptotic inference bootstrap model selection ryan tibshirani alessandro rinaldo rob tibshirani larry wasserman aug carnegie mellon university stanford university abstract recently tibshirani proposed method making inferences parameters defined model selection typical regression setting normally distributed errors study large sample properties method without assuming normality prove test statistic tibshirani asymptotically valid number samples grows dimension regression problem stays fixed asymptotic result holds uniformly wide class nonnormal error distributions also propose efficient bootstrap version test provably asymptotically conservative practice often delivers shorter intervals original approach finally prove test statistic tibshirani enjoy uniform validity setting dimension allowed grow introduction recent surge work conducting formally valid inference regression setting model selection event occurred see berk lockhart tibshirani lee fithian bachoc name interest paper stems particular work tibshirani presented method produce valid confidence intervals adaptively fitted coefficients given step sequential regression procedure like forward stepwise regression least angle regression lar lasso lasso meant thought tracing sequence models along solution path penalty parameter descends authors use statistic carefully crafted pivotal conditioning model selection event idea specific sequential regression setting example broader framework might call selective pivotal inference applicable many settings taylor lee lee taylor loftus taylor reid choi fithian hyun key methodology tibshirani much work selective pivotal inference assumption normality errors fix notation consider regression response predictor variables stacked together columns matrix treat predictors fixed nonrandom assume model unknown mean parameter interest tibshirani assume errors error variance known advantage approach require exact linear combination predictors makes assumptions correlations among predictors far finitesample guarantees concerned normality errors crucial work examine properties test statistic proposed tibshirani truncated gaussian using assumption normal errors assume distribution mean zero essentially restrictions description selective pivotal inference framework sequential regression follows details provided section lar lasso run number steps model selected call lar model always active variables lasso variables added deleted active set step specify linear contrast mean interest one giving coefficient variable interest model step regression onto active variables assuming normal errors examining distribution conditional selected construct confidence interval satisfying model denote given interpretation repeatedly draw run lar lasso steps pay attention cases selected model among cases constructed intervals contain frequency tending conditional perspective selective pivotal inference framework lar lasso unconditional marginal point view also possible describe possible selected model constrast vector specified contrast considered concrete think setup model selected gives coefficient variable model step projection onto active set confidence intervals constructed exactly manner without change conditional coverage models implies following unconditional property vtc interpretation different repeatedly draw run lar lasso intervals contain steps construct confidence intervals respective targets frequency approaching notice construction target may change time draw though give rise selected model terms setting regression contrasts described time draw carry inferential procedure interval covers coefficient possibly different variable active model projection onto active variables figure demonstrates point uniform convergence making asymptotic inferential guarantees paper important clear type guarantee review concepts uniform convergence validity let random vectors joint distribution class distributions example could class could contain product distributions form times notation allows confidence interval repetition figure example conditional unconditional coverage one step variables normalized equivalent one step lar lasso response drawn times model errors different colors denote different active models selected one step active model pair namely variable achieving largest absolute inner product sign inner product across repetitions circles denote target covered segments confidence intervals color green corresponds model repetitions largest among absolute inner products variables green segments denote confidence intervals designed cover contrast similarly red corresponds model blue dotted segments indicate given interval cover target empirical coverage among green intervals among red intervals among blue intervals hence case empirical coverage close nominal level total unconditionally empirical coverage right nominal level general setup one let statistic say converges uniformly distribution provided lim sup sup inequalities meant interpreted componentwise also implicitly assuming limiting distribution continuous otherwise inner supremum restricted continuity points much stronger notion pointwise convergence distribution requires lim sup particular sequence distributions recent article kasy emphasizes importance uniformity asymptotic approximations authors points uniform version continuous mapping theorem follows directly standard proof continuous mapping theorem see theorem van der vaart lemma suppose converges uniformly distribution respect class let map continuous set converges uniformly distribution respect kasy also remarks central limit theorem triangular arrays specifically central limit theorem proposition van der vaart naturally extends uniform case logic roughly speaking uniform convergence equivalent pointwise convergence sequences distributions triangular arrays design different distribution assigned row therefore lindeberg condition holds possible sequence convergence normality lemma let triangular array independent random vectors joint distribution assume mean zero finite variance also assume sequence lim efn lim covfn depend sequence uniformly respect converges distribution work motivating reason study uniform convergence associated property uniform validity asymptotic confidence intervals depends parameter distribution consider confidence set built probability rectangle uniform convergence really rearranging definition implies lim sup sup meanwhile pointwise convergence implies lim sup particular sequence confidence set satisfying given tolerance exists sample size coverage guaranteed least matter underlying distribution class distributions question note necessarily true pointwise confidence set required sample size could depend particular distribution consideration summary main results overview main contributions follows establish statistics typical inferences along lar lasso paths depend data lemmas section important since two quantities asymptotic limits standard asymptotic setup placing mild constraints mean error distribution treating dimension fixed prove test statistic asymptotically pivotal converging standard uniform distribution evaluated true population value pivot argument show holds uniformly wide class distributions errors without real restrictions predictors first part theorem section resulting confidence intervals therefore asymptotically uniformly valid class distributions second part theorem section asymptotic results assume error variance known unknown propose approach replaces statistic simple estimate alternatively efficient bootstrap approach allow conservative asymptotic inference theorem section present detailed numerical experiments support asymptotic validity pvalues confidence intervals inference regression problems nonnormal errors section experiments reveal bootstrap versions also show good performance bootstrap method often deliver substantially shorter intervals based directly statistic experiments also also suggest test statistic bootstrap variants may asymptotically valid even broader settings covered theory problems heteroskedastic errors problems prove statistic exhibit general uniform convergence dimension allowed increase theorem section related work recent paper tian taylor related work authors examine asymptotic distribution statistic nonnormal errors main result proves statistic asymptotically pivotal restrictions model selection events question view work providing complementary perspective consider setting dimension grows place strong regularity conditions selected models adopt basic setting fixed prove broad uniformly valid convergence results pivot free regularity conditions sequence papers leeb potscher prove classical regression setting impossible find distribution estimator underlying coefficients even asymptotically specifically prove estimate underlying coefficient vector quantity form linear transform used inference model selection though made pivotal least asymptotically pivotal chosen appropriately longer true presence selection even dimension fixed sample size approaches furthermore show uniformly consistent estimate distribution either conditionally unconditionally makes unsuitable inference fact essentially manifestation hodges phenomenon selective pivotal inference framework hence paper circumvents problem claim attempt estimate distribution instead make inferences using entirely different pivot constructed via careful conditioning scheme notation paper considers asymptotic regime number samples growing often use subscript mark dependence various quantities sample size exception notation predictors response mean always denote respectively though quantities course vary notation hides dependence simplicity comes probability statements involving drawn write denote probability operator mean vector subscript omitted implicit probability taken also generally write lowercase arbitrary response vector uppercase random response vector drawn intended distinguish statements hold arbitrary statements hold model selection procedure random certain distribution lastly denote associated regression algorithm consideration lar lasso treat mapping space models fixed quantity representing random variable representmodel selected response fixed vector ing model selected response random vector similar notation used related quantities selective inference section review selective pivotal inference framework sequential regression procedures present interpretations inferences conditional unconditional perpsectives sections respectively subsections provide necessary details understanding framework beginning selection events encountered along lar lasso paths model selection consider forward stepwise regression least angle regression lar lasso run number steps arbitrary treated fixed throughout paper procedure defines partition sample space elements denotes selected model given procedure run space selected model may bit abuse common nomenclature possible models calling see describe set selected variables point representation decisions made algorithm across fact one think comprised two things steps define denoting variables given nonzero sequence active sets coefficients steps sequence sign vectors denoting signs nonzero coefficients steps selects one variable add active sets nested across steps active set step however sign vectors since determined least squares active variables step hence defined number possible models steps moreover corresponding partition elements convex cones proof fact difficult requires slight modification arguments tibshirani given appendix completeness result easily seen one step assuming without loss generality unit norm express side intersection passing zero therefore cones form partition convex cone enumerate possible choices figure shows illustration figure example model selection partition one step variables normalized equivalent one step lar lasso colors indicate regions sample space different active variables selected red region contains points maximally aligned lar lasso need modify definition selected model order resulting partition elements convex cones add extra bit model information define list variables play special role construction lar lasso active set step user would typically pay attention truth latter quantity detail included convex cones without partition elements would union cones describe furthermore affect treatment inference follows reason largely ignore minor differences model selection events lar lasso hereafter description proof partition elements cones lar lasso mirrors tibshirani given appendix since one variable added like active sets lar nested active set step lasso necessarily true case variables either added deleted step inference selection review selective pivotal inference approach hypothesis testing model selection lar lasso technical details statistic deferred next two subsections needed understand method used null hypotheses consider form important special case occurs linear contrast gives normalized coefficient regression onto subset variables specific case subset let denote submatrix whose columns correspond elements assumed invertible chosen subset write jth standard basis vector gives therefore test significance jth normalized coefficient linear projection onto written short though normalization denominator irrelevant significance test acts key scaling factor asymptotics section idea using projection parameter inference also appeared berk wasserman lee summary testing framework possible model statistic defined see next subsection whose domain partition element used follows drawn lands partition element model statistic provides test hypothesis concrete case keep mind denoting choice notation jth normalized coefficient regression onto active variables active set step assume errors null hypothesis statistic standard uniform distribution draws land mathematically property pvt probability taken arbitrary mean parameter fact statistic constructed law depends unambiguous order hold course random depend though functions thus serves valid exact finite sample size testing null hypothesis conditional confidence interval obtained inverting test given desired confidence level define set values construction property reiterate assumes errors translates interpretation statement straightforward random interval contains fixed parameter probability conditional truncated gaussian pivot describe truncated gaussian pivotal quantity detail defined section selected model given algorithm lar lasso run steps write convex cone fixed achieveable model hence fixed matrix inequality meant interpreted componentwise define pivot testing several preliminary quantities must introduced min max pivot defined following property stated drawn errors see lemmas pivot uniformly distributed conditional tibshirani proof result confidence intervals null hypothesis seen acts proper conditional defined statistic implicitly aligned power alternative hypothesis therefore seeking test significance say jth coefficient projected linear model actually choose recall sign sign jth coefficient regression onto set active variables orients test meaningful direction hypothesis jth coefficient projection onto nonzero shares sign jth coefficient projection onto choice designed small jth coefficient projection corresponds projected population effect large sign computed coefficient beyond current subsection explicit sign factor discussing contrasts giving regression coefficients projected linear model implicitly understood computing statistic aligned power alternative simply given min purely testing purposes find discussed natural hence serve default hand constructing confidence intervals prefer invert statistics since lead intervals min previously described confidence interval given inverting pivot summarize default work tibshirani consider hypothesis tests intervals two slightly different uses pivot inference selection revisited portrayed selective pivotal inference sequential regression procedures method producing conditional intervals unconditional interpretation framework also possible describe model contrast vector pivot value identified hypothesis tested whenever whenever statistic defined whose domain entire sample space write denote collection contrast vectors pivot values respectively across partition also refer catalogs unconditional statistic defined denotes indicator function partition element defined unconditional statistic used follows response test hypothesis drawn form concrete case keep mind assigns contrast vector model notation usual normalized coefficient projecting onto active variables step assume errors proper hypothesis summing conditional property across partition elements assertion holds parameter use shorthand note full specification across critical order apply relevant null probability within partition element giving rise equality therefore serves valid exact finite sample size testing null hypothesis formally attached exhaustive specification truth carries information models reason actually consider selected one random null hypothesis vtc made precise confidence intervals confidence interval obtained inverting test statistic depends thus given desired confidence level let define set set see confidence interval effectively infinite respect values inverting test yields vtc expression says random interval traps random parameter vtc probability thus supports interpretation vtc null hypothesis underlying unconditional statistic remark pivotal property derived distributional assumption may seem unnatural catalog pivot value large order steps condition possibly many contrasts however worth emphasizing unconditional testing property really useful allows formulate unconditional confidence interval property natural statement coverage single random parameter viewing selective inference unconditional perpsective find natural place focus confidence intervals rather hypothesis testing many ways find former natural two perspectives unconditionally tibshirani fact suggest separate nomenclature unconditional case referring property selection interval rather confidence interval emphasize interval covers moving target master statistic given response predictors description thus far selected model statistics ignored role done simplicity theory come section consider nonrandom asymptotically must course grow help precise dependence selected model statistics emphasize denote quantities dependence define quantity call master statistic name might suggest plays important role normalized coefficients regressing onto subsets variables written terms arbitrary set jth normalized coefficient regression onto depends dependence true turns selected models lar lasso defer proof next lemma proofs paper appendix lemma lar lasso procedures run steps data selected depends master statistic model detail fixed matrix written depends hence lemma asserts master statistic governs model selection performed lar statistic lasso also central pivot procedures denoting depends three quantities third quantity always function lemma chosen normalized coefficient regression onto subset variables first two quantities also functions thus case pivot depends master statistic fact continuous point nonsingular lie boundary model selection event lemma fix model suppose chosen normalized coefficient projecting onto subset variables statistic depends means may write function continous point nonsingular finally show conditional pivotal property statistic phrased entirely terms master statistic lemma assume conditions lemma additionally drawn construct master statistic function thus errors conditional pivotal property statistic reexpressed equipped last two lemmas asymptotic theory test fixed far weak conditions data model central limit theorem tells converges weakly normal random variable converging deterministic matrix continuous mapping theorem provide appropriate asymptotic limit statistic made precise next asymptotic theory treat dimension fixed consider limiting distribution statistic see section case grows throughout matrix treated nonrandom consider sequence predictor matrices satisfying two conditions lim nonsingular matrix lim max denote rows strong conditions nonparametric family distributions specify class distributions working let fixed known constant first define set error distributions xdf first moment condition definition needed make model identifiable second condition used simplicity aside moment conditions class contains small neighborhood say measured total variation metric around essentially every element thus modulo moment assumptions strongly nonparametric sense donoho given let denote distribution given let define class distributions words assigning distribution means drawn model mean errors arbitrary centered distribution variance grows allow underlying mean change place restriction parameter appropriate asymptotic limit specifically consider class sequences mean parameters asymptotic limit lying compact set uniform convergence limit formally write slight abuse notation denote sequence mean parameters let denote set limit points constant require class lim sup sup emphasize vary think columns triangular arrays notation suppresses dependence simplicity uniform convergence results begin result uniform convergence random part master statistic normal distribution marginally conditionally lemma assume asymptotic covariance matrix satisfies normalization condition let class defined sequence mean parameters defined denote converges distribution uniformly uniformly lim sup sup sup given sequence matrices set nonempty interior converges distribution uniformly uniformly lemma combined lemmas last section leads uniform asymptotic theory test remind reader number steps considered fixed next result throughout paper theorem assume conditions lemma suppose lar lasso run steps describe conditional unconditional asymptotic results separately markovic fix model let vector gives normalized coefficient projection onto subset variables let arbitrary pivot value converges distribution conditional statistic uniformly lim sup sup sup moreover define set asymptotically uniformly valid confidence interval lim sup sup sup let catalog vectors yields normalized coefficient projection onto subset variables catalog pivot values results part hold marginally lim sup sup sup defined set lim sup sup sup vtc remark initial version work contained unconditional result part theorem jelena markovic pointed conditional result part also possible thus conditional result also attributed initial current version paper addition revising theorem also revised theorems include appropriate conditional results unknown bootstrap results previous section assumed error variance model known consider two strategies unknown first plugs rather naive estimate usual statistic second computationally efficient bootstrap method show yield asymptotically conservative practice bootstrap often gives shorter confidence intervals based pivot see section simple approach given model contrast vector pivot value consider statistic let abbreviate abm latter two functions defined section notation succintly write statistic abm unknown propose simple approach replaces sample variance denotes sample mean fixed constant explicit consider modified statistic abm scaling factor facilitates theoretical study statistic practically found ignoring setting works perfectly well though choice say seems minor effect anyway mean nonzero sample variance generally large estimate show modified statistic thus yields asymptotically conservative residual based estimates useful setting depend heavily linearity underlying regression model suffer practically close see also discussion start section efficient bootstrap approach alternative method last subsection investigate highly efficient bootstrap scheme rely knowledge general framework far treats fixed bootstrap strategy respect assumption use say pairs bootstrap must perform sampling respect residual bootstrap ruled since assume mean follows linear model leaves consider simple bootstrap sampling components somewhat nonstandard components provides mechanism provably conservative asymptotic inference makes approach computationally efficient given drawn model let denote bootstrap sample denote conditional distribution associated expectation operator shorthand similarly using notation assuming without loss generality last subsection notation abm let motivate bootstrap proposal expressing statistic abm treated fixed probability side taken thus abm denoting random variable main idea approximate truncated normal distribution underlying statistic appropriate one bootstrap samples abm abm recall sample mean denoting vector shifted resulting quantity mimics normal variable mean side nearly defines bootstrap version statistic except technical reasons must make two small modifications particular define bootstrap statistic cvt abm cvt constant small constant found ignoring scaling factor setting works fine practice though choice like cause major differences anyway contrary nonzero choice padding factor like play important practical role since bootstrap probabilities numerator denominator sometimes zero lastly worth emphasizing practical estimation bootstrap probabilities appearing quite easy computational task regression procedure question lar lasso need rerun beyond initial run observed initial run draw say bootstrap samples save realized quantities abm order estimate probabilities computationally expensive moreover estimate multiple trial values invert bootstrap bootstrap confidence interval single common set bootstrap samples needed since shift appropriately bootstrap sample asymptotic theory unknown treating dimension fixed assume previous limiting conditions matrix additionally note already implies little stronger though still strong condition means example satisfied max conditions imply important scaling properties usual choices contrast vectors lemma assume satisfies vector gives normalized regression coefficient projecting onto subset variables specify assumptions distribution similar slightly stronger section constants define set error distributions xdf also define class distributions denotes distribution define class sequences mean parameters satisfies lim sup sup constant recall denotes set limit points also require constants note assumptions much stronger assumptions section require existence two moments error distribution place additional weak condition growth components conditions sufficient prove following helpful lemma lemma assume satisfies let class defined let class fixed lim sup sup uniformly words event probability tending conditional furthermore denoting sample third moment exists sufficiently large sup sup uniformly words conditional last two lemmas allow tie distribution function bootstrap contrast normal random variable lemma assume satisfies let defined let defined let let gives normalized regression coefficient projecting onto subset variables exists sufficiently large sup sup sup use standard normal random variate words sup uniformly conditional ready present uniform asymptotic results bootstrap statistics remind reader number steps treated fixed throughout theorem assume conditions lemma suppose lar lasso run steps conditional statistic conditional bootstrap statistic asymptotically larger distribution uniformly lim sup sup sup pvt lim sup sup sup pvt max denotes positive part given catalog vectors yields normalized coefficient projection onto subset variables results hold marginally remark simplicity analyzed bootstrap statistics simultaneously consequently conditions assumed prove asymptotic properties approach stronger would need study method major differences conditions theorem establishes bootstrap versions statistic asymptotically conservative viewed look broadly distribution test statistics arbitrary value technical barrier arises statistic proof asymptotic conservativeness leverages fact truncated gaussian survival function decreases pointwise sense underlying variance parameter decreases extend results case arbitrary pivot value would need analogous fact hold replace survival function gaussian variate truncated tuncated abm event yet without guarantee abm abm clearly always true arbitrary value longer case decreasing variance always decreases survival functions two truncated gaussians see appendix means confidence intervals given directly inverting either bootstrap statistic provably correct asymptotic coverage properties current analysis arguments proof theorem construct confidence intervals conversative asymptotic coverage forcing include abm pursue details found intervals practically wide interest importantly bootstrap statistics often display excellent empirical properties show next section refined analysis needed establish asymptotic uniformity distribution statistics asymptotic uniformity arbitrary would lead asymptotic coverage guarantees confidence intervals produced inverting statistics leave extension future work examples present empirical examples support theory developed previous sections also suggest much room refine expand current set results first two subsections examine problem setting covered theory last two look substantial departures theoretical framework heteroskedastic settings respectively examples lar algorithm used variable selection associated inferences results lasso paths roughly similar also examples explicitly stated otherwise computed test whether target population value may worth discussing two potentially common reactions experimental setups especially problems described next subsections first statistic uses estimate use estimate full least squares model since would less conservative experiments shown confirm works regression problems estimate becomes number variables grows particularly irrelevant ones obviously applicable problems therefore stick simple estimate always applicable always conservative second determine variable significance problem one could course fit full regression model inspect resulting confidence intervals intervals could even account selection course strategy would possible problem number predictors small enough may work perfectly fine one use complex tools inference important question deserving study topic paper examples follow intended portray robustness selective pivotal inference method nonnormal error distributions meant represent ideal statistical practice given scenario examples begin studying setting defined predictors drawing columns independently according following mixture distribution equal probability column filled entries bern denotes skew normal distribution hagan leonard shape parameter equal scaled columns unit norm underlying mean defined first components equal rest set repetitions drew response errors different choices error distribution normal laplace uniform skew normal case centered error distribution scaled variance skew normal distribution used shape parameter every repetitions predictor matrix regenerated according prescription described figure displays plots testing significance variable entered active model across steps lar plots compare standard uniform distribution computed using statistic statistic estimate bootstrap statistic bootstrap samples used approximate probabilities numerator denominator padding factor scaling factor ignored set bootstrap statistics steps restricted repetitions correct variable selection variable entered active lar model step repetitions incorrect variable selection one variables entered active model since underlying signal fairly strong predictors uncorrelated selections happened majority time specifically displayed steps comprise approximately repetitions respectively steps show reasonable power statistics bootstrap types error distributions also step uniform desired statistics error distributions though guarantees uniform null asymptotic laplace uniform skew normal error distributions asymptotic behavior appears kick quite early distributions sample size plots reveal nonnormal error distributions really farther uniform normal case somewhat remarkable recalling pvalues construction exactly uniform normal errors figure inspects statistic boostrap variants pivot value set true population value set computing statistics data instance step lar figure collects across steps step observed normal errors laplace errors bootstrap expected skewed normal errors expected bootstrap observed bootstrap uniform errors expected laplace errors expected normal errors bootstrap observed observed uniform errors skewed normal errors expected bootstrap observed expected bootstrap observed bootstrap expected expected step bootstrap observed observed uniform errors skewed normal errors expected bootstrap observed expected bootstrap observed bootstrap expected laplace errors expected normal errors step bootstrap observed observed shown steps lar steps pivotal statistics observed observed expected bootstrap skewed normal errors expected bootstrap uniform errors expected bootstrap laplace errors expected normal errors bootstrap observed observed pivotal statistics shown aggregated steps lar figure simulation setup mean nonzero components lar error distribution types according theory distribution pivotal statistics asymptotically uniform clearly supported plots interestingly bootstrap pivotal statistics also appear uniform plots yet case handled asymptotic theory recall theorem fixes pivot value otherwise technical difficulties encountered proof gives empirical evidence idea refined analysis could extend theorem broader setting arbitrary pivot values handled theorem moreover suggests inverting bootstrap statistics yield intervals proper coverage verified next subsection lastly repeated experiments subsection predictors generated way induce population correlation pairs predictor variables results quite similar shown figure hence deferred appendix confidence interval examples stay setting last subsection coefficient vector first components equal rest equal invert bootstrap statistics obtain confidence intervals lar step see table numerical summary coverage refers average fraction intervals contained respective targets repetitions power average fraction intervals excluded zero width median interval width recorded unconditional sense screening repetitions performed based variables selected across steps lar conditional coverages however quite similar table see methods lead accurate coverage around cases see intervals bootstrap statistic shorter statistic cases considerably shorter original statistics steps power bootstrap intervals generally better intervals also par power original statistic step somewhat worse step recall original statistic uses knowledge error variance bootstrap variants boot boot boot boot coverage step power width coverage step power width coverage step power width table summary statistics confidence intervals constructed problem setting figure blocks rows correspond types noise normal laplace uniform skew normal respectively standard errors coverage power width statistics respectively bit surprising bootstrap intervals shorter still worse power original intervals easier understand intervals visualized done figure figure shows sample intervals first lar step normally distributed errors sample intervals error models shown appendix see bootstrap intervals indeed shorter compared original intervals symmetric around target population values original intervals asymmetric often shorter side target value facing results better power repeated experiments predictors generated pairwise correlation comparisons drawn results manner roughly parallels discussions following table however absolute scale methods display decrease power across board correlated predictors clearly make problem difficult details provided appendix heteroskedastic errors setup sections predictors mean generated manner consider heteroskedastic model drawing given laplace uniform skew taking errors denote rows spread error variances ended fairly substantial original statistic computed surrogate common error variance bootstrap variants computed usual brevity plot pivotal statistics aggregated steps lar figure analogous shown figure homoskedastic case steps shown end similar figure power methods generally lower due heteroskedastic errors see pivotal statistics figure look close uniformly distributed desired especially encouraging current problem setup lies outside scope asymptotic theory assumes constant error variance suggests theory could possibly extended accomodate errors unknown nonconstant variance structure examples finally consider regime predictors matrix generated according recipe column equal probability assigned entries bern scaled unit norm mean defined first components equal rest repetitions response generated adding normal laplace uniform skew normal noise error variance every repetitions predictor matrix regenerated figure plots pivotal statistics aggregated first steps lar figure case first lar steps omitted brevity roughly similar figure except display less power due pivotal statistics look quite close uniform desired encouraging especially given current case lies outside scope theory assumes fixed work asymptotic theory pursued see also tian taylor though show next section hope uniform convergence result high dimensions holds generally confidence interval normal errors repetition confidence interval normal errors repetition confidence interval normal errors bootstrap repetition figure confidence intervals draws model figure intervals constructed first step lar uniform distribution noise colors simply visual aid mark selection different variables step open circles denote true population quantity covered coefficient projecting onto first selected variable intervals contain targets drawn dotted segments bootstrap observed expected skewed normal errors expected bootstrap observed uniform errors bootstrap expected laplace errors expected normal errors bootstrap observed observed figure simulation setup heteroskedastic errors shown pivotal statistics aggregated lar steps observed bootstrap observed observed expected skewed normal errors bootstrap expected uniform errors expected bootstrap laplace errors expected normal errors bootstrap observed figure simulation setup shown pivotal statistics lar steps one established theorem low dimensions negative result high dimensions prove statistic fails converge uniform distribution null hypothesis data model nonnormal errors otherwise represents fairly standard setting many means setting write observation model interpret replications dimensions total hence observations denote analyze statistic selection performed based largest inference performed corresponding mean parameter straightforward change notation translate regression problem orthogonal design stick many means formulation problem simplicity assume errors following mixture mixing proportion mean shift scale moreover chosen error variance mentioned consider model selection events form max note exactly selection event first step lar lasso paths run regression version problem orthogonal design hard check statistic conditionally testing given msy max per spirit paper also view statistic unconditionally helpful define denote order statistics see unconditional statistic testing selected mean framework underlying statistic tells errors furany fixed model pivot uniformly distributed conditional ther largest second largest absolute values centered normal random variables variance unconditional pivot uniform large defined order statistics nonnormal random variates statistic case defined extreme tail behavior normal nonuniform next theorem asserts nonuniformity indeed happen asymptotically choose mixture distribution appropriately theorem assume observation model errors drawn let scale manner log let error variance fixed global null hypothesis unconditional statistic converge distribution particular event whose limiting probability least statistic converges results hold conditionally selected model fixed converge distribution conditional statistic least converges event limiting probability conditional expected expected observed observed figure left plot shows plot computed repetitions many means setup exactly described theorem see clearly nonuniform computer precision close theoretically predicted proportion right plot shows model reversed roles also cap see essentially uniform remark assumed condition log requires dimension diverge necessarily number replications though clearly allows diverge sufficiently slow rate hand fixed diverged result theorem would longer true limiting distribution would revert careful would cap mixing probability order mixture make sense since current definition diverges fixed tending fact ensured result theorem reformulating many means problem appropriate regression notation conditions theorem met current setup fixed supported simulation figure remark precise scaling log chosen since implies extreme mixture components probability tending intuitively reasonable property error distribution note scaling important reason proof would still remain correct remark theorem tian taylor authors show statistic converges distribution standard uniform random variable problem setting restrictions sequences selection events allowed one might ask part setup violates conditions results obviously true simultaneously far tell issue lies role assumption tian taylor namely defined error distribution value needed certify third condition assumption work small main assumption theorem hold hence theorem tian taylor apply current setup discussion studied selective pivotal inference framework focus forward stepwise regression least angle regression lar lasso regression problems nonnormal errors shown truncated gaussian pivot asymptotically robust settings departures normality converges distribution pivotal distribution normality uniformly broad class nonnormal error distributions error variance unknown proposed bootstrap versions statistic yield provably conservative asymptotic numerical experiments revealed statistics theoretical investigation generally display excellent performance highly nonnormal error distributions experiments also revealed findings predicted theory bootstrap statistic often produces shorter confidence intervals based statistic even statistic relies error variance three statistics show strong empirical properties classic homoskedastic fixed regression setting presumed theoretically however clearly demonstrated one hope convergence result high dimensions general result obtained low dimensions relatively simple many means problem showed nonconvergence statistic whereas problem fixed statistic converges usual limit still much left terms understanding behavior selective pivotal inference tools constructed exact guarantees normality like statistic tibshirani applied regression settings nonnormal data pivot central cog framework constructed assumption normality creates robustness issues especially worrisome high dimensions appendix provides discussion issues detailed study subject future research acknolwedgements thank jelena markovic jonathan taylor many helpful discussions overall generosity initial version work contained unconditional marginal results main theorems theorems jelena markovic pointed theorem also hold conditionally current version work revised accordingly appendix convex cones lar lasso describe modification conic conditioning set tibshirani version different additionally condition sign every active coefficient every step rather coefficient variable enter model step modifications needed lar lasso conditioning sets made top sets lar lasso given tibshirani follow similarly described hence omit details sorted steps always represent sequence active sets list variables chosen enter model step unfortunately done sequence active signs obey nested structure write signs coefficients corresponding variables step using induction characterize event step rearranged set linear inequalities assume represented event collection linear inequalities represent sbk must append collection inequalities former subevent characterized residual regressing onto residual regression onto projects onto orthocomplement expressing column space rewrite constraints set linear inequalities meanwhile subevent sbk characterized inequalities expressed block form diag completes proof proof lemma prove result results lar lasso paths follows similarly inpsecting form linear inequalities determine selection events consider first step described appendix multiplying see equivalent set inequalities characterize selection event clearly desired form matrix dependent kth step two sets inequalities examined one describes variable enter second describes active signs sbk first set multiplying second set multiplying diag inequalities clearly summarized matrix depends completes proof proof lemma conditions lemma pivot fixed depends master statistic explained lemma dependence pivot quantities turn function master statistic moreover may reexpress statistic functions succinctly note quantities depend smoothly master statistic point nonsingular implies smooth functions point nonsingular lastly thus denominator positive proves desired continuity result proof lemma master statistic note assumed chosen normalized regression coefficient projection onto subset columns may assume without loss generality see must define use denote submatrix rows columns denote subvector entries proof lemma define ith row note independent mean zero components compute cov converges assumption consider seek show converges suffices show maximum expectations summands converges implied assumption max arguments depend sequence verified conditions uniformly hence uniform central limit theorem lemma implies converges distribution uniformly consider writing standard normal cdf density sup sup sup sup sup sup sup sup sup sup sup sup sup second line due triangle inequality third line due simple bound note argument start proof assumption shows converges distribution uniformly lastly establish conditional result repeating arguments uniform central limit theorem condition imply converges uniformly thus along sequence observe rate depend sequence question true numerator denominator converge normal probability counterparts denominator remains bounded away zero since nonempty interior set limits assumed compact since arbitrary distribution continuous lemma van der vaart sup sequence arbitrary shown desired uniform convergence proof theorem begin proof part let also let lemma also converges recall weakly uniformly lemma deterministically also converges uniformly distribution choice specified theorem important two reasons first lemma express function second lemma express function neither depend distribution question function continuous point nonsingular recalling assumed nonsingularity therefore continuous set full probability limiting distribution uniform continuous mapping theorem lemma converges uniformly distributed pivotal property statistic normality proof uniform validity confidence intervals rearrangement uniform asymptotic pivotal statement proof part follows expansion number possible models finite simply apply asymptotic pivotal result part establish asymptotic pivotal property confidence interval result rearrangement pivotal property proof lemma assumption vector written compute denominator converges numerator satisfies bounded converges completes proof proof lemma start proving result event first let study asymptotic probability marginally consider hence var last line used chebyshev inequality recalling lower bound large enough therefore show enough show var uniformly use simple inequality var var follows fact var var also invoke rosenthal inequality rosenthal independent mean zero states ewi max constant depending hence observe var var var var last line used consider individually var var var second line used third used assumptions error distribution also max second line used rosenthal inequality implies var uniformly therefore shown uniformly see take sequence result holds conditional note borrowed notation normal convergence result proof lemma rate convergence last line depend sequence consideration uniform convergence fact denominator bounded away zero since set limits assumed compact arbitrary completes proof first part lemma second part boundedness consider uniformly showed last term satisfies suffices prove exists large enough uniformly markov inequality true long uniformly bounded end use simple inequality compute max second third lines used last line used rosenthal inequality along abbreviations uniform convergence using lemma large enough used upper bound third moment error distribution used lower bound holds uniformly due assumed compactness set limits thus shown uniformly upper bounded similar arguments show uniformly upper bounded assumption uniformly upper bounded completes proof see second part lemma proof lemma let write independent mean zero theorem chen sup side precisely uniformly lemmas imply conditional giving result proof theorem first prove result statistic denoting abm uniconsider event probability approaching conditional formly lemma event monotonicity truncated gaussian survival function variance parameter shown appendix replace increase value statistic verify result appendix indeed applied notice abm left endpoint interval least mean truncated gaussian follows fact design thus write abm uniformly hence pvt pvt remainder term uniform applying part theorem proves conditional result statistic next turn bootstrap result whose proof little involved define function cvt abm abm cvt lemma implies write abm abm uniformly conditional note absorbed role lemma dividing quantity abm abm abm abm abm abm abm abm abm abm abm uniformly conditional precise value may differ line line second line used uniformly similarly third line used fact conditional last line used monotonicity truncated gaussian survival function underlying variance parameter fact uniformly rewriting result last display sup abm abm max denotes negative part particular implies finally means write arbitrary level pvt pvt uniformly conditional therefore pvt pvt term uniform applying part theorem proves conditional result bootstrap statistic unconditional results two modified statistics hold simply marginalization monotonicity truncated gaussian distribution define survival function normal random variable truncated lie interval show following proof similar monotonicity result lemma lee emphasize property true interval lies right without restriction survival function monotone increasing contains generally nonmonotone lies left actually monotone decreasing family distributions forms exponential family natural parameter family gaussian distributions carrier measure changed therefore monotone likelihood ratio sufficient statistic denote truncated gaussian density function fix hence integrating respect obtain integrating respect obtain rearranging gives result examples correlated predictors investigate consequences using correlated predictors simulation setup section constructed preliminary matrix column drawn independently either bern entries equal probability took predictor matrix diagonal entries equal entries equal symmetric square root scaled columns unit norm rest setup section figure shows results format figure lar steps pivotal statistics aggregated lar steps repetitions steps restricted repetitions either variable selected comprising repetitions respectively step restricted repetitions one variables selected comprising repetitions similar display figure see power steps albeit less power uncorrelated case uniform step well uniform pivotal statistics confidence intervals uniform laplace skew normal noise figures show sample confidence intervals problem setting section error distribution uniform laplace skew normal respectively confidence interval summary statistics correlated predictors table gives summary statistics confidence intervals obtained inverting original plugin bootstrap statistics table section correlated predictors setup described section proof theorem let denote number observations jth column data array drawn mixture component similarly let denote number observations jth column drawn mixture component define event step observed normal errors laplace errors bootstrap expected skewed normal errors expected bootstrap observed bootstrap uniform errors expected laplace errors expected normal errors bootstrap observed observed uniform errors skewed normal errors expected bootstrap observed expected bootstrap observed bootstrap expected expected step bootstrap observed observed uniform errors skewed normal errors expected bootstrap observed expected bootstrap observed bootstrap expected laplace errors expected normal errors step bootstrap observed observed shown steps lar steps pivotal statistics observed observed expected bootstrap skewed normal errors expected bootstrap uniform errors expected bootstrap laplace errors expected normal errors bootstrap observed observed pivotal statistics shown aggregated steps lar figure plots figure setup predictor variables pairwise correlation confidence interval laplace errors repetition confidence interval laplace errors repetition confidence interval laplace errors bootstrap repetition figure confidence intervals draws similar figure uniform noise distribution confidence interval uniform errors repetition confidence interval uniform errors repetition confidence interval uniform errors bootstrap repetition figure confidence intervals draws similar figure laplace noise distribution confidence interval skewed normal errors repetition confidence interval skewed normal errors repetition confidence interval skewed normal errors bootstrap repetition figure confidence intervals draws similar figure skew normal noise distribution boot boot boot boot coverage step power width coverage step power width coverage step power width table summary statistics confidence intervals table modified problem setting predictor variables pairwise correlation standard errors roughly coverage power width statistics respectively words event exactly one column observations drawn rest columns least observations calculate second line used construction introduced notation number observations column drawn mixture component third line used markov inequality event intersected event whose probability tends one furthermore max denote standard normals note ultimate bounds sides two lines extremely loose suffice purposes hence using mills ratio bound statistic event consideration exp exp figure left plot shows two densities black red right shows tail functions corresponding colors sufficiently large event straightforward check side bound diverges given assumptions therefore shown event whose probability tends least statistic converges conditional result notice model symmetry well hence condip asymptotically uniform converges tional statistic event whose limiting probability least conditional thoughts instability high dimensions statistic defined ratio normal tail probabilities dimension large case searching large space models large effects often find evaluating pivot far tails point evaluation given linear function data converge gaussian distribution least finite even small amount magnified tails see consider function left plot figure shows two densities nearly indistinguishable right plot shows corresponding tail functions even though close see quite different message inferential method depends heavily extreme tail behavior could unreliable perhaps visually striking plot statistic viewed function fixed shown figure statistic used test used model selection partition elements even matching figure possible fully visualize statistic function function boundaries partition elements corresponding different model selection events technically function continuous interior partition element permits application uniform continuous mapping theorem fixed derivatives boundaries infinite especially problem settings nonnegligible probability near boundary thus small perturbation data could dramatic effect value pivot references bachoc leeb potscher valid confidence intervals predictors arxiv berk brown buja zhang zhao valid inference annals statistics chen goldstein shao normal approximation stein method springer choi taylor tibshirani selecting number principal components estimation true rank noisy matrix arxiv donoho inference functionals density annals statistics fithian sun taylor optimal inference model selection arxv hyun sell tibshirani exact inference changepoint detection generalized lasso problems arxv kasy uniformity delta method unpublished manuscript lee sun sun taylor exact inference application lasso annals statistics lee taylor exact post model selection inference marginal screening advances neural information processing systems leeb potscher distribution estimators uniform versus nonuniform approximations econometric theory leeb potscher one estimate conditional distribution estimators annals statistics leeb potscher one estimate unconditional distribution estimators econometric theory lockhart taylor tibshirani tibshirani significance test lasso annals statistics figure two views statistic pivot value set setup figure statistic plotted function loftus taylor significance test forward stepwise model selection arxiv hagan leonard bayes estimation subject uncertainty parameter constraints biometrika reid taylor tibshirani point interval estimation signal sizes gaussian samples canadian journal statisitcs rosenthal subspaces spanned sequences independent random variables israel journal mathematics taylor loftus tibshirani inference adaptive regression via formula annals statistics tian taylor asymptotics selective inference scandinavian journal statistics tibshirani taylor lockhart tibshirani exact inference sequential regression procedures journal american statistical association van der vaart asymptotic statistics cambridge university press wasserman discussion significance test lasso annals statistics
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pointer analysis strong updates via refinement jan yulei sui school computer science engineering unsw australia jingling xue school computer science engineering unsw australia present new pointer analysis strong updates programs called upa enables computing information via refinement environments small time memory budgets ides formulate upa solving graphreachability problem graph representing program chains efficiently answer client query request variable set upa reasons flow values along chains sparsely rather across program points performing work necessary query rather analyzing whole program particular strong updates performed filter spurious chains refinement long total budget exhausted upa facilitates efficiency precision tradeoffs applying different pointer analyses hybrid analysis framework implemented upa llvm evaluate choosing uninitialized pointer detection major client programs analysis budget increases upa achieves improved precision analysis reaching achieved analysis consuming seconds memory per query average budget edges per query also considered upa twostage analysis becomes precise programs also incurs analysis times upa also amenable parallelization parallel implementation analysis achieves speedup average machine respect sequential version ccs concepts engineering software verification validation software defect analysis computation program analysis additional key words phrases strong updates value flow pointer analysis flow sensitivity introduction pointer analysis one fundamental static program analyses virtually others built goal pointer analysis compute approximation set abstract objects pointer refer pointer analysis respects control flow otherwise contextsensitive distinguishes different calling contexts otherwise strong updates stores overwrite kill previous contents abstract destination objects new values important factor precision pointer analysis hardekopf lin chung case weak updates objects assumed conservatively also retain old contents strong updates possible maintained addition analysis strongly update abstract object written store object exactly one concrete memory address known singleton applying strong updates needed pointer analysis improve precision thereby providing significant benefits many clients change impact analysis acharya robinson bug detection yan security analysis arzt type state verification fink compiler optimization sui symbolic execution blackshear paper introduce pointer analysis investigating perform strong updates effectively framework programs important achieving precision required client applications due strong updates performed contextsensitivity also considered strong updates possible yulei sui jingling xue grams expense analysis times languages like java without strong updates widely used achieving useful precision hendren milanova smaragdakis sun xiao zhang ideally strong updates stores performed analyzing paths independently solving mop problem however even branch conditions ignored problem intractable due potentially unbounded number paths must analyzed landi ramalingam instead traditional pointer analysis hind pioli kam ullman computes solution mfp overapproximation mop solving iterative problem thus facts reach confluence point along different paths merged improving sparse pointer analysis sfs hardekopf lin boosts performance analyzing large programs maintaining strong updates done basic idea first conduct program chains perform propagating facts pointsto information sparsely along chains aka instead program points program graph cfg recently approach chung performing strong updates programs introduced sacrifices precision gain efficiency applying strong updates stores singleton sets available falls back information otherwise nature challenge pointer analysis make judicious tradeoffs efficiency precision virtually prior analyses consider degree analyses precise ones unscalable since must typically consider fscs order maximize number strong updates performed contrast faster ones like chung less precise due missing strong updates propagating information across locations practice client application pointer analysis may require parts program analyzed addition queries may demand precise answers others answered precisely possible small time memory budgets cases performing strong updates blindly across entire program achieving precision programs develop precise efficient pointer analyses focused partial paying closer attention parts programs relevant queries analyses heintze tardieu zhang zheng rugina java shang sridharan yan thus perform strong updates produce precision needed clients oomerang represents recent pointer analysis java however approach performs strong updates store partially updating strongly aliases weakly elsewhere advances analysis exploited form sparsity improve performance hardekopf lin however replicate success analysis unclear finally remains open whether sparse strong update analysis performed avoid paper introduce upa first pointer analysis strong updates designed support flexible yet effective tradeoffs pointer analysis program preanalysis stages efficiency refine queries precision stage stage reachability solver stage select budgets yes fig overview upa ciency precision answering client queries environments small time memory budgets ides shown figure novelty behind upa lies performing strong update analysis precisely refining imprecisely away hybrid analysis framework given query strong updates performed solving problem interprocedural graph captures chains program obtained conservatively obtained applying andersen analysis andersen upa conducts reachability analysis sparsely along precomputed rather addition upa filters imprecise performing strong updates needed loss precision long total analysis budget sufficient precision upa depends degree refinement performed budget spurious upa removes precise facts upa handles large programs staging analyses increasing efficiency decreasing precision hybrid manner currently upa proceeds two stages applying fscs order configurable budget analysis failing answer query stage within alloted budget upa downgrades scalable less precise analysis next stage eventually fall back information stage upa query reusing information found processing current earlier queries increasing budgets used earlier stages fscs upa obtain improved precision via improved refinement summary paper makes following contributions present first pointer analysis strong updates enables computing precise information refining away imprecisely precomputed subject analysis budgets introduce hybrid analysis framework facilitates efficiency precision tradeoffs staging different analyses answering client queries produced implementation upa llvm supa evaluate upa uninitialized pointer detection practical client using total programs analysis budget increases upa achieves improved precision analysis reaching achieved analysis consuming yulei sui jingling xue seconds memory per query average budget edges traversed also considered strong updates also possible upa analysis becomes precise programs expense analysis times present four case studies demonstrate upa effective checking whether variables initialized applications show upa amenable parallelization demonstrate developed parallel implementation upa analysis based intel threading building blocks tbb achieving speedup average machine sequential version rest paper organized follows section provides background information section presents motivating example section introduces formalism upa section discusses analyzes experimental results section contains four case studies section describes parallel implementation upa section describes related work finally section concludes paper background describe represent program interprocedural sparse graph enable pointer analysis via refinement section introduces part relevant pointer analysis section describes put variables ssa form section describes put variables ssa form section constructs sparse graph represents chains variables program perform pointer analysis program balatsouras smaragdakis hardekopf lin chung sui domains llvm instructions relevant pointer analysis given table set variables separated two subsets contains possible abstract objects variables pointer contains variables variables including stack virtual registers symbols starting global variables symbols starting explicit directly accessed variables implicit accessed indirectly llvm load store instructions via variables subset complete llvm instruction set relevant pointer analysis modeled table every function program contains nine types instructions statements including seven types instructions used function body one ntry instruction declarations parameters one xit instruction retf unique return note llvm pass unifyfunctionexitnodes executed pointer analysis order ensure every function one xit instruction let seven types instructions used inside function ddr instruction known allocation site one following objects stack object allocation site via llvm alloca instruction global object global object allocation site program function name dynamically created heap object heap allocation site via malloc call object except function write represent corresponds field pointer analysis initializations global objects take place entry main pointer analysis table domains llvm instructions used pointer analysis analysis domains instruction labels constants field accesses stack virtual registers global variables program functions variables variables program variables llvm instruction set ddr opy ield oad tore ntry xit retf opy denotes casting instruction bitcast llvm standard ssa instruction introduced confluence point cfg select value variable different branches oad tore memory accessing instruction reads write value object handling given pointer aggregate struct array pointer arithmetic used accessing anything aggregate undefined behavior pearce thus modeled model field accesses struct object ield represents getelementptr instruction field offset constant value getelementptr instruction operates field struct modeled opy instructions one every field struct conservatively arrays treated monolithically denotes call instruction either global variable direct call stack virtual register indirect call ssa form variables known partial ssa form since variables ssa form variables explicit directly accessed thus put ssa form using standard ssa construction algorithm cytron instructions inserted confluence points swap code swap partial ssa relations observed runtime swap swap fig swap example partial ssa form let illustrate llvm partial ssa form using example figure figure shows swap program figure gives corresponding partial yulei sui jingling xue ssa form figures depict runtime relations swap operation example note new temporary registers introduced order put program given figure partial ssa form given figure particular decomposed pointer variables ssa form example variables trivially ssa form exactly one definition result chains variables readily available however variables accessed indirectly loads stores via toplevel variables thus ssa form example variable defined implicitly twice variable also defined implicitly twice result chains variables immediately available ssa form variables starting llvm partial ssa form first perform using andersen algorithm andersen implemented svf sui xue put variables memory ssa form using ssa construction algorithm cytron imprecise information computed way refined pointer analysis given variable anderptspvq represents set computed andersen algorithm two steps sui illustrated figures intraprocedurally figures interprocedurally step computing modification reference shown figure every load annotated operator object pointed anderptspqq represent potential use load similarly every store annotated operator object anderptsppq represent potential def use store strongly updated receives whatever points old contents killed otherwise must also incorporate old contents resulting weak update compute function call applying lightweight interprocedural analysis sui given callsite annotated may read modified inside callees discovered andersen pointer analysis addition appropriate operators also added ntry xit instructions callees order mimic passing parameters returning results variables figure gives example modified figure moving four swap instructions function swap read added callsite represent potential uses swap correspondingly swap ntry instruction annotated receive values passed modification added receive potentially modified values returned swap xit instruction annotated step memory ssa renaming variables converted ssa form suggested chow every treated use every treated def use may admit weak update ssa form variables obtained applying standard ssa construction algorithm cytron program annotated figure figure gives memory ssa form similarly figure gives memory ssa form figure pointer analysis swap swap step renaming step adding swap sparse fig memory ssa form sparse constructed intraprocedurally figure obtained andersen analysis anderptsppq tau anderptspqq tcu foo swap swap step adding step renaming swap swap swap foo swap foo sparse fig memory ssa form sparse constructed interprocedurally example modified figure four swap instructions moved separate function called swap correspond ntry xit swap sparse graph variables ssa form chains immediately available shown table discussed variables earlier two variables figure figure depicts chains sparse memory ssa form figure similarly figure gives sparse memory ssa form figure given program sparse graph svfg gvfg directed graph captures chains yulei sui jingling xue table information variables defv sev denotes set definition use instructions variable instruction retf defs uses variables memory ssa form defp defp seq defp seq ser defp seq defp seq seai sep seq defai seai defp seq seri seai defaj seaj defri defai seai sep seai defp sep defai seai anderptspqq anderptspqq retf anderptspqq anderptspqq retf fig construction memory ssa form variables set nodes representing instructions set edges representing potential chains particular edge statement statement signifies potential chain def use refer direct indirect valueflow representation sparse since intermediate program points omitted thereby enabling underlying information gradually refined applying sparse pointer analysis figure gives rules connecting two instructions based defs uses computed table intraprocedural handle variables respectively ssa form every use variable unique definition use identified version annotated unique definition ssa form annotated generated resent potentially thus phi functions introduced variables ignored value versioned let consider interprocedural information table intraprocedural according figure interprocedural constructed represent parameter passing variables operators annotated ntry xit variables pointer analysis connects actual argument call instruction corresponding formal parameter ntry every callee invoked call conversely models xit instruction every callsite invoked like variables build addresstaken variables across functions according annotated note versions ssa variable different functions may different example figure illustrates four obtained applying two rules figure svfg obtained way may contain spurious chains figure andersen pointer analysis fast imprecise however representation allows imprecise information refined performing sparse pointer analysis prior work hardekopf lin nagaraj govindarajan sui paper introduce pointer analysis strong updates answer queries efficiently precisely removing spurious chains svfg iteratively motivating example pointer analysis upa operates svfg program computes queries performing strong updates whenever possible refine away imprecise svfg example program shown figure simple even lines program consists sequence code taken directly figure six new statements added enable highlight key properties upa assume uninitialized initialized svfg embedded figure referred shortly describe upa used prove points initialized object computing query ptpx zyq set program point defined section figure depicts relations six variables ones found end code sequence flowsensitive analysis strong updates like sfs hardekopf lin due multiple solutions pointer maintained example true relations observed end program execution note sfs gives rise figure analyzing figure analyzing also finally figure analyzing points warning issued implying regarded properly initialized figure shows relations figure applying andersen analysis andersen case single solution computed conservatively entire program due lack strong updates analyzing two stores performed swap relations figures merged causing become spurious aliases analyzed seven spurious relations shown dashed arrows figure introduced since points correctly spuriously false alarm issued failing consider andersen analysis precise uninitialization pointer detection client let explain upa shown figure works upa first perform example program build svfg given figure discussed section variables direct defuse chains explicit thus omitted avoid cluttering example three yulei sui jingling xue spurious direct query swap spurious supa analysis resolving traversing backwards program svfg indirect shown indirect relations found hold end program pointers omitted relations pointers omitted fig motivating example illustrating upa stands strong update chains variables nine indirect chains depicted figure let see two chains created points annotated respectively putting ssa form three functions become hence indicating two potential definitions one overwriting one chains built similarly pointer analysis let consider analysis stage stage figure figure shows upa computes ptpx zyq starting performing backward reachability analysis svfg visiting order chains marked formally done illustrated figure chains relevant variables shown filtering four spurious marked upa finds backward reachable thus ptpx zyq tiu warning issued upa differs prior work following three major aspects strong updates analysis like sfs hardekopf lin answer ptpx zyq precisely must accomplish task analyzing statements resulting total six strong updates performed six stores strong updates performed unnecessarily query unfortunately existing fscs even algorithms scale well large programs acharya robinson contrast upa computes ptpx zyq precisely performing three strong updates earlier strong update performed upa reachability analysis fewer number statements traversed performed upa finds points strong update performed upa concludes ptpx zyq tiu refinement pointer analyses shang sridharan yan zhang zheng rugina thus suffer imprecision counterparts absence strong updates many spurious aliases result causing point result false alarm issued discussed earlier however upa performs strong updates filtering four spurious marked points spurious traversed addition strong update enabled rendering spurious finally refined away due another strong update performed thus upa avoided many spurious aliases introduced resulting ptpx zyq tiu precisely thus warning issued precision control balance efficiency precision upa operates hybrid analysis framework asked answer query ptpx zyq budget say maximum sequence three steps traversed upa stop traversal figure fall back results returning ptpx zyq case false positive end reported strong updates introduce pointer analysis strong updates illustrated figure first describe inference rules setting section discuss extension section finally present hybrid analysis framework section analyses thereby enabling strong updates performed struct objects yulei sui jingling xue addr copy phi field load store killp call retf ret compo killp ptpx pyq singletons else ptpx pyq otherwise fig upa analysis strong updates formalism present formalization upa consisting analysis given program upa operate svfg representation gvfg constructed applying andersen analysis andersen discussed section illustrated section let set labeled variables set statement labels defined table upa conducts backward reachability analysis gvfg computing reachability relation formalism signifies def use one multiple paths gvfg object created ddr pointer analysis statement allocation site identified must distinguish accessed elsewhere inference rules abbreviation given query upa computes ptpx vyq set finding reachable target objects defined follows ptpx vyq opu despite formalization figure makes explicit references program points upa operates chains gvfg variable mentioned rule appears point defined let examine rules detail addr object created allocation site backward reachable precisely point direct across variables gvfg always precise copy phi partial ssa form phi exists variables section however indirect across variables gvfg imprecise need refined fly remove spurious aliases thus introduced handling load load traverse backwards def actually used aliased requires reachability relation computed recursively store handled similarly store defined reached backwards aliased load aliased store executed earlier must backward reachable otherwise alias relation established gvfg must spurious thus filtered refinement models strong weak updates store defining kill set killp involves three cases case points one singleton object singletons contains objects except local variables recursion arrays treated monolithically heap objects chung section discuss apply strong updates heap objects strong update possible killing old contents backward travero sal along chain needed thus falsified case set empty traversal must prevented avoid dereferencing null pointer standard hardekopf lin chung case weak update performed old contents preserved thus established implies backward traversal along must continue field handles field access pointer points field object object pointed call ret handle reachability traversal interprocedurally computing call graph program fly instead relying imprecisely precomputed call graph built hardekopf lin svfg interprocedural sinking callee function may come spurious indirect callsite avoid rules ensure function pointer actually points call ret essentially given query indirect callsite pqpq instead analyzing callees found upa recursively computes set discover new callees callsite continues refining ptpx zyq using new callees finally transitive stated compo let try rules first revisiting motivating example strong updates performed example examining weak updates example yulei sui jingling xue addr load addr load compo compo store compo store compo addr compo addr compo store deriving ptpx corresponding figure deriving ptpx corresponding figure load deriving ptpx zyq corresponding figure fig reachability derivations ptpx zyq shown figure reuse cached results inside box pointer analysis example figure shows apply rules upa answer ptpx zyq illustrated figure implicit derivations applied cause strong update store ptpx qyq tcu old contents killed becomes spurious since falsified filtered since falsified finally ignored since points load upa improves performance caching results reduce redundant traversal reuse happening marked boxes figure example figure computed load reused store ptpx tdu indirect andersen direct query fig resolving ptpx zyq weak update example let consider weak update example figure computing ptpx zyq confluence point receives information two branches thus weak update performed two locations let focus applying applying store thus ptpx ayq excludes due strong update performed ptpx qyq tau obtain ptpx zyq unlike chung falls back information weakly updated objects upa handles precisely wholeprogram analysis subject sufficient budget figure due weak update performed ptpx ayq obtained forcing approach adopt ptpx ayq thereafter causing ptpx zyq maintaining strong update applied kill upa obtains ptpx zyq precisely handling cycles compute soundly precisely information cycle svfg upa retraverses whenever new information found fix point reached example figure shows cycle formed compute ptpx zyq must compute ptpx xyq requires aliases load found using ptpx zyq upa computes ptpx zyq analyzing yulei sui jingling xue direct query indirect fig resolving ptpx zyq cycle cycle two iterations first iteration target found since due found aliases found since second iteration another target thus ptpx zyq obtained pointer analysis distinguish different fields struct object consequently gives opportunities performing strong updates struct object struct object may actually represent distinct fields contrast upa truly avoiding two limitations altogether indirect bar foo foo foo bar indirect query foo query fig resolving ptpx ryq tcu example figure illustrates effects computing information without illustrated figure two statements analyzed result strong update possible since represents possibly multiple fields singleton thus ptpx ryq upa answer query compute first applying pointer analysis field load obtain traversing three indirect chains backwards obtain ptpx ryq properties heorem oundnessq upa sound analyzing program long computing svfg program sound roof building svfg program chains variables identified explicitly partial ssa form computing sparse graph program sound chains built variables according inference rules figure upa performs essentially analysis restricting propagation information along precomputed chains falls back sound information computed running given budgets thus upa sound sound heorem recisionq given query ptpx vyq computed upa computed upa successfully resolve query within given budget roof let pts upa vyq ptfs vyq sets computed upa respectively theorem pts upa vyq ptfs vyq since upa demanddriven version thus precise show pts upa vyq ptfs vyq note upa operates svfg program improve efficiency also filtering imprecisely variables direct precise upa proceeds exactly addr copy phi field call ret compo variables upa establishes indirect flowsensitively manner refining away imprecisely load store call ret compo upa complete query within given budget pts upa vyq ptfs vyq thus pts upa vyq ptfs vyq formalism extend formalization considering also enable strong updates especially heap objects solve balancedparentheses problem matching calls returns filter unrealizable interprocedural paths reps shang sridharan yan context stack encoded sequence callsites call instruction denotes operation pushing callsite pops contains top value empty since realizable path may start end different functions statement parameterized additionally context represent instance containing function analyzed labeled variable form representing variable accessed statement context object created ddr statement context also identified opq given query upa computes set applying rules given figure ptpxc vyq opqu yulei sui jingling xue opq opq ldq opq opq killpc retf tpc killpc ptpxc pyq cxtsingletons else ptpxc pyq otherwise fig upa analysis strong updates reachability relation also passing parameters returning results callee invoked callsite handled deals direct indirect valueflows backwards entry instruction callee function callsites based call graph computed fly similarly call figure except likewise deals direct indirect backwards callsite return instruction every callee function upa filter spurious generated andersen analysis thereby producing precise information enable strong updates store kill set contextsensitive strong update applied points singleton pointer analysis bar malloc malloc foo foo indirect foo malloc heap object context query fig resolving ptpr strong updates cxtsingletons singleton defined section heap object concrete context one involved recursion loops example let use example given figure illustrate effects strong updates computing information example adapted real application given figure without upa perform weak update since points passed foo two callsites result found point points also points previously shown avoid cluttering upa finds since points singleton strong update performed causing old contents killed given program sccs strongly connected components call graph constructed fly upa handles sccs program function calls inside scc sridharan supa hybrid analysis facilitate efficiency precision tradeoffs answering queries upa illustrated figure organizes analyses multiple stages sorted increasing efficiency decreasing precision let queries issued successively let stages upa stage stage configured budgets respectively current implementation budget specified maximum number chains traversed svfg program upa answers query applying analyses successively starting stage query answered budget exhausted stage upa query stage eventually falls back results upa caches fully computed information query reuses subsequent queries illustrated figure let set queried variables issued program let set variables reached analysis let queried variable write vyq represent yulei sui jingling xue set variable computed stage budget contextual qualifier stage fscs convention ptn vyq denotes pointsto set obtained stage conceptually resolving vyq stage suppose upa reached variable needs compute represents unknown budget remaining possibly cycle presently upa exploits two types reuse improve efficiency loss precision hybrid manner backward reuse previously cached provided sound representation example stage stage scs scs ptf reused ptf true representing set forward reuse previously computed cached upa also fail since therefore upa exploit second type reuse setting course many schemes possible without precision loss evaluation evaluate upa choosing detection uninitialized pointers major client objective show upa effective answering client queries environments small time memory budgets ides facilitating efficiency precision tradeoffs hybrid analysis framework provide evidence demonstrate good correlation number strong updates performed degree precision achieved analysis implementation implemented upa llvm source files program compiled facilitate detection undefined values zhao clang merged using llvm gold plugin link time produce whole program file compiler option applied promote memory registers otherwise supa perform strong updates memory locations would otherwise promoted registers favoring supa undesirably analyses evaluated positive weight cycles arise processing fields struct objects collapsed pearce arrays considered monolithic elements array distinguished distinct allocation sites ddr statements modeled distinct abstract objects build svfg program based software svf sui xue chains andersen algorithm flow order compute soundly precisely information cycle upa retraverses cycle whenever new information discovered fix point reached compare upa analysis implemented sparse flowsensitive sfs analysis described hardekopf lin also llvm sfs recent solution yielding exactly precision good scalability however differences hardekopf lin sfs pointer analysis mented llvm using imprecisely call graphs representing sets binary decision diagrams bdds paper like upa sfs implemented llvm building program call graph fly section representing sets sparse bit vectors implemented fscs pointer analysis llvm implementation either llvm according acharya robinson existing fscs algorithms scale even order magnitude smaller size programs analyzed andersen algorithm shown sfs already spend hours analyzing programs kloc methodology choose uninitialized pointer detection major client named uninit requires strong update analysis effective common type bugs programs uninitialized pointers dangerous dereferencing cause system crashes security vulnerabilities uninit crucial otherwise strong updates impossible making uninit checks futile show upa answer uninit queries efficiently achieving nearly precision sfs global static variables default initialized local variables order mimic default uninitialization stack heap allocation site uninitialized pointer add special store immediately points unknown abstract object uao given load issue query detect potential uninitialization points may uninitialized performing strong updates often analysis find uao reach pointer thus prove pointers initialized note sfs yield false positives since example path correlations modeled introduce uao local variables involved recursion array objects since strongly updated also ignore stack heap objects created calloc generate meaningful queries one query variable load however ignore query found point uao happens points either objects unmodeled local variables recursion cycles arrays number queries issued program listed last column table iii experimental setup use machine intel xeon cpu memory shown table iii selected total programs variety domains spelling checker numeric processing language quantum chromodynamics terminal pager stream editor quantum chromodynamics sequence similarity searching build automation tool postscript filter parser string searching tar archiving file downloading tool unix shell command language game email server client text editor text editor program table iii lists number lines code statements llvm instructions relevant pointer analysis pointers allocation sites addrof statements queries issued discussed section yulei sui jingling xue table iii program characteristics program total kloc statements pointers allocation sites queries results analysis evaluate upa two configurations upa upa upa analysis considering upa analysis consisting fscs applied order evaluating upa assessing upa consider two different criteria efficiency analysis time memory usage per query precision competitiveness sfs query analysis budget denoted represents maximum number chains traversed consider wide range budgets falling upa highly effectively upa nearly precise sfs consuming seconds memory per query average efficiency figure shows average analysis time per query programs given budget seconds seconds axes logarithmic queries take order magnitude long average cases however queries around across programs take much less average cases take emacs example sfs takes two hours seconds finish contrast upa spends less ten minutes seconds average time memory usage seconds produces answers queries sfs shown figure explained upa lightweight shown table vim taking longest seconds also shared sfs order enable sparse analysis additional time taken sfs analyzing program entirely given last column figure shows average memory usage per query different budgets following common practice measure memory usage reading virtual memory information vmsize linux kernel file memory usage time per query secs pointer analysis analysis time budget memory usage budget fig average analysis time memory usage per query consumed upa different analysis budgets axes logarithmic memory monitor starts measure memory usage answering queries average amount memory consumed per query small even largest budget evaluated upa never uses single query processed ell less maed bis gron wgar gnash sen ail emvim fig percentage queried variables proved initialized upa sfs different budgets yulei sui jingling xue table times taken shared upa sfs analysis times sfs seconds program spell milc less sed hmmer make gzip bison grep tar wget bash gnugo sendmail vim emacs times shared upa sfs andersen analysis svfg total analysis time sfs precision given query ptpx initialized uao pointed potentially uninitialized otherwise measure precision upa terms percentage queried variables proved initialized comparing sfs yields best precision achievable analysis figure reports results increases precision upa generally improves upa answer correctly queries programs results indicate analysis highly accurate even tight budgets programs except bison bash upa produces answers queries sfs three programs upa becomes precise sfs taking average seconds seconds bison seconds bash per query understanding strong updates let examine benefits achieved upa answering client queries applying strong updates program figure shows good correlation number strong updates performed left blue curve number uao reaching uninitialized pointers uao right red curve varying budgets logarithmic number uao reported sfs shown lower bound upa dashed line programs upa performs increasingly strong updates block increasingly uao reach queried variables analysis budget increases upa falls back increasingly less frequently precomputed information increases upa filter spurious svfg obtain precise information thereby enabling strong updates kill uao pointer analysis number strong updates spell uao number uao supa budget uao sed budget uao hmmer uao budget make budget uao milc budget less number uao sfs uao budget gzip uao uao budget budget bison uao budget grep uao uao tar uao budget uao uao budget uao gnugo uao budget sendmail budget bash budget wget budget vim uao emacs uao budget budget budget fig correlating number strong updates number uao upa different analysis budgets upa gives answers sfs programs except bison vim causes upa report respectively yulei sui jingling xue programs spell milc hmmer grep strong updates happen small budgets hmmer example strong updates performed queries issued upa runs three queries fully resolved strong updates observed programs like bison bash gnugo emacs quite strong updates take place two main reasons first programs many indirect call edges bison bash gnugo emacs making call graph construction costly section second many cycles chains occurring cycles bison making constraint resolution costly reach fixed point therefore relatively large budgets needed enable strong updates performed interestingly programs gnugo vim fewer strong updates observed larger budgets used vim number strong updates performed drops due forward reuse described section answering query ptpx vyq two budgets upa reached needs compute ptpx case upa may fall back set resulting strong updates performed part program explored evaluating upa programs regarded important achieving useful high precision however important programs terms obtaining precise information enabling strong updates unfortunately analysis scale well large programs considered section table average analysis times consumed uao reported upa budget stage upa budget total program spell milc less sed hmmer make gzip bison grep tar wget bash gnugo sendmail vim emacs upa time uao upa time uao pointer analysis section demonstrate upa exploit contextsensitivity effectively hybrid analysis framework providing improved precision needed programs table compares upa budget divided evenly fscs stages upa budget single stage maximal depth context stack allowed allocating budgets way investigate additional precision benefits achieved considering general upa longer query response times upa due larger budgets used setting times taken handling milc hmmer bison tar gnugo sendmail upa reports fewer uao upa two reasons first upa perform strong updates stack global objects resulting uao reported upa fscs milc second upa perform strong updates singleton heap objects defined section eliminating uao bison tar sendmail reported upa case studies static void symbol first symbol second symbol tmp first first second second tmp static void query location code snippet int char qcdheader hdr char hdr int char qcdheader hdr int char hdr query sscanf int char qcdheader hdr char hdr sscanf int char qcdheader hdr char hdr sscanf code snippet static struct mark getmark int register struct mark static struct mark switch case break case marks lastmark break return query public void gomark int getmark code snippet static struct hol struct hol hol malloc sizeof struct hol obj return hol static void struct hol hol static struct hol struct hol hol argp cluster hol static void hol argp query hol static void struct hol hol strlen code snippet fig selected code snippets examine real code see client queries answered precisely strong updates four different scenarios figure swap bison line second points singleton stack object passed line strong update applied querying location line upa knows points pointed yulei sui jingling xue fig speedups upa parallelized sequential version two four eight threads figure code fragment less ifile initialized two different branches one recognized due strong update performed store line one due default initialization line according upa ifile line initialized figure code fragment milc line point several stack variables named lines upa finds points one singleton context thus strong update performed stack variable becomes properly initialized queried call sscanf figure code fragment tar hol line points heap object allocated line treated singleton requiring context stack least depth strong update performed line initialize field short options properly parallelizing supa demonstrate upa amenable parallelization analysis parallelized upa using intel threading building blocks tbb concurrent queue used store queries issued program use task group allocate tasks computing queries concurrent queue parallel cached information shared concurrent hash map figure shows speedups achieved parallelization sequential setting eight threads average speedup programs maximum speedup observed time setting excludes time programs enjoy better speedups others three main reasons first programs spell less milc relatively queries issued therefore performance benefits achieved query parallelization small second different queries take different times answer resulting different degrees workload imbalance different programs third different programs suffer different synchronization overheads accessing cached information concurrent hash map related work approaches represent two important solutions pointer analysis problems pointer analysis aims resolve pointers program pointer analysis designed resolve typically small subset set pointers pointer analysis manner work concerned developing wholeprogram pointer analysis rather objective design staged strong update analysis framework facilitates efficiency precision tradeoffs according needs client budgets limit discussion work relevant upa pointer analysis strong updates require pointers analyzed respect program execution order pointer analysis studied extensively literature choi emami hendren gave formulations iterative framework kam ullman wilson lam considered representing procedure summaries partial transfer functions restricted strong updates variables eliminate unnecessary propagation information iterative analysis hardekopf lin form sparsity exploited sparse chains program captured sparse evaluation graphs seg choi ramalingam hind pioli various ssa representations hssa chow partial ssa lattner adve ssi ananian tavares chains pointers put ssa explicitly precisely identified giving rise analysis hardekopf lin later idea staged analysis fink leveraged make pointer analysis variables using fast andersen analysis precise analysis hardekopf lin sui paper first exploit sparsity improve performance analysis strong updates performed programs recently balatsouras smaragdakis balatsouras smaragdakis propose modeling technique performing andersen analysis inferring lazily types heap objects order filter redundant field derivations technique exploited obtain precise improve precision efficiency sparse analysis pointer analysis pointer analyses heintze tardieu zhang zheng rugina java shang sridharan yan formulated terms cfl reachability reps heintze tardieu introduced first pointer analysis later zheng rugina performed alias analysis terms indirect function calls handled conservatively sridharan gave two formulations java initially without considering sridharan later sridharan shang yan investigated summarize information discovered analysis improve performance java programs introduced incremental pointer analysis formulation java demonstrated formulation highly amenable parallelization cpus recently feng focused answering demand queries java programs contextsensitive analysis framework without performing strong updates unlike yulei sui jingling xue insensitive analyses effective many clients like uninit upa perform strong updates flow oomerang represents recent pointer analysis java however analysis performs strong updates partially store updating strongly aliases weakly different variables let explain using following java code corresponding code java code new new new code malloc malloc malloc let consider oomerang first strong update performed make point strong update performed make point weak update performed aliases points also result let consider upa enforced strong update performed pointed respectively thus points hybrid pointer analysis basic idea find right balance efficiency precision programs approach das achieves precision steensgaard andersen analyses applying unification process variables case java programs made effective considering together either alone kastrinis smaragdakis guyer lin adjust analysis precision according client needs discussed zhang focus finding effective abstractions analyses written datalog via abstraction refinement chung chung trades precision efficiency performing strong updates singleton objects falls back information otherwise paper propose carry strong update analysis hybrid analysis framework unlike chung upa achieve precision analysis subject given budget parallel pointer analysis introduced parallel implementation andersen analysis based graph rewriting parallel analysis achieving speedup cpu introduces improvement parallel implementation gpus sparse pointer analysis hardekopf lin also parallelized cpus nagaraj govindarajan gpus nasre speedups cpu paper presents first parallel implementation pointer analysis strong updates programs achieving average speedup cpu conclusion introduced upa pointer analysis enables computing precise information programs strong pointer analysis updates refining away imprecisely subject analysis budgets upa handles large programs effectively allowing pointer analyses different efficiency precision tradeoffs applied hybrid analysis framework upa particularly suitable environments small time memory budgets ides evaluated upa choosing uninitialized pointer detection major client programs upa achieve nearly precision analysis small budgets one interesting future work investigate allocate budgets upa stages improve precision achieved answering queries particular client another direction add stages analysis considering example path correlations references acharya robinson practical change impact analysis based static program slicing industrial software systems icse pages ananian static single information form phd thesis master thesis mit andersen program analysis specialization programming language phd thesis diku university copenhagen arzt rasthofer fritz bodden bartel klein traon octeau mcdaniel flowdroid precise context flow field taint analysis android apps pldi pages balatsouras smaragdakis analysis sas blackshear chang sridharan thresher precise refutations heap reachability pldi pages choi burke carini efficient interprocedural computation aliases side effects popl pages choi cytron ferrante automatic construction sparse data flow evaluation graphs popl pages chow chan liu streich effective representation aliases indirect memory operations ssa form pages cytron ferrante rosen wegman zadeck efficiently computing static single assignment form control dependence graph toplas das pointer analysis directional assignments pldi pages emami ghiya hendren interprocedural analysis presence function pointers pldi pages feng wang dillig lin explorer demanddriven exploration interprocedural control flow properties oopsla pages fink yahav dor ramalingam geay effective typestate verification presence aliasing acm tosem guyer lin pointer analysis sas pages hardekopf lin pointer analysis popl pages hardekopf lin pointer analysis millions lines code cgo pages yulei sui jingling xue heintze tardieu pointer analysis pldi pages hind pioli assessing effects pointer alias analyses sas pages international standard programming languages kam ullman monotone data flow analysis frameworks acta informatica kastrinis smaragdakis hybrid analysis pldi pages landi undecidability static analysis acm letters programming languages systems loplas lattner adve llvm compilation framework lifelong program analysis transformation cgo pages chung analysis efficient strong updates popl pages hendren scaling java analysis using spark pages cifuentes keynes boosting performance analysis using value flow fse pages tan sui xue reflection resolution java ecoop pages shang xie xue incremental analysis mathew pingali parallel analysis oopsla pages milanova rountev ryder parameterized object sensitivity analyses java issta milanova rountev ryder parameterized object sensitivity analysis java acm trans softw eng nagaraj govindarajan parallel pointer analysis pact pages nasre analysis taco heo lee lee design implementation sparse global analyses languages pldi pages pearce kelly hankin efficient pointer analysis acm toplas ramalingam undecidability aliasing acm toplas ramalingam sparse evaluation representations theoretical computer science reps horwitz sagiv precise interprocedural dataflow analysis via graph reachability popl pages shang xie xue dynamic analysis cgo pages smaragdakis bravenboer pick contexts well understanding popl pages ali bodden boomerang pointer analysis java ecoop sridharan analysis java pldi pages pointer analysis sridharan gopan analysis java oopsla pages xue parallel pointer analysis icpp pages xue liao efficient gpu implementation pointer analysis ieee trans parallel distrib sui xue sparse pointer analysis multithreaded programs cgo pages acm sui fan zhou xue pointer analysis automatic simd vectorization lctes pages acm sui xue adaptive heap cloning optimizing compilers cgo pages sui xue svf interprocedural static analysis llvm pages sui xue static memory leak detection using analysis issta pages sui xue detecting memory leaks statically analysis tse sui xue zhang making pointer analysis practical compilers using parameterised summarisation spe sun zhao chen probabilistic analysis java pages supa supa http tavares boissinot pereira rastello parameterized construction program representations sparse dataflow analyses pages springer wilson lam efficient pointer analysis programs pldi pages xiao zhang geometric encoding forging high performance context sensitive analysis java issta pages yan rountev alias analysis java issta pages yan sui chen xue automated memory leak fixing valueflow slices programs proceedings annual acm symposium applied computing sac pages new york usa acm sui xue accelerating dynamic detection uses undefined variables static analysis cgo sui xue selective pointer analysis sas pages xue huo feng zhang level level making flowand pointer analysis scalable millions lines code cgo pages zhang xiao zhang yuan efficient subcubic alias analysis pldi pages zhang mangal grigore naik yang abstraction refinement program analyses datalog pldi pages zhao nagarakatte martin zdancewic formalizing llvm intermediate representation verified program transformations popl pages zheng rugina alias analysis popl pages
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cudnn efficient primitives deep learning dec sharan chetlur cliff woolley philippe vandermersch jonathan cohen john tran nvidia santa clara schetlur jwoolley philippev jocohen johntran bryan catanzaro baidu research sunnyvale bcatanzaro evan shelhamer berkeley berkeley shelhamer abstract present library efficient implementations deep learning primitives deep learning workloads computationally intensive optimizing kernels difficult parallel architectures evolve kernels must reoptimized makes maintaining codebases difficult time similar issues long addressed hpc community libraries basic linear algebra subroutines blas however analogous library deep learning without library researchers implementing deep learning workloads parallel processors must create optimize implementations main computational kernels work must repeated new parallel processors emerge address problem created library similar intent blas optimized routines deep learning workloads implementation contains routines gpus although similarly blas library routines could implemented platforms library easy integrate existing frameworks provides optimized performance memory usage example integrating cudnn caffe popular framework convolutional networks improves performance standard model also reducing memory consumption introduction deep neural networks successful solving many kinds tasks parallel processors gpus played significant role practical implementation deep neural networks computations arise training using deep neural networks lend naturally efficient parallel implementations efficiency provided implementations allows researchers explore significantly higher capacity networks training larger datasets led greatly improved accuracy tasks speech recognition image classification among others example majority entries ilsvrc challenge use gpus implement deep neural networks pioneered improved many others including deep neural networks speech recognition also benefited parallel implementations gpus convolutional neural networks cnns important successful class deep networks convolutional neural networks computed using dense kernels differ traditional dense linear algebra routines accordingly modern deep learning frameworks theano caffe feature suites custom kernels implement basic operations tensor convolutions activation functions pooling routines represent bulk computation training cnn thus account majority execution time deep learning community successful finding optimized implementations kernels underlying architectures evolve kernels must significant investment optimizing kernels requires deep understanding underlying processor architecture careful scheduling data movement memory placement register blocking optimizations order get acceptable performance believe providing library optimized routines computations provide several important benefits firstly deep learning frameworks focus issues rather close optimization parallel kernels specific hardware platforms secondly parallel architectures evolve library providers provide performance portability much way blas routines provide performance portability diverse applications diverse hardware thirdly clearer separation concerns allows specialization library providers take advantage deep understanding parallel architectures provide optimal efficiency goal make much easier deep learning frameworks take advantage parallel hardware library addresses goal providing flexible api deep learning workloads integrates neatly existing frameworks provide immediate efficiency gains rigorously tested maintained order reliable performant across range different processor architectures importantly library designed use minimum possible amount auxiliary memory frees scarce memory larger models datasets also optimize performance across wide range potential use cases including small sizes library one primary goals cudnn enable community neural network frameworks benefit equally apis accordingly users cudnn required adopt particular software framework even data layout rather providing layer abstraction provide computational primitives order simplify integration existing deep learning frameworks abstractions bulk api dedicated functions perform primitive operations data stored buffers keeping api library integrates simply frameworks cudnn supports forward backward propagation variants routines single double precision arithmetic include convolution pooling activation functions library allows variable data layout strides well indexing input images also includes set auxiliary tensor transformation routines allow easy manipulation overview handles library exposes language api requires input output data resident gpu analogously cublas library routines called different host threads convolutional routines forward backward passes use common descriptor encapsulates attributes layer tensors filters represented opaque descriptors flexibility specify tensor layout using arbitrary strides along dimension tensors spatial convolutions important computational primitive convolutional neural networks special form batched convolution parameters governing convolution listed table section describe forward form convolution forms necessary backpropagation closely related two inputs convolution chw forms input data rkcrs forms convolutional filters input data ranges images minibatch input feature maps rows per image columns per image filters range parameter pad pad meaning number images number input feature maps height input image width input image number output feature maps height filter kernel width filter kernel vertical stride horizontal stride height width table convolutional parameters output feature maps input feature maps rows per filter columns per filter output also tensor defined previously pad pad meaning height width output images depends image filter height width along padding striding choices striding parameters allow user reduce computational load computing subset output pixels padding parameters allow users specify many rows columns entries appended image matlab valid convolution mode corresponds setting matlab convolution mode corresponds pad pad matlab full convolution mode corresponds pad pad specifically pad define accessing function account striding padding inverting convolution pad pad forward convolution evaluates convenience define version pad pad seen equation computing convolution involves nested loop four independent loops three accumulation loops many ways implementing computation discuss next section cudnn convolutional routines incorporate implementations convolution well variants functions functions support strides along dimension input output tensors important different frameworks store tensors using different memory layouts example frameworks interleave feature maps others keep separate cudnn allows user specify memory layout makes much simpler integrate existing frameworks cudnn routines also mode either return raw gradients accumulate buffer needed models shared parameters directed acyclic graph structure functions cudnn also provides commonly used functions deep learning example provides three commonly used neuron activation functions sigmoid rectified linear hyperbolic tangent provides softmax routine default uses numerically stable approach scaling element avoid overflow intermediate results softmax may computed per image across feature map height width dimensions per spatial location per image across feature map dimension cudnn provides average max pooling operations well set tensor transformation routines add tensors optional broadcasting goal providing functions reduce amount parallel code required deep learning frameworks providing flexible versions commonly used functions cudnn possible write programs train standard convolutional neural networks without writing parallel code simply using cudnn cublas implementation majority functions cudnn provides straightforward implementations convolution implementation obvious outline motivation reasoning behind design choices several ways implement convolutions efficiently goal provide performance close possible matrix multiplication using auxiliary memory gpu memory high bandwidth low capacity therefore scarce resource training deep networks ideally gpu memory filled data parameters neuron responses auxiliary data structures needed convolution algorithm several approaches computing convolutions require large auxiliary data structures therefore consider approaches cudnn one approach lower convolutions matrix multiplication following done reshaping filter tensor matrix dimensions crs gathering data matrix duplicating original input data matrix dimensions crs computation performed single matrix multiply form output matrix dimension image data filter data figure convolution lowering figure illustrates simple convolution lowered matrix multiplication colors illustration represent input feature maps elements uniquely labeled illustration show participates forming filter matrix dimensions crs data matrix dimensions crs note element duplicated times output matrix dimensions lowering convolutions matrix multiplication efficient since matrix multiplication highly optimized matrix multiplication fast high ratio operations per byte data transferred ratio increases matrices get larger meaning matrix multiplication less efficient small matrices accordingly approach convolution effective creates large matrices multiplication mentioned earlier sizes depend products parameters convolution parameters means performance using approach consistent since algorithm care one parameters small long product large enough example often true early layers convolutional network small large end network large small however product crs usually fairly large layers performance consistently good disadvantage approach forming involves duplicating input data times require prohibitively large temporary allocation work around implementations sometimes materialize piece piece example calling matrix multiplication iteratively element however limits parallelism implementation lead cases matrix multiplications small effectively utilize gpu approach also lowers computational intensity convolutions must written read addition reading requiring significantly memory traffic direct approach accordingly opt use implementation directly although explain implementation related another approach use fast fourier transform compute convolution fft significantly lower work complexity convolutions clever engineering used effectively deep neural networks however fft based approach uses significant amount temporary memory since filters must padded size inputs especially costly filters small compared images often happens first layers convolutional network additionally fft based approach perform efficiently striding parameters greater common many art networks early layers striding reduces computational work convolutions factor computing sparse subset output however nature fft algorithm computing pruned ffts task often slower computing dense fft followed additional subsampling step due drawbacks opted forgo fft approach although agree cases useful another common approach compute convolutions directly efficient requires large number specialized implementations handle many corner cases implicit parameter space convolutions implementations following approach often convolutions certain parts parameter space perform poorly others example performs well batch sizes large poorly batch size falls optimizing maintaining specializations difficult task envision library maintained time ported future architectures searched something simpler would perform robustly across parameter space easier port new architectures approach nvidia provides matrix multiplication routine achieves substantial fraction floatingpoint throughput gpus algorithm routine similar algorithm described fixed sized submatrices input matrices successively read memory used compute submatrix output matrix compute tiles fetching next tiles memory caches memories technique hides memory latency associated data transfer allowing matrix multiplication computation limited time takes perform arithmetic discussed earlier convolutions lowered onto matrix multiplication approach provides simplicity implementation well consistency performance across parameter space although materializing lowered matrix memory costly solution follows approach avoid problems materializing lowered matrix memory lazily materializing memory rather materializing memory tflops cudnn caffe minibatch size figure comparative performance layer layer layer layer layer pad pad table convolutional layer collection calling matrix multiplication routine since tiling required matrix multiplication routine independent parameters convolution mapping tile boundaries convolution problem accordingly approach entails computing mapping using load correct elements memories happens dynamically computation proceeds allows convolution algorithm exploit optimized infrastructure matrix multiplication require additional indexing arithmetic compared matrix multiplication fully leverage computational engine matrix multiplication perform work computation complete perform required tensor transposition store result user desired data layout computing additional indexing requires repeated calculations integer division modulus operations constant divisors make use algorithm integer division modulus presented transform costly operations integer multiplies shifts thus reduce indexing overhead required approach performance convolution routines cudnn provide competitive performance zero auxiliary memory required figure shows performance nvidia tesla three convolution implementations cudnn caffe evaluated implementations using layer configurations shown table commonly used benchmarking convolution performance quote average throughput layers cudnn performance ranges advantage smaller batch sizes compared caffe cudnn performance ranges importantly even small size cudnn performance still maximum performance shows implementation performs well across convolution parameter space table illustrates cudnn performance portability across gpu architectures tesla built using kepler architecture peak throughput tflops layer layer layer layer layer tflops tflops table performance portability geforce gtx built using newer maxwell architecture peak throughput tflops cudnn performance ranges peak tesla peak gtx data illustrates cudnn provides performance portability across gpu architectures need users retune code gpu architectures evolve caffe integration caffe deep learning framework developed expression speed modularity mind architecture mirrors natural modularity deep networks compositional models made collection layers deep network defined plaintext schema layer type implemented according simple protocol setup forward backward steps encapsulate engineering details data derivatives flow layers host device according framework unified memory interface handles allocation communication cudnn integration raises speed memory efficiency framework without sacrificing expression modularity development integration made simple design cudnn handles descriptors function calls together modularity framework core caffe framework unaltered preserving network layer memory interfaces changes isolated new layer definitions implementations helper functions descriptors corresponding tests patch almost purely additive caffe type model operation encapsulated layer layer development comprises declaring implementing layer class defining layer protocol buffer model schema extending layer factory including tests computations carried layer protocol setup forward backward steps cudnn layers fit scheme library handles descriptors configured setup forward backward calls made respective layer methods cudnn primitives yield concise layer implementations cudnn layers replacements standard caffe counterparts caffe standard array memory interface called blob storing communicating data host device blobs hold data gradients parameters particular layer inputs outputs held dimensional blobs cudnn tensor filter descriptors trivially constructed blob arrays flexible support dimension stride coordinating memory solely descriptor caffe retains control memory communication efficiency convolution layer exploits reduced memory consumption cudnn speed execution forward pass parallelizes computation group convolution filtering sets filters restricted channel connectivity backward pass parallelizes computation gradients respect bias filter weights bottom data table illustrates performance improvements gained integrating cudnn caffe overall training time iterations improved training bvlc reference caffenet model using cudnn nvidia tesla backward propagation forward propagation testing update overall caffe seconds seconds speedup table caffe performance user experience cudnn computation transparent user integration model schema framework interfaces completely unchanged setting single compilation flag installation equips caffe cudnn layer implementations sets cudnn default computation engine layers automatically fall back standard caffe functionality cases outside current scope cudnn model execution tuned engine parameters select implementation workflow user virtually identical future cudnn releases integrated like fashion baidu integration several deep learning projects baidu integrated cudnn example integrated paddle baidu internal deep learning framework tesla gpu observed performance benchmark set convolutional layers improves average implementation lowers convolution matrix multiply also using cudnn domains besides image processing speech language cudnn ability convolve inputs asymmetric padding particularly useful domains experience cudnn reduced memory consumption compared matrix multiplication enabled use larger models larger mini batches cudnn simple integrate code need change data structure layout thanks cudnn flexible interface future work considering several avenues expanding performance functionality cudnn firstly although convolution routines competitive available implementations work remains bring performance attained matrix multiplication time hope shrink gap secondly envision adding support primitives example convolutions would useful speech language processing video applications among others local receptive field computations similar convolutions untied weights also useful could added cudnn finally would like library help people use multiple gpus accelerate training conclusion paper presents cudnn library deep learning primitives presented novel implementation convolutions provides reliable performance across wide range input sizes takes advantage matrix multiplication routines provide high performance without requiring auxiliary memory also provide set routines allow users train evaluate complete deep neural networks without need write parallel code manually parallel architectures continue evolve libraries provide increasing value machine learning community library available welcome feedback cudnn references https netlib blas http nvidia cudnn gpu accelerated deep learning https yoshua bengio aaron courville pascal vincent unsupervised feature learning deep learning review new perspectives corr james bergstra olivier breuleux bastien pascal lamblin razvan pascanu guillaume desjardins joseph turian david yoshua bengio theano cpu gpu math expression compiler scipy volume page kumar chellapilla sidd puri patrice simard high performance convolutional neural networks document processing workshop frontiers handwriting recognition adam coates brody huval tao wang david bryan catanzaro andrew deep learning cots hpc systems icml pages ronan collobert koray kavukcuoglu farabet environment machine learning biglearn nips workshop george dahl dong deng alex acero deep neural networks speech recognition audio speech language processing ieee transactions geoffrey hinton deng dong george dahl mohamed navdeep jaitly andrew senior vincent vanhoucke patrick nguyen tara sainath deep neural networks acoustic modeling speech recognition shared views four research groups signal processing magazine ieee yangqing jia evan shelhamer jeff donahue sergey karayev jonathan long ross girshick sergio guadarrama trevor darrell caffe convolutional architecture fast feature embedding arxiv preprint alex krizhevsky https alex krizhevsky ilya sutskever geoffrey hinton imagenet classification deep convolutional neural networks nips pages yann lecun bottou yoshua bengio patrick haffner learning applied document recognition proceedings ieee andrew maas awni hannun daniel jurafsky andrew large vocabulary continuous speech recognition using recurrent dnns arxiv preprint michael mathieu mikael henaff yann lecun fast training convolutional networks ffts arxiv preprint olga russakovsky jia deng hao jonathan krause sanjeev satheesh sean zhiheng huang andrej karpathy aditya khosla michael bernstein alexander berg feifei imagenet large scale visual recognition challenge pierre sermanet david eigen xiang zhang mathieu rob fergus yann lecun overfeat integrated recognition localization detection using convolutional networks arxiv preprint christian szegedy wei liu yangqing jia pierre sermanet scott reed dragomir anguelov dumitru erhan vincent vanhoucke andrew rabinovich going deeper convolutions arxiv preprint guangming tan linchuan sean treichler everett phillips yungang bao ninghui sun fast implementation dgemm fermi gpu supercomputing pages new york usa acm henry warren hacker delight professional
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decentralised adaptive primary controllers distributed secondary control microgrids loads daniel stefano laura gordon apr control intelligent systems group school engineering university college cork ireland united technologies research centre ireland ltd floor penrose business centre cork ireland research centre university college cork ireland technical report march abstract loads notoriously known destabilise power systems microgrids due negative incremental impedance paper equips distributed generation units decentralised adaptive controllers primary level microgrid control hierarchy necessary sufficient conditions provided local controllers overall microgrid stability loads connected advantages architecture conventional heuristic approaches scalable design functionality iii well defined performance robustness guarantees heterogeneous uncertain system avoids need online measurements obtain priori system impedance information proposed primary control architecture evaluated distributed secondary level controls using microgrid consists buck boost converters linear loads stability overall hierarchical control system proven using approximation primary level keywords consensus algorithms loads decentralised distributed control islanded microgrid control scalable design voltage stability research supported irish research council enterprise partnership scheme award collaboration university college cork ireland united technologies research centre ireland email danielokeeffe corresponding author email riverss email albioll email introduction increasing complexities associated interconnected systems lsis led decentralised control architectures conventional centralised approaches fact decentralised controllers become control standard distributed autonomous power systems also known microgrids mgs ieee standards outline guidelines providing flexibility pnp features lsis recently become imperative key challenge decentralised controllers lsis stipulate necessary sufficient stability conditions stability compromised connection loads cpls sense cpls reduce system damping individually stable systems introducing impedance typically mgs experience effects cpls tightly regulated motor drives interfaced inadequately damped power converters multiple mgs clustered together conventional criteria stabilising mgs cpls reviewed approaches employ classical frequency domain analysis predicated determining impedance ratio interconnected power converters known gain limited unidirectional power flows unlike approaches requires priori knowledge total load number effective impedance value moreover stability conditions sufficient unstable criteria lead stable unstable system criterion proposed overcome restrictions considering overall system measuring effective impedance though criterion simple practical tuning guidelines local controllers provided ultimately designs heuristic scalable controllers may require retuning complex lsis proliferate become heterogeneous features flexibility robustness uncertainty pnp operations become increasingly vital consequently restrictive approaches may become prohibitive decentralised pnp control architectures proposed primary secondary control levels distributed generation units dgus guarantee overall voltage current stability power converters irrespective system topology facilitate system scalability reconfiguration control design orientated approach adopted using loadconnected model treats loads exogenous disturbance kron reduction methods subsequently used map interconnections general network topologies among works considered cpl stabilisation within pnp framework local model uses approximation cpl neglects content however local stability conditions provided buck converters tests consider cpls moreover accurate knowledge load required thus restricts compliance flexible systems different owners stakeholders due increasing uncertainty within mgs adaptive controllers implemented primary secondary control levels however strategies based primary droop controllers conventional heuristic method sharing control schemes depend specific models topologies provide transient robustness guarantees moreover pnp operations considered recently decentralised robustadaptive control architectures become attractive due guaranteed robustness fast adaptation compared conventional model reference adaptive control mrac architectures developed scalable decentralised adaptive control architecture augment primary voltage controllers dgus robustness guarantees provided presence arbitrary topology pnp operations uncertain couplings unknown load changes however global asymptotic stability gas achieved conservative fashion computation subsequently formulated distributed architecture guarantee gas scalable pnp manner ultimately approaches predicated ability access hardware software order augment retrofit existing baseline controllers applications might available paper extends previous work designing purely adaptive architecture incorporating cpl stabilisation conditions local controller tuning unlike techniques implement stabilisers converter paper follows whereby dgu equipped stabiliser rationale increasing damping reducing loadside control bandwidth impairs quality moreover generators traditional utility grid act stabilise overall system load changes proposed architecture implemented consisting boost buck converters together speed controlled motors though lowfrequency cpl approximation also used filtering feature inherent architectures means content cpl neglected without adversely affecting transient stability furthermore due adaptive nature architecture mapping coupling parameters design model via kron reduction avoided long parameters contained within uncertainty subset finally proposed primary controllers fitted distributed secondary controllers achieve voltage restoration sharing objectives consensus algorithms become popular distributed systems help decentralised controllers achieve centralisedlike performance consensus allows nodes sparse communications network construct vision global system limited information manner perspective secondary controllers primary level modelled approximation lyapunov functions used demonstrate asymptotic stability using approximations system pnp capabilities resiliency communication faults tested version paper submitted ukacc international conference control load model typical topology automotive marine aircraft applications shown fig figure typical microgrid configuration converters fig shows bus formed via converters converter locally fitted voltage current primary controllers primary control layer typical control hierarchy required provide fast stable voltage current performance response load changes reconfiguration section derives impedance model converters acting cpls decomposing converter impedance low high frequency components using extra element theorem frequencies less bandwidth controller tkl large controller works well impedance converter zkin approximates zkn forced impedance response frequencies greater bandwidth zkin follows impedance zkd represented terms local admittance tkl tkl tkl ckl glk assuming unity sensor actuator gain denotes set number cpls overall admittance written zkin pkl vbus ibus yin cpl expressed pkl vbus ibus represent fixed power variable bus voltage current within bandwidth voltage controller works maintain fixed adjusting duty cycle maintain constant vkbus regardless input voltage disturbances bus typically exhibit good power quality converters tuned fast therefore bus voltage power converter input terminals reduces ibus increase manifestation impedance analytically shown cpl vbus ikbus bus vbus ikbus bus bus ikbus completion transfer function input impedance zkd found power converter models microgrids section defines models power converters boost converters increase input voltages buck converters decrease input voltages considered following control design approach loads treated exogenous inputs topology shows converter circuit topology power lines connect converters figure power converter circuit topologies power line connections neighbours converter modelled form aii dgu state vector control input load disturbance represented exogenous input dgu aij represents coupling denotes set dgus denotes neighbour set dgu matrices boost converter defined vdc ltti lti lti aii aij rij cti rij cti cti aii state matrix aij coupling matrix input vector vini idci rli output vector furthermore vdci effective local load resistance converter terminals however since topology treats loads disturbance rli unknown matrices buck converter defined aii aij rij cti rij cti comparing converter models clear dynamics boost dependent duty cycle operating point phase nmp output voltage control nmp action makes controller tuning difficult particularly coupled unknown power lines addressed introduce cpl model exogenous input altered ili icpl represents current disturbance due loads neighbouring dgus battery banks equivalent line resistances currents terminals dgu must mapped original topology using kirchoff voltage current laws bus vbus rij icpl line resistances connect dgus bus fig furthermore bus voltage varies depending line resistances number dgus present vbus represented terms output voltage dgu vdci using substituting vbus rcpl vdci rcpl vbus bus pkl yields quadratic solution vbus vdci pkl alternatively rkcpl known within bounds adaptation effective cpl rating written rcpl cpl vdcj vdc picpl vdci finally ili must linearised operating point icpl yields picpl vdc result state matrix boost converter model becomes ltti aii picpl rij state matrix buck converter model becomes lti aii rij dci picpl vdc ultimately low frequency approximation cpl longer modelled exogenous input directly influences eigenvalues converters introduces greater uncertainty model loads unknown considering flexible heterogeneous mgs decentralised adaptive primary voltage control last decade efforts improve transient performance robustness guarantees conventional mrac architectures fast adaptation led formulation theory control architecture achieves transient performance bounded state control signal guarantees inserting filter lpf input plant appropriate design lpf typical fast adaptation robustness decoupled theory developed uncertainty disturbances unmodelled dynamics unknown input gains centralised architectures notably successful applications including nasa aircraft manned aircraft unmanned vehicles addition decentralised distributed architectures decentralised schemes formulated implemented augment aircraft baseline controllers unlike consider unmatched interconnections provide robustness constant disturbances via integral action recent criticisms theory found reviewed rebuked plant structure plant known structure unknown parameter values control objective design bounded control input tracks reference voltage convergent state bounded parametric errors presence matched uncertainty unmatched coupling disturbances defining new state ref ref ref ref ref vdc vdci remark initially unmatched coupling term neglected enable local decoupled design subsequently term reintroduced gas conditions provided moreover integrator provides adequate robustness unmatched constant disturbances therefore neglected augmented model matched uncertainty term introduced represent parametric uncertainty dynamics represented ref system measurable state vector control signal unknown parametric uncertainty vector belongs known uniformly bounded convex set matrices defined aii control law control input fusion controller low frequency banded uncertainty compensation signal defined laplace domain usf kii kiv control gain vector represents second butterworth represents differential operator robustness dependent lpf bandwidth subsequently designed generates estimate system states thereafter adaptive law used asymptotically drive uncertain plant dynamics converge desired dynamics without loss generality formulation desired dynamics equal dgus ref hurwitz design matrix specifies renders desired dynamics state matrix designer estimates nominal dynamics without uncertainty order design nominal control gains adaptive law adaptive law generates estimate plant uncertainties defining parametric estimation error vectors dynamics used drive adaptive law defined adaptive law determined lyapunov second stability method quadratic lyapunov candidate defined function terms symmetric matrix pit solution algebraic lyapunov linear inequality atii aii arbitrary qti adaptive gain least negative locally butterworth lpf chosen maximally flat order must least equal order plant ensure bounded control input see section stable since energy along trajectories state estimation errors decreases defining adaptive law proj projection operator defined bounds parametric uncertainty estimate invoking barbalat lemma follows lim hence asymptotic convergence boundedness proven filter design introduced lpfs reduce lead instability section defines conditions local stability ensured lpf inserted dynamics presented laplace domain ref initial state signals bounded however due term determine stability conditions upon insertion lpf prescribe performance specifications linear lti reference model defined uncertainty term known available reference system given ref ref ref ref known estimation decoupled control identification local state vector independent control input performance robustness specifications set independent estimation process subsequently section shows converges lti reference model uniform decoupled performance bounds adaptation increases using lti reference model conditions local stability due insertion lpf determined following lemma lemma following theorem lemma reference system stable respect initial conditions reference output proof definition reference system follows ref ref stable transfer functions following bound holds ref ref since uniformly bounded reference states bounded long denominator equal zero defined worst case adaptation bound max determined required gain eigenvalues uncertain maximal deviation plant desired eigenvalues represents boundary projection estimating parameters using adaptation law finally reference states remain bounded following condition must satisfied consequently barbalat lemma used show lim asymptotic tracking maintained inserting lpf designing lpf bandwidth minimise guarantees uniform transient performance bounds remark order neglect component cpl bandwidth smaller natural frequency overall impedance zin design considerations section derives local stability conditions cpls augment dynamics dgu uniform decoupled performance bounds input output signals system lti reference model global stability conditions desired dynamics matrix boost converters defined kiv vdci ltdci lti lti cpl boost rij cti cti cti vdc desired dynamics matrix buck converters defined picpl rij vdc cti cti necessary sufficient conditions stability local subsystems result control gains must satisfy det trace following conditions order preserve local stability cpls conditions boost converters iti rti vdci vdci cpl kiv rti kii vdci iti rij vdci lti conditions buck converters cpl kii cti rij vdci lti cpl kiv rti kii rij vdci convergence adaptive system uncertainty free linear reference model shown subtracting uncertain plant bound represented lemma laplace transform initialisation rewritten bound real lti reference control inputs define ref uref following laplace transform using lemma implies exists arbitrary define strictly proper stable transfer function written laplace domain using becomes uref performance bound written uref uref remark clear relative degree least equal relative degree plant first term strictly proper result unbounded control inputs therefore transient guarantees provided also standard mrac architectures guarantees provided remark inversion results unstable poles systems boost converters output voltage control therefore desired dynamics must carefully designed done transforming local systems form performed finally lim lim shows local plant converges linear reference model performance bounds state control inputs decrease adaptive gain increases conservative global asymptotic stability conditions derived offline using collective lyapunov functions overall lyapunov function candidate describes global system written assumption assume local controllers exploit plant dynamics converged desired dynamics derivative matrix satisfies lyapunov inequality equation diag diag represents overall desired dynamics represents coupling dynamics dgu designed locally asymptotically stable matrices negative definite therefore design performed iteratively offline ensure typically results detuned controller gains expected due conservative requirements decentralised systems furthermore method suffer system size expands retuning becomes difficult despite advantage designing desired dynamics dgus controller gains tuned optimally using lqr guidelines provided conditions depending converter topology satisfied decentralised adaptive primary control architecture seen figure purely adaptive architecture decentralised adaptive controller coordinated secondary control coordination among multiple dgus within imperative dynamic operation system management two key control objectives include voltage restoration sharing reference bus voltage directly controlled since unknown voltage drops occur across unknown secondary control level required compute appropriate voltage references primary control level order maintain dgu voltage within prescribed range reference secondary controllers required also provide voltage reference ensures dgus share power delivery independent topology mathematical definitions follows definition sharing achieved output current dgu equals average iout hil idci buck converters iln idci boost converters local unknown load current vector operator denotes average value vector iout iout ref ref assumption secondary control voltage references identical vbus vbus definition assumption voltage restoration achieved average voltage dgu ref output voltages equals vbus ref hvi vbus vdcn remark consider network communication links consisting set nodes connected edges network described graph node represents dgu network edges represent communication links information exchange laplacian matrix describes graph matrix form defined adjacent deg otherwise deg represents number adjacent nodes connected secondary control level typically implemented slower bandwidth primary level utilises communication network order coordinate manage voltage levels within overall distributed controller design distributed system bus voltage directly controlled instead primary level voltage reference controlled average local voltages equals estimate bus secondary controllers utilised fullyconnected communications lbc network whereby average voltage estimate determined measuring every dgu however clearly places restrictions network result lbc network proposed whereby dynamic consensus algorithms implemented estimate global information limited number nodes dynamic consensus algorithm used estimate bus voltage denoted vdci vdci aij aij aij connected otherwise aij subsequently estimate bus voltage used drive average voltages equal bus voltage reference achieve voltage reference primary level controlled ensure equal current sharing current injected equals average total current injected within consensus algorithm similar also used iout refi iout iout aij aij using remark global consensus algorithm represented vector form depending vector constructed similarly iref depending vector constructed ref finally following controlled correction terms added vbus ref vbus kivi ref vbus iout iout irefi irefi kii equal generated controllers represented laplace domain kivi kiii controllers tuned representing primary level approximation finally voltage reference sent proposed primary control level achieve given ref ref vbus vdc overall distributed hierarchical control structure seen stability analysis section uses lyapunov stability analysis show effect secondary control stability overall system maintained dgu output currents voltages converge satisfied following approximations primary level made subsequently stability proven figure overall hierarchical control architecture decentralised adaptive primary controller distributed secondary controllers approximation approximating primary adaptive control loops ideal unitary gains reasonable one since loop shaped lpf global relationship primary level based secondary control level represented vref bus overall adjustment signals controllers defined vector form kip kii diag kpi diag kii diag kpi diag kii global secondary control error dynamics defined vref bus bus iref lemma symmetric laplacian matrix positive definite proof arbitrary vector always written subspace composed vectors zero average subspace orthogonal equivalent following cases therefore positive definite convenience prove convergence output current loop first isolating currentsharing adjustment term expressed iref pri iref pri diag following lyapunov function candidate considered derivative equates eti kip kii derivative derivative yields therefore written kii eti kip prove convergence barbalat lemma invoked derivative kii akip akii since bounded uniformly continuous lim individual terms positive definite must converge lim result output currents converge reference remark trivial solution given riref case loadsharing result therefore next following lyapunov function considered prove voltage stability convergence derivative equates using similar analysis etv prove asymptotic stability invoke barbalat lemma bounded second derivative etv etv leads uniformly continuous lim similar average output voltage converges bus voltage reference results simulations performed firstly show instability induced negativeincremental impedance two power converters using controllers subsequently proposed architecture evaluated using consisting dgus power electronic loads load instability lightly damped boost converter stepping interfaced tightly regulated buck converter powers resistive load fig highlights family unstable stable eigenvalues effective incremental impedance boost converter load varied shows poorly damped bus voltage response step source dgu voltage imaginary axis load load real axis time effect cpl instability bus voltage family eigenvalues ricpl varies red tightly regulated converter blue bandwidth detuned figure load instability also shows detuning converter controller bus voltage response improves however previously mentioned would cost decreased load performance stable voltage restoration sharing next dgus similar equipped dgu dgu dgu dgu dgu dgu boost converters buck converter dgu provides power bus loaded linear resistive load controlled motor load interfaced bus via buck converters bus voltage stepped system parameters detailed table shown section must least match smallest relative degree plant therefore butterworth lpf designed according frequency sweep rads figure calculation appropriate lpf bandwidth performed bandwidth selected arbitrarily however remark order allow zin approximated resistance bandwidth provided compares bode plot overall load impedance zin overall load impedance overall high frequency content cpl loads zin asymptotically approximates small cpl levels load equal zin asymptotically approximates large cpl levels load equal bode diagram bode diagram phase deg phase deg magnitude magnitude frequency frequency pil pil figure bode plots zin varying pil result conservatively chosen general assumed power levels unknown selected using priori estimate smallest expected load power within overall therefore dgu dgu dgu dgu powered initially dgus ref powers local load bus voltage reference vbus set operation steps input voltage topology change represented following laplacian matrices shows response dgus communications network used shows response sparse connected communications network optimal link redundancy used dgu output voltage dgu output voltages dgu dgu dgu dgu dgu dgu dgu dgu dgu dgu dgu dgu time time communications network sparse communications network figure voltage restoration microgrid bus voltage clear dgus restore output voltages average equals secondary control response adequately fast settling times sparse network compares well network shows maintenance sharing test fullyconnected network sparse network compared dgu dgu dgu dgu dgu dgu dgu output current dgu output currents dgu dgu dgu dgu dgu dgu time time communications network sparse communications network figure sharing maintained dgu dgu dgu dgu dgu dgu dgu dgu dgu dgu dgu average voltage estimate total load current versus number dgus ratio joins network dgu output current injections quickly converge current though performance similar networks settling time particular large overshoot approximately transient response dgu result undershoots amount order satisfy convergence rates topology reconfigures changes plotted time time convergence dgu current estimates sparse communications network figure convergence dgu current voltage estimates resiliency finally system resiliency communication link failures investigated though topology physical system remains topology communication network changes edges fail exchange information dgu laplace changes response change communication topology shown shows system fault tolerant capability long node communication network neighbour information exchanged throughout system either directly indirectly system irrespective communication faults ultimately topology change reflected laplacian matrices affects transient response general less nodes connected slower response larger shoots seen conclusion paper develops novel scalable purely adaptive architecture stabilise dgus cpls architecture draws control orientated model enables scalability treats unknown loads exogenous inputs due adaptive nature architecture topology need mapped loadconnected topology long parameters exist within uncertainty bound destabilising current disturbances induced cpl dgu terminal approximated content due lpf architecture necessary sufficient stability conditions provided boost buck converters using nominally expected dgu dgu dgu dgu dgu dgu dgu output current dgu dgu dgu dgu dgu dgu dgu output voltage time time dgu output current response dgu output voltage response figure output current voltage responses communications links fail load power subsequently smallest priori expected load total used determine upperbound designing bandwidth architecture lpf result fast asymptotic convergence local state estimation errors theoretically proven demonstrated busconnected overall system stability primary level conservative offline approach used proposed adaptive primary controllers cascaded distributed secondary controllers global knowledge average bus voltage load current estimated using limited local information asymptotic stability hierarchical control system shown using lyapunov functions voltage restoration sharing responses shown fast stable pnp communication link failure tests references anuradha massoud ieee vision smart grid controls beyond roadmap ieee css andreasson dimarogonas sandberg johansson distributed controllers multiterminal hvdc transmission systems ieee transactions control network systems vol stoustrup plug play control control technology towards new challenges european journal control vol shafiee dragicevic vasquez guerrero modeling stability analysis active stabilization multiple clusters energycon ieee international energy conference riccobono santi comprehensive review stability criteria power distribution systems ieee transactions industry applications vol dragicevic vasquez guerrero microgrids part review control strategies stabilization techniques ieee transactions power electronics vol riccobono santi novel stability criterion pbsc 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differentiable learning logical rules knowledge base reasoning nov fan yang zhilin yang william cohen school computer science carnegie mellon university zhiliny wcohen abstract study problem learning probabilistic logical rules knowledge base reasoning learning problem difficult requires learning parameters continuous space well structure discrete space propose framework neural logic programming combines parameter structure learning logical rules differentiable model approach inspired differentiable logic called tensorlog inference tasks compiled sequences differentiable operations design neural controller system learns compose operations empirically method outperforms prior work multiple knowledge base benchmark datasets including freebase wikimovies introduction large body work machine learning considered problem learning models composed sets logical rules example rules shown figure logical rules useful representations knowledge base reasoning tasks interpretable provide insight inference results many cases interpretability leads robustness transfer tasks example consider scenario figure new facts companies locations added knowledge base rule hasofficeincountry still usefully accurate without retraining might true methods learn embeddings specific knowledge base entities done transe new york uber cityincountry usa new york country uber usa cityincountry paris cityincountry france paris france figure using logical rules shown box knowledge base reasoning learning collections relational rules type statistical relational learning learning involves proposing new logical rules often called inductive logic programming often underlying logic probabilistic logic markov logic networks proppr advantage using probabilistic logic equipping logical rules probability one better model statistically complex noisy data unfortunately learning problem quite difficult requires learning structure particular sets rules included model parameters confidence associated rule determining conference neural information processing systems nips long beach usa structure discrete optimization problem one involves search potentially large problem space many past learning systems thus used optimization methods interleave moves discrete structure space moves parameter space paper explore alternative approach completely differentiable system learning models defined sets rules allows one use modern programming frameworks optimization methods inductive logic programming task approach inspired differentiable probabilistic logic called tensorlog tensorlog establishes connection inference using rules sparse matrix multiplication enables certain types logical inference tasks compiled sequences differentiable numerical operations matrices however tensorlog limited learning system learns parameters rules order learn parameters structure simultaneously differentiable framework design neural controller system attention mechanism memory learn sequentially compose primitive differentiable operations used tensorlog stage computation controller uses attention softly choose subset tensorlog operations performs operations contents selected memory call approach neural logic programming neural experimentally show neural performs well number tasks improves performance knowledge base completion several benchmark datasets obtains performance recent challenging variant neural also performs well standard benchmark datasets statistical relational learning including datasets biomedicine kinship relationships since good performance many datasets obtained using short rules also evaluate neural synthetic task requires longer rules finally show neural perform well answering partially structured queries query posed partially natural language particular neural also obtains results version iki ovies dataset knowledge base addition show logical rules recovered executing learned controller examples tracking attention summarize contributions paper include following first describe neural knowledge first differentiable approach learning parameters also structure logical rules second experimentally evaluate neural several types knowledge base reasoning tasks illustrating new approach inductive logic programming outperforms prior work third illustrate techniques visualizing neural model logical rules related work structure embedding popular approach reasoning knowledge base approach usually learns embedding maps knowledge base relations cityincountry entities usa tensors vectors latent feature spaces though neural system used similar tasks structure embedding methods quite different structure embedding focuses learning representations relations entities neural learns logical rules addition logical rules learned neural applied entities seen training time achievable structure embedding since reasoning ability relies representations neural differs prior work logical rule learning system differentiable thus enabling gradient based optimization prior work involves discrete search problem space instance kok domingos interleave beam search using discrete operators alter rule set parameter learning via numeric methods rule confidences lao cohen introduce rules restricted set use regression select subset predictive rules wang use iterative structural gradient algorithm alternate search parameters probabilistic logic proppr structural additions suggested parameter gradients recent work neural program induction used attention mechanism softly choose differentiable operators attentions simply approximations binary choices main difference work attentions treated confidences logical rules semantic meanings words neural learns distribution logical rules instead approximation particular rule therefore use hardmax replace softmax inference time framework knowledge base reasoning knowledge bases collections relational data format relation head tail head tail entities relation binary relation entities examples data tuple hasofficeincity new york uber cityincountry usa new york knowledge base reasoning task consider consists entity tail query entity head answer query goal retrieve ranked list entities based query desired answer head ranked high possible reason knowledge base query interested learning weighted logical rules following form similar stochastic logic programs query confidence associated rule relations knowledge base inference given entity score defined sum confidence rules imply query return ranked list entities higher score implies higher ranking tensorlog reasoning next introduce tensorlog operators describe used reasoning given knowledge base let set entities set binary relations map entities integers entity associated encoded vector entry tensorlog defines operator relation concretely matrix entry knowledge base entity similarly draw connection tensorlog operations restricted case logical rule inference using operators described imitate logical rule inference entity performing matrix multiplications words entries vector equals set exists though describe case rule length two straightforward generalize connection rules length using tensorlog operations want learn query shown equation mrk indexes possible rules confidence associated rule ordered list relations particular rule inference given entity score retrieved entity equivalent entries vector shown equation mrk score vyt summarize interested following learning problem query max score max mrk entity pairs satisfy query learned work notion query refers relations differs conventional notion query usually contains relation entity figure neural controller system learning logical rules describe differentiable rule learning process including learnable parameters model architecture shown equation query need learn set rules imply confidences associated rules however difficult formulate differentiable process directly learn parameters structure parameter associated particular rule enumerating rules inherently discrete task overcome difficulty observe different way write equation interchange summation product resulting following formula different parameterization akt mrk max length rules number relations knowledge base key parameterization difference equation equation latter associate relation rule weight combines rule enumeration confidence assignment however parameterization equation sufficiently expressive assumes rules length address limitation equation introduce recurrent formulation similar equation recurrent formulation use auxiliary memory vectors initially memory vector set given entity step described equation model first computes weighted average previous memory vectors using memory attention vector model softly applies tensorlog operators using operator attention vector formulation allows model apply tensorlog operators previous partial inference results instead last step akt mrk finally model computes weighted average memory vectors thus using attention select proper rule length given recurrent formulation learnable parameters query describe neural controller system learn operator memory attention vectors use recurrent neural networks fit recurrent formulation also likely current step attentions dependent previous steps every step network predicts operator memory attention vectors using equation input query special end token update input softmax softmax system performs computation equation stores memory memory holds step partial inference results figure shows overview system final inference result last vector memory discussed equation objective maximize vyt particular maximize log vyt nonlinearity empirically improves optimization performance also observe normalizing memory vectors unit length sometimes improves optimization recover logical rules neural controller system query write rules confidences terms attention vectors based relationship equation equation recover rules following equation keep track coefficients front matrix mrk detailed procedure presented algorithm algorithm recover logical rules attention vectors input attention vectors notation let set partial rules step rule represented pair described equation confidence ordered list relation indexes initialize placeholder storing intermediate results initialize rule update store updated rule initialize rule update akt append add updated rule else return experiments test reasoning ability neural conduct experiments statistical relation learning grid path finding knowledge base completion question answering knowledge base tasks data used experiment divided three files facts train test facts file used knowledge base construct tensorlog operators mrk train test files contain query examples query head tail unlike case learning embeddings require entities train test overlap since system learns rules entity independent system implemented tensorflow trained using gradient methods recurrent neural network used neural controller long memory hidden state dimension optimization algorithm use adam batch size learning rate initially set maximum number training epochs validation sets used early stopping statistical relation learning conduct experiments two benchmark datasets statistical relation learning first dataset unified medical language system umls biomedicine entities biomedical concepts disease antibiotic relations like treats diagnoses second dataset kinship contains kinship relationships among members alyawarra tribe central australia datasets statistics shown table randomly split datasets facts train test files described ratio evaluation metric hits experiment results shown table comparing iterative structural gradient isg neural achieves better performance datasets conjecture mainly optimization strategy used neural isg optimization alternates structure parameter search table datasets statistics umls kinship data relation entity table experiment results indicates maximum rule length isg umls kinship figure accuracy grid path finding neural grid path finding since previous tasks rules learned length three design synthetic task test neural learn longer rules experiment setup includes knowledge base contains location information grid north southeast query randomly generated combining series directions train test examples pairs start end locations generated randomly choosing location grid following queries classify queries four classes based path length hamming distance start end ranging two ten figure shows inference accuracy task learning logical rules using isg neural path length learning difficulty increase results show neural accurately learn rules length task robust isg terms handling longer rules knowledge base completion also conduct experiments canonical knowledge base completion task described task query tail part missing data tuple goal retrieve related head example hasofficeincountry usa uber missing knowledge base goal reason existing data tuples retrieve usa presented query hasofficeincountry uber represent query continuous input neural controller jointly learn embedding lookup table query embedding dimension randomly initialized unit norm vectors knowledge bases experiments wordnet freebase use datasets introduced also considered challenging dataset constructed removing inverse relations use split prior work augment data files reversed data tuples relation add inverse order create use implementation isg available https wang isg compared statistical relational learning methods different experiment setup isg superior several methods including markov logic networks facts file used knowledge base split original train file facts train ratio dataset statistics summarized table table knowledge base completion datasets statistics dataset facts train test relation entity attention vector step default applied relations knowledge base sometimes creates unnecessarily large search space experiment use subset operators query subsets chosen including top relations share common entities query datasets max rule length evaluation metrics use mean reciprocal rank mrr hits mrr computes average reciprocal rank desired entities hits computes percentage many desired entities ranked among top ten following protocol bordes also use filtered rankings compare performance neural several models summarized table table knowledge base completion performance comparison transe neural tensor network results extracted results neural tensor network transe ist ult implicit reasonets neural mrr hits mrr hits mrr hits neural gives results results close noted many relations inverse also defined makes easy learn challenging dataset neural substantially improves performance achieves similar performance ist ult terms mrr note since test entities rarely directly linked knowledge base models need reason explicitly compositions relations logical rules learned neural naturally capture compositions examples rules learned neural shown table number front rule normalized confidence computed dividing maximum confidence rules relation examples see neural successfully combines structure learning parameter learning induce multiple logical rules capture complex structure knowledge base also learn distribute confidences rules demonstrate inductive learning advantage neural conduct experiments training testing use disjoint sets entities create setting first randomly select subset test tuples test set secondly filter train set excluding tuples share entities selected test tuples table shows experiment results inductive setting also make minimal adjustment ensure query relations test appear least train entities train test also facts also ensure entities train directly linked facts table examples logical rules learned neural letters ungrounded logic variables contains contains contains contains contains contains nationality contains nationality contains table inductive knowledge base completion metric hits transe neural expected inductive setting results huge decrease performance transe uses transductive learning approach three datasets hits drops near zero one could expect contrast neural much less affected amount unseen entities achieves performance scale setting emphasizes neural model advantage able transfer unseen entities question answering knowledge base also conduct experiments knowledge reasoning task query partially structured query posed partially natural language example partially structured query would country office given entity instead hasofficeincountry neural handles queries sort naturally since input neural controller vector encode either structured query natural language text use iki ovies dataset miller dataset contains knowledge base pairs question query entity answers sets entities knowledge base train examples test examples knowledge base movie related entities nine relations subset dataset shown table table subset iki ovies dataset knowledge base blade runner ridley scott blade runner philip dick blade runner harrison ford blade runner sean young questions year movie blade runner released writer film blade runner process dataset match input format neural question identity tail entity checking words match entities knowledge base also filter words question keeping top frequent words length question limited six words represent query natural language continuous input neural controller jointly learn embedding lookup table words appearing query query representation computed arithmetic mean embeddings words use implementation transe available https compare neural several embedding based models main difference methods neural embed knowledge base instead learns compose operators defined knowledge base comparison summarized table experiment results extracted miller table performance comparison memory network system model figure visualization learned logical rules accuracy memory network system memory network neural visualize learned model randomly sample questions test dataset compute embeddings question use tsne reduce embeddings two dimensional space plot figure learned logical rules consist one relation knowledge base use different colors indicate different relations label clusters relation experiment results show neural successfully handle queries posed natural language jointly learning word representations well logical rules conclusions present differentiable method learning parameters well structure logical rules knowledge base reasoning method neural inspired recent probabilistic differentiable logic tensorlog empirically neural improves performance several knowledge base reasoning datasets future plan work problems logical rules essential complementary pattern recognition acknowledgments work funded nsf google research references jacob andreas marcus rohrbach trevor darrell dan klein learning compose neural networks question answering proceedings pages kurt bollacker colin evans praveen paritosh tim sturge jamie taylor freebase collaboratively created graph database structuring human knowledge proceedings acm sigmod international conference management data pages acm antoine bordes nicolas usunier alberto jason weston oksana yakhnenko translating embeddings modeling data advances neural information processing systems pages antoine bordes sumit chopra jason weston question answering subgraph embeddings arxiv preprint william cohen tensorlog differentiable deductive database arxiv preprint woodrow denham detection patterns alyawara nonverbal behavior phd thesis university washington lise getoor introduction statistical relational learning mit press alex graves greg wayne malcolm reynolds tim harley ivo danihelka agnieszka sergio colmenarejo edward grefenstette tiago ramalho john agapiou hybrid computing using neural network dynamic external memory nature sepp hochreiter schmidhuber long memory neural computation adam kilgarriff christiane fellbaum wordnet electronic lexical database diederik kingma jimmy adam method stochastic optimization arxiv preprint stanley kok pedro domingos statistical predicate invention proceedings international conference machine learning pages acm lao william cohen relational retrieval using combination random walks machine learning lao tom mitchell william cohen random walk inference learning large scale knowledge base proceedings conference empirical methods natural language processing pages association computational linguistics laurens van der maaten geoffrey hinton visualizing data using journal machine learning research nov alexander miller adam fisch jesse dodge karimi antoine bordes jason weston memory networks directly reading documents arxiv preprint george miller wordnet lexical database english communications acm stephen muggleton ramon otero alireza inductive logic programming volume springer stephen muggleton stochastic logic programs advances inductive logic programming arvind neelakantan quoc ilya sutskever neural programmer inducing latent programs gradient descent arxiv preprint arvind neelakantan quoc martin abadi andrew mccallum dario amodei learning natural language interface neural programmer arxiv preprint matthew richardson pedro domingos markov logic networks machine learning yelong shen huang chang jianfeng gao implicit reasonet modeling structured relationships shared memory arxiv preprint richard socher danqi chen christopher manning andrew reasoning neural tensor networks knowledge base completion advances neural information processing systems pages kristina toutanova danqi chen observed versus latent features knowledge base text inference proceedings workshop continuous vector space models compositionality pages william yang wang kathryn mazaitis william cohen programming personalized pagerank locally groundable probabilistic logic proceedings acm international conference information knowledge management pages acm william yang wang kathryn mazaitis william cohen structure learning via parameter learning cikm jason weston sumit chopra antoine bordes memory networks arxiv preprint bishan yang yih xiaodong jianfeng gao deng embedding entities relations learning inference knowledge bases iclr
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restaurant process mixture model connectivity based parcellation cortex mar daniel moyer boris gutman neda jahanshad paul thompson imaging genetics center university southern california moyerd abstract one primary objectives human brain mapping division cortical surface functionally distinct regions parcellation generally agreed different regions cortex different functions exact number configuration regions known methods discovery regions thus important particularly volume available information grows towards end present parcellation method based bayesian mixture model cortical connectivity keywords human connectome cortical parcellation bayesian nonparametrics introduction historically researchers proposed investigated regional brain parcellations manual dissection qualitative description rise noninvasive coupled advances computing computer vision allowed exploration automated parcellation methods fitting existing atlases data discovery functionally structurally cohesive parcels success former propelled rise interest analyses brain connectomics last decade connectomics topic interest within scales neuroscience often defined discrete networks cortical gray matter regions nodes weighted binary edges connecting structural focus edge weights usually based counts estimated structural connections recovered using diffusion mri tractography sometimes weighted microstructural measure number papers focused converse using connection profile either voxels vertices case functional mri pairwise signal correlation vertex pair define parcellation cortex see references therein connectivity based parcellation cbp methods using structural connectivity two modeling choices required spatial resolution grid connections defined criteria clusters formed almost every existing method connectivity measures first estimated high resolution grid atomic units choice one voxels vertices atomic units combined spatial constraints optimizing desiderata choice two first decision essentially division connectivity two scales level higher resolution level task learn former regions latter second modeling choice criteria atoms clustered many popular choices come general clustering literature sum squares explained variance statistical distances mixture model likelihoods criteria use connectivity profiles atom without regard network structure induce words treat vertex voxel data point associated vector row adjacency matrix cluster based vector space one vertex changed one group another methods generally quality groups though connective profiles would changed present work address choices present method framed context generative models specifically bayesian mixture models place priors possible partitions higher resolution grid require number clusters predefined one large classes priors restaurant processes used implement continuum form connectivity mixture components leverage conjugate relationship produce closed form marginal likelihoods network interactions allowing efficient sampling paper organized follows first define terminology rigorously define parcellation task describe model whole describe components closer detail present results two datasets discuss model relation existing models methods model let white matter interface inner cortical surface acknowledgment general composed two disjoint sheets boundary medial wall fix coordinate system define parcellation set regions regions almost disjoint assume exists latent parcellation accurately describes cortical surface respect underlying neuroanatomical structure objective recovery specifically using structural connectivity information without specifying exact number regions order accomplish construct joint generative model parcellations connectivity start choosing model partitions use distance dependent chinese restaurant process ddcrp variant popular chinese restaurant process crp bayesian models crp models commonly used mixture models providing prior possible bel assignments number pairs main assumption crp exchangeability data ddcrp removes exchangeability assumption allowing topologies dependence data points discussed section detail briefly ddcrp allows use style mixture models mesh grids assume priori neighboring patches dependent example assume spatial discrete manifold mesh practically speaking ddcrp component responsible merging splitting parcels clusters general configuration next choose mixture component model distribution chosen generate observed network estimated regions ddcrp choose follow style infinite relational model model interactions clusters instead profiles clusters thus need separate parameter pair regions diving however important consider form connectivity data structural connectivity estimated using streamlines tractography usually via identifying tracts intersect cortical surface two locations thus evidence connectivity set endpoint pairs traditional connectivity analysis endpoints counted region pair graph formed resulting count statistics representations abstract away knowledge region geometry surface area curvature etc well topological information region adjacency information using ddcrp model possible ignore conflicting motives directly kluge graph spatial patch model instead attempt retain spatial intuition connectivity representation consider set possible tract endpoints intersecting cortical surface model observation pairs endpoints spatial point process assuming tract independently process poisson point process region pair number tract rendpoint pairs observed region pair poisson distributed parameter dydx rate function assumed integrable poisson processes completely characterizes process discuss poisson point process section view overall model important note one convenient property disjoint regions independent counts moving back mixture components make following simplifying assumption form tract endpoint process region pair interacts homogenous manner assume constant pair parcels thus finite configuration parcels order scalar parameters estimate parameters rate parameters poisson spatial processes generating evidence connectivity choose use gamma distribution model parameters pair parcels draws rate gamma prior shown important make distinction physical fascicles recovered tracts define latter reconstructed tractography section conjugacy gamma distribution homogenous poisson process allows closed form marginal distributions thus efficient collapsed sampling methods also choose use mesh faces elements connectivity usually defined intersections tracts areal units ddcrp well poisson process naturally operate regions putting together leaving meaning next section model follows ddcrp adj gamma dij poisson point process distance dependent chinese restaurant process ddcrp suggested name distance directed chinese restaurant process variant chinese restaurant process crp often used bayesian mixture models prior possible mixture components distribution distributions let positive constant concentration parameter let prior distribution mixture component parameters gamma distribution original crp mixture model describes endless stream customers data entering restaurant infinite number tables clusters customer either chooses prescribed probability dependent sit existing table particular component distribution sit unoccupied table draw new component distribution prior indexing tables finite number observations number clusters configuration cluster associations possible original crp data assumed exchangeable joint likelihood observations invariant permutations observation indices however spatial context topology face adjacencies permutations face indexes thus faces exchangeable model ddcrp allows customer choose another customer possibly sit based dependencies forms directed graph seating choices table assignments made group customers chosen sit connected component seating choice graph mixture components drawn table actual data drawn mixture component clearly two stage procedure context means face choose cluster one neighbors let set mesh faces let corresponding assignments draw conditioned adjacency information adj follows adjacent otherwise denote cluster faces number clusters due restriction indices faces adjacent contiguous region set groups forms valid parcellation note original ddcrp defined general distance functions mixture components evidence pairwise interaction regions structural connectivity set tract endpoints since regional clusters defined discrete grids areal atoms mesh faces naturally aggregated count measures pair pair regions define dij model counts using poisson process fixed intensity area contains random count distributed oisson using independence assumption tract endpoints likelihood configuration tract endpoints written exp dxdy use gamma prior parameters conjugate prior poisson distribution using gamma distribution allows derive simple closed form marginal distribution dij integrates leaving likelihood terms prior parameters follows dij dij dij exp exp homogeneous point process exp gamma prior gamma posterior combined model collapsed sampling scheme estimate model via collapsed gibbs sampling specifically using closed form integral avoid sampling interaction parameters starting iteration update following conditional likelihood dij since assume restricted mesh topology small number options evaluate denote seating graph edge critical edge node exists path face index let gold previous component gcrit component without edge critical component without loss generality may order neighbor groups using definitions triangle meshes write possible scenarios critical neighbors component update simply choose via difference respect induced components critical surrounded component asking essentially previously induced component join one yet independent neighbors thus gold neighbor gold critical neighboring component including component gold neighbors gcrit iteratively update face associates using collecting samples every pass generates posterior distribution simply take maximum posteriori map estimate selected parcellation general updates made gibbs sampling algorithms done sequentially strong dependencies concurrent updates destabilize samplers however cases low dependence approximate asynchronous parallel updates used empirically strong results called hogwild updates case updates either within components small components correspondingly small interdependencies small degree parallelism possible practice use compromise serial algorithm parallel version use shared memory parallel sampler calculating likelihoods small batch make serial updates based likelihoods allows roughly linear number threads used though slowly scaling cost serial update implementation notes fixing coordinate system common split white matter interface two spheres null region corpus callosum bridges longitudinal fissure thus easy system constructed using spherical coordinates marker hemisphere symmetry tract endpoints requires careful consideration avoid double counting intuition model understood without thinking symmetry data evaluating joint probabilities important include data point achieved evaluating dij computing parallel updates experience much efficient keep threads active idle simply single thread serial update avoids overhead repeated thread spawns costly procedure results order test proposed model use two open datasets one composed subjects scanned twice institute psychology chinese academy sciences ipcas subset consortium reliability reproducibility corr dataset composed subjects human connectome project hcp release differs slightly datasets account different imaging parameters general hcp dataset higher resolution voxel size angular resolution leading different tractographies dataset compare performance proposed method two recommended alternatives ward method greedy hierarchical clustering method spectral clustering preprocessing tractography ipcas diffusion weighted dwi images obtained siemens triotim original investigators using head coil directions subject scanned twice roughly two weeks apart images processed freesufer pipeline obtain triangle mesh matter boundary registered shared spherical space resample space geodesic grid face approximately equal area total faces computing tract intersections surface probabilistic streamline tractography conducted using dwi isotropic mni space using dipy implementation constrained spherical deconvolution csd harmonic order tractography streamlines seeded random locations white matter voxel labeled fsl fast streamline tracking followed directions randomly proportion orientation function sample point steps starting bidirectionaly seed point restarts per seed per dipy anatomically constrained tractography act retained tracts longer endpoints likely gray matter hcp used minimally preprocessed diffusion weighted dwi images rigidly aligned mni space briefly preprocessing images included motion correction eddy current correction dwi linear nonlinear alignment betweek dwi used hcp pipeline version freesurfer protocol run optimized version pipeline computes surface meshes higher resolution isotropic space resample space geodesic grid computing tract intersections surface tractography conducted using dwi native isotropic voxel size mni space probabilistic streamline tractography performed ipcas fitting results fit proposed method using parallel sampling scheme using passes sampler parallel threads approximately updates per subject using use map estimate results fitted ward clustering maximizing explained variance search every possible merge spectral clustering method use exponential kernel using normalized cosine distance metric use number eigenvectors equal number clusters baselines take vector connections feature vector clustering schemes specify number clusters equal proposed method assess cluster quality using based measure take number tracts face region objective distribution measure well approximated average number tracts form two matrices dimension normalize matrices sum one measure divergence cluster well represented average connectivity profile divergence low ipcas dataset additional measure reproducibility measured normalized mutual information nmi measures cluster similarity without requiring similar numbers clusters let binary matrix cluster assignments fig plots divergence based goodness fit measure three methods datasets lower better row entry zij cluster zero otherwise nmi defined mutual information entropy cluster assignments first second scan respectively uses convention customary information theory log nmi also invariant permutations labels seen figure figure proposed method performing well compared baseline methods hcp dataset uses clusters around per hemisphere ipcas dataset around per hemisphere difference may due part higher resolution hcp dataset leading greater resolving power respect regional connections averages upper range number suggested van essen discussion model draws wide range previously proposed methods connectivity based parcellation several bayesian methods proposed particular two excellent works jbabdi baldassano use conjugations mixture components baldassano use special case models also enjoy closed form marginal distributions infinite divisibility distributions model spatial processes jbabdi whose work predates ddcrp use dirichlet process spatial priors partition prior define hierarchical process top links multiple subjects baldassano use ddcrp directly model voxel connections without aid spatial process instead model aggregate connectivity coming normal distribution fig left normalized mutual information scans higher better right histograms number clusters selected subject ddcrp similar markov random field model strong spatial prior models successful obtaining parcellations functional connectivity though used bayesian frame reference leads toward traditional computer vision tasks pixel labeling many cases surface parcellation framed vertex parcellation small relatively important conceptual difference pixel models areal units vertex parcellations graphs infinitesimal points intuition former leads toward use spatial processes similar spatial process viewpoint connectivity proposed moyer discovery new parcellations discussed poisson count processes network interactions also explored literature infinite relational variants though usually context network clustering via stochastic blockmodel clustering regions usually ignore spatial constraints alternative methods bayesian models usually specify number clusters note parisot subsequent work authors propose spectral methods parcellation task augmented local agglomeration papers note propensity spectral clustering form clusters seen figure method form groups thus may case lower number clusters spectral clustering may perform better rich body functional anatomical knowledge regarding cortex parcellations based connectivity information alone would need proper neuroanatomical histological functional validation information sources would ideally used optimize parcellations model presented uses spatial constraints connectivity estimate fig exemplar parcellation hcp subject region colors random feasible parcellations based recoverable structural connections imaging however believe modeling techniques explored easily imputed larger models general improvements made may increase accuracy reproducibility studies connectivity patterns critical furthering understanding living human brain acknowledgements work supported nih grant well nsf graduate research fellowship program authors would like thank reviewers well greg ver steeg multiple helpful conversations references baldassano beck parcellating connectivity spatial maps peerj blei frazier distance dependent chinese restaurant processes journal machine learning research aug clarkson malone modat leung ryan alexander fox ourselin framework using diffusion weighted imaging improve cortical parcellation international conference medical image computing intervention springer eickhoff thirion varoquaux bzdok parcellation critique implications human brain mapping fischl freesurfer neuroimage fischl intersubject averaging coordinate system cortical surface human brain mapping garyfallidis dipy library analysis diffusion mri data front neuroinform hinne probabilistic clustering human connectome identifies communities hubs plos one honnorat grasp geodesic segmentation shape priors functional parcellation cortex neuroimage jbabdi woolrich behrens parcellation using hierarchical dirichlet process mixture models neuroimage johnson analyzing hogwild parallel gaussian gibbs sampling advances neural information processing systems kemp learning systems concepts infinite relational model moyer mixed membership stochastic blockmodels human connectome bayesian graphical models biomedical imaging moyer continuous model cortical connectivity international conference medical image computing intervention springer parisot groupwise parcellation cortex international conference information processing medical imaging springer parisot parcellation cortex spectral clustering neuroimage pitman combinatorial stochastic processes ryali parcellation scheme based von distributions markov random fields segmenting brain regions using fmri neuroimage smith tractography improved diffusion mri streamlines tractography effective use anatomical information neuroimage sporns tononi human connectome structural description human brain plos comput biol tournier resolving crossing fibres using constrained spherical deconvolution validation using imaging phantom data neuroimage van essen consortium human connectome project overview neuroimage van essen glasser dierker harwell coalson parcellations hemispheric asymmetries human cerebral cortex analyzed surfacebased atlases cerebral cortex yeo organization human cerebral cortex estimated intrinsic functional connectivity journal neurophysiology zilles amunts centenary brodmann fate nature reviews neuroscience zuo open science resource establishing reliability reproducibility functional connectomics scientific data
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limited communication analysis design decentralized estimation jan andreea alexandru pequito paper pertains analysis design decentralized estimation schemes make use limited communication briefly schemes equip sensors scalar states iteratively merge measurements state sensors used state estimation contrarily commonly used distributed estimation schemes information exchanged scalars one common communication estimation retrieval state system sensors achieved extend previous work general setup provide necessary sufficient conditions required communication sensors enable use limited communication decentralized estimation schemes additionally discuss cases sensors memoryless sensors might capacity discern contributions sensors based conditions fact communication channels incur cost cast problem finding minimum cost communication graph enables limited communication decentralized estimation schemes integer programming problem ntroduction sensors often geographically deployed collect measurements networked dynamical systems used state estimators implementing state observers retrieve estimate overall state system estimate provided actuator implements controller steer dynamical system desired state estimators reduced extended state observers implicitly explore communication estimation context distributed estimation additional information used state estimators shared improve quality estimate example share either estimate error predicted state observation measurement innovation used part state estimation process information shared resorting communication different sensors computational units communication capabilities impact ability retrieve estimate state therefore fundamental understand communication required ensure successful recovery system state observers implemented systems require large amount information exchanged work supported part terraswarm research center one six centers supported starnet phase focus center research program fcrp semiconductor research corporation program sponsored marco darpa department electrical systems engineering school engineering applied science university pennsylvania department civil environmental engineering institute data systems society massachusetts institute technology ali jadbabaie george pappas communication state error measurement predicted state observation furthermore estimators shown asymptotically stable always arbitrary error decay might restrict actuation performance context networked dynamical systems overcome limitations proposed approach equips sensors scalar states exchanged sensors together sensors measurements suffice retrieve state networked dynamical system sensors refer limited communication decentralized estimation scheme subsequently information exchanged sensors reduced bare minimum communication topologies analyzed designed ensure state recovery paper seek better understand restrictions limited communication decentralized estimation scheme specifically main contributions paper waive implicit assumptions made communication scheme performed sensors general sufficient emphasize remark explore implications two different setups sensors memoryless keep track previous state sensors might capacity discern contributions state sensors relying radio technology iii leverage new conditions cast problem determining minimum communication cost required deploy limited communication decentralized estimation scheme integer programming problem roblem tatement let evolution possibly networked dynamical system captured state system consider sensors measurements described follows output vector describing contributions different observed state variables assume observable necessarily observable specific sensor necessarily observable limited communication decentralized estimation scheme consider sensors possess scalar state communicate process share states enables retrieval state networked dynamical system sensors communication capabilities captured directed communication graph set vertices labels sensors edge translates capability sensor receive data sensor besides sensor computes linear combination scalar measurement scalar data received neighboring sensors described follows wij indices sensor given communication graph subsequently write using following compact representation augmented system state dynamics sensors induced communication graph wij zero otherwise worth noticing weights wij may set zero particular wii dealing memoryless sensors work relays addressed setup explored additionally augmented system output described follows containing rows identity matrix indices particular linear combination incoming sensor states performed locally sensor sensors capacity discern contributions state sensors relying radio technology latter case addressed setup explored paper seek solutions following problems problem characterize necessary sufficient conditions must satisfied subsequently ensuring observable particular provide characterization required memoryless sensor scenario case sensor access state next propose determine communication topologies ensure necessary sufficient conditions required solve previous problem minimizing communication cost different sensors problem let communication cost incurred establishing communication link sensors obtain communication graph aim determine solves following optimization problem satisfies conditions problem min iii erminology revious esults follows rely structural systems theory assess system theoretical properties considering states sensors one system property structural observability considers sparsity binary patterns entry matrices zero direct dependency two state sensor variables one otherwise pair structurally observable exists observable pair zero entries also zero subsequently proved observable pair exists almost possible pairs satisfying sparsity pattern also observable furthermore structural properties structural observability necessary ensure properties observability therefore section rely structural systems ensure first necessary conditions show fact also sufficient one key features structural systems theory interpret sparsity patterns directed state graph graph vertices labeled states sensors edges capture state sensor variables follows use brevity additionally use notions paths cycles address structural properties particular characterize structural observability introduce notion bipartite graph associated state graph graph state bipartite graph bipartite graph consists two sets graphically interpreted refer left right set vertices edges left right set vertices encode dependencies described directed state graph directed graph also due correspondence edges paths cycles graph captured subsets edges state bipartite graph referred matchings subset largest number edges referred maximum matching consequently left right vertices state bipartite graph belong edge matching referred vertices accordingly following result lemma consider digraph let maximum matching associated bipartite graph digraph comprises disjoint union cycles elementary paths vertices vertices span moreover decomposition minimal sense spanning subgraph decomposition elementary paths cycles contains strictly fewer elementary paths different concepts come together assess structural observability follows theorem let denote digraph bipartite representation pair structurally observable following two conditions hold path every state vertex output vertex exists maximum matching associated vertices therefore previously mentioned build upon results analyze design limited communication decentralized estimation scheme provided necessary sufficient conditions needs satisfy ensure observability following simplifying implicit assumption implicit assumption sensor retains previous state always weighted sensor dynamics wii implies diag words excludes case memoryless sensors address paper also explore setups sensors able differentiate individual contributions sensors due technology used next state two main results ease comparison main results attained work waive implicit assumption stated theorem let state digraph corresponds labels state vertices labels sensors states addition let set vertex representing sensor following two conditions necessary sufficient ensure generically observable every must exist directed path every must exist set vertices associated maximum matching bipartite representation hence theorem used obtain next result problem implicit assumption stated theorem observable structurally observable almost realizations ensure observable brevity sake use shortened notation rest paper imited ommunication nalysis esign section introduce main results paper lemma shows sensor capabilities impose strong constraints network structure required ensure structural observability technical result plays key role understanding theorem states necessary sufficient conditions required address problem specifically provides conditions communication graph observable next consider design communication graphs attain former conditions minimizing total cost incurred communication sensors problem particular cast problem integer programming problem solved solvers start showing structural observability pair often assessed graph properties captured theorem enforce particular structure certain sensing capabilities lemma let canonical one position remaining entries equal zero given structured adjacency matrix graph pairs structurally observable associated state digraph strongly connected spanned disjoint union cycles proof necessity assume structurally observable suppose contradiction strongly connected strongly connected digraph also strongly connected directed acyclic representation contains number strongly connected components since output vertex one vertex connected vertex readily follows condition theorem hold next prove spanned disjoint union cycles consider condition theorem exists maximum matching vertex two possibilities maximum matchings case means perfect matching lemma spanned disjoint union cycles want prove case happen structurally observable let first address case let represent set vertices graph described consider maximum matching vertices since strongly connected exists neighbor let another maximum matching vertex lemma exists maximum matching however also maximum matching fact perfect matching leads contradiction fact maximum matching therefore set vertices empty meaning maximum matching also perfect matching leading fact spanned disjoint union cycles case iteratively find augmented paths construct larger cardinality maximum matchings thus reducing cardinality set vertices respect matchings sufficiency assume strongly connected spanned disjoint union cycles follows also strongly connected condition theorem satisfied since spanned disjoint union cycles lemma exists perfect matching also maximum matching implies condition remark proof lemma technical challenge show state vertices bipartite graph always matched edge whose sensor vertex state graph strongly connected implies states need always matched edges whose state vertices therefore leveraging lemma follows state graph spanned cycles esults present solution former problem need review notion linking see instance vertices state digraph vertices communication digraph graph associated augmented system specifically linking set simple paths vertices vertices additionally sensor denote linking starting ending vertices simple paths belong communication graph set sensor vertices belong edges maximum matching subset sensor equal cardinality particular many sensor vertices vertices maximum matching associated due observability structural observability prescribed theorem consequently solution problem formally stated follows theorem consider system observable pair observable strongly connected exists linking spanned disjoint union cycles proof necessity enough show one condition theorem theorem hold spanned disjoint union cycles strongly connected therefore assume structurally observable strongly connected proof follows along lines proof necessity lemma since condition theorem fails strongly connected assume structurally observable spanned disjoint union cycles using second part proof lemma corresponding strongly connected components proves contradiction condition theorem spanned disjoint union cycles sufficiency assume spanned disjoint union cycles strongly connected order prove sufficiency follow similar steps proof show conditions satisfied general necessity sufficiency condition sufficiency condition theorem follow since assumed strongly connected original system also structurally observable theorem know exists maximum matching associated every vertex distinct sensor associated expanding maximum matching augmented system bipartite state graph match vertices distinct sensor measuring yields possible vertices hence sensors path previously unmatched state vertex spanned disjoint cycles assumption either vertices already sensor following similar procedure proof lemma find another maximum matching next show exists realization ensures observability leverage proof theorem specifically invoke criterion assess observability system result must following equalities hold rank rank imi structure allow arbitrary placing eigenvalues opposed proof theorem however able prove eigenvalues place affect rank eigenvalues associated cycles arbitrarily placed cycle composed edges weights wrj number cycles associated eigenvalues values wrj eigenvalues placed associated paths zero analysis holds zero eigenvalues associated paths spanned cycles specifically vertices paths exactly vertices respect maximum matching digraph order match vertices paths extended links described sensors states communication graph let number vertices minimum length paths spanned cycles corresponding zero eigenvalues suitable permutations separate blocks associated disjoint cycles denoted symbolically blocks associated linkings respectively eigenvalues chosen different eigenvalues different zero hence eigenvalues moreover linkings measured outputs given imi therefore pair imi observable popov criterion rank imi completes proof rank conditions theorem sufficient remark theorem accounts scenarios sensors memoryless retain previous state integrate overall dynamics extends results revisited section iii specifically case sensors readily memoryless leads case communication graph strongly connected subgraph spanned disjoint union cycles since access sensor state incorporation overall dynamics corresponds communication graph elementary cycle remark context limited communication decentralized estimation schemes employ sensors discern contributions coming neighbors due technology used follows rank specifically one possible ending sensor vertex implying state bipartite graph maximum matching one vertex finally given necessary sufficient conditions communication graph provided theorem aim formulate problem designing minimum cost communication graph stated problem integer programming problem end leverage problems minimum cost maximum matching problem minimum cost spanning trees also need formulate conditions communication graph require encode minimum cardinality linkings better visualize results write communication graph represents set sensor states set communication links sensors let communication cost incurred establishing link sensor sensor want obtain communication graph dealing memoryless sensors prescribe obtain finite cost graph feasible solution briefly constraints satisfied described following algorithm steps addressed simultaneously every sensor find number vertices find minimum cost linking run minimum cost maximum matching algorithm select edges compose add minimum cost edges graph strongly connected leverage insights provided heuristic algorithm provided obtain integer programming problem formulation without explicitly computing sensor add virtual output part sensing technology role intermediate step according imi denote set vertices next let bipartite graph system virtual outputs associated digraph expand cost structure follows assign weights sensor edges communication graph setup ensures minimum cost maximum matching algorithm former bipartite graph return matching partitions digraph paths cycles incurring minimum cost specifically paths contain described rest digraph spanned disjoint cycles furthermore notice since edges communication digraph state digraph cycle spanning vertices vertices state integer programming formulation let denote binary variable associated existence edge set cutset represents subset edges start vertex end vertex given set edges set vertices hence constraints given matching problem digraph strong connectivity communication graph imposed via rooted minimum spanning tree vertex brevity denote thus obtain following formulation problem min otherwise edges minimum cost communication graph retrieved design problem proposed problem since contains particular instance design problem addressed hence straightforward greedy algorithm implemented sequentially performing steps described nonetheless solution depend initial point since iteration previous selected edges set one guarantee final configuration indeed minimum cost consequently leverage integer fig plant state nodes black sensors deployed blue interconnections state nodes depicted black measurement terminals state nodes sensors depicted red programming formulation also known general solve resorting highly optimized software toolboxes yalmip efficiently deployed practice dealing complex problems llustrative example consider example figure associate following cost matrix communication links minimum topology communication graph decentralized observability sensors ensured depicted figure since two leftunmatched vertices sensor least two illustrate decentralized observability sensor achieved need find linking spanned disjoint union cycles case pick maximum matching composed edges linking matched sensors sensor chosen obtain trivially spanned union disjoint cycles vii onclusions paper extended limited communication decentralized estimation schemes cope general scenarios provided necessary sufficient conditions communication graph retrieval state system sensors possible particular present extension enables deployment limited communication decentralized estimation schemes scenarios sensors memoryless sensors capacity discern contributions state sensors furthermore cast design problem communication costs problem determining minimum cost communication graph required implement limited communication decentralized estimation scheme integer programming problem formulation enables use software toolboxes reliable practice dealing complex problems fig minimum communication topology ensure decentralized observability plant figure future research focus proposing communication protocols sensors subject constrained energy budgets towards goal important understand communication information contained sensors states particular aim quantify classify role sensors state dimension estimation process accuracy key adding communication infrastructure prohibitive findings suggest number dimensions exchanged states increases fewer communication links required guarantee decentralized observability moreover one design dimension sensors memory additional links necessary eferences gupta distributed estimation control networked systems dissertation california institute technology huang werner huang kashyap gupta state estimation electric power grids meeting new challenges presented requirements future grid ieee signal processing magazine vol kar large scale networked dynamical systems distributed inference dissertation carnegie mellon university khan consensus networks theory applications dissertation carnegie mellon university wang morse distributed observer linear system american control conference acc ieee mitra sundaram distributed observers lti systems arxiv preprint garin schenato survey distributed estimation control applications using linear consensus algorithms networked control systems springer das moura distributed kalman filtering dynamic observations consensus ieee transactions signal processing vol subbotin distributed decentralized estimation dissertation university california santa barbara khan jadbabaie coordinated networked estimation strategies using structured systems theory conference decision control european control conference ieee khan kar jadbabaie moura connectivity observability stability distributed estimation ieee conference decision control ieee alexandru pequito jadbabaie pappas decentralized observability limited communication sensors conference decision control ieee dion commault van der woude generic properties control linear structured systems survey automatica vol lin structural controllability ieee transactions automatic control vol shields pearson structural controllability linear systems ieee transactions automatic control vol pequito kar aguiar framework structural control configuration selection systems ieee transactions automatic control vol diestel graph theory ser graduate texts mathematics springer coates solutions linear algebraic equations ire transactions circuit theory vol reinschke multivariable control graph theoretic approach nemhauser wolsey integer programming combinatorial optimization springer vol lofberg yalmip toolbox modeling optimization matlab international symposium computer aided control systems design ieee
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interplay homological dimensions complex right derived section jul cyrus jalali abstract let commutative noetherian local ring proper ideal prove respectively idr idr respectively fdr fdr next proved right derived section functor complex necessarily local computed via genuine complex gorenstein injective modules show dualizing complex gfdr gfdr gidr gidr also show relative respect respectively dimension gfdr hahtm gfdr respectively gidr gidr results generalize known results provide characterizations gorenstein rings introduction throughout paper commutative noetherian ring proper ideal category denoted use subscripts denote genuine boundedness conditions full subcategory bounded complexes see definition also derived category denoted use subscripts denote homological boundedness conditions see definition symbol sign isomorphism quasiisomorphisms also use superscript signify homology modules degreewise finitely generated semiinjective functor homr converts injective quasiisomorphisms surjective quasiisomorphisms semiinjective resolution semiinjective complex quasiisomorphism injective dimension idr defined idr inf several main results paper involve hypothesis dualizing complex complex dualizing finite injective dimension canonical morphism rhomr isomorphism dualizing complex may consider functor rhomr notion gorenstein injective module introduced enochs jenda said gorenstein injective exists homr exact acyclic complex injective ker gorenstein injective dimension gidr defined infimum set integers exists complex consisting gorenstein injective modules satisfying also mathematics subject classification key words phrases flat dimension injective dimension gorenstein injective dimension derived local cohomology jalali semiflat functor preserves injective quasiisomorphisms flat dimension fdr defined semif lat fdr inf said gorenstein flat exists exact acyclic complex flat ker gorenstein flat dimension gfdr defined infimum set integers exists complex consisting gorenstein flat modules satisfying let right derived section functor complex defined semiinjective resolution see generating set ideal corresponding complex see theorem shown theorem right derived section functor support ideal sends complexes finite flat dimension respectively finite injective dimension complexes finite flat dimension respectively finite injective dimension section prove local ring idr idr see theorem shows following statements equivalent gorenstein idr dim ideal iii idr ideal provides characterization gorenstein rings recovers corollary next prove local ring idr idr notice result generalization theorem also flat version demonstrated indeed shown theorem local ring fdr fdr proved theorem gfdr gfdr moreover dualizing complex implication may reversed jacobson radical show local ring gfdr gfdr gfdr admits dualizing complex gfdr gfdr corollary prove relative respect defined grade gfdr hna gfdr section first prove see theorem implies right derived section functor complex computed via genuine complex gorenstein injective modules also main result show local ring admitting dualizing complex gidr gidr see theorem shows following statements equivalent gorenstein gorenstein injective gorenstein flat dimensions gidr dim ideal iii gidr ideal provides characterization gorenstein rings improves corollary theorem may prove without assuming cohenmacaulay next theorem prove complex version improves theorem corollary deduce gidr hnm gidr wherever local ring dimr right derived section functor injective dimension gorenstein flat dimension following lemma immediate consequence corollary proposition determines bass number see definition lemma let local ring let particular every equality following theorem one main results section provides comparison injective dimensions complex right derived section functor theorem let local ring let idr idr proof let idr view lemma lemma spec therefore follows lemma idr opposite inequality let idr theorem infrhomr cyclic hence view proposition isomorphisms rhomr rhomr therefore idr theorem following corollary recovers corollary immediate consequence previous theorem corollary let local ring following statements equivalent gorenstein idr dim ideal iii idr ideal following theorem immediate consequence theorem generalization theorem theorem let local ring suppose idr idr jalali proof let since equality idr idr equivalent module category rmodules may identify hna hence idr hna idr desired equality follows theorem following use notion semifree resolution semifree resolution semifree complex see definition quasiisomorphism lemma let local ring let proof let semifree resolution residue field let generating set ideal complex respect proof lemma exists quasiisomorphism result follows since following lemma immediate consequence corollary determines betti number see definition lemma let local ring let particular every equality theorem let local ring let proper ideal suppose fdr fdr proof let fdr view lemma lemma spec therefore follows lemma fdr opposite inequality let fdr theorem supt cyclic hence view lemma isomorphisms therefore supk fdr theorem following theorem immediate consequence theorem flat version theorem let local ring suppose fdr fdr proof straightforward verification similar proof theorem rest section make comparison gorenstein flat dimensions complex right derived section functor proposition suppose gfdr gfdr gorenstein injective gorenstein flat dimensions proof notice gfdr nothing prove may assume gfdr hence follows theorem gfdr theorem exists gfdr depthrp depthrp depthrp depthrp follows theorem gfdr depthrp depthrp gfdr desired proposition let local ring let gfdr gfdr gfdr proof theorem gfdr hence theorem equalities sup depthr depthr depthr depthr sup since sup gfdr result follows fact gfdr sup see corollary following theorem gorenstein flat version theorem theorem let local ring let gfdr gfdr gfdr admits dualizing complex gfdr gfdr proof straightforward application proposition proposition view part may assume gfdr hence theorem gfdr desired equality follows proposition proposition corollary let local ring suppose relative respect grade gfdr hna gfdr local ring admitting dualizing complex proof notice grade relative respect hence view theorem may assume complete dualizing complex result therefore follows theorem right derived section functor gorenstein injective dimension section category denoted recall exercise local cohomology modules respect calculated resolution first prove complex version definition see let left exact functor assume injective resolution defined define tot jalali lemma let two let morphism suppose injective resolutions respectively exists sequence morphisms proof straightforward application theorem lemma let left exact functor let two quasiisomorphism induces isomorphism proof straightforward verification similar proof corollary following theorem one main results section enables prove interesting results theorem let left exact functor let assume injective every injective every semiinjective resolution proof let semiinjective resolution supm let quasiisomorphism theorem exist cartaneilenberg injective resolutions hence view lemma sequence morphisms diagram commutes lemma natural morphism tot quasiisomorphism similarly natural morphism tot quasiisomorphism thus proposition exists quasiisomorphism tot since injective lemma isomorphisms hence morphism tot tot quasiisomorphism tot tot therefore quasiisomorphisms tot tot desired gorenstein injective gorenstein flat dimensions next theorem offers application previous theorem complex version exercise theorem let next corollary shows computed via genuine complex gorenstein injective modules also notice theorem immediate consequence fact corollary let let gorenstein injective modules proof since lemma every gorenstein injective module result follows theorem proved corollary admits dualizing complex gidr gidr following proposition together theorem recover result proposition suppose gidr gidr gidr proof notice gidr nothing prove may assume gidr theorem exists gidr depthrp widthrp widthrp widthrp follows theorem gidr depthrp widthrp gidr desired proposition let local ring let gidr gidr gidr proof proposition inequality gidr depthr widthr widthr widthr thus gidr depthr widthr depthr inf result therefore follows corollary following theorem gorenstein injective version theorem one main results section theorem let local ring admitting dualizing complex let gidr gidr proof straightforward application theorem proposition tion next theorem gorenstein injective version theorem recovers theorem jalali theorem let local ring admitting dualizing complex suppose gidr gidr proof follows theorem similar proof theorem following corollary improves corollary consequence previous theorem corollary let local ring let dimr following statements hold gidr gidrb gidr gidr local ring admitting dualizing complex proof notice dimension straightforward application theorem theorem inequality gid gidrb gidr result view lemma gidrb hnmb follows part next corollary provides characterization gorenstein rings together corollary show theorem corollary hold without assuming corollary let local ring admitting dualizing complex following statements equivalent gorenstein gidr dim ideal iii gidr ideal proof straightforward application theorem proposition references brodmann sharp local cohomology algebraic introduction geometric applications cambridge university press cambridge christensen foxby hyperhomological algebra applications commutative rings dec christensen foxby holm beyond totally reflexive modules back noetherian perspectives edited fontana kabbaj olberding swanson springer media llc new york christensen frankild holm gorenstein projective injective flat functorial descripotion aplications algebra christensen transfer gorenstein dimensions along ring homomorphisms pure appl algebra enochs jenda relative homological algebra gruyter expositions mathematics walter gruyter new york foxby bounded complexes flat modules pure appl algebra gelfand manin homological algebra berlin iyengar local homomorphisms applications frobenius endomorphism illinois gorenstein injective gorenstein flat dimensions lipman lectures local cohomology duality local cohomology applications volume lecture notes pure appl dekker new york pages mahmood schenzel invariants endomorphism rings certain local cohomology modules algebra northcott introduction homological algebra cambridge university press rotman introduction homological algebra springer media llc sazeedeh gorenstein injective section functor forum mathematicum schenzel proregular sequences local cohomology completions math spaltenstein resolutions unbounded complexes compositio weibel introduction homological algebra cambridge university press new york yassemi generalization theorem bass comm algebra yoshizawa gorenstein injective top local cohomology modules proc amer math zargar zakeri injective gorenstein injective dimensions local cohomology modules appear algebra colloquium jalali faculty mathematical sciences computer kharazmi university taleghani avenue tehran iran address
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flexible design communications cellular networks apr ahmad alammouri hesham elsawy alouini abstract backward compatibility essential ingredient success new technologies context communication base stations bss support users equipment ues without sacrificing foreseen gains paper presents flexible tractable modeling framework cellular networks bss ues presented model based stochastic geometry accounts intrinsic vulnerability uplink transmissions results show ues necessarily required harvest rate gains bss particular results show adding ues bss offers maximum rate gain bss ues case diversity exploited marginal gain compared burden required implement transceivers ues side end shed light practical scenarios ues operation bss outperforms operation bss ues find closed form expression critical value attenuation power required ues outperform ues index terms full duplex half duplex stochastic geometry network interference network rate network topology ntroduction time division duplexing tdd frequency division duplexing fdd commonly used techniques protect receivers overwhelming implies authors computer electrical mathematical sciences engineering cemse divison king abdullah university science technology kaust thuwal makkah province saudi arabia email part work presented ieee international conference communications icc resources time frequency divided forward reverse links creates performance cancellation sic eliminates via communication gives forward reverse links opportunity simultaneously utilize complete set resources transceivers capable sufficiently attenuating interference simultaneously transmit receive channel offers higher bandwidth fdd systems longer transmission time tdd systems consequently communication improves performance forward reverse links improvement depends efficiency sic leveraging communication networks bottleneck due increased mutual interference compared case link contains two active transmitters link contains one active transmitter one passive receiver therefore rigorous studies capture effect network interference communication required draw legitimate conclusions operation largescale setup context stochastic geometry used model operation large scale networks understand behavior stochastic geometry succeeded provide systematic mathematical framework modeling cellular networks despite higher interference injected network recent studies shown communications outperform communications large scale setup sufficient sic achieved instance asymptotic study shows maximum improvement rate gain monotonically decreases link distance communication case realistic network setup shows offers average rate gain compared operation case cellular networks shows around improvement total rate compared case authors show increase aggregate interference networks creates average spectral efficiency coverage probability however reveals gains cellular networks mainly confined due high disparity uplink downlink transmission powers furthermore authors show constrained power control employed communication gains may come expense high degradation authors advise use communications small cell tiers users equipment ues base stations bss comparable transmit powers operation macro tiers high disparity transmit powers authors advocate using pulse shaping along partial overlap spectrum neutralize interference avoid deteriorating rate pulse shaping partial overlap shows simultaneous improvement respectively ought mentioned addition transmit power disparity asymmetric traffic naturally exists practical cellular networks imposes another challenge operation harvest aforementioned gains transceivers required sides link however cellular networks operators upgrade bss direct access upgrade ues furthermore high cost transceivers terms complexity power consumption price may impedes penetration ues domain therefore techniques achieve gains cellular networks bss ues required context topology proposed harvest gains serving two ues within bss sic capabilities simultaneously serve users channels merge channel pair larger channel reuse channel serve users simultaneously studies show potential harvest gains however results based simulations results based simplistic system models paper present unified mathematical framework based stochastic geometry model bss users topology bss ues cellular networks proposed mathematical framework used conduct rigorous comparison different presented system model accounts explicit performance cell center users ccus cell edge users ceus cellular network also captures realistic system parameters accounting matched filtering power control maximum power constraint ues ues scheduling different bss characteristics network tier compared proposed framework considers network different topologies flexible association different pathloss exponents different network elements incorporate uncertainties sic however exploit duplexing strategy proposed allows partial overlap channels denoted scheme parameter controls amount overlap channels captures special cases beside used optimize spectrum allocation parameter shows gradual effect interference induced via communication system performance optimize amount overlap channels results show achieve close performance within compared ues efficient sic diversity ues scheduling exploited hand ues poor sic achieves better performance cases evident network operators need bear burden implementing sic ues harvest gains rest paper organized follows section present system model methodology analysis section iii analyze performance system numerical simulation results discussion presented section presenting conclusion section notations denotes expectation random variables rvs inside denotes expectation respect denotes indicator function takes value statement true otherwise denotes convolution operator denotes complex conjugate denotes laplace transform probability density function pdf italic letters used distinguish variables constants ystem odel network model cellular network considered tier modeled via independent homogeneous poisson point processes ppps intensity location ith tier denoted beside simplifying analysis ppp assumption abstracting cellular bss verified several experimental studies ues distributed according ppp work assume bss ues equipped single antenna combining mimo transmitters covered independent bss locations intensity bss transmit constant power however value within tier varies across different tiers contrast ues employ truncated channel inversion power control maximum transmit power constraint compensates maintain tierspecific target average power level serving ues maintain threshold transmit maximum power ues keep threshold denoted cell center users ccus ues transmit maximum power denoted cell edge users ceus power transmitted signals experiences power law path loss attenuation exponent due different relative antenna heights propagation environments discriminate path loss exponent paths two bss interference two ues interference interference respectively denoted shown fig assuming channel reciprocity path loss exponent interference denoted equivalent one interference hence symbols used also rayleigh fading channels assumed channels power gains independent identically distributed exponential rvs unit operation modes spectrum allocation consider fine grained scheme allows partial overlap channels captures special cases denote bws used case respectively buhd bdhd buhd bdhd necessarily equal avoid adjacent channel interference bss utilize guard band pair bands min bdhd buhd shown fig used bdhd buhd note assume exponents direction different equivalent tiers assuming equal exponents tiers common simplifying assumption literature extending results capture fading models done following scheme proposed captured setting zero since guard bands assumed fig channel allocation interference types exponents frequency bands allocation ues scheduling fig frequncy bands allocation ues scheduling parameter controls partial overlap frequency bands also modes captured special cases setting respectively assumed tier duplexing parameter used bss within tier without loss generality assume two pairs channels universally reused across network simplicity assume two channel pairs sufficiently separated frequency domain avoid adjacent channel interference different pairs worth noting idealized rectangular frequency domain pulse shapes shown fig used illustration however discussed later use pulse shapes impose adjacent channel interference due band ripples frequency domain network ues transceivers use belonging pair operation contrast ues transceivers transmit receive overlapping channels hence user assigned channels two different pairs shown fig fig consequently ues benefit larger channels without note bss cases well ues would experience shown fig contrast experience interference direction due partial overlap channel one channel end assume bss exploit diversity control interference imposing minimum separation angle constraint users scheduled channel shown fig sectored bss value estimated certain accuracy depending number sectors bss estimate angeles users set zero refer case random scheduling bss ues denote attenuation power respectively positive constants representing mean sic power values respectively follows general unit mean distribution pdf given fhs represents uncertainty sic three special cases interest fhs considered namely constant attenuation fhs degenerate distribution random attenuation fhs exponential distribution rician fading captures previous two cases special cases shown distributions leads performance trends ues bss association consider biased association scheme biasing used enable flexible load balancing tiers encouraging ues connect lower power bss balance average load served tiers across network define distance dependent biasing factor assume bss within tier biasing factor hence connects tier used association scheme captures different association strategies special cases example set value tiers closest association considered advanced sophisticated scheduling diversity techniques postponed future work decoupled association analyzed using stochastic geometry traditional network extending analysis decoupled association postponed future work fig realization associations areas assuming different association factors green squares diamonds circles represent macro micro pico bss respectively connects providing highest received signal strength rss note different association schemes changes relative bss association areas across tiers shown fig three tiers network shown macro bss micro bss pico fig nearest association considered hence association areas represented voronoi tessellation fig connects according rss case association areas construct multiplicative weighted voronoi tessellation also denoted circular tessellation pulse shaping employ pulse denoted unit energy pulse bandwidth indicate pulse types used tier respectively assume flexible pulse shaping scheme tier pulse shapes however bss within tier use pulse shapes unified effective values mode kept equal channel hence pulse shapes also functions parameter values transmit powers based work focus pulse shapes avoid isi since including effect isi complicate analysis much however nyquist pulses root raised cosine also used since protect nodes isi information effect different pulse shapes scheme refer signal representation sake simple presentation use denote duplexing factor pulse shape pulse shape respectively tier also use indicate desired transmission respectively bss tier index exploiting notation received baseband signal input matched filter test transceiver ith tier expressed iju idj pvo pvo denote intended symbol transmit power link distance channel power gains respectively set interfering bss tier ijd interference tier set interfering ues tier iju interference connected tier isv term affecting direction white complex gaussian noise zero mean power spectral density downlink uplink interference given ijd iju hdj puj huj exp rdj exp ruj denote interfering symbol interfering symbol tier following interpretation subscripts superscripts defined interfering symbols hdj huj denote interfering channel gains puj denote interfering transmit powers rdj ruj denote interfering link distances denote center frequencies interfering frequency bands see fig note index removed transmit power bss tier transmit power similarly indices removed center frequencies elements tier employ overlap parameter term given exp exp isd represents average attenuation power average attenuation power affecting ith tier hence tier different sic capability depending bss sizes receivers complexity puo transmit power tagged represents difference center frequencies ith tier note difference also depends chosen tier since tier different duplexing factor leads different bws center frequencies methodology analysis analysis conducted test transceiver located origin operating test channel pair according slivnyak theorem loss generality assumption also loss generality focus test channel pair interferences different bands statistically equivalent asses impact communication via outage probability transmission rate outage probability defined sinr transmission rate assume nodes transmit fixed rate regardless state channel hence transmission rate defined sinr degraded sinr compensated increased linear term hence used fairly assess performance operation shown outage probability transmission rate independent symbol structure depend sinr consequently transmitted symbols abstracted independent complex gaussian random variables abstracting symbols via gaussian random variables negligible effect sinr distribution shown analysis start modeling effect matched filtering baseband signal based signal format filtering expressions sinr different cases obtained performance metrics expressed terms pdf interference obtained later evaluate iii erformance nalysis received signal first convolved conjugated pulse shape template passed filter sampled baseband signal filtering sampling input decoder given pvj hvj rvj pvo combined matched filter impulse response transceiver ith tier frequency domain representation given elsewhere represents used pulse shape discussed section factors represent vice versa effective received energy factors respectively expressions bit error probability derived using obtained sinr next section following expressing convolution operation frequency domain pulse shaping filtering factors obtained bvz bvz noted although mode links use similar pulse shapes tier effective energy received transmitters unity shown includes combined impulse response matched filters extracts desired frequency range received signal consequently energy outside desired discarded energy contained within pulse shape longer unity also interference factor strictly less unity due filtering possibly different pulse shapes partial overlap channels let rvj hvj pvo pvj conditioning sinr given sinr pvo pvj hvj rvj residual power normalized expressed sinr used next section evaluate outage provability rate discussed section performance metrics outage probability link ith tier written pvo general pvj hvj written pvo pvo rvj exploiting exponential distribution pvo pvo expectation pvo since pvo depends ues type ccu ceu discussed section present explicit study type serving distances ccus ceu characterized via following lemma lemma serving distance distribution randomly selected ccu ceu given connected ith tier denoted respectively given following equations exp exp exp proof refer appendix lemma straightforward find probability randomly selected ith tier ccu ceu given ccu exp ceu exp law total probability average outage probability expressed via ccus outage probability ceus outage probability denoted ovc ove respectively ovc ccu ove ceu ovc ove represented specific parameters related ccus ceus clear aggregate interference tier required evaluate ovc ove aggregate interference hence depends spatial distribution set interfering bss ues tier respectively set interfering bss tier original set bss excluding transmitting serving hence ppp intensity ues associations intensity interfering ues certain channel tier also however ppp one use channel impose correlations among positions interfering ues channel violates ppp assumption furthermore employed association makes set interfering ues set interfering bss correlated interfering ues ues bss impede model tractability hence maintain tractability ignore correlations used assumptions keep model tractability formally stated assumption set interfering ues tier ppp intensity assumption point process interfering bss point process interfering ues tier independent assumption point processes represent interfering ues connected different tiers independent remark previous assumptions necessary maintain model tractability assumption used validated assumption assumption important mention assumptions ignore mutual correlations interfering sources however correlations interfering sources test receiver captured proper calculation interference exclusion region enforced association power control accuracy developed model assumptions validated via independent monte carlo simulation section based assumptions aggregated interference always generated ppp different parameters interference exclusion regions interferers intensity transmit power distribution brevity present following unified lemma aggregate interference generated homogeneous ppp general parameters use obtain lts lemma let aggregate interference generated ppp network intensity transmit powers unit means exponentially distributed channel power gains protection region ball centered origin radius given exp function expectation transmitted power sources general path loss exponent special case equation reduces csc exp proof refer appendix due expectation power distribution expression integral function interference exclusion distance around test receiver independent transmit powers expression given simplified expression given following lemma lemma let aggregate interference generated ppp network intensity transmit powers unit means exponentially distributed channel power gains interference protection region ball centered origin radius assuming independent exp proof refer appendix lemma precludes necessity integrate pdf transmit power interfering sources give containing first moment transmit power reduces computational complexity lts sake simplified expressions always use bound whenever applicable verified section using lemma lemma lts aggregated interference ccus ceus given following lemma lemma let represent lts aggregate interference generated tier affecting ccu ceu serving given ith tier lts given exp sro exp exp exp csc exp spd exp exp exp interference case expressed ccu rro rro cos ceu cos lower incomplete gamma function proof refer appendix note equations represents interference effect case tier tagged transceiver belongs sake simplified expressions also present approximation following lemma lemma interference given equation approximated ccu sin ceu sin erf exp exp erfc exp erf erfc error function complementary error function respectively proof substituting cos average values using results lemmas along outage probabilities types connections depicted system model characterized via following theorem theorem outage probabilities ith tier ccus ceus given ouc oue usiu odc dro dro dro usid usiu ode usid usiu exp hpd fhs usid exp fhs given equations fhs distribution sic power respectively lts given lemma proof refer appendix special case interest leads simple forms outage probability presented following corollary corollary interference limited dense single tier cellular network unbinding transmit power outage probability assuming exp given exp arctan unt exp arctan unt exp rro rro cos arctan dro proof expressions follow theorem lemmas considering single tier network setting following rate expressed terms outage probability follows hence general expressions rate network obtained directly substituting outage probability expressions theorem equation sake brevity list rate expressions special case interest following corollary corollary interference limited dense single tier cellular network unbinding transmit power average rate assuming exp given arctan exp arctan unt dro exp arctan unt exp rro rro cos proof follows corollary equation last corollary find critical sic outperforms function serving distance value given following corollary corollary interference limited dense single tier cellular network unbinding transmit power approximate minimum value required outperform function serving distance assuming channels exp given table parameters values parameter value parameter value mhz dbm buhd bdhd proof follows equation equation expresses critical value function get insights critical value respect bss intensity assume tagged located average serving distance assumption reduces next section theorem lemmas corollaries used analyze performance cellular network operations imulations umerical esults throughout section verify developed mathematical paradigm via independent system level simulations bss realized via ppp area ues distributed uniformly area least two ues within association area randomly selects two ues serve angle separation illustrated fig satisfied sinr calculated summing interference powers ues bss multiplying effective interference factors transmit powers ues set according power control discussed section results taken bss closest origin avoid edge effect unless otherwise stated parameters values table used note average sic power maximum reported value according hence set consider bss likely powerful sic capabilities dbm rdc rde rate mbps rate mbps rdc rde ruc rue rate mbps dbm throughput throughput throughput fig rates pulse shaping consider two basic pulse shapes namely sinc pulse fts given sinc sinc sinc sinc sinc sinc pulse considered pulse considered unless otherwise stated sic power distribution fhs assumed exponentially distributed unit mean fig shows rate variation versus represent cases respectively solid lines represent analytical results obtained theorem exact lts lemma diamonds represent results obtained simulations squares fig found using approximation intracell interference given lemma squares fig using bounds lts given lemma close match analysis approximations simulation results validates developed mathematical model verifies accuracy assumptions section well bound presented lemmas several insights obtained fig instance figure shows ccus better performance compared ceus cases intuitive due larger service distances lead higher attenuation ceus compared ccus note employing designing sophisticated pulse shapes specific purposes left future work approx rde mbps rdc mbps mbps average rate rdc rdc rdc approx ccu rate rde rde rde approx ceu rate fig rates different network topology denotes denotes ceus sufficient power invert direction hence received power serving less leads deteriorated ceu performance compared ccu case figure also shows exist optimal value partial overlap maximizes transmission hence despite efficient sic neither optimal case due prominent interference hand performance mainly affected sic rather particularly efficient sic full overlap best strategy hand partial overlap better inefficient sic worth mentioning prominent effect ceu ccu nearly nullifies rate high value efficient sic ceus transmit maximum power makes residual power prominent compared ccus comparing fig fig see achieves close performance sufficient sic outperforms poor sic note ues operates mode hence affected sic shown fig fig plots transmission rate attenuation power figure performance maximized due orthogonality used pulse shapes particular value details refer case perfect low values dbm degradation sinr due interference since negligible compared interference realistic set network parameters increase linearly overcome decrease sinr results approximately linear curve shows rate outperforms rate cases fig also shows critical value outperforms critical value interpreted point experienced ues becomes significant interference experienced found closed form approximation corollary interestingly gain offered low values significant compared hence interference limiting parameter words network operators harvest gains ues almost similar gains harvested ues efficient sic capabilities figure also shows case poor sic ues offer significant gains specially ceus compared case study effect serving distance performance plot fig figure plots minimum required outperform serving distance along pdf serving distance sold dashed lines obtained exact approximate expression interference given equation close match exact approximate results validates approximation given figure shows appealing farther ues tighter constraint sic required outperform may require sophisticated expensive transceivers two reasons result first large serving distance implies interferer average due enforced scheduling technique reduces negative effect interference second longer service distance implies larger transmit power due employed power control hence powerful sic required useful design insight fig bss select mode operation ues based distances along sic get insights network operation different intensities plot fig based equation expected becomes appealing dense cellular networks self interference less prominent smaller cell finally study rate gains fig expected increasing distance two scheduled ues increases hence power fig focuses comparing different intensities effect intensity gain cellular network covered network transmitters covered mbps dbm dbm meters rcd rcd red red critical values critical values bss rate illustrated fig ing distance intesnity fig critical values serving distance bss intesnity using equations rate illustrated fig operation decreases moreover figure shows necessity ues scheduling diversity otherwise rate loss compared case random scheduling onclusion paper presents mathematical paradigm cellular networks bss ues presented model captures detailed system parameters including pulse shaping filtering imperfect cancellation partial overlap uplink power control limited users transmit powers association ues scheduling end unified rate expressions topology users topology users presented used compare performance results show exist critical value cancellation performance outperforms moreover closed form approximations critical value function serving distance bss intensity obtained results also show even efficiently canceled offer significant gains compared operation diveristy users scheduling exploited implies network operators harvest gains implementing transceivers bss regardless state users ppendix roof emma exploiting independence network tiers using null probability ppp cumulative distribution function cdf ith service distance given denominator given fri nominator given frj drj dri frj drj dri fri exp substituting equations results exp exp pdf given exp given ccu pdf truncated according channel inversion power control let denote serving distance test ccu connected ith tier pdf given exp exp exp exp similarly pdf service distance ceus exp exp exp ppendix roof emma proof follows ehj exp exp iii exp exp rdr follows independence using probability generation functional pgfl ppp iii using evaluating integral ppendix roof emma lemma proved showing second derivative function appears inside expectation exponent positive hence function interest convex result lemma follows jensen inequality section let function interest denoted expressed second derivative given found using eqs mathematical simplifications owing fact positive positive prove completed proving using integral definition hypergeometric function projecting case iii follows follows fact iii proved sequel taking first derivative integrand shows decreasing function hence minimum occurs boundary iii follows integral minimum value integrand multiplied integration region hence second derivative positive completes prove ppendix roof emma based lemma need determine interference exclusion region ier tier case due association rule section always satisfied hence ier defined substituting lemma final expression found based power inversion ccus following based power inversion ceus following using lemma final expression found ppp assumption bss location implies ier cases ccus ccus hence assume tagged collocated associated hence effect interference considered also let denote interference let denote transmitted power interfering user channel gain two users distance distance interfering serving respectively interfering power expressed cos follows using cosine rule fig uniformly distributed ccu probability ccu ceu probability ceu substituting values averaging expression found ppendix roof heorem starting outage probability ccus transmitted power equal ceus transmitted power set maximum substituting values get equations except usiu found substituting value given averaging conditioning similar steps followed find outage direction eferences alammouri elsawy alouini harvesting rate gains cellular networks user terminals proc ieee international conference communications icc bharadia mcmilin katti full duplex radios proc acm sigcomm conf sigcomm ser sigcomm new york usa acm online available http hong brand choi jain mehlman katti levis applications cancellation beyond ieee wireless commun vol sabharwal schniter guo bliss rangarajan wichman wireless challenges opportunities ieee sel areas vol kim lee hong survey transmission perspective phy mac layers ieee commun surveys vol fourthquarter alves lima nardelli demo souza average spectral efficiency networks proc international conf cognitive radio oriented wireless networks communications crowncom jun goyal liu panwar difazio yang bala full duplex cellular systems doubling interference prevent doubling capacity ieee commun vol may elsawy hossain haenggi stochastic geometry modeling analysis design cognitive cellular wireless networks survey ieee commun surveys vol third quarter haenggi stochastic geometry wireless networks cambridge university press cambridge books online online available http andrews baccelli ganti tractable approach coverage rate cellular networks ieee trans wireless vol renzo stochastic geometry modeling cellular networks analysis simulation experimental validation corr vol online available http xie zhang double capacity wireless networks proc ieee infocom apr tong haenggi throughput analysis wireless networks imperfect cancellation ieee trans vol nov lee quek hybrid system analysis heterogeneous wireless networks ieee trans wireless vol may goyal galiotto marchetti panwar throughput coverage mixed full half duplex small cell network corr online available http randrianantenaina elsawy alouini limits capacity full duplex communication uplink cellular networks ieee global commun conf globecom workshops san diego usa alammouri elsawy amin alouini scheme cellular networks stochastic geometry approach ieee trans wireless submitted online available http randrianantenaina elsawy dahrouj alouini interference management partial spectrum overlap proc ieee int conf commun icc accepted elsawy alammouri amin alouini uplink transmissions survive cellular environments proc european wireless conference mahmood berardinelli mogensen frederiksen throughput analysis full duplex communication asymmetric traffic small cell systems proc eleventh int conf wireless mobile commun icwmc sundaresan khojastepour chai rangarajan without strings enabling halfduplex clients proc annu int conf mobile computing networking ser mobicom acm lima alves nardelli hybrid communications correlated lognormal shadowing proc ieee vehicular technology conf vtc spring may goyal liu hua panwar analyzing cellular system proc annu conf information sciences systems ciss mar mohammadi suraweera krikidis tellambura radio transmission spatial randomness proc ieee int conf commun icc london jun psomas krikidis outage analysis architectures cellular networks vehicular technology conf vtc spring ieee may atzeni kountouris mimo networks performance analysis proc ieee global commun conf globecom dec psomas mohammadi krikidis suraweera directional antennas interference management cellular networks corr online available http guo haenggi spatial stochastic models metrics structure base stations cellular networks ieee trans wireless vol elsawy hossain stochastic geometry modeling cellular uplink transmission truncated channel inversion power control ieee trans wireless vol alammouri elsawy alouini modeling uplink cellular networks environment proc ieee pimrc washington usa dhillon ganti baccelli andrews modeling analysis downlink heterogeneous cellular networks ieee sel areas vol april chandrasekhar andrews uplink capacity interference avoidance femtocell networks ieee trans wireless vol july renzo guidotti corazza average rate downlink heterogeneous cellular networks generalized fading channels stochastic geometry approach ieee tans vol jul alammouri elsawy renzo alouini modeling cellular networks fading environments dominant specular components proc ieee int conf commun icc online available http yun intra resource management heterogeneous cellular networks ieee trans mobile singh zhang andrews joint rate sinr coverage analysis decoupled biased cell associations hetnets ieee trans wireless andrews singh lin dhillon overview load balancing hetnets old myths open problems ieee trans wireless vol apr elsawy hossain kim hetnets cognitive small cells user offloading distributed channel access techniques ieee commun vol jun jain katiyar agrawal hierarchical cellular structures highcapacity cellular communication systems ijacsa international journal advanced computer science vol ash bolker generalized dirichlet tessellations geometriae dedicata vol zhang letaief throughput energy efficiency analysis small cell networks base stations ieee trans wireless vol may afify elsawy alouini influence gaussian signaling approximation error performance cellular networks ieee commun lett vol dec giorgetti chiani influence fading gaussian approximation bpsk qpsk asynchronous cochannel interference ieee trans wireless vol renzo eid approach analysis cellular networks using stochastic geometry ieee commun lett vol may abramowitz stegun handbook mathematical functions formulas graphs mathematical tables new york dover wang venkateswaran zhang exploring gains wireless networks spatial stochastic framework ieee conf computer commun infocom apr boyd vandenberghe convex optimization new york usa cambridge university press
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sep adapting resilient propagation deep learning alan mosca george magoulas department computer science information systems birkbeck university london malet street london united kingdom email department computer science information systems birkbeck university london malet street london united kingdom email gmagoulas resilient propagation rprop algorithm popular backpropagation training multilayer neural networks various applications standard rprop however encounters difficulties context deep neural networks typically happens learning algorithms paper propose modification rprop combines standard rprop steps special drop technique apply method training deep neural networks standalone components ensemble formulations results mnist dataset show proposed modification alleviates standard rprop problems demonstrating improved learning speed accuracy ntroduction deep learning techniques generated many models reached impressive results benchmark datasets like mnist models usually trained variations standard backpropagation method stochastic gradient descent sgd field shallow neural networks several developments training algorithms sped convergence paper aims bridge gap field deep learning advanced training methods combining resilient propagation rprop dropout deep neural networks ensembles rprop resilient propagation weight update rule initially introduced possible solution vanishing gradients problem depth complexity artificial neural network increase gradient propagated backwards standard sgd backpropagation becomes increasingly smaller leading negligible weight updates slow training considerably rprop solves problem using fixed update value increased decreased multiplicatively iteration asymmetric factor respectively depending whether gradient respect wij changed sign two iterations backtracking allows rprop still converge local minima acceleration provided multiplicative factor helps skip flat regions much quickly avoid double punishment backtracking phase rprop artificially forces gradient product following iteration skipped illustration rprop found algorithm algorithm rprop pick sgn min sgn wij wij else max else sgn wij wij end end dropout dropout regularisation method random selection nodes network updated training iteration final evaluation stage whole network used selection performed sampling dropout mask bernoulli distribution mutedi mutedi probability node muted weight update step backpropagation dropout rate usually middle layers input layers output layer convenience dropout mask represented weight binary matrix covering weights network used multiply network obtain called thinned network current training iteration weight wij zeroed based probability parent node muted remainder paper structured follows section explain using dropout causes incompatibility rprop propose modification solve issue section iii show experimental results using mnist dataset first highlight rprop able converge much quickly initial epochs use speed training stacked ensemble finally section look work extended evaluation development prop ropout section explain zero gradient problem propose solution adapting rprop algorithm aware dropout problem order avoid double punishment change sign gradient rprop artificially sets gradient product associated weight next iteration condition checked following iteration true updates weights wij learning rate performed using gradient product indication skip iteration acceptable normal gradient descent occurrence would learning terminated dropout introduced additional number events produce zero values neuron skipped dropout mask weights wij going layer value neuron layer skipped gradient propagated back weights wij also additional events force additional skipped training iterations missed learning rate adaptations slow training unnecessarily adaptations rprop making rprop aware dropout mask able distinguish whether event occurs signal skip next weight update whether occurs different reason therefore updates allowed new version rprop update rule weight shown algorithm use indicate current training example previous training example next training example value appears intended initial value notation used original rprop error function case negative log likelihood current update value weight index current weight update value index particular conditions line line providing necessary protection additional zerogradients implementing correctly recipe prescribed dropout completely skipping every weight dmij means neuron dropped therefore gradient necessarily expect methodolgy extended variants rprop limited jrprop algorithm rprop adapted dropout pick sgn dmij else min sgn wij wij else max else sgn wij wij else end end end end iii valuating mnist section describe initial evaluation performance mnist dataset experiments use deep neural network dnn five middle layers neurons respectively dropout rate drmid middle layers dropout inputs dropout rate shown optimal choice mnist dataset similar architecture used produce results however authors used entire training set validation graphical transformations said set training added transformations led virtually infinite training set size whereby every epoch new training set generated much larger validation set original images test set remains original image test set explanation transformations provided also confirms important practice getting training set large possible expand training set adding new form distorted data therefore attribute big improvements transformations applied found primary goal replicate additional transformations obtain results instead focused utilising untransformed compared sgd results table see modified version rprop able much quicker reaches error value close minimum much quickly sgd reaches higher error value much longer time although overall error improvement significant speed gain using rprop appealing allows save large number iterations could used improving model different ways rprop obtains best validation error epochs whilst sgd reached minimum illustration first epochs seen figure stays consistently reaches minimum also unmodified version reach final error modified version starts overtraining much sooner reach better error sgd table shows detail performance two methods compares first epochs unmodified rprop modified rprop modified unmodified validation error dataset using images training validation testing subsequently performed search using validation set indicator find optimal hyperparameters modified version rprop found best results reached trained models maximum allowed epochs measured error validation set every epoch could used select model applied test set also measured time took reach best validation error report approximate magnitude use comparison orders magnitude results presented average repeated runs limited maximum training epochs training epoch fig validation error unmod mod rprop using modified rprop speed training deep learning ensembles compared unmodified rprop increase speed convergence make practical produce ensembles deep neural networks time train member dnn considerably reduced without undertraining network able train ensembles less hours total singlegpu desktop system trained different ensemble types report final results table methods used bagging stacking member dnns member trained maximum epochs bagging ensemble method several different training sets created random resampling original training set used train new classifier entire set trained classifiers usually aggregated taking average majority vote reach single classification decision stacking ensemble method different classifiers aggregated using additional learning algorithm uses inputs classifiers learn information reach better classification result additional learning algorithm called classifier case stacking final classifier another dnn two middle layers respectively size number dnns ensemble trained maximum epochs see figure modified version rprop faster unmodified version used nvidia graphics card core processor programmed theano python sgd modified rprop modified rprop sgd validation error training epoch fig validation error sgd mod rprop method sgd rprop mod rprop min val err epochs time min min min test err epoch table simulation results modified rprop used original train validation test sets collected average repeated runs results still comparable presented consistent observations importance dataset transformations however note able improve error less time took train single network sgd wilcoxon signed ranks test shows increase performance obtained using ensembles size compared ensemble size significant confidence level method bagging bagging stacking stacking size test err time min min min min table ensemble performance onclusions uture ork highlighted many training methods used shallow learning may adapted use deep learning looked rprop appearance training side effect dropout poses challenge learning proposed solution allows rprop train dnns better error still much faster standard sgd backpropagation showed increase training speed used train effectively ensemble dnns commodity desktop system reap added benefits ensemble methods less time would take train deep neural network sgd remains assessed work whether improved methodology would lead new error applying dataset enhancements used methods improvements rprop ported numerous variants acknowledgement authors would like thank school business economics informatics birkbeck college university london grant received support research eferences wan zeiler zhang cun fergus regularization neural networks using dropconnect proceedings international conference machine learning ciresan meier schmidhuber deep neural networks image classification proceedings ieee conference computer vision pattern recognition cvpr ieee press ciresan meier gambardella schmidhuber deep big simple neural nets handwritten digit recognition neural computation vol lecun cortes mnist database handwritten online available http riedmiller braun direct adaptive method faster backpropagation learning rprop algorithm proceeding ieee international conference neural networks ieee anastasiadis magoulas vrahatis new globally convergent training scheme based resilient propagation algorithm neurocomputing vol hinton srivastava krizhevsky sutskever salakhutdinov improving neural networks preventing coadaptation feature detectors corr vol igel improving rprop learning algorithm proceedings second international icsc symposium neural computation vol citeseer srivastava hinton krizhevsky sutskever salakhutdinov dropout simple way prevent neural networks overfitting journal machine learning research vol simard steinkraus platt best practices convolutional neural networks applied visual document analysis online available http breiman bagging predictors machine learning vol stacked generalization neural networks vol
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reductions approximate counting holger john jul saarland university cluster excellence mmci hdell oxford university july abstract main problems complexity orthogonal vectors problem triangle problem closely related shortest path paper consider approximate counting version problem thus instead simply deciding whether witness exists attempt multiplicatively approximate number witnesses case provide reduction approximate counting form usual decision form example algorithm solves prove algorithm approximately count satisfying assignments similarly get exponential time hypothesis eth equivalent approximate counting version mirrors result sipser stoc stockmeyer sicomp proved result classical setting similar result due iwpec fpt setting algorithm problems applies general setting approximately count edges bipartite graph limited access particular means applied problem variants significant improvements conjectured running time bounds already known example orthogonal vectors problem constant solved time poly result williams soda result implies approximately count number orthogonal pairs essentially running time moreover overhead polylogarithmic applied subpolynomial improvements exp log time algorithm triangle problem due williams stoc introduction approximate counting decision settings clearly least hard count objects decide existence often harder example valiant defined class natural counting variant part work done authors visiting simons institute theory computing research leading results received funding european research council european union seventh framework programme erc grant agreement paper reflects authors views views erc european commission european union liable use may made information contained therein toda proved contains entire polynomial hierarchy decision counterparts many problems example counting perfect matchings detecting one however situation changes substantially consider approximate counting rather exact counting real say holds clearly computing least hard deciding whether holds surprisingly many settings harder indeed sipser stockmeyer proved implicitly every problem randomised algorithm using result later proved explicitly valiant vazirani foundational result wider complexity theory polynomial approximate counting initiated dyer goldberg greenhill jerrum another example arises parameterised complexity usual goal determine whether instance size parameter solved fpt time poly computable function hardness results normally presented using see example proved problem randomised algorithm using runs instances time poly computable also proves analogous results rest hierarchy consider setting popular randomised exponential time hypothesis eth introduced impagliazzo paturi asserts exists randomised algorithm solve instance time prove eth equivalent seemingly weaker approximate counting version theorem eth true exists randomised algorithm run instances time note usual argument valiant vazirani apply setting without modification adds clauses linear width instance proof take similar approach making use sparse hashing technique due calabro impagliazzo kabanets paturi give detail section results exponential setting complexity concerned classification algorithms broad categories polynomial fpt subexponential precise running times analogue eth known strong exponential time hypothesis seth see impagliazzo paturi zane asserts exists randomised algorithm solve instances time analogue theorem seth implicit thurley provides randomised approximate counting algorithm makes use decision oracle seth true exists randomised algorithm run instances time however result ideal perspective guarantee solving approximating similar time complexities limit indeed given algorithm running time thurley approximation algorithm running time exponential savings exhaustive search goes decision using streamline discussion ignore detail papers allow deterministic algorithms throughout paper require randomised algorithms success probability least unless otherwise specified thurley algorithm traxler proved allow clause width log instead considering savings achieved approximate counting decision strengthen traxler result applies setting seth theorem let suppose randomised algorithm runs instances time randomised algorithm runs instances time particular seth false theorem yields efficient algorithm approximating sufficiently large note particular reason believe efficient decision algorithm would yield efficient counting algorithm directly indeed efficient known algorithms run time decision due hertli time counting due thurley time exact counting due kutzkov remains open interesting question whether result analogous theorem holds fixed whether deciding approximating time complexity subexponential factor large fixed decision counting exact counting algorithms due paturi saks zane thurley impagliazzo matthews paturi respectively running time progressively worse constants exponent reduction overhead log get improved approximate counting algorithms fixed results setting alongside sat perhaps important problems complexity orthogonal vectors shortest paths apsp three problems admit notions hardness sense many problems reduce equivalent reductions known reduce one another see williams recent survey prove analogues theorem clear canonical counting version apsp nevertheless prove analogue theorem triangle problem nwt equivalent apsp subcubic reductions results together previous decision algorithms immediately imply three new approximate counting algorithms however believe two theorems may also derived directly modifying known decision algorithms asks given three integer lists total length whether exists tuple frequently input taken single list rather three two versions equivalent easy see solved operations sorting iterating pairs conjectured admits randomised algorithm obtain analogue theorem natural counting version approximate number tuples see section details model computation theorem integers randomised algorithm log log randomised algorithm note normally assumed case algorithm polylogarithmic overhead decision thus independently whether conjecture true conclude say essentially time complexity chan lewenstein proved conjecture fails problem restricted instances elements one list somewhat clustered interesting special case several applications including monotone coordinates see introduction overview chan lewenstein algorithm combined analogue theorem obtain following result theorem randomised algorithm running time instances integers least one may covered intervals length next consider asks given two lists total length vectors whether exists orthogonal pair easy see solved operations iterating pairs conjectured log admits randomised algorithm conjecture implied seth abboud williams proved fails log obtain analogue theorem natural counting version approximate number orthogonal pairs theorem vectors dimensions randomised algorithm randomised log algorithm note impossible decide time polylogarithmic usual assumption algorithm polylogarithmic overhead decision thus result able turn log algorithm approximate counting algorithm chan williams already gave deterministic exact counting algorithm similar complexity version real vectors replaced arbitrary vectors finite fields rings also studied efficient randomised algorithms due williams algorithms immediately generalise counting analogue theorem nevertheless obtain efficient approximate counting algorithms theorem let constant prime power randomised algorithm instances resp time resp note dependence may close best possible seth log resp solved time resp finitely many values finally study triangle problem nwt deciding whether edgeweighted tripartite graph contains triangle negative total weight williams williams shown equivalent apsp reductions easy see nwt solved operations checking every possible triangle conjectured nwt admits randomised algorithm obtain analogue theorem natural counting version nwt approximate number triangles theorem nwt graphs weights randomised time algorithm randomised log log algorithm nwt note impossible decide nwt time polynomially bounded algorithm polylogarithmic overhead decision note also provides subcubic reduction listing triangles nwt although polynomial overhead imply result together algorithm williams theorem implies following theorem randomised runs instances nwt polynomially bounded weights time log techniques first discuss theorems prove section polynomial setting standard reduction approximating deciding due valiant vazirani runs follows formula solutions using standard argument one count number solutions calls otherwise one may form new formula conjoining uniformly random xor clauses relatively easy see long number sat satisfying assignments substantially greater sat concentrated around sat thus choosing appropriately one count sat exactly multiply obtain estimate sat unfortunately argument requires modification exponential setting variables uniformly random xor length therefore expressed cnf without introducing new variables follows example contain variables blowup acceptable polynomial setting exponential one example given algorithm would yield useless randomised approximate counting algorithm afford add xors general result concentration number solutions therefore make use hashing scheme developed calabro impagliazzo kabanets paturi related problem reducing choose subset uniformly random large constant choose variables binomially random within set still yield concentration number solutions turns variance sufficiently low remedy summing many slightly stronger hashes results section follow general theorem consider problem approximately counting edges arbitrary bipartite graph limited access particular allow adjacency queries independence queries adjacency query checks whether edge exists two given vertices graph independence query checks whether given set vertices independent set graph standard approach used thurley would use random adjacency queries handle instances many edges independence queries handle instances edges approach requires polynomially many independence queries many establish tight relationship approximate counting decision required results section contrast main algorithm theorem approximates number edges graph time makes many independence queries using algorithm obtain results problems straightforward way example proof theorem vertices input vectors edges correspond orthogonal pairs adjacency query corresponds orthogonality check done time dimensions independence query corresponds call decision oracle takes time assumption since independence queries occur theorem follows algorithm theorem works roughly follows let bipartite graph whose edges trying count let vertex classes using binary search together independence oracle quickly find vertices contains vertices standard argument used theorems quickly determine exactly suppose contains many vertices every vertex contained small proportion edges approximately halve passing uniformly random subset proceed similarly valiant vazirani however general case number edges resulting graph concentrated must therefore detect remove problematic vertices procedure use quite technical forms bulk proof defer explanation section preliminaries write set positive integers positive integer use denote set use log denote logarithm denote logarithm notation write set positive integers positive integer use denote set use log denote logarithm denote logarithm consider graphs undirected write use denote neighbourhood convenience shall generally present bipartite graphs triple partition stating quantitative bounds running times algorithms assume standard machine model words assume lists functions problem input presented natural way array using least one word per entry general shall overly concerned logarithmic factors running times shall write constant log similarly write constant require problem inputs given finite binary strings write set strings randomised approximation scheme function randomised algorithm takes input instance rational error tolerance outputs rational number random variable depending coin tosses made algorithm every instance approximate counting algorithms randomised approximation schemes probability theory require results probability theory collate reference first state chebyshev inequality lemma let random variable mean let var also use following concentration result due mcdiarmid lemma suppose real function independent random variables let suppose exist pairs differing ith coordinate finally use following chernoff bounds proved example corollaries remark janson rucinski lemma suppose binomial hypergeometric random variable mean decision approximate counting section prove results satisfiability cnf formulae formally defined follows problem input formula task decide satisfiable problem input formula task compute number sat satisfying assignments also define technical intermediate problem say matrix every row contains entries following definition constants problem input boolean formula form formula matrix task decide satisfiable define growth rate infimum randomised algorithm runs time outputs correct answer probability least main reduction encapsulated following theorem theorem let let suppose randomised approximation scheme given formula approximation error parameter runs time prove theorem let derive theorems immediate corollaries theorem restated eth true exists randomised algorithm run instances time proof first note may use randomised approximation scheme decide success probability least taking outputting yes result thus backward implication theorem immediate conversely suppose eth false result impagliazzo paturi zane lemma implies constant randomised algorithm decide time success probability least hence constant natural reduction max obtain result therefore follows theorem theorem restated let suppose randomised algorithm runs instances time randomised algorithm runs instances time proof suppose specified theorem statement constant natural reduction max thus result follows theorem proof theorem given access oracle decides satisfiability queries compute exact number solutions formula solutions using standard argument given see also lemma algorithm sparse given instance variables access oracle algorithm computes sat sat otherwise outputs fail query oracle unsatisfiable return variables left contains variables return branch recurse let formulae obtained setting first free variable respectively sparse sparse return sum otherwise abort entire computation return fail lemma sparse correct runs time min sat min sat calls oracle moreover oracle query formula variables proof consider recursion tree sparse inputs vertex algorithm takes time compute issues single oracle call convenience call leaves tree sparse returns respectively let number path root otherwise would paths vertices total finally every sibling parent would total overall tree vertices easy induction using implies certainly sat running time oracle access bounds satisfied correctness likewise follows straightforward induction input formula many solutions apply sparse efficiently first reduce number solutions hashing particular use hash functions calabro based random sparse matrices formally defined follows definition let random matrix defined follows row let uniformly random subset choose values independently uniformly random set entries zero intuition suppose formula set satisfying assignments holds small easy see uniformly random number satisfying assignments expected value see lemma concentrated around expectation choosing appropriate value could reduce number solutions apply sparse count exactly multiply result obtain approximation usual approach pioneered valiant vazirani exponential setting however afford take means general concentrated around expectation limited concentration needed require strong concentration achieve rather counting satisfying assignments single formula sum many formulae first bound variance individual suitably large analysis similar calabro although concerned probability least one solution remains hashing give bounds variance lemma let let suppose let suppose let let uniformly random independent let var proof let indicator variable event exposing implies hence bound second moment convenient partition terms sum according hamming distance denote write binary entropy function write left inverse let follows immediately denote projection vector onto xri since whenever xri yri follows xri yri xri yri since equal number subsets exposing follows xri yri xri yri particular implies since ball hamming radius contains points follows suppose since definition xri yri hence follows combining obtain var since follows var since result follows state algorithm use prove theorem use lemmas prove correctness following definition rational constant algorithm given instance rational number access oracle algorithm computes rational number probability least sat sat instances solve problem brute force return result satisfying assignments count exactly let apply sparse return result equal fail try larger larger equation systems set maximum number solutions find explicitly let prepare query independently sample uniformly random vector let ask oracle using subroutine let output sparse bad hash small fail next outer otherwise set return estimate return failed return garbage return lemma correct runs time moreover oracle called instances variables proof let instance variables let running time algorithm dominated clearly takes time lemma inner number controls maximum running time willing spend particular lemma running time one iteration inner fail otherwise bounded remaining iterations inner loop skipped easy see holds point inner loop hence overall running time required likewise oracle access requirements satisfied remains prove correctness terminates correctness immediate suppose holds set solutions satisfies let max note formulas oblivious execution algorithm analysis may view sampled advance let set solutions let following event thus implies lemma applied var since independent follows lemma thus union bound implies probability least event occurs simultaneously suppose happens particular reaches iteration none calls sparse fail iteration thus returns estimate moreover since occurs estimate satisfies required thus behaves correctly probability least result follows theorem restated let let suppose randomised approximation scheme given formula approximation error parameter runs time proof solve instance exactly brute force time suppose definition exists randomised algorithm failure probability running time lemma constant applying algorithm times outputting common result may reduce failure probability apply using procedure place take sufficiently large lemma union bound overall failure probability running time required general result first define setting result definition let bipartite graph define independence oracle function indg indg independent set define adjacency oracle function adjg adjg think edges corresponding witnesses decision problem example correspond pairs orthogonal vectors thus calling adjg correspond verifying potential witness calling indg correspond solving decision problem main result following theorem randomised algorithm following properties given input two disjoint sets rational number bipartite graph access independence adjacency oracles iii returns rational number holds probability least runs time makes log calls independence oracle throughout rest section take bipartite graph theorem statement moreover define briefly compare performance algorithm theorem standard approach sampling deal dense instances combined brute force counting deal sparse instances used thurley suppose constant evaluate indg time evaluate adjg time input graph contains edges sampling requires time brute force enumeration sort used sparse requires time worst case arises case algorithm requires time however algorithm theorem requires time cases thus polylogarithmic overhead deciding whether graph contains edges similarly section shall obtain approximation repeatedly approximately halving number edges graph remain counting remaining edges exactly halving step rather hashing current graph induced shall simply delete half vertices chosen uniformly random however single vertex incident large proportion remaining edges edge count resulting graph around expectation approach fail prove essentially obstacle definition given say set every vertex degree lemma let suppose suppose log let random subset formed including vertex independently probability probability least log proof first note since lemma let log vertex let indicator random variable event note function changing single indicator variable alters exactly moreover therefore follows lemma exp let define function suppose maximises subject constraints must one value otherwise could increase value perturbing thus reordering since follows follows result therefore follows union bound make balanced need find reasonable approximation set vertices captured following definition definition given say satisfies following properties every vertex degree least contained every vertex degree least call otherwise call find use following procedure note vertices remaining instead return exactly algorithm findcore algorithm takes input set log log returns otherwise returns set probability least log use adjg enumerate edges incident return total number let uniformly random ordering let log recursively define sequence max indg using binary search find let uki use adjg enumerate edges incident return total number otherwise use adjg find set vertices adjacent least vertices return lemma findcore correct moreover runs time makes log calls independence oracle makes calls adjacency oracle proof let log first note requires log time log calls adjg requires log time requires log time calls indg together require time calls adjg thus running time guarantees statement correct also immediate algorithm behaves correctly log note min indg min thus consists least elements random ordering thus algorithm behaves correctly uniformly random subset suppose remains prove probability least let set vertices degree least follows hypergeometric distribution mean log lemma follows conversely let set vertices degree less follows hypergeometric distribution mean lemma yields log obtain required upper bound failure probability algorithm union bound following lemma implies finding small allow either apply lemma halve possibly removing halve size removing lemma let let set let either contains proof suppose either every vertex degree since every vertex incident edge suppose instead least vertices degree least hence since every vertex satisfies follows thus holds suppose suppose let arbitrary moreover since follows holds prove main result algorithm edgecount given sets rational return rational number holds probability least let log let let count exactly using adjg output apply findcore findcore returns output otherwise returns subset remove element independently probability increment using adjg find apply findcore argument findcore returns output otherwise returns subset remove remove element independently probability add increment otherwise remove add theorem randomised algorithm following properties given input two disjoint sets rational number bipartite graph access independence adjacency oracles iii returns rational number holds probability least runs time makes log calls independence oracle proof take edgecount modified return arbitrary value running time oracle access exceeds allowed thus every desired property immediate except iii suppose without loss generality integer let value start ith iteration algorithm terminates point define fashion let output findcore ith iteration algorithm terminates point let event log log let event edgecount terminates start ith iteration let note let log suppose occur first claim occurs occurs immediate suppose instead occurs log either uxm edgecount terminates thus must occur cases lemma requires time calls adjacency oracle iteration requires time log calls independence oracle calls adjacency oracle iteration requires time iteration requires time calls adjacency oracle iteration requires time since log follows occur edgecount satisfies time oracle access restrictions terminate early let max occur occur edgecount outputs follows log since moreover log log hence log log log log log therefore follows occur required iii remains prove let event ith iteration one following events occurs invocation findcore fails edgecount loops either log invocation findcore fails edgecount loops either log claim conditioned suppose occurs note implies occurs must prove occurs split cases depending behaviour edgecount ith iteration case edgecount loops case since occurs moreover log log log log likewise log thus occurs case edgecount loops case similar case place omit case edgecount loops since occur must since edgecount loop follows likewise since occur edgecount loop output findcore must follows lemma since occurs follows moreover since occurs also holds occurs therefore shown conditioned claimed follows correctness findcore lemma occur edgecount loops must since occur lemma moreover since edgecount halt log thus lemma likewise edgecount loops ith iteration log output findcore must follows union bound hence log last inequality follows since applications theorem formally define problems follows problem input three lists integers task decide whether exists tuple problem input three lists integers task count number tuples theorem restated integers randomised time algorithm log log randomised algorithm proof let instance let solve problem exactly time log log suppose let let note sorted adjg evaluated time log log using binary search moreover indg instance indg evaluated failure probability time recall otherwise proof theorem constant indg evaluated failure probability log time therefore sort log time apply algorithm theorem return result theorem restated randomised algorithm running time instances integers least one may covered intervals length proof say set covered intervals length note checked quasilinear time whether set let instance least one negating permuting sets necessary may assume exactly proof theorem randomised algorithm instances yields log log randomised approximation scheme particular note remains instance restricted problem chan lewenstein corollary provide randomised algorithm instances result follows orthogonal vectors formally define problems follows problem input two lists vectors task decide whether exists pair problem input two lists vectors task count number pairs theorem restated vectors dimensions randomised algorithm randomised log algorithm proof let instance let solve problem exactly time suppose let let bipartite graph note adjg evaluated time moreover indg instance indg evaluated failure probability time proof theorem follows indg evaluated sufficiently low failure probability time therefore apply algorithm theorem return result following definitions constant finite ring problem input two lists vectors task decide whether exists pair problem input two lists vectors task count number pairs theorem restated let constant prime power randomised algorithm instances resp time resp proof exactly proof theorem randomised algorithm yields log randomised approximation scheme result therefore follows theorems respectively williams triangles formally define problems follows problem nwt input tripartite graph symmetric function task decide whether exists triangle abc problem nwt input tripartite graph symmetric function task count number triangles abc theorem restated nwt graphs weights randomised algorithm randomised log log time algorithm nwt proof let instance nwt let vertex classes let solve problem exactly log time suppose let let let let bipartite graph note evaluated log time moreover define graph let instance nwt evaluated failure probability time proof theorem follows evaluated sufficiently low failure probability time therefore apply algorithm theorem return result problem apsp input directed graph function task output matrix minimum weight path theorem restated randomised runs instances nwt polynomially bounded weights time log instance apsp polynomially bounded proof williams theorem edge weights solved time log reduction nwt apsp constant overhead give explicitly following paragraph result therefore theorem noting polylogarithmic overhead subsumed log term let instance nwt writing form instance apsp follows let let let thus paths correspond exactly triangles containing let output apsp yes instance nwt checked time references amir abboud richard ryan williams huacheng applications polynomial method algorithm design proceedings annual symposium discrete algorithms soda san diego usa january pages chris calabro russell impagliazzo valentine kabanets ramamohan paturi complexity unique 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konstantin kutzkov new upper bound problem inf process colin mcdiarmid method bounded differences surveys combinatorics invited papers twelfth british combinatorial conference pages moritz randomized approximations parameterized counting problems parameterized exact computation second international workshop iwpec switzerland september proceedings pages ramamohan paturi pavel michael saks francis zane improved exponentialtime algorithm acm may michael sipser complexity theoretic approach randomness proceedings fifteenth annual acm symposium theory computing stoc pages new york usa larry stockmeyer approximation algorithms siam marc thurley approximation algorithm international symposium theoretical aspects computer science stacs february march paris france volume lipics pages seinosuke toda hard hierarchy siam october patrick traxler relative exponential time complexity approximate counting satisfying assignments parameterized exact computation international symposium ipec wroclaw poland september revised selected papers pages leslie valiant complexity computing permanent theor comput leslie valiant vijay vazirani easy detecting unique solutions theor comput ryan williams new algorithm optimal satisfaction implications theor comput ryan williams faster shortest paths via circuit complexity symposium theory computing stoc new york usa may june pages ryan williams huacheng finding orthogonal vectors discrete structures proceedings annual symposium discrete algorithms soda portland oregon usa january pages virginia williams hardness easy problems basing hardness popular conjectures strong exponential time hypothesis international symposium parameterized exact computation ipec september patras greece pages virginia williams ryan williams subcubic equivalences path matrix triangle problems annual ieee symposium foundations computer science focs october las vegas nevada usa pages
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jan free energy minimization using cluster variation method initial code verification validation thm ajm alianna maren northwestern university school professional studies master science data science program themasis associates date version abstract new approach general artificial intelligence gai building neural network deep learning architectures make use one hidden layers ability continuously reach free energy minimum even input stimulus removed allowing variety possible behaviors one reason approach developed lack suitable free energy equation one would avoid difficulties known neural networks cluster variation method cvm offers means characterizing local pattern distributions configuration variables provides free energy formalism terms configuration variables equilibrium distribution configuration variables defined terms single interaction enthalpy parameter case equiprobable distribution bistate ensemble units distributions equilibrium distribution characterized providing fixed value fraction units active state corresponding influence activation enthalpy together pairwise interaction enthalpy parameter paper provides verification validation code computes configuration variable thermodynamic values cvm grids characterized different interaction enthalpy parameters means working foundation experimenting hidden layer alternative responding strictly inputs also independently come free energy minimum system also return free state perturbed enable range behaviors hitherto available use cvm grid characterizing different kinds patterns terms corresponding together respective fraction units means quantitatively characterizing different kinds neural topographies allows connect topographic descriptions terms local patterns free energy minimization allowing approach characterizing topographies building new computational engines keywords artificial intelligence neural networks deep learning statistical thermodynamics free energy free energy minimization cluster variation method entropy brain networks neural connectivity introduction overview article documents verification validation results first two stages code development free energy minimization within cluster variation method cvm system intention cvm system free energy minimized independent use process ultimately cvm system inserted single layer neural network creating new form computational engine call cortecon standing temporallyconnected neural network first described presented early results using cvm work described focuses cvm grid operate hidden layer independent functional unit ability achieve free energy minimum extraneous signal coming layer shown figure notion using cvm computational engine hidden layer advances ideas originally proposed along incorporates makes practical ideas put forth karl friston whose notation adopted figure specifically figure illustrates computational engine using friston notion set computational representational units separated external system markov blanket also allows variational bayes free energy minimization approach described friston earlier beal brief friston building work beal proposes computational system markov blanket separates computational representational elements engine external events shown figure communication external system elements denoted representational system denoted mediated two distinct layers components markov blanket sensing elements action ones article provides first two code development stages computing values configuration variables system various values interaction enthalpy parameter computing thermodynamic quantities associated system given set configuration variables possible compute enthalpy entropy free energy figure illustration cluster variation method cvm computational engine markov blanket sensing active units corresponds input output layers see friston unique approach advanced computational layer composed cvm free energy equation explicitly written free energy minimum found either analytically computationally depending parameters used cvm layer comprises internal representational units communicate external field shown two parts visualization purposes however units within representational layer receive inputs sensory units send signals active units sensory units receive inputs external stimulus send signals representational units active units receive inputs representational units send signals external output units notional view advanced friston cit broader set interactions allowed simplicity engine interaction pathways streamlined crucially code incorporates free energy minimization process initial pattern created adjusted process achieve desired specification allows implicitly enfold nominal activation energy relationship parameter explicitly stated time achieve free energy minimization given set typically cvm grid needs state changes various units achieve free energy minimum configuration variables first aspect task documented ensure configuration variables grid counted correctly configuration variable definitions including counted cvm grid cvm grid specifications size results configuration variable counts select examples introducing configuration variables cluster variation method introduced kikuchi uses entropy term includes distribution simple states also distribution local patterns configurations illustrated following figures cvm characterized set configuration variables collectively represent single unit pairwise combination triplet values configuration variables denoted single units pairs pairs triplets configuration variables illustrated single zigzag chain figure figure single zigzag chain created arranging two staggered sets units configuration variables shown single units triplets table configuration variables cluster variation method name variable instances unit triplet bistate system one units either state state six different ways triplet configuration variables constructed shown figure also table notice within figure triplets two possible configurations means degeneracy factor triplets degeneracy factors number ways constructing given configuration variable shown figure constructed either similarly triplets two ways constructing triplets degeneracy factors set counting configuration variables experiment cvm system constructed various grids units illustrated figure figure six ways configurations constructed decided use grid several reasons sufficient variety local patterns able construct grids illustrated several distinct kinds topographies corresponding different sufficient nodes extrema could explored detail countability needed able manually count configuration values given grid configuration match results program crucial step one final advantage grid layout different grid configurations large enough show diversity small enough could create figure illustrating activation state figure illustration configuration variables cluster variation method showing ways configurations variables constructed together degeneracy factors node thus illustrating detailed particulars configuration design began grid configurations two shown figure two configurations correspond somewhat notions topographies observed various neural communities references please consult two different grid configurations early attempts characterize identified grids different total configuration variable values following section discuss context free energy equation systems created constraint equiprobable occurrence units states done facilitate next step discussed section thus configurations shown figure grids nodes units state state figure illustration two different grids experiments cvm system configuration left figure effort build system notion system kind pattern replicates throughout various scales observation system thus configuration shown figure created design originally intended symmetrical around central axis dihedral symmetry specifically left right sides identical sense ease design system focused creating pattern one side duplicating used base pattern order create greater dispersion values across triplets wanted minimize islands would yield little way triplets respectively practical limitation attempting fit various islands nodes black surrounding sea nodes white meant quite enough nodes act borders around compact sets nodes thus pattern right half grid bit compressed originally planned original plan nodes grid half would right half left nodes side side nodes state black nodes per side sixteen nodes would used create large island remaining nodes would used islands two islands eight nodes etc plan shown figure notation center refers placement various islands largest islands placed center respective left right sides grid remaining islands situated around primary respective large islands resulting patterns close original plan although exactly details see figure even though changes made original design plan original constraint number units states would identical nodes kept details shown figure validation step stage code development manually count configuration variables several different configuration grids ones shown figure counts grid shown figure shown figure suffices say results manual counting configuration variables created computer code identical held true across several different grids different node configurations note achieve fractional variables shown figure also table following relations used accounting degeneracy occurs accounting degeneracy occurs accounting degeneracy occur note exact details row counts difficult read figures original diagrams corresponding slidedeck figure cvm system equal number state state nodes nodes available associated github repository see details end document second configuration configuration shown figure contrast grid configuration used figures created second configuration one large compact region nodes state grid envelope shown figure configuration designed maximize number pairwise triplet configurations put previous configuration shown figure direction purpose configurations different dispersions among configuration variable values would figure cvm system equal number state state nodes nodes putatively yield different correspond different points equilibrium curve free energy equation case equiprobable units states analytic results free energy minimum curve equilibrium point free energy different interaction enthalpy values would serve useful experiment test results discussed following section initial stage code development ascertaining configuration variable counts complete accompanying slidedeck documents code block structure provides important elements code documentation also available github see end figure cvm system equal number state state nodes nodes document details far complex element configuration variable counting code counting triplets step ensured counts wrapping around right left top bottom creating envelope initial grid performed desired expected verification validation computing thermodynamic variables two primary means obtaining validation code computing thermodynamic variables correct comparison analytic equiprobable case case equiprobable distribution among variables developed analytic solution gives means comparing results expected analytic results comparison analytic case interaction enthalpy zero second means check codegenerated results case distributuion values equiprobable however interaction enthalpy set zero thus exact distribution configuration values precisely computed allowing exact analytic computation thermodynamic variables previous section described patterns generated validation configuration variable counting interesting see thermodynamic variables emerged systems described however illustrated certain system equilibrium even though equiprobable distribution values results realm theory less discussed elsewhere realization patterns would necessarily equilibrium meant needed test cases patterns would indeed equilibrium required random generation patterns also modified associated free energy values achieved minimum process described thoroughly following section validation support analytic solution analytic solution case found using full interaction enthalpy term solution similar limited enthalpy equation used well predecessor work free energy equation cvm system including configuration variables entropy term lagrange multipliers set note full derivation cvm free energy presented preceding equation corresponds equations reference also single enthalpy parameter enthalpy parameter unit activation implicitly set zero earlier intention solve equation analytic solution possible case meaning enthalpy activation parameter enthalpy term used previously approach using currently take enthalpy equation originally advocated kikuchi gives equations found using equivalence relations specifically previously found analytic solution equation condition complete enthalpy expression used viz analytic solution becomes note full derivation results published separately results data sets correspond analytic results neighborhood reason range limited analytic solution makes use equivalence relations expressed resulting solution divergences comparison shown following figure figure column table marked corresponds results eqns current approach next column corresponds results eqns previous approach graph shown following figure divergent behavior analytic solution likely due use equivalence relationships identified eqn validation support basic thermodynamic results following figure shows results case results conform analytic solution corresponding figure presents data table supporting figure verification validation free energy minimization enough simply compute thermodynamic variables given grid configuration important mechanism figure data table giving results reaching free energy minimum analytic results two different formulations enthalpy expression case pattern node activations grid adjust order reach free energy minimum accomplished writing code two stages bring close desired value adjust configuration variables achieve free energy minimum adjusting total number nodes achieve desired code initial specification desired value main randomly generates cvm grid according probabilistic assignment state state units grid however probability random number generation set specified value say mean resulting total state nodes precisely total number nodes nodes thus nodes flipped order bring actual number nodes figure graph giving results reaching free energy minimum analytic results two different formulations enthalpy expression case discussed body work analytic solution diverges denominator becomes zero state closer desired value code specifies tolerance value close actual value needs desired value runs function randomly select flip unit values needed going right direction continues resulting actual within tolerance desired value validation printing actual values ensuring within tolerance desired value adjusting configuration variables achieve free energy minimum figure configuration variable thermodynamic values case interaction enthalpy parameter ranges see detailed explanation results following section nature similar results guarantee current version code free energy minimum actually met instead code run entire process generating new grid adjusting within tolerance adjusting units free energy progressively decreased specified number trials debug phase number trials could closely monitor process actual runs trials typically much variability results goal process keep adjusting grid units free energy decreased run constant value means nodes flipped program randomly find node state another node state flip two state state vice versa compute new free energy requires recomputing entire set configuration variable values held constant process likely figure data table configuration variable thermodynamic values case interaction enthalpy parameter ranges configuration variables change program computes new free energy value using new configuration variable values well tested run free energy lower change units kept units reverted back original values trials strictly probabilistic generation code development attempt find nodes whose topographic position sets neighbors triplets would likely produce free energy decrease node change one version code designed less run multiple trials collect print plot thermodynamic variables series attempts flip nodes test resulting free energy one validation step visual observation thermodynamic variables course one trials noting free energy fact decrease another validation step interaction energy case final configuration variable values close probabilistic likelihoods thus example expect etc thus possible compare actual resultant configuration variable values probabilistic expectancies final validation step compare resulting behaviors theoretical expectations discussed fully following subsection validation support exemplar code run example shown following figure data actually perturbation run grid established previously described perturbed given amount case fraction existing nodes flipped taken free energy minimum second time figure configuration variable thermodynamic values case interaction enthalpy parameter ranges see detailed explanation results within section results obtained program run friday parameter settings total twenty trials numt rials data table run shown figure reported results configuration variable thermodynamic values averages numt rials runs numt rials validation support results values observed conform expectations figure shown green order bring values within visual range results expected result true expected results observed value average twenty trials deviance theoretical expectation acceptable values greater values smaller fact ranges expected results separate document address theoretical expectations detail brief meaning interaction enthalpy parameter negative negative enthalpy decreased increasing interaction enthalpy multiplies term thus maximizing expected limit far increased presumably approach however would mean units arranged strict checkerboard manner instances rather difficult achieve creation systems particular code uses simplistic strategy previously noted values smaller case system enthalpy decreased made smaller thus system moves towards configuration increasing maximizing size various islands decreasing size borders minimizing practical limit far decreased always border area state islands even single massive state continent surrounding sea state figure data table containing configuration variable thermodynamic values case interaction enthalpy parameter ranges see detailed explanation results within section units means get close zero actual practical limit actually depend total system size total number nodes border area progressively decrease although disappear islands join become continents thus value occurs surprising pushed suitably small value becomes increasingly difficult simple strategy randomly find nodes flip accomplish free energy reduction likely general stability values beyond simply many nodes flip much good keeping mind two nodes different state flipped order maintain value thus preliminary conclusion free energy minimization accomplished values behaving expected validation support delta results referencing figure examine curve delta shown cyan defined actual term multiplied achieve interaction enthalpy term delta curve shown dark green figure curve behaves expected particular note nearly linear behavior region delta according data table shown figure delta would expect would purely probabilistic distribution units configurations thus expect mentioned earlier would actual value delta acceptably close similar arguments hold expected observed values delta preceding discussion note values delta level increases beyond many units simple strategy easily find particular note value typically around small particular observe value indicates pushed system limit minimizing configuration value indicates border rather large island state units sea units specifically system subject investigation triplets involve border units around islands continents state approximately total number units available suggests pushed system far course visual inspection resulting grid would enormously useful confirming assessments included subsequent document validation support thermodynamic results enthalpy maximum expected minimize free energy minimize enthalpy soon introduce interaction enthalpy opportunity adjust configuration values specifically discussed lower enthalpy entropy similarly maximum minimum negative entropy increases values expected particularly note vicinity generally range variances entropy enthalpy approximately scale one appreciably dwarf move beyond find enthalpy term strongly dominates entropy thus dominates free energy increasing value interaction enthalpy coefficient gaining appreciable difference configuration values noted previous discussions values stabilized range thus increasing beyond serve useful value suggesting practical bound kind system actual practical choices based kind behavior want see configuration values modeling systems likely want induces clustering seems characterize certain neural collectives code included description perturbation analysis main values numt rials many parameters code availability code referenced made available public github repository https short delay initial publication document related code supported extensive documentation also placed github repository anyone desiring access original code prior placement public github repository contact maren alianna copyright code referenced independently develeoped maren maren holds copyright code document arxiv granted irrevocable license distribute article references maren free energy driving function neural networks symposium nonlinear theory applications hawaii december maren schwartz seyfried configurational entropy stabilizes pattern formation neural network ieee int conf smc chicago october maren cluster variation method primer neuroscientists brain sciences vol friston levin sengupta pezzulo knowing one place approach pattern regulation soc interface vol available online http friston principle unified brain theory nat rev vol friston life know journal royal society interface vol beal variational algorithms approximate bayesian inference phd thesis university college london pdf http kikuchi theory cooperative phenomena phys vol maren cluster variation method grid zigzag chains basic theory analytic solution free energy variable distributions midpoint tech thm ajm themasis kikuchi brush improvement cluster variation method chem vol
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nov efficient exploration double uncertain value networks thomas moerland joost broekens catholijn jonker department computer science delft university technology netherlands abstract paper studies directed exploration reinforcement learning agents tracking uncertainty value available action identify two sources uncertainty relevant exploration first originates limited data parametric uncertainty second originates distribution returns return uncertainty identify methods learn distributions deep neural networks estimate parametric uncertainty bayesian return uncertainty propagated bellman equation gaussian distribution identify jointly estimated one network call double uncertain value network policy directly derived learned distributions based thompson sampling experimental results show types uncertainty may vastly improve learning domains strong exploration challenge introduction reinforcement learning dominant class algorithms learn sequential data start zero prior knowledge need actively collect data therefore settle policy early instead trying actions properly explored yet however neither want continue exploring actions already know best challenge known reinforcement learning implementations use undirected forms exploration boltzmann exploration methods act point estimates mean actionvalue usually applying random perturbation avoid selecting currently optimal action however undirected methods known highly inefficient osband tracking point estimates mean value algorithms lack information example discriminate action never tried requires exploration action tried extensively deemed avoided natural solution problem originates tracking intuition limited data large uncertainty reason explore narrow distributions naturally transfer exploitation see appendix detailed illustration work identify two types interesting exploration parametric uncertainty classical statistical uncertainty function number available data points cardinal example posterior distribution mean return uncertainty distribution returns pair given policy work focus deterministic domains makes return distribution entirely induced exploratory stochastic policy conference neural information processing systems nips long beach usa argue deterministic environments explore acting probabilistically optimal respect distributions see section identify neural network methods estimate separately subsequently show combined one network call double uncertain value network duvn best knowledge first distinguish uncertainty due limited data parametric uncertainty return distribution propagate bellman equation track neural networks function approximators use improve remainder paper organized follows section provide general introduction bayesian deep learning distributional reinforcement learning section discuss parametric return uncertainty identify potential exploration section discusses implementations policy evaluation neural networks also discusses derive policy learned distributions based thompson sampling sections show experimental results discuss future work draw conclusions respectively bayesian deep learning bayesian neural networks mackay represent uncertainty model posterior distributions model parameters assume observe random variables interested conditional distribution introduced neural network parameters estimate conditional distribution bayesian setting treat model parameters random variables given observed dataset may use posterior distribution model parameters obtain posterior predictive distribution new observed datapoint neural networks practical interest posterior distribution analytically intractable gal ghahramani showed wellknown empirical procedure actually produces approximation providing samples posterior predictive distribution simply retaining test time prediction use technique paper discuss alternative methods bayesian inference neural networks future work section distributional reinforcement learning reinforcement learning sutton barto agents studied interact unknown environment goal optimize performance measure framework adopts markov decision process mdp given tuple every observe state pick action available discrete actions mdp follows transition dynamics returns rewards work assume discrete action space deterministic transition reward functions act mdp according stochastic policy discounted return pair random process given discount factor emphasize return random variable distribution induced stochastic policy assume deterministic environment side contribution introduce initial return entropy ire measure task exploration difficulty see appendix may rewrite equation recursive form known distributional bellman equation omitting superscript note equality sign represents distributional equality engel ready define function denote expectation traces induced policy applying operator defines value applying operator gives known bellman equation sutton barto papers actually present current introduction emphasize mean action value quantity estimate sampling underlying return distribution approximate distribution deep neural network write network predicting point estimate network approximating entire return distribution learn value algorithms follow variants scheme known generalized policy iteration gpi sutton barto gpi iterates policy evaluation calculate new estimates value based new sample data sarsa policy improvement use estimate improve policy whether policy gradient algorithm distributional perspective exploration argue probabilistic perspective value functions exploration two distributions might useful exploration point view statistical parametric uncertainty mean action value distribution return parametric uncertainty mean given policy value scalar number definition expectation possible future traces however statistical point view makes sense treat estimate random variable need approximate finite number samples call parametric uncertainty parametric exploration acting optimistic respect uncertainty mean successful bandit setting however sparsely applied see appendix related work believe due fundamental complication regarding uncertainties identified dearden bandits decision problems originating stationary distribution makes value approximation ordinary supervised learning problem however target distribution highly standard target like falsely assumes known actually uncertain therefore repeatedly visiting pair makes certain value still uncertain next words value certainty depends future policy certainty standard parametric uncertainty account problem local parametric uncertainty converge supervised learning somehow need propagate uncertainty future pairs value global uncertainty back bellman equation illustration seen fig right uncertainty influences current future value estimates ignoring distributions graph need propagate parametric uncertainty timesteps already applies learn mean empirically observe shape return distribution strongly differs domains matters shape return distribution also influences easily estimate expectation quantity like upper confidence bound samples example long thin right tail return distribution frequently case good traces may give mean estimate high variance would actually need importance sampling appendix visualize return distributions domains also introduce initial return entropy measure task exploration difficulty figure three types neural networks different distributions circles probabilistic nodes left parametric uncertainty mean middle propagating return distributions point estimate parameters right parametric uncertainty propagating distribution double uncertain value network illustration propagating distributions subscripts identify unique pairs initialize pairs prior parametric uncertainty prior output distribution new observed transition want update estimates current state action pair propagating distribution next node bellman operator instead propagating mean work consider two quantities propagate return distribution next node point estimate parametric uncertain return distribution next node arrows point backwards focus direction uncertainty runs different direction exploration return distribution standard also parametric uncertainty introduced usually deal mean however exploration point view makes sense learn full return distribution note still focus deterministic environments therefore distribution returns solely induced policy may modify policy makes sense act optimistically respect return distribution observe illustration consider pair particular mean action value estimate really matters whether average originates highly varying return consistently return matters policy may influence shape distribution highly varying returns may actively transform distribution towards good returns words really care deterministic domains best return upper end return turns challenges focus around propagating either parametric uncertainty return distributions bellman equation fig overall idea memorize propagating global mdp uncertainties neural network makes locally available action selection time thereby avoid need forward planning get global information approach entirely double uncertain value networks policy evaluation discuss three probabilistic policy evaluation approaches incorporate uncertainties introduced previous section local parametric uncertainty return distribution combined respective network structures illustrated fig implementation details provided appendix parametric uncertainty estimate parametric uncertainty may use type bayesian inference method suitable neural networks paper consider bayesian dropout gal simple practical implementation see sec gives sample posterior predictive distribution mean action value associated network structure visualized figure left stochastic domains return distribution additional noise want act expectation return distribution next consider problem learning return distributions instead mean work assume return distribution approximated gaussian therefore modify neural network output distribution parameters clearly note network parameters point estimates associated network structure visualized figure middle policy evaluation need estimate distributional targets instead point estimate targets construct bootstrap estimators based distributional bellman equation derivation mean standard focus propagating return standard deviation distributional bellman equation assume next state distributions independent may ignore see standard deviation linear combination standard deviations one timestep ahead reweighted policy probabilities shrunken approximate sum policy probabilities sampling policy usual solution right expectation multiple traces network may trained move current predictions closer targets example squared loss usual fix bootstrap predictions training gradients blocked approach seen form analytic approximate return propagation heuristic distributional loss see appendix distributional losses similar ideas approximate return propagation recently explored discrete network output distributions bellemare may also accommodate propagating multimodality second simple propagation method also experimented propagation setting sample values push bellman operator construct train network collection samples maximum likelihood loss may require samples less accurate also work complicated network output distributions like deep generative models analytic propagation projection infeasible results approach shown comparable results approximate return propagation shown section parametric uncertainty return distributions finish observation ideas may actually naturally combined one function approximator fig right note propagate return distribution parametric uncertainty next timestep effectively propagating uncertain return distributions parametric uncertainty network output distribution starting sampled transition want propagate return distribution weighted parametric uncertainty next timestep besides distributional bellman propagating machinery applies refer general mechanism uncertainty propagation parametric return bellman uncertainty appearance network uncertainty network parameters output distribution track propagating uncertain return distributions makes refer random variables scalar constants var var var cov sample next make network predictions bellman propagation repeated sampling monte carlo integration numerical integration like dearden infeasible neural network setting double uncertain value network duvn fig right intuition early learning mostly propagating uncertainty converging distributions eventually start propagating true return distributions summary identified three types probabilistic policy evaluation algorithms three associated network structures visualized fig local parametric uncertainty mean value propagating distribution return point parameters propagating uncertain return distr policy improvement describe use distributions naturally balance exploration versus exploitation based thompson sampling thompson see appendix well generalize notation introduce new random variable distribution capture ofq three policy evaluation distributions introduced previous section write joint distribution state assume posterior distributions per action independent thompson sampling selects action probability equal notational convention words choose action probability equal probability specific action optimal one averaging uncertainty joint distribution practical implementation thompson sampling simple may sample every argmax values sample equivalently dropout mask sample select arg maxa consider parametric uncertainty ignore first sampling step use current parameter point estimates consider bellman uncertainty replace second sampling step deterministic prediction thompson sampling possible choice make decisions uncertainty shown good empirical performance bandit literature chapelle naturally performs policy improvement gradually starts prefer better actions distributions start thereby hope improve instability greedy policy improvement see also bellemare undirected exploration ideally uncertain return distribution would gradually narrow deterministic environment eventually converge dirac distribution optimal value function experiments evaluate different types probabilistic policy evaluation combination thompson sampling exploration refer thompson sampling three types discussed policy evaluation parametric exploration return exploration uncertain return exploration experimental details provided appendix first consider chain domain appendix figure domain consists chain states length two available actions state trace giving positive could think another algorithm propagate probabilistic network output propagates parametric uncertainty mean next entire return distribution come algorithm concurrently work donoghue focus problem see related work figure learning curves chain domain thompson sampling parametric uncertainty return distribution uncertain return distribution versus exploration plots progress increased depth chain increased exploration difficulty note correct action state chain initialized random always action visualization fig results averaged repetitions reward select correct action every step correct action per step determined domain initialization sampling uniform bernoulli domain strong exploration challenge grows exponentially length chain see appendix learning curves chain domain shown fig different lengths chain first note strategy learn domain even short length three probabilistic approaches explore best performance uncertain return exploration longest chain length probabilistic exploration methods also get trouble solving domain however see rewards makes hypothesize could issue stabilization exploration work see appendix results correct action always original variants problem osband next test method set tasks openai gym repository fig see exploration methods manage learn domains achieved end policies reflect good policies problem exploration bit unstable domains cartpole lunarlander generally performs reasonable well note uncertainty exploration methods completely different exploration mechanism compared exploration never really perform worse domains hypothesize domains much structure challenging enough show exploration difference seen chain domain future work address challenging exploration problems also want stress probabilistic exploration always outperform undirected methods especially domains relatively simple exploration uncertainty methods generally create cautious agent first wants properly verify parts domain contrast undirected exploration agents may exploit sooner beneficial domains exploration quick rewards future work identify several directions future work types bayesian inference neural networks parametric uncertainty hypothesize bayesian may unstable tedious tune sometimes observed experiments well potentially different methods approximate posterior network parameters welling teh may improve estimation parametric uncertainty expressive output distributions bellman uncertainty propagation work experimented gaussian distributions propagation recently bellemare studied return distribution propagation categorical distributions naturally accommodate see related work well another extension could involve expressive continuous network distributions based conditional variational inference moerland figure learning curves parametric exploration return exploration uncertain return exploration different openai gym environments results averaged repetitions continuous current implementations focussed discrete action spaces thompson sampling easily applied maintaining distribution per action enumerating actions action selection extension continuous setting would require either directly propagating policy uncertainty learning parametric policy whose distribution mimics uncertainty value function stochastic environments paper entirely focussed domains deterministic reward transition functions makes return distribution induced stochastic policy stochastic domains return distribution additional noise want act expectation prevent continuing act optimistically respect something influence finally want stress algorithms paper entirely uncertainty theme also appears track uncertainty estimated transition reward function deisenroth rasmussen depeweg parametric model uncertainty different parametric uncertainty studied work ideas may extended setting well would add another source uncertainty conclusion paper introduced double uncertain value networks duvn best knowledge first algorithm distinguishes uncertainty due limited data parametric uncertainty return distribution propagates bellman equation tracks neural networks function approximators uses improve exploration implemented duvn algorithm bayesian dropout parametric uncertainty gaussian distribution bellman uncertainty propagation main appeal implementation simplicity deep implementation easily extended work adding neural network layers specifying gaussian output distribution instead prediction take lines code automatic differentiation software packages showed even vanilla implementation least match improve undirected exploration performance variety problems drastically improve performance exploration heavy domain chain believe improvements distributional approach expressive network output distributions capture promising direction exploration research references auer fischer analysis multiarmed bandit problem machine learning bellemare srinivasan ostrovski schaul saxton munos unifying countbased exploration intrinsic motivation advances neural information processing systems pages bellemare dabney munos distributional perspective reinforcement learning arxiv preprint blundell cornebise kavukcuoglu wierstra weight uncertainty neural networks arxiv preprint brafman tennenholtz general polynomial time algorithm reinforcement learning journal machine learning research oct chapelle empirical evaluation thompson sampling advances neural information processing systems pages dearden friedman andre model based bayesian exploration proceedings fifteenth conference uncertainty artificial intelligence pages morgan kaufmann publishers dearden friedman russell bayesian pages deisenroth rasmussen pilco approach policy search proceedings international conference machine learning pages depeweg udluft uncertainty decomposition bayesian neural networks latent variables arxiv preprint engel mannor meir bayes meets bellman gaussian process approach temporal difference learning proceedings international conference machine learning pages engel mannor meir reinforcement learning gaussian processes proceedings international conference machine learning pages acm gal uncertainty deep learning phd thesis phd thesis university cambridge gal ghahramani dropout bayesian approximation representing model uncertainty deep learning international conference machine learning pages gal mcallister rasmussen improving pilco bayesian neural network dynamics models machine learning workshop volume page ghavamzadeh engel bayesian algorithms proceedings international conference machine learning pages acm ghavamzadeh engel bayesian policy gradient algorithms advances neural information processing systems pages guez silver dayan efficient reinforcement learning using search advances neural information processing systems pages houthooft chen duan schulman turck abbeel vime variational information maximizing exploration advances neural information processing systems pages kearns singh reinforcement learning polynomial time machine learning kocsis bandit based planning ecml volume pages springer mackay information theory inference learning algorithms cambridge university press mannor simester sun tsitsiklis bias variance value function estimation proceedings international conference machine learning page acm mannor simester sun tsitsiklis bias variance approximation value function estimates management science mannor tsitsiklis optimization markov decision processes arxiv preprint matthias rein prafulla szymon richard tamim pieter marcin parameter space noise exploration arxiv preprint moerland broekens jonker learning multimodal transition dynamics reinforcement learning arxiv preprint morimura sugiyama kashima hachiya tanaka parametric return density estimation reinforcement learning arxiv preprint donoghue osband munos mnih uncertainty bellman equation exploration arxiv preprint osband blundell pritzel van roy deep exploration via bootstrapped dqn advances neural information processing systems pages osband van roy wen generalization exploration via randomized value functions arxiv preprint rasmussen kuss gaussian processes reinforcement learning nips volume page sobel variance discounted markov decision processes journal applied probability stadie levine abbeel incentivizing exploration reinforcement learning deep predictive models arxiv preprint sutton barto reinforcement learning introduction volume mit press cambridge tamar castro mannor learning variance journal machine learning research thompson likelihood one unknown probability exceeds another view evidence two samples biometrika van hoof tanneberg peters generalized exploration policy search machine learning welling teh bayesian learning via stochastic gradient langevin dynamics proceedings international conference machine learning pages white mean variance probabilistic criteria finite markov decision processes review journal optimization theory applications related work exploration widely studied topic reinforcement learning discuss work based parametric uncertainty return add context exploration approaches motivation uncertainty methods uncertainty parametric uncertainty mean research direction uses uncertainty mean action value either frequentist bayesian direction direct exploration exploration usually based optimism uncertainty parametric uncertainty extensively studied bandit setting environments single state multiple actions unknown stochastic rewards succesful approaches ucb auer acts upper confidence bound frequentist confidence interval thompson sampling thompson also studied paper extensions ideas setting first occurrence parametric uncertainty seems bayesian dearden authors use tabular normal distributions conjugate updating also ones explicitly identify necessity propagate parametric uncertainty future states exploration based either thompson sampling call sampling also consider myopic value perfect information vpi another exploration strategy osband extended ideas linear function approximation setting randomized value iteration rlsvi neural network context parametric uncertainty based variational inference studied bandits blundell gal ghahramani studied use dropout uncertainty parametric value uncertainty similar work consider propagation distribution returns osband also studied parametric exploration neural networks using bootstrap frequentist approach uncertainty estimation confused use term bootstrapping concurrently present paper donoghue also identified need propagate parametric uncertainty timesteps approach based variance estimate similar role gaussian uncertainty propagation neural network implementation derives local parametric uncertainty estimates linearity last network layer frequentist uncertainty estimates known linear regression contrasts bayesian approach parametric uncertainty moreover still propagate uncertainty mean action value consider returns paper work track uncertainty policy evaluation use policy improvement exploration focussed gaussian process regression engel rasmussen uses two gaussian processes one track parametric uncertainty value second one model uncertainty transition model however paper still uses greedy policy improvement gaussian process approach also extended continuous action spaces ghavamzadeh engel policy search ghavamzadeh engel algorithms track uncertainty gradients stabalize update direct exploration idea exploration based parametric uncertainty also connects difference action space parameter space exploration matthias van hoof undirected exploration methods like boltzmann inject exploration noise action space level however beneficial inject noise parameter level instead usually allows retain particular noise setting multiple steps entire episode risk exploration noise agent redecide every timestep therefore stick exploration decision effect might jittering behaviour exploration exploitation steps also identified challenge ensuring deep exploration osband considered problem paper could example implemented fixing dropout mask entire episode also want note exact exploration problem occurs classical tree search succesful monte carlo tree search mcts algorithms like upper confidence bounds trees uct kocsis act upper confidence bound frequentist confidence interval value pair overlap reinforcement learning modelbased search identified long sutton barto assume known environment model usually includes parametric function approximator represent search stores tree structure technically effective sparse form tabular representation besides exploration themes appear fields return uncertainty distributional bellman equation certainly new sobel white nearly research focussed mean papers study underlying return distribution study variance return engel learned distribution return gaussian processes use exploration tamar studied variance return linear function approximation mannor tsitsiklis theoretically studies policies bound variance return variance return need used optimism uncertainty actually frequently considered several scenarios may want avoid incidental large negative desastrous robot financial portfolio morimura studied parametric return distribution propagation well exploration softmax exploration quantile also known var financial management literature distribution losses based including normal laplace skewed laplace distributions could better distributional loss heuristic loss however implementations remain tabular setting recently bellemare theoretically studied distributional bellman operator authors show operator still contraction policy evaluation setting contraction distribution metric control setting hypothesize due inherent instability greedy updates bellman optimality operator algorithm called uses categorical distribution propagate returns distributions may easily accommodate multimodality compared gaussian distribution used work complete bellman distributions use exploration methods nevertheless improves previous deep atari games exploration intrinsic motivation exploration uses slightly different incentive exploration focussing rewarding regions visited often ideas extensively studied tabula rasa setting brafman tennenholtz exploit kearns singh guez explicitly plans ahead using monte carlo tree search uncertain transition dynamics models applications domains include stadie bellemare intrinsic motivation generalizes notion novelty internal reward characteristics next external reward function example rewarding actions decreases parametric uncertainty transition model houthooft alltogether class exploration methods usually depends ability learn good transition models limited data problem trivial deisenroth rasmussen depeweg moerland theoretical problem intrinsic motivation approaches change objective example bonuses novelty might make agent continue visit region value functions already certain yet states frequently visited yet like continuously walking around room view angles gives new visual state time nevertheless intrinsic approaches hold challenging exploration problems like montezuma revenge bellemare uncertainty work paper considered however similar issues uncertainty appear learning transition reward function known dearden first address problem tabular setting mannor studied two environment sources variance may influence return distribution internal variance due stochastic environment parametric model variance due bias environment model neither considered work may added ideas studied neural networks depeweg instead uses terms empistemic uncertainty model bias aleatoric uncertainty inherent environment approach infers distributions neural network parameters capture model bias uses expressive output distributions capture true environment stochasticity actually similar structure double uncertain value network sense learn double uncertain transition network course transition model learning involve uncertainty propagation supervised learning problem finally gal used bayesian considered work parametric value uncertainty track parametric model uncertainty initial return entropy ire measure initial exploration difficulty define initial return distribution ird pinit distribution trace returns sampling initial state sinit initial state distribution following uniform random policy undirected exploration uniform policy best policy specify start encountering varying returns define initial return entropy ire entropy distribution pinit log pinit propose ire interesting measure domain exploration difficulty lower values indicate higher exploration challenge figure shows ird ire various domains openai gym repository see quite large differences shape distributions importantly domains hard exploration example atari game montezuma revenge show spiked ird therefore low ire challenge domains nearly initial traces give return makes hard ever find first indication many domains nearly traces give reward example mountain car shows traces giving reward penalty per timestep gym caps mountaincar episodes length default entropy return distribution course robust reward function translations making stable measure initial exploration difficulty propose measure domain exploration difficulty example wellknown exploration challenge choosing small suboptimal exploring obtain potential higher reward type exploration challenge accurately reflected ire simple early rewards spread initial return distribution may falsely suggest domain easy course many dimensions influence task difficulty like state action space cardinality ire nicely illustrates task like chain appendix actually quite challenging figure return distributions initial state different environments uniform random policy histogram produced traces maximum steps first domains directly taken openai gym chain domain introduced appendix orange line kernel density estimate vertical dashed line empirical mean monte carlo estimate uniform random policy display initial return entropy ire estimate domain illustration undirected versus directed exploration quickly elaborate difference undirected exploration methods like boltzmann exploration directed methods like thompson sampling theoretical example consider two available actions given observed data posterior actionvalue distribution figure shows two scenario left right difference scenario uncertainty value action second scenario much uncertain true value compare boltzmann thompson sampling act scenario main point undirected methods leverage uncertainty information figure example posterior value distributions two available actions scenario left action blue solid line action green dashed line scenario right except exploration uses distribution means act scenarios preferring action selecting action small probability boltzmann exploration usually seen subtle gradually preferring actions higher boltzmann consider numerical scale action means including difference something ignores however still acts scenario boltzmann approximates distribution actions still considering means although softmax returns probability distribution actions interpreted another problem boltzmann action selection translation reward function making tedious tune therefore many undirected implementations still prefer exploration thompson sampling directed exploration method uses example normal random variables analytically calculate define still normal distribution standard laws probability applying example gives scenario left scenario right note thompson sampling naturally assigns extra probability mass action scenario much uncertain potential value similar phenomenon happens softmax loss classification tasks outputs softmax also frequently falsely interpreted measure uncertainty classes gal however extrapolate far away observed data one classes usually gets high probability thereby appears certain since observed data region input space actually uncertain illustrates point estimates discrete set transformed uncertainties need entire per class output illustration exploration challenge chain domain study chain domain example mdp fig illustrates difficulty exploration sparse rewards domain also empirically studied results section paper mdp consists chain states time step agent two available actions left right every step one actions correct one deterministically moves agent one step chain wrong action terminates episode states zero reward except final chain state variants problem studied frequently osband ordered implementation correct action always optimal policy always walk right variant illustrated fig well figure chain domain example mdp undirected exploration highly inefficient based osband osband studied expected regret variant scenario present different illustration show expected time first visit terminal state first trace domain example let denote number episodes first reach state clearly reach first time seen reward information undirected exploration follow uniform random policy probability trace reaching state uniform policy therefore number episodes first reach follows negative binomial distribution success probability follows example shows undirected exploration point estimates required number exploratory episodes scales exponentially exploration depth although clearly simplified domain important note setting actually representative exploration problem sparse reward domains well visible fig see chain domain similar initial return distributions example montezuma revenge game notorious challenging exploration additional results ordered chain experiments section paper discusses unordered chain correct action every step randomized compare ordered chain correct action every step always although standard implementation literature believe systematic bias domain makes really exponentially challenging exploration problem optimal policy network predicting every step happen easily function approximator like neural network due natural tendency generalize show results ordered chain fig first compared results fig see exploration indeed much easier ordered problem example also solves problem something would expect exponential exploration time discussed nevertheless see probabilistic exploration methods still outperform especially length chain increases figure learning curves ordered chain domain implementation details network architecture consists layer network discrete action nodes hidden layer relu activations parametric uncertainty hidden layer applied output pkeep probability keep node use separate subnetworks per action explicitly separate uncertainty larger problems initial representation layers may shared learning rates fixed experiments optimization performed stochastic gradient descent using adam updates tensorflow experiments parametric exploration parametric uncertainty train standard squared loss new target predicted mean action value first half use target network replay database replay times prioritized way maintaining separate prioritized replay queue based total temporal difference error previous time trace trained domains except chain taken openai gym repository available https experiments fixed throughout learning rates either pkeep parametric uncertainty pkeep uncertain returns experiments used sarsa updates eligibility traces parameter sutton barto except mountaincar experiments use note ideas eligibility traces cutting traces equally apply propagation distributions allow quicker propagation multiple timesteps always risk propagating exploratory results thompson sampling uses policy exploration evaluation sense proper uncertainty policies somewhat blur line policy differs policy reasonable probabilistic policy incorporating uncertainty nevertheless could consider thompson sampling exploration evaluating policy act mean value believe policies like thompson sampling may also benefit work cutting traces trace cutting usually based importance sampling ratios exploratory policy target policy thompson sampling may provide realistic probabilities exploratory actions may allow natural trace cutting example always strongly cuts trace every exploratory step matter whether exploratory action close best one known bad contrast probabilistic policies cut traces possible actions state much uncertainty left indeed stop speed challenge neural network implementation naturally samples next action associated probability action directly available course small discrete action space could approximate repeatedly sampling policy
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structured matrix completion forecasting time series analysis jonathan gillarda konstantin usevichb feb cardiff school mathematics cardiff university senghennydd road cardiff lorraine cran umr campus sciences france cnrs cran umr france abstract paper consider matrix completion problem specific application forecasting time series analysis briefly matrix completion problem problem imputing missing values matrix rank constraint consider matrix completion problem hankel matrices convex relaxation based nuclear norm based new theoretical results number numerical real examples investigate cases proposed approach work results highlight importance choosing proper weighting scheme known observations keywords hankel matrices matrix completion forecasting nuclear norm introduction representations approximations shown useful tool time series forecasting one popular approaches singular spectrum analysis ssa forecasting embeds time series hankel matrix uses approximation continuation compute next values time series ssa uses fact many time series well approximated class time series finite rank despite many successful examples ssa forecasting number disadvantages paper develop method based hankel matrix completion follow approach proposed embed time series hankel matrix missing data forecasted stored bottom corner matrix method based minimising nuclear norm provides convex relaxation matrix completion problem general see example nuclear norm sum singular values popular convex surrogate rank similar using sparse approximation shown successful tool imputing missing values matrix see example nuclear norm relaxation popular tool spectral estimation recommender systems system identification advantage nuclear norm relaxation corresponding author email addresses gillardjw jonathan gillard konstantin usevich preprint submitted elsevier february considered paper ability build complex models represent potentially complex behavior observed time series important question convex relaxation solves original matrix completion problem much famous research conducted topic available research assumes position missing entries matrix random often known entries also random mostly unstructured matrices considered therefore results applicable due special arrangement missing data due hankel structure problem moreover noted case structured matrices much challenging available results completion hankel matrices fixed pattern missing values special case analysed square hankel matrices nearly half values missing shown nuclear norm relaxation gives correct completion embedded time series written sum decreasing exponentials analysis extended case pattern missing values paper several contributions first consider general case rectangular hankel matrices potentially fewer missing values show missing values convex relaxation matrix completion using nuclear norm give identical solutions without using convex relaxation time series undamped exponentially increasing periodic components establish bounds number missing values also study question choosing optimal shape hankel matrix parameterized socalled window length second suggest new relative formulation matrix completion problem hankel matrices allows possibility allocate past observations different weights particular exponential weighting designed overcome problems related performance nuclear norm time series expressed sum increasing exponentials empirical comparisons show proper choice weights novel formulation performs well relative number classical techniques numerical examples paper use cvx matlab package specifying solving convex programs paper following structure section formally define problems considered first define exact matrix completion considering approximate version section also describes settings used throughout paper known theoretical results necessary stated reviewed section first time series finite rank recalled solution exact minimal rank completion summarized next known results time series finite rank recalled section contains main results paper first give theoretical bounds matrix completion case arbitrary shape matrix number missing values check tightness bounds based numerical experiments second establish connection exponential weighting preprocessing time series finally examples forecasting real model time series demonstrating advantages proposed methodology provided section problem statement hankel matrices vector window length hankel matrix defined pfq follows going pose problem forecasting given time series matrix completion hankel matrix formally let vector length follows number observations forecasted length given time series wish forecast use notation first elements next let integers matrix structure sppq parameterized consider sppq ppq values known others missing hankel matrix structure belongs class affine matrix structures form sppq mqu given linearly independent basis matrices particular hankel matrix structure basis matrices matrices given following subsections describe formal statement matrix completion problem nuclear norm relaxation approach propose exact matrix completion let given vector observations time series given matrix structure structured matrix completion slrmc problem posed arg min rank sppq subject pprpn implicit assumption hankel matrix corresponds class time series finite rank described section matrix completion problem general matrix structures see convex relaxation problem based nuclear norm became increasingly popular recently formally matrix nuclear norm defined minpl pxq singular values convex relaxation obtained replacing rank nuclear norm arg min sppq subject pprpn intuition behind relaxation using norm compressed sensing nuclear norm expected force singular values zero solution remark hankel matrix case solution known given section still performance nuclear norm relaxation important understanding behavior forecasting approximate case introduced next subsection approximate matrix completion let vector length denote vector weights approximate rank minimization posed follows given find arg min rank sppq subject pprpn parameter controls precision approximation two extreme cases distinguished equivalent exact matrix completion problem approximation approximation given vector forecast unlike problem known solution fact dual problem structured approximation known difficult optimization problem order circumvent complexity problem consider following relaxation using nuclear norm arg min sppq subject pprpn remark alternative ways extend problem approximate versions example consider following equivalent formulations min subject sppq min sppq pprpn pprpn regularisation parameters formulations fact shown similarly result equivalence lasso formulations problems equivalent following sense value exist solutions coincide however relation equivalent known priori see example choice weights several natural choices weights defining approximation problem trapezoid weighting take vector weights case vector given uniform weighting take vector weights exponential weighting take vector weights number make following comments regarding possible choices weights given vector weights enforced upon observations frobenius norm hankel matrix used frobenius norm matrix optimization problems commonly traditionally used due classical results optimality low rank approximations achieved truncating singular value decomposition however unintended consequence using frobenius norm vector observations placed hankel matrix receive weights given weights may unnatural forecasting example recent observations given declining weight time natural correction weights offered giving observation equal weight mentioned earlier purposes forecasting may advantages adopting weights given recent observations receive larger weight time series finite rank section recall class time series finite rank solution exact rank minimization problem hankel matrix structure time series finite rank linear recurrent formulae first recall basic properties time series finite informally speaking time series finite rank time series hankel matrix low rank known chapter class time series given sums products cosines exponential polynomial functions pkq real polynomials degrees positive integers usj distinct pairs qusj also distinct shown time series form particularly suitable model trends periodicities modulated periodicities time series analysis number called finite difference dimension rank time series equal rank hankel matrix sppq given see chapter time series finite rank compactly represented case results paper formulated time series next recall summary based results consider infinite time series pkq complex polynomials degrees positive integers note time series special cases time series indeed take known corollary time series satisfy minimal linear recurrent formula complex coefficients coefficients characteristic polynomial qpzq number called finite difference dimension time series fact consider class time series finite difference dimension however coincides time series finite rank time series infinite corollary use term time series finite rank paper solution rank minimization problem time series finite rank solution rank minimization problem hankel structure known equivalent problem minimal rank extension hankel matrices solved square matrices rectangular matrices summarize results form theorem proof given appendix theorem let minpl consider hankel matrix structure given time series finite rank form vector defined solution rank minimization problem arg min rank sppq subject ppcpn unique given computed using linear recurrent formula remark theorem implies holds case time series finite rank time series form vector defined solution rank minimization problem unique given variants theorem case also used framework ssa fact continuation linear recurrent formula core forecasting methods ssa remark parametric form time series known minimal rank completion given formula however advantage recursion parametric form derived due reason matrix completion approaches referred nuclear norm forecasting method proposed paper advantage performance nuclear norm known results performance nuclear norm solution coincides solution studied recently special case structure sppq matrix square values main antidiagonal hankel matrix sppq missing first result appeared refined treats case given theorem theorem let vector given solution also solution arg min sppq subject ppcpn particular solution unique unique solution given theorem illustrated fig note exponential needs decreasing damped order nuclear norm give solution convex relaxation similar result proved case exponentials decreasing sufficiently fast decreasing exponential increasing exponential figure forecasts red given nuclear norm time series theorem theorem fix structure exists number time series finite rank solution unique coincides solution given theorem theorem implies result holds problems note theorem requires roots characteristic polynomial within disk radius complex plane theorem impose conditions separation roots multiplicities remark theorem proves case matrices radius equal general case theorem give good estimate radius theoretical results nuclear norm forecasting square matrices fewer missing values subsection show radius obtained theorems also directly applicable case fewer missing values typical case forecasting corollary let case fewer missing values theorem matrix sppq form sppq let radius theorem time series form solution nuclear norm minimization structure coincides solution rank minimization problem rectangular matrices case section aim improve results previous subsection give explicit bound single complex exponential rectangular matrices proposition let arbitrary minpl maxpl let time series given satisfies solution nuclear norm minimization matrix coincides solution rank minimization problem illustration proposition figure contains plots different figure contains plots different fig see small number missing values nuclear norm minimization gives correct solution complex exponential undamped increasing magnitude fig shows fixed limiting number missing data nuclear norm gives correct solution fact find number explicitly given corollary figure plot different figure plot different corollary let fixed time series given solution nuclear norm minimization matrix coincides solution log define next look question choosing optimal window length next corollary shows performance nuclear norm maximised hankel matrix square almost square corollary fixed minimised case odd case even figure plot different confirms conclusions corollary indeed curves approximately square shape hankel matrix lower others figure figure plot different next would like see whether bound given corollary tight take time series test performance nuclear norm different values frobenius norm minimal rank completion minimal nuclear norm completion nuclear norm completion declared successful results plotted figure figure see bound given corollary optimal small rho figure success nuclear norm white success black failure red curve bound given corollary nuclear norm works higher values rectangular matrices case first provide improved result corollary valid rectangular matrices stronger may reduce radius proposition let theorem consider matrix structure fix number minpl define let radius theorem matrices size time series form solution nuclear norm minimization full matrix sppq coincides solution rank minimization problem holds problem statements next verify numerically maximum number missing entries nuclear norm gives correct solution case different values consider time series drawn uniformly realisations calculate empirical probability success based criterion previous example results plotted fig figure probability success nuclear norm greyscale white black figure see higher rank smaller number missing values correctly imputed nuclear norm decaying exponentials nuclear norm recovers correct solution large number missing values ranks undamped increasing exponentials special care taken results discussion approximate matrix completion previous subsections suggest nuclear norm heuristic works sufficiently damped sinusoids hence may beneficial preprocess time series introducing artificial damping solving formally given define scaled matrix structure ssc ppq sppsc psc exp exp exp arg min ssc ppq subject scaled time series multiplied decreasing exponential simplicity consider case uniform weights solve pprpn fact show next proposition problem equivalent exponentially weighted problem scaled time series proposition solution problem given construct scaled initial data vector exp exp exp solve problem psc arg min sppsc subject psc pprpn scale back weighted approximation psc exp psc exp psc exp hence proposition exponential weighting may help overcome potential problems increasing exponentials time series examples fortified wine section consider classical time series fortified wine observations depict monthly volumes wine sales australia thousands litres period january december denote vector observations consider forecast thus take figure contains plot vector three approximations approximations obtained weighting scheme sppq define prq min rankpxq given value prq found using singular value decomposition see example sect motivation taking prq ensure approximation least good would obtained unstructured low rank approximation consider three values prq figure fortified wine solid line three approximations obtained long dash short dash figure contains plots square root first fifteen singular values matrix solution three values described minimizing sum singular values results many individual singular values going close zero parameter used control complexity approximation smaller value closer approximation given vector figure plot square root first singular values matrix solution forecasting deaths section consider forecasting famous death series recording monthly accidental deaths usa data studied many authors found number time series data libraries wish replicate exercise given aimed forecast final six values series time series contains total observations truncate series first observations forecast remaining six observations denote series observations consider table contains forecasts final six data points data series several methods along square root mean square error results taken full details fitted models found within summary model model examples sarima models described model given qzi model given realisation white noise zero mean variance backward shift operator defined hws represents model fitted seasonal algorithm arar represents model fitted transforming data prior fitting autoregressive model original data model model hws arar table forecasted data using four different models along square root mean square error consider following exercise take consider three forms defined section weighting scheme take solve three weighting schemes select solutions obtained least close solutions obtained unstructured rank approximations precicely weight vector rank take solution arg min rankpxq arg min ppq pprn prq particular weighting scheme equivalent taking obtained orthogonal projection space hankel also note original data rank rank rank rank rank rank rank rank rank table forecasted data using nine different models along square root mean square error matrices hence computed diagonal averaging sec fact ssa approximation results given table figure contains plots along approximations obtained defined effects different weighting schemes clearly seen also effect decreasing visible figure contains plot logarithm square root mean square error forecast weighting scheme smallest square root mean square error obtained make remarks rank increases formally increases quality forecast improves appears worst weighting scheme recall case weights given seen weighting unfortunate characteristic observations towards end vector contradictory argument recent observations important forecasting weighting scheme gives increased weight recent observations seen weighting scheme gives best forecast sense provides forecast smallest mean square error smaller best mean square error found simulation study consider simulated example consider time series observations form denoted case denoted case white noise gaussian error term zero mean standard deviation truncate series first terms aim study forecast remaining observations take example figures contain square root mean square errors obtained forecasting remaining observations using two possible weighting schemes case case consider three weighting schemes defined section select rank rank rank rank rank rank rank rank rank figure plot data solid line approximation forecasts dashed line solutions obtained least close unstructured rank approximation weighting schemes given way subsection remark weighting scheme provides better forecasts especially case given form time series case result expected time series case grows exponentially time gives indication merits alternative weighting schemes opposed traditional weighting scheme given conclusion paper considered matrix completion tool forecasting time series analysis formulated nuclear norm relaxation structured matrix completion suitable purpose demonstrated practical potential towards end paper shown time series sufficiently damped exponentials increasing fast nuclear norm approach work becomes rootmse tau alpha figure logarithm square root mean square error forecast weighting scheme figure case square root mean square errors number forecasted observations three weighting schemes taken simulations figure case square root mean square errors number forecasted observations three weighting schemes taken simulations ticularly important rank time series number values forecasted large approach based exponential weighting proposed shown equivalent preprocessing time series numerical experiments indicate use exponential weighting improves performance nuclear norm forecasting appendix structured matrices optimality conditions proofs theorems based following necessary sufficient condition global optimum lemma appendix generalization proposition let compact svd spp given spp let uuh vvh uvh following statements hold true global minimizer minimizer spp real exists matrix holds unique minimizer minimizer addition spp real matrix lemma appendix called optimality certificate fact proofs theorems proceed constructing optimality certificate appendix proofs proof theorem proof follows theorem applied submatrix sppq precisely conditions theorem matrix sppq contains matrix rank known entries missing values theorem particular exists one extension matrix preserves rank extension would increase rank exactly completion given proof corollary let solution given theorem solution unique minimiser matrix moreover proof theorem see also lemma appendix remark exists matrix equation holds hence particular equation holds lemma appendix point also unique minimiser structure given proof proposition solution given continuing formula let show point optimal denote matrix since matrix compact svd form therefore matrix cabh let find explicitly terms compact svd assume imaginary unit singular vectors compact svd found hence singular value expressed addition matrices defined lemma appendix given aah bbh follows use optimality condition given lemma appendix let find matrix satisfies truncating matrix using theorem truncated matrix consider lower right submatrix matrix ppq matrix form yyt singular vectors compact svd chosen nqqq hence singular value equal denote matrix found let construct optimality certificate theorem theorem exists matrix cpm mqu also next define matrix follows since qmqt qmqt first term found hence qmqt satisfies since construction lemma appendix proof complete proof corollary fixed value minimized clk maximized clear quantity geometric mean two quantities monotonically depend maximised close possible proving proposition need technical lemma bases subspaces lemma appendix let rank rank let orthonormal bases column spaces respectively let matrices containing last rows respectively holds proof first need prove first row removed case proved recursively removing first row times second quantities depend particular bases subspaces due invariance frobenius norm unitary transformations hence assume example obtained thin factorizations consider case full factorization without loss generality assume form one element first row nonzero next use fact obtained removing first row sec structure pqgr vector appropriate givens rotations rotating plane indices full factorization finally since rotation matrices affect first columns cpr reduced givens rotation matrices hence vector last elements completes proof proof proposition going prove exists certificate matrix satisfies similarly proof proposition going construct certificate form rewritten xbred bred cpm lower right submatrix projector column space lower right submatrix sppq defined ppq going look matrix satisfies extra constraints added eqn rewrite constraints vecpbred vecpsn vecpsn matrices eqn next proof lemma going find minimal frobenius norm satisfies matrix given vecpbred vecpbred bred eqn projector subspace spanned first unit vectors eqn note bred bred prove lemma bred bred bounded right hand side eqn result hold true analogously theorem relies eqn proof lemma bred bred qbred bred contain last rows svd factors sppq respectively finally lemma appendix right hand side bounded left factor svd ppq defined last equation eqn fact hence proof complete proof corollary case easy consider case evident denote condition becomes condition becomes statement follows properties logarithm proof proposition consider change variables obvious equivalent indeed follows fact psc scaling needed perform change variables inverse references decomposition smoothing forecasting sparse functional data tech rep working paper monash university department econometrics 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analysis forecasting control john wiley sons
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